“Wir müssen wissen, wir werden wissen.” (We must know, we will know.)

— D. Hilbert, Address to the Society of German Scientists and Physicians, Königsberg (September 08, 1930).

One century of peering into Einstein’s field equations has given us elegant and simple solutions, and shown how they behave when slightly displaced from equilibrium. We were rewarded with a beautiful mathematical theory of black holes (BHs) and their perturbations, and a machinery that is able to handle all weak-field phenomena. After all, one hundred years is not a very long time to understand a theory with such conceptual richness. Left behind, as an annoying nuisance, was the problem of dynamical strong-field effects such as the last stages of BH mergers.

In the last few decades, it gradually became clear that analytical or perturbative tools could only go so far: gravitational-wave (GW) detectors were promising to see the very last stages of BH-binary inspirals; fascinating developments in String/M theory (SMT) were hinting at a connection between gauge theories and strong gravity effects; extensions of the standard model of particle physics were conjecturing the existence of extra dimensions, which only gravity had access to, and were predicting BH formation at accelerators! This, and more, required the ability to solve Einstein’s equations (numerically) in full generality in the nonlinear regime. The small “annoying nuisance” rapidly grew to become an elephant in the room that had to be tamed.

But necessity is the mother of inventions. In 2005, several groups achieved the first long-term stable evolutions of BH-binaries in four-dimensional, asymptotically flat spacetimes, starting a phase transition in the field. It is common to refer to such activity — numerically solving Einstein’s equations

$${R_{\mu \nu}} - {1 \over 2}R{g_{\mu \nu}} = {{8\pi G} \over {{c^4}}}{T_{\mu \nu}},$$

or extensions thereof — as “numerical relativity” (NR). In practice, any numerical procedure is a means to an end, which is to know. In this sense, NR is a gray area which could lie at the intersection between numerical analysis, general relativity (GR) and high-energy physics. Many different numerical techniques have been used to solve the field equations in a variety of contexts. NR usually entails solving the full set of nonlinear, time-dependent Einstein-type equations.

This is a review on NR. We will cover all aspects of the main developments in the last decade, focusing for the most part on evolutions of BH spacetimes. The numerical resolution of Einstein’s equations in a computer has a five-decade long history and many important ingredients. In fact, NR is sufficiently complex that a number of outstanding review works have already been dedicated to specific aspects, like construction of initial data, finding horizons in numerical spacetimes, evolving the field equations in the presence of matter, etc. We will not attempt to cover these in any detail; we refer the reader to the relevant section of Living ReviewsFootnote 1 for this and to textbooks on the subject at large [21, 79, 111, 364]. The present work is mostly intended to make the reader familiar with new developments, which have not and could not have been covered in those works, given the pace at which the field is evolving.

A few words about the range and applicability of NR methods are in order, as they help clarify the content of this review work. NR is but one, albeit important and complex, tool that helps us to get through solving and understanding certain processes. Traditionally, the two-body problem in GR for instance, was approached via a slow-motion, large separation post-Newtonian expansion. The PN expansion breaks down when the distances between the bodies are small and the velocities are large. BH perturbation theory on the other hand, can handle the two-body problem for any separation and velocity, but as long as there is a decoupling of mass scales, i.e, one of the objects must be much more massive than the other. The remaining is NR turf: large velocities, small separations, strong field and similar masses. This is depicted in Figure 1, which we have extended to allow for generic situations. NR methods typically break down (due to large computational requirements) when there are extremely different scales in the problem, i.e., when extremely large or small dimensionless quantities appear. For instance, the two-body problem in GR can be handled for a relatively short timescale, and as long as the two bodies do not have extreme mass ratios. In spacetimes with other lengthscales, for instance AdS, NR encounters difficulties when the binary lengthscale is much smaller than the AdS lengthscale for example. While such simulations can in principle be done, they may not capture the relevant physics associated with the AdS boundary.

Figure 1
figure 1

Range of various approximation tools (”UR” stands for ultra-relativistic). NR is mostly limited by resolution issues and therefore by possible different scales in the problem.

To conclude this discussion, neither NR nor perturbative techniques are paradisiac islands in isolation; input and interplay from and with other solutions is often required. As such, we will also discuss in some detail some of the perturbative tools and benchmarks used in the field.

NR has been crucial to answer important questions in astrophysics, GW physics, high-energy physics and fundamental physics, and as such we thought it convenient — and fun — to start with a timeline and main theoretical landmarks that have stimulated research in the last years. This will hopefully help the reader getting started by understanding which are the main breakthroughs and where exactly do we stand.


Numerical solving is a thousand-year-old art, which developed into modern numerical analysis several decades ago with the advent of modern computers and supercomputers. For a compelling account of the early history of numerical analysis and computing we refer the reader to Goldstine [359, 360].

It is impossible to summarize all the important work on the subject in this review, but we find it instructive to list a chronogram of several relevant milestones taking us to 2014, in the context of GR. The following is a list — necessarily incomplete and necessarily biased — of works which, in our opinion, have been instrumental to shape the evolution of the field. A more complete set of references can be found in the rest of this review.

  • 1910 — The analysis of finite difference methods for PDEs is initiated with Richardson [648].

  • 1915 — Einstein develops GR [293, 294].

  • 1916 — Schwarzschild derives the first solution of Einstein’s equations, describing the gravitational field generated by a point mass. Most of the subtleties and implications of this solution will only be understood many years later [687].

  • 1917 — de Sitter derives a solution of Einstein’s equations describing a universe with constant, positive curvature A. His solution would later be generalized to the case Λ < 0 [255].

  • 1921, 1926 — In order to unify electromagnetism with GR, Kaluza and Klein propose a model in which the spacetime has five dimensions, one of which is compactified on a circle [463, 476].

  • 1928 — Courant, Friedrichs and Lewy use finite differences to establish existence and uniqueness results for elliptic boundary-value and eigenvalue problems, and for the initial-value problem for hyperbolic and parabolic PDEs [228].

  • 1931 — Chandrasekhar derives an upper limit for white dwarf masses, above which electron degeneracy pressure cannot sustain the star [193]. The Chandrasekhar limit was subsequently extended to NSs by Oppenheimer and Volkoff [590].

  • 1939 — Oppenheimer and Snyder present the first dynamical collapse solution within GR [589].

  • 1944 — Lichnerowicz [515] proposes the conformal decomposition of the Hamiltonian constraint laying the foundation for the solution of the initial data problem.

  • 1947 — Modern numerical analysis is considered by many to have begun with the influential work of John von Neumann and Herman Goldstine [763], which studies rounding error and includes a discussion of what one today calls scientific computing.

  • 1952 — Choquet-Bruhat [327] shows that the Cauchy problem obtained from the spacetime decomposition of the Einstein equations has locally a unique solution.

  • 1957 — Regge and Wheeler [641] analyze a special class of gravitational perturbations of the Schwarzschild geometry. This effectively marks the birth of BH perturbation theory, even before the birth of the BH concept itself.

  • 1958 — Finkelstein understands that the r = 2M surface of the Schwarzschild geometry is not a singularity but a horizon [320]. The so-called “golden age of GR” begins: in a few years there would be enormous progress in the understanding of GR and of its solutions.

  • 1961 — Brans and Dicke propose an alternative theory of gravitation, in which the metric tensor is non-minimally coupled with a scalar field [128].

  • 1962 — Newman and Penrose [575] develop a formalism to study gravitational radiation using spin coefficients.

  • 1962 — Bondi, Sachs and coworkers develop the characteristic formulation of the Einstein equations [118, 667].

  • 1962 — Arnowitt, Deser and Misner [47] develop the canonical 3 + 1 formulation of the Einstein equations.

  • 1963 — Kerr [466] discovers the mathematical solution of Einstein’s field equations describing rotating BHs. In the same year, Schmidt identifies the first quasar (quasi-stellar radio source) [681]. Quasars are now believed to be supermassive BHs, described by the Kerr solution.

  • 1963 — Tangherlini finds the higher-dimensional generalization of the Schwarzschild solution [740].

  • 1964 — Chandrasekhar and Fock develop the post-Newtonian theory [194, 325].

  • 1964 — First documented attempt to solve Einstein’s equations numerically by Hahn & Lindquist [385]. Followed up by Smarr & Eppley about one decade later [710, 311].

  • 1964 — Seymour Cray designs the CDC 6600, generally considered the first supercomputer. Speeds have increased by over one billion times since.

  • 1964 — Using suborbital rockets carrying Geiger counters new sources of cosmic X-rays are discovered. One of these X-ray sources, Cygnus X-1, confirmed in 1971 with the UHURU orbiting X-ray observatory, is soon accepted as the first plausible stellar-mass BH candidate (see, e.g., [110]). The UHURU orbiting X-ray observatory makes the first surveys of the X-ray sky discovering over 300 X-ray “stars”.

  • 1965 — Penrose and Hawking prove that collapse of ordinary matter leads, under generic conditions, to spacetime singularities (the so-called “singularity theorems”) [608, 401]. A few years later, Penrose conjectures that these singularities, where quantum gravitational effects become important, are generically contained within BHs — The cosmic censorship conjecture [610, 767].

  • 1965 — Weber builds the first GW detector, a resonant alluminium cylinder [771, 772].

  • 1966 — May and White perform a full nonlinear numerical collapse simulation for some realistic equations of state [543].

  • 1967 — Wheeler [661, 778] coins the term black hole (see the April 2009 issue of Physics Today, and Ref. [779] for a fascinating, first-person historical account).

  • 1967, 1971 — Israel, Carter and Hawking prove that any stationary, vacuum BH is described by the Kerr solution [453, 188, 403, 406]. This result motivates Wheeler’s statement that “a BH has no hair” [661].

  • 1968 — Veneziano proposes his dual resonance model, which will later be understood to be equivalent to an oscillating string [759]. This date is considered the dawn of SMT.

  • 1969 — Penrose shows that the existence of an ergoregion allows to extract energy and angular momentum from a Kerr BH [610]. The wave analogue of the Penrose process is subsequently shown to occur by Zeldovich, who proves that dissipative rotating bodies (such as Kerr BHs, for which the dissipation is provided by the horizon) amplify incident waves in a process now called superradiance [827, 828].

  • 1970 — Zerilli [829, 830] extends the Regge-Wheeler analysis to general perturbations of a Schwarzschild BH. He shows that the problem can be reduced to the study of a pair of Schröodingerlike equations, and applies the formalism to the problem of gravitational radiation emitted by infalling test particles.

  • 1970 — Vishveshwara [762] studies numerically the scattering of GWs by BHs: at late times the waveform consists of damped sinusoids (now called ringdown waves, or quasi-normal modes).

  • 1971 — Davis et al. [250] carry out the first quantitative calculation of gravitational radiation emission within BH perturbation theory, considering a particle falling radially into a Schwarzschild BH. Quasi-normal mode (QNM) ringing is excited when the particle crosses the maximum of the potential barrier of the Zerilli equation, located close to the unstable circular orbit for photons.

  • 1973 — Bardeen, Carter and Hawking derive the four laws of BH mechanics [74].

  • 1973 — Teukolsky [743] decouples and separates the equations for perturbations in the Kerr geometry using the Newman-Penrose formalism [575].

  • 1973 — York [808, 809] introduces a split of the extrinsic curvature leading to the Lichnerowicz-York conformai decomposition, which underlies most of the initial data calculations in NR.

  • 1973 — Thorne provides a criterium for BH formation, the hoop conjecture [750]; it predicts collapse to BHs in a variety of situations including very high-energy particle collisions, which were to become important in TeV-scale gravity scenarios.

  • 1974 — Hulse and Taylor find the first pulsar, i.e., a radiating neutron star (NS), in a binary star system [447]. The continued study of this system over time has produced the first solid observational evidence, albeit indirect, for GWs. This, in turn, has further motivated the study of dynamical compact binaries and thus the development of NR and resulted in the 1993 Nobel Prize for Hulse and Taylor.

  • 1975 — Using quantum field theory in curved space, Hawking finds that BHs have a thermal emission [405]. This result is one of the most important links between GR and quantum mechanics.

  • 1977 — NR is born with coordinated efforts to evolve BH spacetimes [708, 287, 711].

  • 1978 — Cunningham, Price and Moncrief [229, 230, 231] study radiation from relativistic stars collapsing to BHs using perturbative methods. QNM ringing is excited.

  • 1979 — York [810] reformulates the canonical decomposition by ADM, casting the Einstein equations in a form now commonly (and somewhat misleadingly) referred to as the ADM equations.

  • 1980 — Bowen & York develop the conformai imaging approach resulting in analytic solutions to the momentum constraints under the assumption of maximal slicing as well as conformal and asymptotic flatness [121].

  • 1983 — Chandrasekhar’s monograph [195] summarizes the state of the art in BH perturbation theory, elucidating connections between different formalisms.

  • 1985 — Stark and Piran [724] extract GWs from a simulation of rotating collapse to a BH in NR.

  • 1985 — Leaver [504, 505, 506] provides the most accurate method to date to compute BH QNMs using continued fraction representations of the relevant wavefunctions.

  • 1986 — McClintock and Remillard [547] show that the X-ray nova A0620-00 contains a compact object of mass almost certainly larger than 3 M, paving the way for the identification of many more stellar-mass BH candidates.

  • 1986 — Myers and Perry construct higher-dimensional rotating, topologically spherical, BH solutions [565].

  • 1987 —’ t Hooft [736] argues that the scattering process of two point-like particles above the fundamental Planck scale is well described and calculable using classical gravity. This idea is behind the application of GR for modeling trans-Planckian particle collisions.

  • 1989 — Echeverria [290] estimates the accuracy with which one can estimate the mass and angular momentum of a BH from QNM observations. The formalism is substantially refined in Refs. [97, 95].

  • 1992 — The LIGO detector project is funded by the National Science Foundation. It reaches design sensitivity in 2005 [6]. A few years later, in 2009, the Virgo detector also reaches its design sensitivity [10].

  • 1992 — Bona and Massó show that harmonic slicing has a singularity-avoidance property, setting the stage for the development of the “l+log” slicing [115].

  • 1992 — D’Eath and Payne [256, 257, 258, 259] develop a perturbative method to compute the gravitational radiation emitted in the head-on collision of two BHs at the speed of light. Their second order result will be in good agreement with later numerical simulations of high-energy collisions.

  • 1993 — Christodoulou and Klainerman show that Minkowski spacetime is nonlinearly stable [219].

  • 1993 — Anninos et al. [37] first succeed in simulating the head-on collision of two BHs, and observe QNM ringing of the final BH.

  • 1993 — Gregory and Laflamme show that black strings, one of the simplest higher-dimensional solutions with horizons, are unstable against axisymmetric perturbations [367]. The instability is similar to the Rayleigh-Plateau instability seen in fluids [167, 162]; the end-state was unclear.

  • 1993 — Choptuik finds evidence of universality and scaling in gravitational collapse of a massless scalar field. “Small” initial data scatter, while “large” initial data collapse to BHs [212]; first use of mesh refinement in NR.

  • 1994 — The “Binary Black Hole Grand Challenge Project”, the first large collaboration with the aim of solving a specific NR problem (modeling a binary BH coalescence), is launched [542, 213].

  • 1995, 1998 — Through a conformal decomposition, split of the extrinsic curvature and use of additional variables, Baumgarte, Shapiro, Shibata and Nakamura [695, 78] recast the ADM equations as the so-called BSSN system, partly building on earlier work by Nakamura, Oohara and Kojima [569].

  • 1996 — Brü gmann [140] uses mesh refinement for simulations of BH spacetimes in 3 + 1 dimensions.

  • 1997 — Cactus 1.0 is released in April 1997. Cactus [154] is a freely available environment for collaboratively developing parallel, scalable, high-performance multidimensional component-based simulations. Many NR codes are based on this framework. Recently, Cactus also became available in the form of the Einstein Toolkit [521, 300].

  • 1997 — Brandt & Brügmann [126] present puncture initial data as a generalization of Brill-Lindquist data to the case of generic Bowen-York extrinsic curvature.

  • 1997 — Maldacena [536] formulates the AdS/CFT duality conjecture. Shortly afterward, the papers by Gubser, Klebanov, Polyakov [372] and Witten [798] establish a concrete quantitative recipe for the duality. The AdS/CFT era begins. In the same year, the correspondence is generalized to non-conformal theories in a variety of approaches (see [15] for a review). The terms “gauge/string duality”, “gauge/gravity duality” and “holography” appear (the latter had been previously introduced in the context of quantum gravity [737, 734]), referring to these generalized settings.

  • 1998 — The hierarchy problem in physics — the huge discrepancy between the electroweak and the Planck scale — is addressed in the so-called braneworld scenarios, in which we live on a four-dimensional subspace of a higher-dimensional spacetime, and the Planck scale can be lowered to the TeV [46, 40, 638, 639].

  • 1998 — First stable simulations of a single BH spacetime in fully D = 4 dimensional NR within a “characteristic formulation” [508, 362], and two years later within a Cauchy formulation [23].

  • 1998 — The possibility of BH formation in braneworld scenarios is first discussed [45, 69]. Later work suggests BH formation could occur at the LHC [279, 353] or in ultra-high energy cosmic ray collisions [315, 33, 304].

  • 1999 — Friedrich & Nagy [335] present the first well-posed formulation of the initial-boundary-value problem (IBVP) for the Einstein equations.

  • 2000 — Brandt et al. [127] simulate the first grazing collisions of BHs using a revised version of the Grand Challenge Alliance code [227].

  • 2000 — Shibata and Uryū [698] perform the first general relativistic simulation of the merger of two NSs. More recent simulations [62], using a technique developed by Baiotti and Rezzolla that circumvents singularity excision [64], confirm that ringdown is excited when the merger leads to BH formation. In 2006, Shibata and Uryū perform NR simulations of BH-NS binaries [699].

  • 2001 — Emparan and Reall provide the first example of a stationary asymptotically flat vacuum solution with an event horizon of non-spherical topology — the “black ring” [307].

  • 2001 — Horowitz and Maeda suggest that black strings do not fragment and that the end-state of the Gregory-Laflamme instability may be an inhomogeneous string [440], driving the development of the field. Non-uniform strings are constructed perturbatively by Gubser [371] and numerically by Wiseman, who, however, shows that these cannot be the end-state of the Gregory-Laflamme instability [789].

  • 2003 — In a series of papers [479, 452, 480], Kodama and Ishibashi extend the Regge-Wheeler-Zerilli formalism to higher dimensions.

  • 2003 — Schnetter et al. [684] present the publically available Carpet mesh refinement package, which has constantly been updated since and is being used by many NR groups.

  • 2005 — Pretorius [629] achieves the first long-term stable numerical evolution of a BH binary. Soon afterwards, other groups independently succeed in evolving merging BH binaries using different techniques [159, 65]. The waveforms indicate that ringdown contributes a substantial amount to the radiated energy.

  • 2007 — First results from NR simulations show that spinning BH binaries can coalesce to produce BHs with very large recoil velocities [363, 161].

  • 2007 — Boyle et al. [122] achieve unprecedented accuracy and number of orbits in simulating a BH binary through inspiral and merger with a spectral code that later becomes known as “SpEC” and uses multi-domain decomposition [618] and a dual coordinate frame [678].

  • 2008 — The first simulations of high-energy collisions of two BHs are performed [719]. These were later generalized to include spin and finite impact parameter collisions, yielding zoom-whirl behavior and the largest known luminosities [697, 720, 717, 716].

  • 2008 — First NR simulations in AdS for studying the isotropization of a strongly coupled \({\mathcal N} = 4\) supersymmetric Yang-Mills plasma through the gauge/gravity duality [205].

  • 2009 — Dias et al. show that rapidly spinning Myers-Perry BHs present zero-modes, signalling linear instability against axially symmetric perturbations [272], as previously argued by Emparan and Myers [305]. Linearly unstable modes were subsequently explored in Refs. [271, 270].

  • 2009 — Shibata and Yoshino evolve Myers-Perry BHs nonlinearly and show that a non-axisymmetric instability is present [701].

  • 2009 — Collisions of boson stars show that at large enough energies a BH forms, in agreement with the hoop conjecture [216]. Subsequent investigations extend these results to fluid stars [288, 647].

  • 2010 — Building on previous work [215], Lehner and Pretorius study the nonlinear development of the Gregory-Laflamme instability in five dimensions, which shows hints of pinch-off and cosmic censorship violation [511].

  • 2010, 2011 — First nonlinear simulations describing collisions of higher-dimensional BHs, by Zilhao et al., Witek et al. and Okawa et al. [841, 797, 587].

  • 2011 — Bizoń and Rostworowski extend Choptuik’s collapse simulations to asymptotically AdS spacetimes [108], finding evidence that generic initial data collapse to BHs, thereby conjecturing a nonlinear instability of AdS.

  • 2013 — Collisions of spinning BHs provide evidence that multipolar structure of colliding objects is not important at very large energies [716].

Strong Need for Strong Gravity

The need for NR is almost as old as GR itself, but the real push to develop these tools came primarily from the necessity to understand conceptual issues such as the end-state of collapse and the two-body problem in GR as well as from astrophysics and GW astronomy. The breakthroughs in the last years have prompted a serious reflexion and examination of the multitude of problems and fields that stand to gain from NR tools and results, if extended to encompass general spacetimes. The following is a brief description of each of these topics. The range of fundamental issues for which accurate strong-gravity simulations are required will hopefully become clear.


Gravitational wave astronomy

GWs are one of the most fascinating predictions of GR. First conceived by Einstein [294, 296], it was unclear for a long time whether they were truly physical. Only in the 1960s were their existence and properties founded on a sound mathematical basis (see [450, 451] and references therein). In the same period, after the seminal work of Weber [770], the scientific community was starting a growing experimental effort to directly detect GWs. The first detectors were resonant antennas; their sensitivity was far too low to detect any signal (unless a nearby galactic supernova exploded when the detector was taking data), and they were eventually replaced by interferometric detectors. The first generation of such detectors (LIGO, Virgo, GEO600, TAMA) did not reveal any gravitational signal, but the second generation (Advanced LIGO/Virgo [517, 761]) should be operative by 2015 and is expected to make the first detection of GWs. In parallel, Pulsar Timing Arrays are promising to detect ultra-low frequency GWs [507], whereas the polarization of the cosmic microwave background can be used as a detector of GWs from an inflationary epoch in the very early universe [690, 369, 725, 659, 7]. In the subsequent years more sensitive detectors, such as the underground cryogenic interferometer KAGRA [462] (and, possibly, ET [299]) and possibly a space-based detector such as LISA/eLISA [302], will allow us to know the features of the signal in more detail, and then to use this information to learn about the physics of the emitting sources, and the nature of the gravitational interaction.

Soon after the beginning of the experimental efforts to build a GW detector, it became clear that the detection of GWs emitted by astrophysical sources would open a new window of observational astronomy, in addition to the electromagnetic spectrum, neutrinos, cosmic rays, etc. The impact of such a detection would be similar to that of X-rays from astrophysical sources, i.e., the birth of a new branch of astronomy: “GW astronomy” [628, 370, 686]. In this new field, source modelling is crucial, since a theoretical understanding of the expected GW sources is needed to enhance the chances of detection and to extract the relevant physics. Indeed, template-matching techniques — frequently used in data analysis — can be helpful to extract the signal from the detector noise, but they require an a-priori knowledge of the waveforms [752].

A wide scientific community formed, with the aim to model the physical processes that are expected to produce a detectable GW signal, and to compute the emitted gravitational waveform (which depends on the unknown parameters of the source and of the emitting process). Together with the understanding of the two-body problem in GR, this effort was one of the main driving forces leading to the development of NR. Indeed, many promising GW sources can only be modeled by solving the fully nonlinear Einstein equations numerically.

Ground-based interferometers are (and are expected to be in the next decades) sensitive to signals with frequencies ranging from some tens of Hz to about one kHz. Space-based interferometers would be sensitive at much lower frequencies: from some mHz to about one tenth of Hz. GW astronomy, of course, is presently concerned with sources emitting GWs in these frequency bands.

Many astrophysical processes are potential sources for GW detectors. In the following, we shall briefly discuss only some of them, i.e., those that require NR simulations to be modeled: compact binary inspirals, and instabilities of rotating NSs. We shall not discuss supernova core collapse — one of the first GW sources that have been studied with NR, and one of the most problematic to model — since it will be discussed in Section 3.1.2.

Compact binary inspirals, i.e., the inspiral and merger of binary systems formed by BHs and/or NSs, are the most promising GW sources to be detected. Advanced LIGO/Virgo are expected to detect some tens of these sources per year [5]. While the inspiral phase of a compact binary system can be accurately modeled through PN approaches, and the final (”ringdown”) phase, when the BH resulting from the coalescence oscillates in its characteristic proper modes, can be accurately described through perturbative approaches, the intermediate merger phase can only be modeled by NR. This task has posed formidable theoretical and computational challenges to the scientific community.

The numerical simulation of the merger phase of a BH-BH binary coalescence, and the determination of the emitted gravitational waveform, had been an open problem for decades, until it was solved in 2005 [629, 159, 65]. This challenge forced the gravitational community to reflect on deep issues and problems arising within Einstein’s theory, such as the role of singularities and horizons, and the possible ways to locally define energy and momentum.

BH-NS and NS-NS binary coalescences pose a different sort of problems than those posed by BH-BH coalescences. They are not a “clean” system such as purely vacuum BH spacetimes, characterized by the gravitational interaction only. An accurate numerical modeling involves various branches of physics (nuclear physics, neutrinos, electromagnetic fields), and requires the understanding of many different processes. Typically, NR simulations of BH-NS and NS-NS mergers make simplifying assumptions, both because taking into account all aspects at the same time would be too complicated, and because some of them are not fully understood. Currently, the behaviour of matter in the inner core of a NS is one of the challenges to be tackled. Indeed, nuclear physicists still do not understand which is the equation of state of matter at such extreme conditions of density and temperature (see, e.g., [501] and references therein). This uncertainty reflects our ignorance on the behaviour of the hadronic interactions in the non-perturbative regime. On the other hand, understanding the NS equation of state is considered one of the main outcomes expected from the detection of a GW signal emitted by NSs, for instance in compact binary coalescences [583, 640, 80, 738].

Neutron star oscillations are also a candidate GW source for ground-based interferometers. When perturbed by an external or internal event, a NS can be set into non-radial damped oscillations, which are associated to the emission of GWs. The characteristic frequencies of oscillation, the QNMs, are characterized by their complex frequency ω = σ + i/τ, where σ is the pulsation frequency, and σ is the damping time of the oscillation (for detailed discussions on the QNMs of NSs and BHs see [487, 580, 316, 95] and references therein).

If a NS rotates, its oscillations can become unstable. In this case, the oscillation grows until the instability is suppressed by some damping mechanism or by nonlinear effects; this process can be associated to a large GW emission (see, e.g., [34] and references therein). These instabilities may explain the observed values of the NS rotation rates [101]. Their numerical modeling, however, is not an easy task. Perturbative approaches, which easily allow one to compute the QNMs of non-rotating NSs, become very involved in the presence of rotation. Therefore, the perturbation equations can only be solved with simplifying assumptions, which make the model less accurate. Presently, NR is the only way to model stationary, rapidly rotating NSs (see, e.g., [728] and references therein), and it has recently been applied to model their oscillations [842].

Collapse in general relativity

Decades before any observation of supermassive compact objects, and long before BHs were understood, Chandrasekhar showed that the electron degeneracy pressure in very massive white dwarfs is not enough to prevent them from imploding [193]. Similar conclusions were reached later by Oppenheimer and Volkoff, for neutron degeneracy pressure in NSs [590]. We can use Landau’s original argument to understand these results [498, 499, 691]: consider a star of radius R composed of N fermions, each of mass mF. The momentum of each fermion is pFħn1/3, with n = N/R3 the number density of fermions. In the relativistic regime, the Fermi energy per particle then reads EF = pFc = ħcN1/3/R. The gravitational energy per fermion is approximately \({E_G} \sim - Gm_F^2/R\), and the star’s total energy is thus,

$$E \equiv {E_F} + {E_G} = {{\hbar c{N^{1/3}} - GNm_F^2} \over R}.$$

For small N, the total energy is positive, and we can decrease it by increasing R. At some point the fermion becomes non-relativistic and \({E_F} \sim p_F^2 \sim 1/{R^2}\). In this regime, the gravitational binding energy EG dominates over EF, the total energy is negative and tends to zero as R. Thus there is a local minimum and the star is stable. However, for large N in the relativistic regime the total energy is negative, and can be made even more negative by decreasing R: it is energetically favoured for the star to continually collapse! The threshold for stability occurs at a zero of the total energy, when

$${N_{\max}} > {\left({{{\hbar c} \over {Gm_F^2}}} \right)^{3/2}},$$
$${M_{\max}} \sim {N_{\max}}{m_F} \sim {\left({{{\hbar c} \over {Gm_F^{4/3}}}} \right)^{3/2}}.$$

for neutrons, stars with masses above ∼ 3 M cannot attain equilibrium.

What is the fate of massive stars whose pressure cannot counter-balance gravity? Does the star’s material continually collapse to a single point, or is it possible that pressure or angular momentum become so important that the material bounces back? The answer to these questions would take several decades more, and was one of the main driving forces to develop solid numerical schemes to handle Einstein’s equations.

Other developments highlighted the importance of understanding gravitational collapse in GR. One was the advent of GW detectors. The strongest sources of GWs are compact and moving relativistically, and supernovae are seemingly ideal: they occur frequently and are extremely violent. Unfortunately, Birkhoff’s theorem implies that spherically symmetric sources do not radiate. Thus a careful, and much more complex analysis of collapse is required to understand these sources.

In parallel, BH physics was blooming. In the 1970s one key result was established: the uniqueness theorem, stating that — under general regularity assumptions — the only stationary, asymptotically flat, vacuum solution of Einstein’s field equations is the Kerr BH. Thus, if a horizon forms, the final stationary configuration is expected to be of the Kerr family. This important corollary of Einstein’s field equations calls for a dynamical picture of BH formation through collapse and an understanding of how the spacetime multipolar structure dynamically changes to adapt to the final Kerr solution as a BH forms.


It has been known since the early 1960s that GWs emitted by accelerated particles do not only carry energy but also momentum away from the system on which thus is imparted a kick or recoil. This effect was first studied by Bonnor & Rotenberg [119] for the case of a system of oscillating particles, and has been identified by Peres [612] to be at leading order due to the interference of the mass quadrupole radiation with the mass octupole or flow quadrupole.

From an astrophysical point of view, the most important processes generating such gravitational recoil are the collapse of a stellar core to a compact object and the inspiral and merger of compact binaries. Supermassive BHs with masses in the range of 105 M to 1010 M in particular are known to reside at the centre of many galaxies and are likely to form inspiralling binary systems as a consequence of galaxy mergers. Depending on the magnitude of the resulting velocities, kicks can in principle displace or eject BHs from their hosts and therefore play an important role in the formation history of these supermassive BHs.

The first calculations of recoil velocities based on perturbative techniques have been applied to gravitational collapse scenarios by Bekenstein [84] and Moncrief [556]. The first analysis of GW momentum flux generated by binary systems was performed by Fitchett [322] in 1983 for two masses in Keplerian orbit. The following two decades saw various (semi-)analytic calculations for inspiraling compact binary systems using the particle approximation, post-Newtonian techniques and the close-limit approach (see Section 5 for a description of these techniques and main results). In conclusion of these studies, it appeared likely that the gravitational recoil from non-spinning binaries was unlikely to exceed a few hundred km/s. Precise estimates, however, are dependent on an accurate modeling of the highly nonlinear late inspiral and merger phase and therefore required NR simulations. Furthermore, the impact of spins on the resulting velocities remained essentially uncharted territory until the 2005 breakthroughs of NR made possible the numerical simulations of these systems. As it turned out, some of the most surprising and astrophysically influencial results obtained from NR concern precisely the question of the gravitational recoil of spinning BH binaries.

Astrophysics beyond Einstein gravity

Although GR is widely accepted as the standard theory of gravity and has survived all experimental and observational (weak field) scrutiny, there is convincing evidence that it is not the ultimate theory of gravity: since GR is incompatible with quantum field theory, it should be considered as the low energy limit of some, still elusive, more fundamental theory. In addition, GR itself breaks down at small length scales, since it predicts singularities. For large scales, on the other hand, cosmological observations show that our universe is filled with dark matter and dark energy, of as yet unknown nature.

This suggests that the strong-field regime of gravity — which has barely been tested so far — could be described by some modification or extension of GR. In the next few years both GW detectors [786, 826] and astrophysical observations [635] will provide an unprecedented opportunity to probe the strong-field regime of the gravitational interaction, characterized by large values of the gravitational field \(\sim {{GM} \over {r{c^2}}}\) or of the spacetime curvature \(\sim {{GM} \over {{r^3}{c^2}}}\) (it is a matter of debate which of the two parameters is the most appropriate for characterizing the strong-field gravity regime [635, 826]). However, our present theoretical knowledge of strong-field astrophysical processes is based, in most cases, on the a-priori assumption that GR is the correct theory of gravity. This sort of theoretical bias [825] would strongly limit our possibility of testing GR.

It is then of utmost importance to understand the behaviour of astrophysical processes in the strong gravity regime beyond the assumption that GR is the correct theory of gravity. The most powerful tool for this purpose is probably NR; indeed, although NR has been developed to solve Einstein’s equations (possibly coupled to other field equations), it can in principle be extended and modified, to model physical processes in alternative theories of gravity. In summary, NR can be applied to specific, well motivated theories of gravity. These theories should derive from — or at least be inspired by — some more fundamental theories or frameworks, such as for instance SMT [366, 624] (and, to some extent, Loop Quantum Gravity [657]). In addition, such theories should allow a well-posed initial-value formulation of the field equations. Various arguments suggest that the modifications to GR could involve [826] (i) additional degrees of freedom (scalar fields, vector fields); (ii) corrections to the action at higher order in the spacetime curvature; (iii) additional dimensions.

Scalar-tensor theories for example (see, e.g., [337, 783] and references therein), are the most natural and simple generalizations of GR including additional degrees of freedom. In these theories, which include for instance Brans-Dicke gravity [128], the metric tensor is non-minimally coupled with one or more scalar fields. In the case of a single scalar field (which can be generalized to multi-scalar-tensor theories [242]), the action can be written as

$$S = {1 \over {16\pi G}}\int {{{\rm{d}}^4}x} \sqrt {- g} [F(\phi)R - 8\pi Gz(\phi){g^{\mu \nu}}{\partial _\mu}\phi {\partial _\nu}\phi - U(\phi)] + {S_m}({\psi _m},{g_{\mu \nu}})$$

where R is the Ricci scalar associated to the metric gμν, F, Z, U are arbitrary functions of the scalar field ϕ, and Sm is the action describing the dynamics of the other fields (which we call “matter fields”, ψm). A more general formulation of scalar-tensor theories yielding second order equations of motion has been proposed by Horndeski [435] (see also Ref. [260]).

Scalar-tensor theories can be obtained as low-energy limits of SMT [342]; this provides motivation for studying these theories on the grounds of fundamental physics. An additional motivation comes from the recently proposed “axiverse” scenario [49, 50], in which ultra-light axion fields (pseudo-scalar fields, behaving under many respects as scalar fields) arise from the dimensional reduction of SMT, and play a role in cosmological models.

Scalar-tensor theories are also appealing alternatives to GR because they predict new phenomena, which are not allowed in GR. In these theories, the GW emission in compact binary coalescences has a dipolar ( =1) component, which is absent in GR; if the scalar field has a (even if extremely small) mass, superradiant instabilities occur [183, 604, 794], which can determine the formation of floating orbits in extreme mass ratio inspirals [165, 824], and these orbits affect the emitted GW signal; last but not least, under certain conditions isolated NSs can undergo a phase transition, acquiring a nontrivial scalar-field profile (spontaneous scalarization [242, 243]) while dynamically evolving NSs — requiring full NR simulations to understand — may display a similar effect (dynamical scalarization [73, 596]). A detection of one of these phenomena would be a smoking gun of scalar-tensor gravity.

These theories, whose well-posedness has been proved [669, 670], are a perfect arena for NR. Recovering some of the above smoking-gun effects is extremely challenging, as the required timescales are typically very large when compared to any other timescales in the problem.

Other examples for which NR can be instrumental include theories in which the EinsteinHilbert action is modified by including terms quadratic in the curvature (such as R2, RμνRμν, \({R_{\mu \nu \alpha \beta}}{R^{\mu \nu \alpha \beta}},{\epsilon _{\mu \nu \alpha \beta}}{R^{\mu \nu \rho \sigma}}{R^{\alpha \beta}}_{\rho \sigma}\)), possibly coupled with scalar fields, or theories which explicitly break Lorentz invariance. In particular, Einstein-Dilaton-Gauss-Bonnet gravity and Dynamical Chern-Simons gravity [602, 27] can arise from SMT compactifications, and Dynamical Chern-Simons gravity also arises in Loop Quantum Gravity; theories such as Einstein-Aether [456] and “Horava-Lifshitz” gravity [433], which break Lorentz invariance (while improving, for instance, renormalizability properties of GR), allow the basic tenets of GR to be challenged and studied in depth.

Fundamental and mathematical issues

Cosmic censorship

Spacetime singularities signal the breakdown of the geometric description of the spacetime, and can be diagnosed by either the blow-up of observer-invariant quantities or by the impossibility to continue timelike or null geodesics past the singular point. For example, the Schwarzschild geometry has a curvature invariant RabcdRabcd = 48 G2M2/(c4r6) in Schwarzschild coordinates, which diverges at r = 0, where tidal forces are also infinite. Every timelike or null curve crossing the horizon necessarily hits the origin in finite proper time or affine parameter and, therefore, the theory breaks down at these points: it fails to predict the future development of an object that reaches the singular point. Thus, the classical theory of GR, from which spacetimes with singularities are obtained, is unable to describe these singular points and contains its own demise. Adding to this classical breakdown, it is likely that quantum effects take over in regions where the curvature radius becomes comparable to the scale of quantum processes, much in the same way as quantum electrodynamics is necessary in regions where EM fields are large enough (as characterized by the invariant E2B2) that pair creation occurs. Thus, a quantum theory of gravity might be needed close to singularities.

It seems therefore like a happy coincidence that the Schwarzschild singularity is cloaked by an event horizon, which effectively causally disconnects the region close to the singularity from outside observers. This coincidence introduces a miraculous cure to GR’s apparently fatal disease: one can continue using classical GR for all practical purposes, while being blissfully ignorant of the presumably complete theory that smoothens the singularity, as all those extra-GR effects do not disturb processes taking place outside the horizon.

Unfortunately, singularities are expected to be quite generic: in a remarkable set of works, Hawking and Penrose have proved that, under generic conditions and symmetries, collapse leads to singularities [608, 402, 408, 570]. Does this always occur, i.e., are such singularities always hidden to outside observers by event horizons? This is the content of Penrose’s “cosmic censorship conjecture”, one of the outstanding unsolved questions in gravity. Loosely speaking, the conjecture states that physically reasonable matter under generic initial conditions only forms singularities hidden behind horizons [767].

The cosmic-censorship conjecture and the possible existence of naked singularities in our universe has triggered interest in complex problems which can only be addressed by NR. This is a very active line of research, with problems ranging from the collapse of matter to the nonlinear stability of “black” objects.

Stability of black hole interiors

As discussed in Section 3.1.2, the known fermionic degeneracies are unable to prevent the gravitational collapse of a sufficiently massive object. Thus, if no other (presently unkown) physical effect can prevent it, according to GR, a BH forms. From the uniqueness theorems (cf. Section 4.1.1), this BH is described by the Kerr metric. Outside the event horizon, the Kerr family — a 2-parameter family described by mass M and angular momentum J — varies smoothly with its parameters. But inside the event horizon a puzzling feature occurs. The interior of the J = 0 solution — the Schwarzschild geometry — is qualitatively different from the J = 0 case. Indeed, inside the Schwarzschild event horizon a point-like, spacelike singularity creates a boundary for spacetime. Inside the 0 < J < M2 Kerr event horizon, by contrast, there is a ring-like, timelike singularity, beyond which another asymptotically flat spacetime region, with r < 0 in Boyer-Lindquist coordinates, may be reached by causal trajectories. The puzzling feature is then the following: according to these exact solutions, the interior of a Schwarzschild BH, when it absorbes an infinitesimal particle with angular momentum, must drastically change, in particular by creating another asymptotically flat region of spacetime.

This latter conclusion is quite unreasonable. It is more reasonable to expect that the internal structure of an eternal Kerr BH must be very different from that of a Kerr BH originating from gravitational collapse. Indeed, there are arguments, of both physical [609] and mathematical nature [198], indicating that the Cauchy horizon (i.e., inner horizon) of the eternal charged or rotating hole is unstable against small (linear) perturbations, and therefore against the accretion of any material. The natural question is then, what is the endpoint of the instability?

As a toy model for the more challenging Kerr case, the aforementioned question was considered in the context of spherical perturbations of the RN BH by Poisson and Israel. In their seminal work, the phenomenon of mass inflation was unveiled [621, 622]: if ingoing and outgoing streams of matter are simultaneously present near the inner horizon, then relativistic counter-streamingFootnote 2 between those streams leads to exponential growth of gauge-invariant quantities such as the interior (Misner-Sharp [552]) mass, the center-of-mass energy density, or curvature scalar invariants. Since this effect is causally disconnected from any external observers, the mass of the BH measured by an outside observer remains unchanged by the mass inflation going on in the interior. But this inflation phenomenon causes the spacetime curvature to grow to Planckian values in the neighbourhood of the Cauchy horizon. The precise nature of this evolution for the Kerr case is still under study. For the simpler RN case, it has been argued by Dafermos, using analytical methods, that the singularity that forms is not of space-like nature [234]. Fully nonlinear numerical simulations will certainly be important for understanding this process.

Most luminous events in the universe

The most advanced laser units on the planet can output luminosities as high as ∼ 1018 W [301], while at ∼ 1026 W the Tsar Bomba remains the most powerful artificial explosion ever [732]. These numbers pale in comparison with strongly dynamical astrophysical events: a γ-ray burst, for instance, reaches luminosities of approximately ∼ 1045 W. A simple order of magnitude estimate can be done to estimate the total luminosity of the universe in the EM spectrum, by counting the total number of stars, roughly 1023 [443]. If all of them have a luminosity equal to our Sun, we get a total luminosity of approximately ∼ 1049 W, a number which can also be arrived at through more careful considerations [781]. Can one possibly surpass this astronomical number?

In four spacetime dimensions, there is only one constant with dimensions of energy per second that can be built out of the classical universal constants. This is the Planck luminosity \({\mathcal L_G}\),

$${\mathcal L}_G \equiv {{{c^5}} \over G} = 3.7 \times {10^{52}}{\rm{W}}.$$

The quantity \({\mathcal L_G}\) should control gravity-dominated dynamical processes; as such it is no wonder that these events release huge luminosities. Take the gravitational collapse of a compact star with mass M and radius RGM/c2. During a collapse time of the order of the infall time, \(\tau \sim R/\sqrt {GM/R} \sim GM/{c^3}\), the star can release an energy of up to Mc2. The process can therefore yield a power as large as \({c^5}/G = {\mathcal L_G}\). It was conjectured by Thorne [751] that the Planck luminosity is in fact an upper limit for the luminosity of any process in the universe.Footnote 3 The conjecture was put on a somewhat firmer footing by Gibbons who has shown that there is an upper limit to the tension of c4/(4G), implying a limit in the luminosity of \({{\mathcal L}_G}/4\) [349].

Are such luminosities ever attained in practice, is there any process that can reach the Planck luminosity and outshine the entire universe? The answer to this issue requires once again a peek at gravity in strongly dynamical collisions with full control of strong-field regions. It turns out that high energy collisions of BHs do come close to saturating the bound (6) and that in general colliding BH binaries are more luminous than the entire universe in the EM spectrum [719, 720, 717, 716].

Higher dimensions

Higher-dimensional spacetimes are a natural framework for mathematicians and have been of general interest in physics, most notably as a tool to unify gravity with the other fundamental interactions. The quest for a unified theory of all known fundamental interactions is old, and seems hopeless in four-dimensional arenas. In a daring proposal however, Kaluza and Klein, already in 1921 and 1926 showed that such a programme might be attainable if one is willing to accept higher-dimensional theories as part of the fundamental picture [463, 476] (for a historical view, see [283]).

Consider first for simplicity the D-dimensional Klein-Gordon equation □ϕ(xμ,zi) = 0 (ϕ = 0,…,3, i = 4,…, D − 1), where the (D − 4) extra dimensions are compact of size L. Fourier decompose in \({z^j},\;\;{\rm{i}}{\rm{.e}},\;\;\phi ({x^\mu},{z^j}) = \sum\nolimits_n {\psi ({x^\mu})} {e^{in{z^j}/L}}\), to get \(\square \psi - {{{n^2}} \over {{L^2}}}\psi = 0\), where here □ is the four-dimensional d’Alembertian operator. As a consequence,

  1. i)

    the fundamental, homogenous mode n = 0 is a massless four-dimensional field obeying the same Klein-Gordon equation, whereas

  2. ii)

    even though we started with a higher-dimensional massless theory, we end up with a tower of massive modes described by the four-dimensional massive Klein-Gordon equation, with mass terms proportional to n/L. One important, generic conclusion is that the higher-dimensional (fundamental) theory imparts mass terms as imprints of the extra dimensions. As such, the effects of extra dimensions are in principle testable. However, for very small L these modes have a very high-energy and are very difficult to excite (to “see” an object of length L one needs wavelengths of the same order or smaller), thereby providing a plausible explanation for the non-observation of extra dimensions in everyday laboratory experiments.

The attempts by Kaluza and Klein to unify gravity and electromagnetism considered five-dimensional Einstein field equations with the metric appropriately decomposed as,

$${\rm{d}}{\hat s^2} = {e^{\alpha \phi}}{\rm{d}}{s^2} + {e^{- 2\alpha \phi}}{({\rm{d}}z + {\mathcal A})^2}.$$

Here, ds2 = gμν dxμ dxν is a four-dimensional geometry, \(\mathcal A = {A_\mu}\;{\rm{d}}{x^\mu}\) is a gauge field and ϕ is a scalar; the constant α can be chosen to yield the four-dimensional theory in the Einstein frame. Assuming all the fields are independent of the extra dimension z, one finds a set of four-dimensional Einstein-Maxwell-scalar equations, thereby almost recovering both GR and EM [283]. This is the basic idea behind the Kaluza-Klein procedure, which unfortunately failed due to the presence of the (undetected) scalar field.

The idea of using higher dimensions was to be revived decades later in a more sophisticated model, eventually leading to SMT. The development of the gauge/gravity duality (see Section 3.3.1 below) and TeV-scale scenarios in high-energy physics (see Section 3.3.2) highlighted the importance of understanding Einstein’s equations in a generic number of dimensions. Eventually, the study of Einstein’s field equations in D-dimensional backgrounds branched off as a subject of its own, where D is viewed as just another parameter in the theory. This area has been extremely active and productive and provides very important information on the content of the field equations and the type of solutions it admits. Recently, GR in the large D limit has been suggested as a new tool to gain insight into the D dependence of physical processes [309].

The uniqueness theorems, for example, are known to break down in higher dimensions, at least in the sense that solutions are uniquely characterized by asymptotic charges. BHs of spherical topology — the extension of the Kerr solution to higher dimensions — can co-exist with black rings [307]. In fact, a zoo of black objects are known to exist in higher dimensions, but the dynamical behavior of this zoo (of interest to understand stability of the solutions and for collisions at very high energies) is poorly known, and requires NR methods to understand.

One other example requiring NR tools is the instability of black strings. Black strings are one of the simplest vacuum solutions one can construct, by extending trivially a four-dimensional Schwarzschild BH along an extra, flat direction. Such solutions are unstable against long wavelength perturbations in the fifth dimension, which act to fragment the string. This instability is known as the Gregory-Laflamme instability [367]. The instability is similar in many aspects to the Rayleigh-Plateau instability seen in fluids, which does fragment long fluid cylinders [167]. However, the same scenario in the black string case would seem to lead to cosmic censorship violation, since the pinch-off would be accompanied by (naked) regions of unbounded curvature.Footnote 4 Evidence that the Gregory-Laflamme does lead to disruption of strings was recently put forward [511].

High-energy physics

The gauge/gravity duality

The gauge/gravity duality, or AdS/CFT correspondence, is the conjecture, first proposed by Maldacena in 1998 [536], and further developed in [798, 372], that string theory on an AdS spacetime (times a compact space) is dual (i.e., equivalent under an appropriate mapping) to a CFT defined on the boundary of the AdS space. Since its proposal, this conjecture has been supported by impressive and compelling evidence, it has branched off to, e.g., the AdS/Condensed Matter correspondence [396], and it has inspired other proposals of duality relations with a similar spirit, such as the dS/CFT correspondence [731] and the Kerr/CFT correspondence [373]. All these dualities are examples of the holographic principle, which has been proposed in the context of quantum gravity [737, 734], stating that the information contained in a D-dimensional gravitational system is encoded in some boundary of the system. The paradigmatic example of this idea is a BH spacetime, whose entropy is proportional to the horizon area.

These dualities — in which strong gravity systems play a crucial role — offer tools to probe strongly coupled gauge theories (in D − 1 dimensions) by studying classical gravity (in D dimensions). For instance, the confinement/deconfinement phase transition in quantum chromodynamicslike theories has been identified with the Hawking-Page phase transition for AdS BHs [799]. Away from thermal equilibrium, the quasi-normal frequencies of AdS BHs have been identified with the poles of retarded correlators describing the relaxation back to equilibrium of a perturbed dual field theory [439, 104]. The strongly coupled regime of gauge theories is inaccessible to perturbation theory and therefore this new tool has created expectations for understanding properties of the plasma phase of non-Abelian quantum field theories at non-zero temperature, including the transport properties of the plasma and the propagation and relaxation of plasma perturbations, experimentally studied at the Relativistic Heavy Ion Collider and now also at the LHC [189]. Strong coupling can be tackled by lattice-regularized thermodynamical calculations of quantum chromo-dynamics, but the generalization of these methods beyond static observables to characterizing transport properties has limitations, due to computational costs. An example of an experimentally accessible transport property is the dimensionless ratio of the shear viscosity to the entropy density. Applying the gauge/gravity duality, this property can be computed by determining the absorption cross section of low-energy gravitons in the dual geometry (a BH/black brane) [490], obtaining a result compatible with the experimental data. This has offered the holographic description of heavy ion collisions phenomenological credibility. An outstanding theoretical challenge in the physics of heavy ion collisions is the understanding of the ‘early thermalization problem’: the mechanism driving the short — less than 1 fm/c [414] — time after which experimental data agrees with a hydrodynamic description of the quark-gluon plasma. Holography uses \({\mathcal N} = 4\) Super Yang-Mills theory as a learning ground for the real quark-gluon plasma. Then, the formation of such plasma in the collision of high-energy ions has been modeled, in its gravity dual, by colliding gravitational shock waves in five-dimensional AdS space [205]. These strong gravity computations have already offered some insight into the early thermalization problem, by analyzing the formation and settling down of an AdS BH in the collision process. But the use of shock waves is still a caricature of the process, which could be rendered more realistic, for instance, by colliding other highly boosted lumps of energy or BHs in AdS.

Another example of gauge/gravity duality is the AdS/Condensed Matter correspondence, between field theories that may describe superconductors and strong gravity [396, 437, 397]. The simplest gravity theory in this context is Einstein-Maxwell-charged scalar theory with negative cosmological constant. The RN-AdS BH solution of this theory, for which the scalar field vanishes, is unstable for temperatures T below a critical temperature Tc. If triggered, the instability leads the scalar field to condense into a non-vanishing profile creating a scalar hair for the BH and breaking the U (1)-gauge symmetry spontaneously. The end point of the instability is a static solution that has been constructed numerically and has properties similar to those of a superconductor [398]. Thus, this instability of the RN-AdS BH at low temperature was identified with a superconducting phase transition, and the RN-AdS and hairy BHs in the gravitational theory, respectively, were identified with the normal and superconducting phases of a holographic superconductor, realized within the dual field theory. Holographic superconductors are a promising approach to understanding strongly correlated electron systems. In particular, non-equilibrium processes of strongly correlated systems, such as superconductors, are notoriously difficult and this holographic method offers a novel tool to tackle this longstanding problem. In the gauge/gravity approach, the technical problem is to solve the classical dynamics of strong gravitational systems in the dual five-dimensional spacetime. Using the AdS/CFT dictionary, one then extracts the dynamics of the phase transition for the boundary theory and obtains the time dependence of the superconducting order parameter and the relaxation time scale of the boundary theory.

Theories with lower fundamental Planck scale

As discussed in Section 3.2.4, higher-dimensional theories have been suggested since the early days of GR as a means to achieve unification of fundamental interactions. The extra dimensions have traditionally been envisaged as compact and very small (∼ Planck length), in order to be compatible with high energy experiments. Around the turn of the millennium, however, a new set of scenarios emerged wherein the extra dimensions are only probed by the gravitational interaction, because a confining mechanism ties the standard model interactions to a 3 + 1-dimensional subspace (which is called the “brane”, while the higher-dimensional spacetime is called the “bulk”). These models — also called “braneworld scenarios” — can be considered SMT inspired. The main ideas behind them are provided by SMT, including the existence of extra dimensions and also the existence of subspaces, namely Dirichlet-branes, on which a well defined mechanism exists to confine the standard model fields.

Our poor knowledge of the gravitational interaction at very short scales (below the millimeter at the time of these proposals, below ≲ 10−4 meters at the time of writing [802, 783]), allows large [40, 46, 279] or infinitely large extra dimensions [638, 639]. The former are often called ADD models, whereas the latter are known as Randall-Sundrum scenarios. Indeed these types of extra dimensions are compatible with high energy phenomenology. Besides being viable, these models (or at least some of them) have the conceptual appeal of providing an explanation for the “hierarchy problem” of particle physics: the large hierarchy between the electroweak scale (∼ 250 GeV) and the Planck scale (∼ 1019 GeV), or in other words, why the gravitational interaction seems so feeble as compared to the other fundamental interactions. The reason would be that whereas nuclear and electromagnetic interactions propagate in 3 +1 dimensions, gravity propagates in D dimensions. A 3 + 1-dimensional application of Gauss’s law then yields an incomplete account of the total gravitational flux. Thus, the apparent (3 + 1-dimensional) gravitational coupling appears smaller than the real (D-dimensional) one. Or, equivalently, the real fundamental Planck energy scale becomes much smaller than the apparent one. An estimate is obtained considering the D-dimensional gravitational action and integrating the compact dimensions by assuming the metric is independent of them:

$${\mathcal S}{\propto}{1 \over {{G_D}}}\int {{{\rm{d}}^D}x\sqrt {{}^Dg}} {}^DR = {{{V_{D - 4}}} \over {{G_D}}}\int {{{\rm{d}}^4}x} \sqrt {{}^4g} {}^4R,$$

thus the four-dimensional Newton’s constant is related to the D-dimensional one by the volume of the compact dimensions G4 = GD/VD−4.

In units such that c = ħ =1 (different from the units G = c =1 used in the rest of this paper), the mass-energy Planck scale in four dimensions \(E_{{\rm{Planck}}}^{(4)}\) is related to Newton’s constant by \({G_4} = {(E_{{\rm{Planck}}}^{(4)})^{- 2}}\), since \(\int {\rm d}^{4}x \sqrt {\ ^{4}g}\ ^{4}R\) has the dimension of length squared; similar dimensional arguments in Eq. (8) show that in D dimensions \({G_D} = {(E_{{\rm{Planck}}}^{(D)})^{- (D - 2)}}\). Therefore, the D-dimensional Planck energy \(E_{{\rm{Planck}}}^{(D)}\) is related to the four-dimensional one by

$${{E_{{\rm{Planck}}}^{(D)}} \over {E_{{\rm{Planck}}}^{(4)}}} = {\left({{1 \over {{{(E_{{\rm{Planck}}}^{(4)})}^{D - 4}}{V_{D - 4}}}}} \right)^{{1 \over {D - 2}}}} = {\left({{{{{(L_{{\rm{Planck}}}^{(4)})}^{D - 4}}} \over {{V_{D - 4}}}}} \right)^{{1 \over {D - 2}}}},$$

where we have defined the four-dimensional Planck length as \(L_{{\rm{Planck}}}^{(4)} = 1/E_{{\rm{Planck}}}^{(4)}\). For instance, for D = 10 and taking the six extra dimensions of the order of the Fermi, Eq. (9) shows that the fundamental Planck scale would be of the order of a TeV. For a more detailed account of the braneworld scenario, we refer the reader to the reviews [658, 532].

The real fundamental Planck scale sets the regime in particle physics beyond which gravitational phenomena cannot be ignored and actually become dominant [736]; this is the trans-Planckian regime in which particle collisions lead to BH formation and sizeable GW emission. A Planck scale at the order of TeV (TeV gravity scenario) could then imply BH formation in particle accelerators, such as the LHC, or in ultra high-energy cosmic rays [69, 279, 353]. Well into the trans-Planckian regime, i.e., for energies significantly larger than the Planck scale, classical gravity described by GR in D-dimensions is the appropriate description for these events, since the formed BHs are large enough so that quantum corrections may be ignored on and outside the horizon.

In this scenario, phenomenological signatures for BH formation would be obtained from the Hawking evaporation of the micro BHs, and include a large multiplicity of jets and large transverse momentum as compared to standard model backgrounds [1]. Preliminary searches of BH formation events in the LHC data have already been carried out, considering pp collisions with center-of-mass energies up to 8 TeV; up to now, no evidence of BH creation has been found [201, 3, 202, 4]. To filter experimental data from particle colliders, Monte Carlo event generators have been coded, e.g., [336], which need as input the cross section for BH formation and the inelasticity in the collisions (gravitationally radiated energy). The presently used values come from apparent horizon (AH) estimates, which in D = 4 are known to be off by a factor of 2 (at least). In D-dimensions, these values must be obtained from numerical simulations colliding highly boosted lumps of energy, BHs or shock waves, since it is expected that in this regime ‘matter does not matter’; all that matters is the amount of gravitational charge, i.e., energy, carried by the colliding objects.

Exact Analytic and Numerical Stationary Solutions

Any numerical or analytic analysis of dynamical processes must start with a careful analysis of the static or stationary solutions underlying those dynamics. In GR this is particularly relevant, as stationary solutions are known and have been studied for many decades, and important catalogs have been built. Furthermore, stationary solutions are also relevant in a NR context: they can be used as powerful benchmarks, initial data for nonlinear evolutions, and as a final state reference to interpret results. We now briefly review some of the most important, and recent, work on the subject directly relevant to ongoing NR efforts. This Section does not dispense with the reading of other reviews on the subject, for instance Refs. [727, 308, 438, 790].

Exact solutions

Four-dimensional, electrovacuum general relativity with Λ

Exact solutions of a nonlinear theory, such as GR, provide invaluable insights into the physical properties of the theory. Finding such solutions analytically and through a direct attack, that is by inserting an educated ansatz into the field equations, can be a tour de force, and, in general, only leads to success if a large isometry group is assumed from the beginning for the spacetime geometry. For instance, assuming spherical symmetry, in vacuum, leads to a fairly simple problem, whose general solution is the Schwarzschild metric [687]. This simplicity is intimately connected with the inexistence of a spherically symmetric mode for gravitational radiation in Einstein gravity, which means that, in vacuum, a spherically symmetric solution must be static, as recognized by Birkhoff [103]. On the other hand, assuming only axial symmetry leads to a considerably more difficult problem, even under the additional assumption of staticity. This problem was first considered by Weyl [776] who unveiled a curious and helpful mapping from these solutions to axially symmetric solutions of Newtonian gravity in an auxiliary 3-dimensional flat space; under this mapping, a solution to the latter problem yields a solution to the vacuum Einstein equations: a Weyl solution. For instance, the Schwarzschild solution of mass M can be recovered as a Weyl solution from the Newtonian gravitational field of an infinitely thin rod of linear density 1/2 and length 2 M. As we shall discuss in Section 4.1.2, the generalization of Weyl solutions plays an important role in the construction of qualitatively new solutions to the higher-dimensional Einstein equations.

Within the axially symmetric family of solutions, the most interesting case from the astrophysical viewpoint is the solution for a rotating source, which could describe the gravitational field exterior to a rotating star or the one of a rotating BH. An exact solution of Einstein’s equations describing the exterior of a rotating star has not been found (rotating stars are described using perturbative and numerical approaches [728]),Footnote 5 but in the case of a rotating BH, such a solution does exist. To obtain this stationary, rather than static, geometry, the Weyl approach by itself is unhelpful and new methods had to be developed. These new methods started with Petrov’s work on the classification of the Weyl tensor types [616]. The Weyl tensor determines four null complex ‘eigenvectors’ at each point, and the spacetime is called ‘algebraically special’ if at least two of these coincide. Imposing the algebraically special condition has the potential to reduce the complicated nonlinear PDEs in two variables, obtained for a vacuum axially symmetric stationary metric, to ordinary differential equations. Using the (then) recently shown Goldberg-Sachs theorem [357], Kerr eventually succeeded in obtaining the celebrated Kerr metric in 1963 [466]. This family of solutions was generalized to include charge by Newman et al. — the Kerr-Newman solution [576] — and to include a cosmological constant by Carter [187]. In Boyer-Lindquist coordinates, the Kerr-Newman-(A)dS metric reads:

$${\rm{d}}{s^2} = - {{{\Delta _r}} \over {{\rho ^2}}}{\left[ {{\rm{d}}t - {{a{{\sin}^2}\theta} \over \Sigma}{\rm{d}}\phi} \right]^2} + {{{\rho ^2}} \over {{\Delta _r}}}{\rm{d}}{r^2} + {{{\rho ^2}} \over {{\Delta _\theta}}}{\rm{d}}{\theta ^2} + {{{\Delta _\theta}{{\sin}^2}\theta} \over {{\rho ^2}}}{\left[ {a\;{\rm{d}}t - {{{r^2} + {a^2}} \over \Sigma}{\rm{d}}\phi} \right]^2},$$


$${\rho ^2} = {r^2} + {a^2}{\cos ^2}\theta, \quad \Sigma = 1 + {{{a^2}\Lambda} \over 3},$$
$${\Delta _r} = ({r^2} + {a^2})\left({1 - {{{r^2}\Lambda} \over 3}} \right) - 2Mr + {Q^2} + {P^2},\quad {\Delta _\theta} = 1 + {{{a^2}\Lambda} \over 3}{\cos ^2}\theta.$$

Here, M, aM, Q, P, Λ are respectively, the BH mass, angular momentum, electric charge, magnetic charge and cosmological constant.

At the time of its discovery, the Kerr metric was presented as an example of a stationary, axisymmetric (BH) solution. The outstanding importance of the Kerr metric was only realized some time later with the establishment of the uniqueness theorems [188, 654]: the only asymptotically flat, stationary and axisymmetric, electrovacuum solution to the Einstein equations, which is nonsingular on and outside an event horizon is the Kerr-Newman geometry. Moreover, Hawking’s rigidity theorem [406] made the axisymmetric assumption unnecessary: a stationary BH must indeed be axisymmetric. Although the stability of the Kerr metric is not a closed subject, the bottom line is that it is widely believed that the final equilibrium state of the gravitational collapse of an enormous variety of different stars is described by the Kerr geometry, since the electric charge should be astrophysically negligible. If true, this is indeed a truly remarkable fact (see, however, Section 4.2 for “hairy” BHs).

Even if we are blessed to know precisely the metric that describes the final state of the gravitational collapse of massive stars or of the merger of two BHs, the geometry of the time-dependent stages of these processes seems desperately out of reach as an exact, analytic solution. To understand these processes we must then resort to approximate or numerical techniques.

Beyond four-dimensional, electrovacuum general relativity with Λ

As discussed in Section 3 there are various motivations to consider generalizations of (or alternative theories to) four-dimensional electrovacuum GR with A. A natural task is then to address the exact solutions of such theories. Here we shall briefly address the exact solutions in two different classes of modifications of Einstein electrovacuum gravity: i) changing the dimension, D ≠ 4; ii) changing the equations of motion, either by changing the right-hand side — i.e., theories with different matter fields, including non-minimally coupled ones —, or by changing the left-hand side — i.e., higher curvature gravity. We shall focus on relevant solutions for the topic of this review article, referring to the specialized literature where appropriate.

  • Changing the number of dimensions: GR in D ≠ 4. Exact solutions in higher-dimensional GR, D > 4, have been explored intensively for decades and an excellent review on the subject is Ref. [308]. In the following we shall focus on the vacuum case.

    The first classical result is the D > 4 generalization of the Schwarzschild BH, i.e., the vacuum, spherically — that is SO(D − 1) — symmetric solution to the D-dimensional Einstein equations (with or without cosmological constant), obtained by Tangherlini [740] in the same year the Kerr solution was found. Based on his solution, Tangherlini suggested an argument to justify the (apparent) dimensionality of spacetime. But apart from this insight, the solution is qualitatively similar to its four-dimensional counterpart: an analog of Birkhoff’s theorem holds and it is perturbatively stable.

    On the other hand, the existence of extra dimensions accommodates a variety of extended objects with reduced spherical symmetry — that is SO(D − 1 − p) — surrounded by an event horizon, generically dubbed as p-branes, where p stands for the spatial dimensionality of the object [441, 285]. Thus, a point-like BH is a 0-brane, a string is a 1-brane and so on. The charged counterparts of these objects have played a central role in SMT, especially when charged under a type of gauge field called ‘Ramond-Ramond’ fields, in which case they are called Dp-branes or simply D-branes [284]. Here we wish to emphasize that the Gregory-Laflamme instability discussed in Section 3.2.4 was unveiled in the context of p-branes, in particular black strings [367, 368]. The understanding of the nonlinear development of such instability is a key question requiring numerical techniques.

    The second classical result was the generalization of the Kerr solution to higher dimensions, i.e., a vacuum, stationary, axially — that isFootnote 6 \(SO{(2)^{[{{D - 1} \over 2}]}}\) — symmetric solution to the D-dimensional Einstein equations, obtained in 1986 by Myers and Perry [565] (and later generalized to include a cosmological constant [351, 350]). The derivation of this solution was quite a technical achievement, made possible by using a Kerr-Schild type ansatz. The solution exhibits a number of new qualitative features, in particular in what concerns its stability. It has \([{{D - 1} \over 2}]\) independent angular momentum parameters, due to the nature of the rotation group in D dimensions. If only one of these rotation parameters is non-vanishing, i.e., for the singly spinning Myers-Perry solution, in dimensions D ≥ 6 there is no bound on the angular momentum J in terms of the BH mass M. Ultra-spinning Myers-Perry BHs are then possible and their horizon appears highly deformed, becoming locally analogous to that of a p-brane. This similarity suggests that ultra-spinning BHs should suffer from the Gregory-Laflamme instability. Entropic arguments also support the instability of these BHs [305] (see Section 7.4 for recent developments).

    The third classical result was the recent discovery of the black ring in D = 5 [307], a black object with a non-simply connected horizon, having spatial sections that are topologically S2 × S1. Its discovery raised questions about how the D = 4 results on uniqueness and stability of vacuum solutions generalized to higher-dimensional gravity. Moreover, using the generalization to higher dimensions of Weyl solutions [306] and of the inverse scattering technique [394], geometries with a non-connected event horizon — i.e., multi-object solutions — which are asymptotically flat, regular on and outside an event horizon have been found, most notably the black Saturn [303]. Such solutions rely on the existence of black objects with non-spherical topology; regular multi-object solutions with only Myers-Perry BHs do not seem to exist [425], just as regular multi-object solutions with only Kerr BHs in D = 4 are inexistent [574, 424].

    Let us briefly mention that BH solutions in lower dimensional GR have also been explored, albeit new ingredients are necessary for such solutions to exist. D = 3 vacuum GR has no BH solutions, a fact related to the lack of physical dimensionality of the would be Schwarzschild radius MG(3), where G(3) is the 3-dimensional Newton’s constant. The necessary extra ingredient is a negative cosmological constant; considering it leads to the celebrated Bañados-Teitelboim-Zanelli (BTZ) BH [68]. In D = 2 a BH spacetime was obtained by Callan, Giddings, Harvey and Strominger (the CGHS BH), by considering GR non-minimally coupled to a scalar field theory [156]. This solution provides a simple, tractable toy model for numerical investigations of dynamical properties; for instance see [55, 54] for a numerical study of the evaporation of these BHs.

  • Changing the equations: Different matter fields and higher curvature gravity.

    The uniqueness theorems of four-dimensional electrovacuum GR make clear that BHs are selective objects. Their equilibrium state only accommodates a specific gravitational field, as is clear, for instance, from its constrained multipolar structure. In enlarged frameworks where other matter fields are present, this selectiveness may still hold, and various “no-hair theorems” have been demonstrated in the literature, i.e., proofs that under a set of assumptions no stationary regular BH solutions exist, supporting (nontrivial) specific types of fields. A prototypical case is the set of no-hair theorems for asymptotically flat, static, spherically symmetric BHs with scalar fields [546]. Note, however, that hairy BHs, do exist in various contexts, cf. Section 4.2.

    The inexistence of an exact stationary BH solution, i.e., of an equilibrium state, supporting (say) a specific type of scalar field does not mean, however, that a scalar field could not exist long enough around a BH so that its effect becomes relevant for the observed dynamics. To analyse such possibilities dynamical studies must be performed, typically involving numerical techniques, both in linear and nonlinear analysis. A similar discussion applies equally to the study of scalar-tensor theories of gravity, where the scalar field may be regarded as part of the gravitational field, rather than a matter field. Technically, these two perspectives may be interachanged by considering, respectively, the Jordan or the Einstein frame. The emission of GWs in a binary system, for instance, may depend on the ‘halo’ of other fields surrounding the BH and therefore provide smoking guns for testing this class of alternative theories of gravity.

    Finally, the change of the left-hand side of the Einstein equations may be achieved by considering higher curvature gravity, either motivated by ultraviolet corrections to GR, i.e., changing the theory at small distance scales, such as Gauss-Bonnet [844] (in D ≥ 5), Einstein-Dilaton-Gauss-Bonnet gravity and Dynamical Chern-Simons gravity [602, 27]; or infrared corrections, changing the theory at large distance scales, such as certain f (R) models. This leads, generically, to modifications of the exact solutions. For instance, the spherically symmetric solution to Gauss-Bonnet theory has been discussed in Ref. [120] and differs from, but asymptotes to, the Tangherlini solution. In specific cases, the higher curvature model may share some GR solutions. For instance, Chern-Simons gravity shares the Schwarzschild solution but not the Kerr solution [27]. Dynamical processes in these theories are of interest but their numerical formulation, for fully nonlinear processes, may prove challenging or even, apart from special cases (see, e.g., [265] for a study of critical collapse in Gauss-Bonnet theory), ill-defined.

State of the art

  • D ≠ 4: The essential results in higher-dimensional vacuum gravity are the Tangherlini [740] and Myers-Perry [565] BHs, the (vacuum) black p-branes [441, 285] and the Emparan-Reall black ring [307]. Solutions with multi-objects can be obtained explicitly in D = 5 with the inverse scattering technique. Their line element is typically quite involved and given in Weyl coordinates (see [308] for a list and references). The Myers-Perry geometry with a cosmological constant was obtained in D = 5 in Ref. [407] and for general D and cosmological constant in [351, 350]. Black rings have been generalized, as numerical solutions, to higher D in Ref. [472]. Black p-branes have been discussed, for instance, in Ref. [441, 285]. In D = 3, 2 the best known examples of BH solutions are, respectively, the BTZ [68] and the CGHS BHs [156].

  • Changing the equations of motion: Hawking showed [404] that in Brans-Dicke gravity the only stationary BH solutions are the same as in GR. This result was recently extended by Sotiriou and Faraoni to more general scalar-tensor theories [712]. Such type of no-hair statements have also been proved for spherically symmetric solutions in GR (non-)minimally coupled to scalar fields [85] and to the electromagnetic field [546]; but they are not universal: for instance, a harmonic time dependence for a (complex) scalar field or a generic potential (together with gauge fields) are ways to circumvent these results (see Section 4.2 and e.g. the BH solutions in [352]). BHs with scalar hair have also been recently argued to exist in generalized scalar-tensor gravity [713].

Numerical stationary solutions

Given the complexity of the Einstein equations, it is not surprising that, in many circumstances, stationary exact solutions cannot be found in closed analytic form. In this subsection we shall very briefly mention numerical solutions to such elliptic problems for cases relevant to this review.

The study of the Einstein equations coupled to nonlinear matter sources must often be done numerically, even if stationarity and spatial symmetries — typically spherical or axisymmetry — are imposed.Footnote 7 The study of numerical solutions of elliptic problems also connects to research on soliton-like solutions in nonlinear field theories without gravity. Some of these solitons can be promoted to gravitating solitons when gravity is included. Skyrmions are one such case [107]. In other cases, the nonlinear field theory does not have solitons but, when coupled to gravity, gravitating solitons arise. This is the case of the Bartnik-McKinnon particle-like solutions in Einstein-Yang-Mills theory [77]. Moreover, for some of these gravitating solitons it is possible to include a BH at their centre giving rise to “hairy BHs”. For instance, in the case of Einstein-Yang-Mills theory, these have been named “colored BHs” [105]. We refer the reader interested in such gravitating solitons connected to hairy BHs to the review by Bizoń [106] and to the paper by Ashtekar et al. [51].

A particularly interesting type of gravitating solitons are boson stars (see [685, 516] for reviews), which have been suggested as BH mimickers and dark matter candidates. These are solutions to Einstein’s gravity coupled to a complex massive scalar field, which may, or may not, have self-interactions. Boson stars are horizonless gravitating solitons kept in equilibrium by a balance between their self-generated gravity and the dispersion effect of the scalar field’s wave-like character. All known boson star solutions were obtained numerically; and both static and rotating configurations are known. The former ones have been used in numerical high energy collisions to model particles and test the hoop conjecture [216] (see Section 7.3 and also Ref. [599] for earlier boson star collisions and [561] for a detailed description of numerical studies of boson star binaries). The latter ones have been shown to connect to rotating BHs, both for D = 5 Myers-Perry BHs in AdS [275] and for D = 4 Kerr BHs [422], originating families of rotating BHs with scalar hair. Crucial to these connections is the phenomenon of superradiance (see Section 7.5), which also afflicts rotating boson stars [182]. The BHs with scalar hair branch off from the Kerr or Myers-Perry-AdS BHs precisely at the threshold of the superradiant instability for a given scalar-field mode [423], and display new physical properties, e.g., new shapes of ergo-regions [419].

The situation we have just described, i.e., the branching off of a solution to Einstein’s field equations into a new family at the onset of a classical instability, is actually a recurrent situation. An earlier and paradigmatic example — occurring for the vacuum Einstein equations in higher dimensions — is the branching off of black strings at the onset of the Gregory-Laflamme instability [367] (see Section 3.2.4 and Section 7.2) into a family of non-uniform black strings. The latter were found numerically by Wiseman [789] following a perturbative computation by Gubser [371]. We refer the reader to Ref. [470] for more non-uniform string solutions, to Refs. [11, 790] for a discussion of the techniques to construct these numerical (vacuum) solutions and to [442] for a review of (related) Kaluza-Klein solutions. Also in higher dimensions, a number of other numerical solutions have been reported in recent years, most notably generalizations of the Emparan-Reall black ring [474, 472, 473] and BH solutions with higher curvature corrections (see, e.g., [131, 475, 132]). Finally, numerical rotating BHs with higher curvature corrections but in D = 4, within dilatonic Einstein-Gauss-Bonnet theory, were reported in [471].

In the context of holography (see Section 3.3.1 and Section 7.8), numerical solutions have been of paramount importance. Of particular interest to this review are the hairy AdS BHs that play a role in the AdS-Condensed matter duality, by describing the superconducting phase of holographic superconductors. These were first constructed (numerically) in [398]. See also the reviews [396, 436] for further developments.

In the context of Randall-Sundrum scenarios, large BHs were first shown to exist via a numerical calculation [318], and later shown to agree with analytic expansions [8].

Finally, let us mention, as one application to mathematical physics of numerical stationary solutions, the computation of Ricci-flat metrics on Calabi-Yau manifolds [409].

Approximation Schemes

The exact and numerically-constructed stationary solutions we outlined above are, as a rule, objects that can also have interesting dynamics. A full understanding of these dynamics is the subject of NR, but before attempting fully nonlinear evolutions of the field equations, approximations are often useful. These work as benchmarks for numerical evolutions, as order-of-magnitude estimates and in some cases (for example extreme mass ratios) remain the only way to attack the problem, as it becomes prohibitively costly to perform full nonlinear simulations, see Figure 1. The following is a list of tools, techniques, and results that have been instrumental in the field. For an analysis of approximation schemes and their interface with NR in four-dimensional, asymptotically flat spacetimes, see Ref. [502].

Post-Newtonian schemes

Astrophysical systems in general relativity

For many physical phenomena involving gravity, GR predicts small deviations from Newtonian gravity because for weak gravitational fields and low velocities Einstein’s equations reduce to the Newtonian laws of physics. Soon after the formulation of GR, attempts were therefore made (see, e.g., [295, 254, 522, 298, 324, 606, 619, 194, 292]) to express the dynamics of GR as deviations from the Newtonian limit in terms of an expansion parameter ϵ. This parameter can be identified, for instance, with the typical velocities of the matter composing the source, or with the compactness of the source:

$$\epsilon \sim {v \over c} \sim \sqrt {{{GM} \over {r{c^2}}}},$$

which uses the fact that, for bound systems, the virial theorem implies υ2GM/r. In this approach, called “post-Newtonian”, the laws of GR are expressed in terms of the quantities and concepts of Newtonian gravity (velocity, acceleration, etc.). A more rigorous definition of the parameter ϵ can be found elsewhere [109], but as a book-keeping parameter it is customary to consider ϵ = υ/c. The spacetime metric and the stress-energy tensor are expanded in powers of ϵ and terms of order ϵn are commonly referred to as (n/2)-PN corrections. The spacetime metric and the motion of the source are found by solving, order by order, Einstein’s equations.

Strictly speaking, the PN expansion can only be defined in the near zone, which is the region surrounding the source, with dimensions much smaller than the wavelength λGw of the emitted GWs. Outside this region, and in particular in the wave zone (e.g., at a distance ≫ λgw from the source), radiative processes make the PN expansion ill-defined, and different approaches have to be employed, such as the post-Minkowskian expansion, which assumes weak fields but not slow motion. In the post-Minkowskian expansion the gravitational field, described by the quantities \({h^{\alpha \beta}} = {\eta ^{\alpha \beta}} - \sqrt {- g} {g^{\alpha \beta}}\) (in harmonic coordinates, such that \({h^{\mu \nu}}_{,\nu} = 0\)) is formally expanded in powers of Newton’s constant G. Using a variety of different tools (PN expansion in the near zone, post-Minkowskian expansion in the wave zone, multipolar expansions, regularization of point-like sources, etc.), it is possible to solve Einstein’s equations, and to determine both the motion of the source and its GW emission. Since each term of the post-Minkowskian expansion can itself be PN-expanded, the final output of this computation has the form of a PN expansion; therefore, these methods are commonly referred to as PN approximation schemes.

PN schemes are generally used to study the motion of N-body systems in GR, and to compute the GW signal emitted by these systems. More specifically, most of the results obtained so far with PN schemes refer to the relativistic two-body problem, which can be applied to study compact binary systems formed by BHs and/or NSs (see Section 3.1.1). In the following we shall provide a brief summary of PN schemes, their main features and results as applied to the study of compact binary systems. For a more detailed description, we refer the reader to one of the many reviews that have been written on the subject; see e.g. [109, 620, 676, 454].

Two different but equivalent approaches have been developed to solve the relativistic two-body problem, finding the equations of motion of the source and the emitted gravitational waveform: the multipolar post-Minkowskian approach of Blanchet, Damour and Iyer [109], and the direct integration of the relaxed Einstein’s equations, developed by Will and Wiseman [785]. In these approaches, Einstein’s equations are solved iteratively in the near zone, employing a PN expansion, and in the wave zone, through a post-Minkowskian expansion. In both cases, multipolar expansions are performed. The two solutions, in the near and in the wave zone, are then matched. These approaches yield the equations of motion of the bodies, i.e., their accelerations as functions of their positions and velocities, and allow the energy balance equation of the system to be written as

$${{{\rm{d}}E} \over {{\rm{d}}t}} = - {\mathcal L}.$$

Here, E (which depends on terms of integer PN orders) can be considered as the energy of the system), and \({\mathcal L}\) (depending on terms of half-integer PN orders) is the emitted GW flux. The lowest PN order in the GW flux is given by the quadrupole formula [297] (see also [553]), \(\mathcal L = G/(5{c^5})({Q_{ab}}{Q_{ab}} + O(1/{c^3}))\) where Qab is the (traceless) quadrupole moment of the source. The leading term in C is then of 2.5-PN order (i.e., ∼ 1/c5), but since Qab is computed in the Newtonian limit, it is often considered as a “Newtonian” term. A remarkable result of the multipolar post-Minkowskian approach and of the direct integration of relaxed Einstein’s equations, is that once the equations are solved at n-th PN order both in the near zone and in the wave zone, E is known at n-PN order, and \({\mathcal L}\) is known at n-PN order with respect to its leading term, i.e., at (n + 2.5)-PN order. Once the energy and the GW flux are known with this accuracy, the gravitational waveform can be determined, in terms of them, at n-PN order.

Presently, PN schemes determine the motion of a compact binary, and the emitted gravitational waveform, up to 3.5-PN order for non-spinning binaries in circular orbits [109], but up to lower PN-orders for eccentric orbits and for spinning binaries [48, 148]. It is estimated that Advanced LIGO/Virgo data analysis requires 3.5-PN templates [123], and therefore some effort still has to go into the modeling of eccentric orbits and spinning binaries. It should also be remarked that the state-of-the-art PN waveforms have been compared with those obtained with NR simulations, showing a remarkable agreement in the inspiral phase (i.e., up to the late inspiral stage) [122, 389].

An alternative to the schemes discussed above is the ADM-Hamiltonian approach [676], in which using the ADM formulation of GR, the source is described as a canonical system in terms of its Hamiltonian. The ADM-Hamiltonian approach is equivalent to the multipolar post-Minkowskian approach and to the direct integration of relaxed Einstein’s equations, as long as the evolution of the source is concerned [246], but since Einstein’s equations are not solved in the wave zone, the radiative effects are only known with the same precision as the motion of the source. This framework has been extended to spinning binaries (see [726] and references therein). Recently, an alternative way to compute the Hamiltonian of a post-Newtonian source has been developed, the effective field theory approach [358, 149, 627, 340], in which techniques originally derived in the framework of quantum field theory are employed. This approach was also extended to spinning binaries [626, 625]. ADM-Hamiltonian and effective field theory are probably the most promising approaches to extend the accuracy of PN computations for spinning binaries.

The effective one-body (EOB) approach developed at the end of the last century [147] and recently improved [247, 600] (see, e.g., [240, 249] for a more detailed account) is an extension of PN schemes, in which the PN Taylor series is suitably resummed, in order to extend its validity up to the merger of the binary system. This approach maps the dynamics of the two compact objects into the dynamics of a single test particle in a deformed Kerr spacetime. It is a canonical approach, so the Hamiltonian of the system is computed, but the radiative part of the dynamics is also described. Since the mapping between the two-body system and the “dual” one-body system is not unique, the EOB Hamiltonian depends on a number of free parameters, which are fixed using results of PN schemes, of gravitational self-force computations, and of NR simulations. After this calibration, the waveforms reproduce with good accuracy those obtained in NR simulations (see, e.g., [240, 249, 600, 61]). In the same period, a different approach has been proposed to extend PN templates to the merger phase, matching PN waveforms describing the inspiral phase, with NR waveforms describing the merger [17, 673]. Both this “phenomenological waveform” approach and the EOB approach use results from approximation schemes and from NR simulations in order to describe the entire waveform of coalescing binaries, and are instrumental for data analysis [584].

To conclude this section, we mention that PN schemes originally treated compact objects as point-like, described by delta functions in the stress-energy tensor, and employing suitable regularization procedures. This is appropriate for BHs, and, as a first approximation, for NSs, too. Indeed, finite size effects are formally of 5-PN order (see, e.g., [239, 109]). However, their contribution can be larger than what a naive counting of PN orders may suggest [557]. Therefore, the PN schemes and the EOB approach have been extended to include the effects of tidal deformation of NSs in compact binary systems and on the emitted gravitational waveform using a set of parameters (the “Love numbers”) encoding the tidal deformability of the star [323, 248, 760, 102].

Beyond general relativity

PN schemes are also powerful tools to study the nature of the gravitational interaction, i.e., to describe and design observational tests of GR. They have been applied either to build general parametrizations, or to determine observable signatures of specific theories (two kinds of approaches that have been dubbed top-down and bottom-up, respectively [636]).

Let us discuss top-down approaches first. Nearly fifty years ago, Will and Nordtvedt developed the PPN formalism [784, 581], in which the PN metric of an N-body system is extended to a more general form, depending on a set of parameters describing possible deviations from GR. This approach (which is an extension of a similar approach by Eddington [291]) facilitates tests of the weak-field regime of GR. It is particularly well suited to perform tests in the solar system. All solar-system tests can be expressed in terms of constraints on the PPN parameters, which translates into constraints on alternative theories of gravity. For instance, the measurement of the Shapiro time-delay from the Cassini spacecraft [99] yields the strongest bound on one of the PPN parameters; this bound determines the strongest constraint to date on many modifications of GR, such as Brans-Dicke theory.

More recently, a different parametrized extension of the PN formalism has been proposed which, instead of the PN metric, expands the gravitational waveform emitted by a compact binary inspiral in a set of parameters describing deviations from GR [825, 203]. The advantage of this so-called “parametrized post-Einsteinian” approach — which is different in spirit from the PPN expansion, since it does not try to describe the spacetime metric — is its specific design to study the GW output of compact binary inspirals which are the most promising sources for GW detectors (see Section 3.1.1).

As mentioned above, PN approaches have also been applied bottom-up, i.e., in a manner that directly calculates the observational consequences of specific theories. For instance, the motion of binary pulsars has been studied, using PN schemes, in specific alternative theories of gravity, such as scalar-tensor theories [244]. The most promising observational quantity to look for evidence of GR deviations is probably the gravitational waveform emitted in compact binary inspirals, as computed using PN approaches. In the case of theories with additional fundamental fields, the leading effect is the increase in the emitted gravitational flux arising from the additional degrees of freedom. This increase typically induces a faster inspiral, which affects the phase of the gravitational waveform (see, e.g., [91]). For instance, in the case of scalar-tensor theories a dipolar component of the radiation can appear [787]. In other cases, as in massive graviton theories, the radiation has ℓ ≥ 2 as in GR, but the flux is different. For further details, we refer the interested reader to [782] and references therein.

State of the art

The post-Newtonian approach has mainly been used to study the relativistic two-body problem, i.e., to study the motion of compact binaries and the corresponding GW emission. The first computation of this kind, at leading order, was done by Peters and Mathews for generic eccentric orbits [614, 613]. It took about thirty years to understand how to extend this computation at higher PN orders, consistently modeling the motion and the gravitational emission of a compact binary [109, 785]. The state-of-the-art computations give the gravitational waveform emitted by a compact binary system, up to 3.5-PN order for non-spinning binaries in circular orbits [109], up to 3-PN order for eccentric orbits [48], and up to 2-PN order for spinning binaries [148]. An alternative approach, based on the computation of the Hamiltonian [676], is currently being extended to higher PN orders [726, 457, 399]; however, in this approach the gravitational waveform is computed with less accuracy than the motion of the binary.

Recently, different approaches have been proposed to extend the validity of PN schemes up to the merger, using results from NR to fix some of the parameters of the model (as in the EOB approach [249, 600, 61, 240]), or matching NR with PN waveforms (as in the “phenomenological waveform” approach [17, 673]). PN and EOB approaches have also been extended to include the effects of tidal deformation of NSs [323, 248, 760, 102].

PN approaches have been extended to test GR against alternative theories of gravity. Some of these extensions are based on a parametrization of specific quantities, describing possible deviations from GR. This is the case in the PPN approach [784, 581], most suitable for solar-system tests (see [782, 783] for extensive reviews on the subject), and in the parametrized post-Einsteinian approach [825, 203], most suitable for the analysis of data from GW detectors. Other extensions, instead, start from specific alternative theories and compute — using PN schemes — their observational consequences. In particular, the motion of compact binaries and the corresponding gravitational radiation have been extensively studied in scalar-tensor theories [244, 787, 30].

Spacetime perturbation approach

Astrophysical systems in general relativity

The PN expansion is less successful at describing strong-field, relativistic phenomena. Different tools have been devised to include this regime and one of the most successful schemes consists of describing the spacetime as a small deviation from a known exact solution. Systems well described by such a perturbative approach include, for instance, the inspiral of a NS or a stellar-mass BH of mass μ into a supermassive BH of mass Mμ [354, 32], or a BH undergoing small oscillations around a stationary configuration [487, 316, 95].

In this approach, the spacetime is assumed to be, at any instant, a small deviation from the background geometry, which, in the cases mentioned above, is described by the Schwarzschild or the Kerr solution here denoted by \(g_{\mu \nu}^{(0)}\). The deformed spacetime metric \({g_{\mu \nu}}\) can then be decomposed as

$${g_{\mu \nu}} = g_{\mu \nu}^{(0)} + {h_{\mu \nu}},$$

where hμν ≪ 1 describes a small perturbation induced by a small object or by any perturbing event.Footnote 8 Einstein’s equations are linearized around the background solution, by keeping only first-order terms in hμν (and in the other perturbation quantities, if present).

The simple expansion (15) implies a deeper geometrical construction (see, e.g., [730]), in which one considers a family of spacetime manifolds \({\mathcal M}_{\lambda}\), parametrized by a parameter λ; their metrics g(λ) satisfy Einstein’s equations, for each λ. The λ = 0 element of this family is the background spacetime, and the first term in the Taylor expansion in λ is the perturbation. Therefore, in the spacetime perturbation approach it is the spacetime manifold itself to be perturbed and expanded. However, once the perturbations are defined (and the gauge choice, i.e., the mapping between quantities in different manifolds, is fixed), perturbations can be treated as genuine fields living on the background spacetime \({\mathcal M_o}\). In particular, the linearized Einstein equations can be considered as linear equations on the background spacetime, and all the tools to solve linear differential equations on a curved manifold can be applied.

The real power of this procedure comes into play once one knows how to separate the angular dependence of the perturbations hμν. This was first addressed by Regge and Wheeler in their seminal paper [641], where they showed that in the case of a Schwarzschild background, the metric perturbations can be expanded in tensor spherical harmonics [541], in terms of a set of perturbation functions which only depend on the coordinates t and r. They also noted that the terms of this expansion belong to two classes (even and odd perturbations, sometimes also called polar and axial), with different behaviour under parity transformations (i.e., θ → π − θ, ϕ → ϕ + π). The linearized Einstein equations, expanded in tensor harmonics, yield the dynamical equations for the perturbation functions. Furthermore, perturbations corresponding to different harmonic components or different parities decouple due to the fact that the background is spherically symmetric. After a Fourier transformation in time, the dynamical equations reduce to ordinary differential equations in r.

Regge and Wheeler worked out the equations for axial perturbations of Schwarzschild BHs; later on, Zerilli derived the equations for polar perturbations [830]. With their gauge choice (the “Regge-Wheeler gauge”, which allows us to set to zero some of the perturbation functions), the harmonic expansion of the metric perturbation is

$${h_{\mu \nu}}(t,r,\theta, \phi) = \sum\limits_{l,m} {\int\nolimits_{- \infty}^{+ \infty} {{e^{- i\omega t}}} [h_{\mu \nu}^{{\rm{ax}},lm}(\omega, r, \theta, \phi) + h_{\mu \nu}^{{\rm{pol}},lm}(\omega, r, \theta, \phi)]\;{\rm{d}}\omega}$$


$$h_{\mu \nu}^{{\rm{ax}},lm}\;{\rm{d}}{x^\mu}\;{\rm{d}}{x^\nu} = 2\;\left[ {h_0^{lm}(\omega ,r)\;{\rm{d}}t + h_1^{lm}(\omega ,r)\;{\rm{d}}r} \right]\;\,[\csc \theta {\partial _\phi}{Y_{lm}}(\theta ,\phi)\;{\rm{d}}\theta - \sin \theta {\partial _\theta}{Y_{lm}}(\theta ,\phi)\;{\rm{d}}\phi ]$$
$$\begin{array}{*{20}c} {h_{\mu \nu}^{{\rm{pol}},lm}\;{\rm{d}}{x^\mu}\;{\rm{d}}{x^\nu} = \left[ {f(r)H_0^{lm}(\omega ,r)\;{\rm{d}}{t^2} + 2{H_1}{{(\omega ,r)}^{lm}}\;{\rm{d}}t\;{\rm{d}}r + H_2^{lm}(\omega ,r)\;{\rm{d}}{r^2}} \right.\quad \;\;} \\ {\left. {+ {r^2}{K^{lm}}(\omega ,r)({\rm{d}}{\theta ^2} + {{\sin}^2}\theta \;{\rm{d}}{\phi ^2})} \right]\;{Y_{lm}}(\theta ,\phi),} \\ \end{array}$$

where f(r) = 1 − 2M/r, and Ylm(θ, ϕ) are the scalar spherical harmonics.

It turns out to be possible to define a specific combination \(Z_{{\rm{RW}}}^{lm}(\omega, r)\) of the axial perturbation functions \(h_{\rm{0}}^{lm},h_{\rm{1}}^{lm}\), and a combination \(Z_{{\rm{Zer}}}^{lm}(\omega, r)\) of the polar perturbation functions \(H_{0,1,2}^{lm},\;{K^{lm}}\), Klm which describe completely the propagation of GWs. These functions, called the Regge-Wheeler and the Zerilli function, satisfy Schrödinger-like wave equations of the form

$${{{{\rm{d}}^2}{\Psi _{{\rm{RW}},{\rm{Zer}}}}} \over {{\rm{d}}r_\ast^2}} + ({\omega ^2} - {V_{{\rm{RW}},{\rm{Zer}}}}){\Psi _{{\rm{RW}},{\rm{Zer}}}} = {{\mathcal S}_{{\rm{RW}},{\rm{Zer}}}}.$$

Here, r* is the tortoise coordinate [553] and S represents nontrivial source terms. The energy flux emitted in GWs can be calculated straightforwardly from the solutions ΨRW, Zer.

This approach was soon extended to general spherically symmetric BH backgrounds and a gauge-invariant formulation in terms of specific combinations of the perturbation functions that remain unchanged under perturbative coordinate transformations [555, 346]. In the same period, an alternative spacetime perturbation approach was developed by Bardeen, Press and Teukolsky [75, 744], based on the Newman-Penrose formalism [575], in which the spacetime perturbation is not described by the metric perturbation hμν, but by a set of gauge-invariant complex scalars, the Weyl scalars, obtained by projecting the Weyl tensor Cαβγδ onto a complex null tetrad \(\ell, \;k,\;m,\;\bar m\) defined such that all their inner products vanish except \(- k \;\cdot \;\ell = 1 = m \;\cdot \;\bar m\). One of these scalars, Ψ4, describes the (outgoing) gravitational radiation; it is defined as

$${\Psi _4} \equiv - {C_{\alpha \beta \gamma \delta}}{\ell ^\alpha}{\bar m^\beta}{\ell ^\gamma}{\bar m^\delta}.$$

In the literature one may also find Ψ4 defined without the minus sign, but all physical results derived from Ψ4 are invariant under this ambiguity. We further note that the Weyl and Riemann tensors are identical in vacuum. Most BH studies in NR consider vacuum spacetimes, so that we can replace Cαβγδ in Eq. (20) with Rαβγδ.

In this framework, the perturbation equations reduce to a wave equation for (the perturbation of) which is called the Teukolsky equation [743]. For a general account on the theory of BH perturbations (with both approaches) see Chandrasekhar’s book [195].

The main advantage of the Bardeen-Press-Teukolsky approach is that it is possible to separate the angular dependence of perturbations of the Kerr background, even though such background is not spherically symmetric. Its main drawback is that it is very difficult to extend it beyond its original setup, i.e., perturbations of Kerr BHs. The tensor harmonic approach is much more flexible. In particular, spacetime perturbation theory (with tensor harmonic decomposition) has been extended to spherically symmetric stars [753, 518, 266, 196] (the extension to rotating stars is much more problematic [330]). As we discuss in Section 5.2.3, spacetime perturbation theory with tensor harmonic decomposition can be extended to higher-dimensional spacetimes. It is not clear whether such generalizations are possible with the Bardeen-Press-Teukolsky approach.

The sources \({\mathcal S_{{\rm{RW,Zer}}}}\) describe the objects that excite the spacetime perturbations, and can arise either directly from a non-vanishing stress-energy tensor or by imposing suitable initial conditions on the spacetime. These two alternative forms of exciting BH spacetimes have branched into two distinct tools, which can perhaps be best classified as the “point particle” [250, 179, 569, 93] and the “close limit” approximations [634, 637].

In the point particle limit the source term is a nontrivial perturbing stress-tensor, which describes for instance the infall of a small object along generic geodesics. The “small” object can be another BH, or a star, or even matter accreting into the BH. While the framework is restricted to objects of mass μM, it is generically expected that the extrapolation to μ ∼ M yields at least a correct order of magnitude. Thus, the spacetime perturbation approach is in principle able to describe qualitatively, if not quantitatively, highly dynamic BHs under general conditions. The original approach treats the small test particle moving along a geodesic of the background spacetime. Gravitational back-reaction can be included by taking into account the energy and angular momentum loss of the particle due to GW emission [232, 445, 548]. More sophisticated computations are required to take into account the conservative part of the “self-force”. For a general account on the self-force problem, we refer the interested reader to the Living Reviews article on the subject [623]. In this approach μ is restricted to be a very small quantity. It has been observed by many authors [37, 718] that promoting μ/M to the symmetric mass ratio M1M2/(M1 + M2) describes surprisingly well the dynamics of generic BHs with masses M1, M2.

In the close limit approximation the source term can be traced back to nontrivial initial conditions. In particular, the original approach tackles the problem of two colliding, equal-mass BHs, from an initial separation small enough that they are initially surrounded by a common horizon. Thus, this problem can be looked at as a single perturbed BH, for which some initial conditions are known [634, 637].

A universal feature of the dynamics of BH spacetimes as given by either the point particle or the close limit approximation is that the waveform Ψ decays at late times as a universal, exponentially damped sinusoid called ringdown or QNM decay. Because at late times the forcing caused by the source term \({\mathcal S}\) has died away, it is natural to describe this phase as the free oscillations of a BH, or in other words as solutions of the homogeneous version of Eq. (19). Together with the corresponding boundary conditions, the Regge-Wheeler and Zerilli equations then describe a freely oscillating BH. In vacuum, such boundary conditions lead to an eigenvalue equation for the possible frequencies ω. Due to GW emission, these oscillations are damped, i.e., they have discrete, complex frequencies called quasi-normal mode frequencies of the BH [487, 316, 95]. Such intuitive picture of BH ringdown can be given a formally rigorous meaning through contour integration techniques [506, 95].

The extension of the Regge-Wheeler-Zerilli approach to asymptotically dS or AdS spacetimes follows with the procedure outlined above and decomposition (16); see also Ref. [176]. It turns out that the Teukolsky procedure can also be generalized to these spacetimes [192, 277, 276].

Beyond electrovacuum GR

The Regge-Wheeler-Zerilli approach has proved fruitful also in other contexts including alternative theories of gravity. Generically, the decomposition works by using the same metric ansatz as in Eq. (16), but now augmented to include perturbations in matter fields, such as scalar or vector fields, or further polarizations for the gravitational field. Important examples where this formalism has been applied include scalar-tensor theories [668, 165, 824], Dynamical Chern-Simons theory [175, 554, 603], Einstein-Dilaton-Gauss-Bonnet [602], Horndeski gravity [477, 478], and massive theories of gravity [135].

Beyond four dimensions

Spacetime perturbation theory is a powerful tool to study BHs in higher-dimensional spacetimes. The tensor harmonic approach has been successfully extended by Kodama and Ishibashi [479, 452] to GR in higher-dimensional spacetimes, with or without cosmological constant. Their approach generalizes the gauge-invariant formulation of the Regge-Wheeler-Zerilli construction to perturbations of Tangherlini’s solution describing spherically symmetric BHs.

Since many dynamical processes involving higher-dimensional BHs (in particular, the collisions of BHs starting from finite distance) can be described in the far field limit by a perturbed spherically symmetric BH spacetime, the Kodama and Ishibashi approach can be useful to study the GW emission in these processes. The relevance of this approach therefore extends well beyond the study of spherically symmetric solutions. For applications of this tool to the wave extraction of NR simulations see for instance [797].

In the Kodama and Ishibashi approach, the D-dimensional spacetime metric is assumed to have the form \({g_{\mu \nu}} = g_{\mu \nu}^{(0)} + {h_{\mu \nu}}\) where \(g_{\mu \nu}^{(0)}\) is the Tangherlini solution and hμν represents a small perturbation. Decomposing the D-dimensional spherical coordinates into \({x^\mu} = (t,r,\vec \phi)\) with D − 2 angular coordinates \(\vec \phi = {\{{\phi ^a}\} _{a = 1, \ldots D - 2}}\), the perturbation hμν can be expanded in spherical harmonics, as in the four-dimensional case (see Section 5.2.1). However, the expansion in D > 4 is more complex than its four-dimensional counterpart: there are three classes of perturbations called the “scalar”, “vector” and “tensor” perturbations. The former two classes correspond, in D = 4, to polar and axial perturbations, respectively. These perturbations are decomposed into scalar \(({\mathcal S^{ll{\prime} \ldots}})\), vector \((\mathcal V_a^{ll{\prime} \ldots})\) and tensor \((\mathcal T_{ab}^{ll{\prime} \ldots})\) harmonics on the (D − 2)-sphere SD−2 and their gradients, as follows:

$${h_{\mu \nu}}(t,r,\vec \phi) = \sum\limits_{ll{\prime} \ldots} {\int\nolimits_{- \infty}^{+ \infty} {{e^{- i\omega t}}\left[ {h_{\mu \nu}^{{\rm{S}},ll{\prime} \ldots}(\omega, r, \vec \phi) + h_{\mu \nu}^{{\rm{V}},ll{\prime} \ldots}(\omega, r, \vec \phi) + h_{\mu \nu}^{{\rm{T}},ll{\prime} \ldots}(\omega, r, \vec \phi)} \right]} \,\;{\rm{d}}\omega,}$$

where ll′… denote harmonic indices on SD−2 and the superscripts S,V,T refer to scalar, vector and tensor perturbations, respectively. Introducing early upper case Latin indices A, B, … = 0, 1 and xA = (t,r), the metric perturbations can be written as

$$\begin{array}{*{20}c} {h_{\mu \nu}^{{\rm{S}},ll\prime \ldots}(\omega ,r,\vec \phi)\;{\rm{d}}{x^\mu}\;{\rm{d}}{x^\nu} = \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\left[ {f_{A\,B}^{{\rm{S}}\;ll\prime \ldots}(\omega ,r)\;{\rm{d}}{x^A}\;{\rm{d}}{x^B} + H_L^{{\rm{S}}\;ll\prime \ldots}(\omega ,r)\;{\Omega _{ab}}\;{\rm{d}}{\phi ^a}\;{\rm{d}}{\phi ^b}} \right]\;{{\mathcal S}^{ll\prime \ldots}}(\vec \phi)\quad \quad} \\ {+ f_A^{{\rm{S}}\;ll\prime \ldots}(\omega ,r)\;{\rm{d}}{x^A}{\mathcal S}_a^{ll\prime \ldots}(\vec \phi)\;{\rm{d}}{\phi ^a} + H_T^{{\rm{S}}\;ll\prime \ldots}(\omega ,r){\mathcal S}_{ab}^{ll\prime \ldots}(\vec \phi)\;{\rm{d}}{\phi ^a}\;{\rm{d}}{\phi ^b}\quad} \\ {h_{\mu \nu}^{{\rm{V}},ll\prime \ldots}(\omega ,r,\vec \phi)\;{\rm{d}}{x^\mu}\;{\rm{d}}{x^\nu} = \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\left[ {f_A^{{\rm{V}}\;ll\prime \ldots}(\omega ,r)\;{\rm{d}}{x^A}} \right]\;{\mathcal V}_a^{ll\prime \ldots}(\vec \phi)\;{\rm{d}}{\phi ^a} + H_T^{{\rm{V}}\;ll\prime \ldots}(\omega ,r){\mathcal V}_{ab}^{ll\prime \ldots}(\vec \phi)\;{\rm{d}}{\phi ^a}\;{\rm{d}}{\phi ^b}\;} \\ {h_{\mu \nu}^{{\rm{T}},ll\prime \ldots}(\omega ,r,\vec \phi)\;{\rm{d}}{x^\mu}\;{\rm{d}}{x^\nu} = H_T^{{\rm{T}}\;ll\prime \ldots}(\omega ,r){\mathcal T}_{ab}^{ll\prime \ldots}(\vec \phi)\;{\rm{d}}{\phi ^a}\;{\rm{d}}{\phi ^b},\quad \quad \quad \;} \\ \end{array}$$

where \(f_{AB}^{S,ll{\prime} \ldots}(\omega, r),f_A^{S,ll{\prime} \ldots}(\omega, r), \ldots\) are the spacetime perturbation functions. In the above expressions, Ωab is the metric on SD−2, Sa = −S,a/k, Sab = S:ab/k2 minus trace terms, where κ2 = l(l + D − 3) is the eigenvalue of the scalar harmonics, and the “:” denotes the covariant derivative on SD2; the traceless \({\mathcal V_{ab}}\) is defined in a similar way.

A set of gauge-invariant variables and the so-called “master functions”, generalizations of the Regge-Wheeler and Zerilli functions, can be constructed out of the metric perturbation functions and satisfy wave-like differential equations analogous to Eq. (19). The GW amplitude and its energy and momentum fluxes can be expressed in terms of these master functions.

For illustration of this procedure, we consider here the special case of scalar perturbations. We define the gauge-invariant quantities

$$F = {H_L} + {1 \over {D - 2}}{H_T} + {1 \over r}{X_A}{\hat D^A}r,\quad {F_{AB}} = {f_{AB}} + {\hat D_B}{X_A} + {\hat D_A}{X_B},$$

where we have dropped harmonic indices,

$${X_A} \equiv {r \over k}\left({{f_A} + {r \over k}{{\hat D}_A}{H_T}} \right),$$

and \({\hat D_A}\) denotes the covariant derivative associated with (t,r) sub-sector of the background metric. A master function Φ can be conveniently defined in terms of its time derivative according to

$${\partial _t}\Phi = (D - 2){r^{{{D - 4} \over 2}}}{{- {F^r}_t + 2r{\partial _t}F} \over {{k^2} - D + 2 + {{(D - 2)(D - 1)} \over 2}{{R_{\rm{S}}^{D - 3}} \over {{r^{D - 3}}}}}}.$$

From the master function, we can calculate the GW energy flux

$${{{\rm{d}}{E_{\ell m}}} \over {{\rm{d}}t}} = {1 \over {32\pi}}{{D - 3} \over {D - 2}}{k^2}({k^2} - D + 2){({\partial _t}{\Phi _{\ell m}})^2}.$$

The total radiated energy is obtained from integration in time and summation over all multipoles

$$E = \sum\limits_{\ell = 2}^\infty {\sum\limits_{m = - \ell}^\ell {\int\nolimits_{- \infty}^\infty {{{{\rm{d}}{E_{\ell m}}} \over {{\rm{d}}t}}} \;{\rm{d}}t.}}$$

In summary, this approach can be used, in analogy with the Regge-Wheeler-Zerilli formalism in four dimensions, to determine the quasi-normal mode spectrum (see, e.g., the review [95] and references therein), to determine the gravitational-wave emission due to a test source [98, 94], or to evaluate the flux of GWs emitted by a dynamical spacetime which tends asymptotically to a perturbed Tangherlini solution [797].

The generalization of this setup to higher-dimensional rotating (Myers-Perry [565]) BHs is still an open issue, since the decoupling of the perturbation equations has so far only been obtained in specific cases and for a subset of the perturbations [564, 496, 481].

Spacetime perturbation theory has also been used to study other types of higher-dimensional objects as for example black strings. Gregory and Laflamme [367, 368] considered a very specific sector of the possible gravitational perturbations of these objects, whereas Kudoh [495] performed a complete analysis that builds on the Kodama-Ishibashi approach.

State of the art

  • Astrophysical systems. Perturbation theory has been applied extensively to the modelling of BHs and compact stars, either without source terms, including in particular quasi-normal modes [487, 316, 95], or with point particle sources. Note that wave emission from extended matter distributions can be understood as interference of waves from point particles [400, 693, 615]. Equations for BH perturbations have been derived for Schwarzschild [641, 830], RN [831], Kerr [744] and slowly rotating Kerr-Newman BHs [601]. Equations for perturbations of stars have been derived for spherically symmetric [753, 518, 196] and slowly rotating stars [197, 482].

    Equations of BH perturbations with a point particle source have been studied as a tool to understand BH dynamics. This is a decades old topic, historically divided into investigations of circular and quasi-circular motion, and head-ons or scatters.

    Circular and quasi-circular motion. Gravitational radiation from point particles in circular geodesics was studied in Refs. [551, 252, 130] for non-rotating BHs and in Ref. [267] for rotating BHs. This problem was reconsidered and thoroughly analyzed by Poisson, Cutler and collaborators, and by Tagoshi, Sasaki and Nakamura in a series of elegant works, where contact was also made with the PN expansion (see the Living Reviews article [675] and references therein). The emission of radiation, together with the self-gravity of the objects implies that particles do not follow geodesics of the background spacetime. Inclusion of dissipative effects is usually done by balance-type arguments [445, 446, 733, 338] but it can also be properly accounted for by computing the self-force effects of the particle motion (see the Living Reviews article [623] and references therein). EM waves from particles in circular motion around BHs were studied in Refs. [252, 130, 129].

    Head-on or finite impact parameter collisions: non-rotating BHs. Seminal work by Davis et al. [250, 251] models the gravitational radiation from BH collisions by a point particle falling from rest at infinity into a Schwarzschild BH. This work has been generalized to include head-on collisions at non-relativistic velocities [660, 317, 524, 93], at exactly the speed of light [179, 93], and to non-head-on collisions at non-relativistic velocities [269, 93].

    The infall of multiple point particles has been explored in Ref. [96] with particular emphasis on resonant excitation of QNMs. Shapiro and collaborators have investigated the infall or collapse of extended matter distributions through superpositions of point particle waveforms [400, 693, 615].

    Electromagnetic radiation from high-energy collisions of charged particles with uncharged BHs was studied in Ref. [181] including a comparison with zero-frequency limit (ZFL) predictions. Gravitational and EM radiation generated in collisions of charged BHs has been considered in Refs. [459, 460].

    Head-on or finite impact parameter collisions: rotating BHs. Gravitational radiation from point particle collisions with Kerr BHs has been studied in Refs. [484, 483, 485, 486]. Suggestions that cosmic censorship might fail in high-energy collisions with near-extremal Kerr BHs, have recently inspired further scrutiny of these scenarios [71, 72] as well as the investigation of enhanced absorption effects in the ultra-relativistic regime [376].

    Close Limit approximation. The close limit approximation was first compared against nonlinear simulations of equal-mass, non-rotating BHs starting from rest [634]. It has since been generalized to unequal-mass [35] or even the point particle limit [524], rotating BHs [494] and boosted BHs at second-order in perturbation theory [577]. Recently the close limit approximation has also been applied to initial configurations constructed with PN methods [503].

  • Beyond electrovacuum GR. The resurgence of scalar-tensor theories as a viable and important prototype of alternative theories of gravity, as well as the conjectured existence of a multitude of fundamental bosonic degrees of freedom, has revived interest in BH dynamics in the presence of fundamental fields. Radiation from collisions of scalar-charged particles with BHs was studied in Ref. [134]. Radiation from massive scalar fields around rotating BHs was studied in Ref. [165] and shown to lead to floating orbits. Similar effects do not occur for massless gravitons [464].

  • Beyond four-dimensions and asymptotic flatness. The gauge/gravity duality and related frameworks highlight the importance of (A)dS and higher-dimensional background spacetimes. The formalism to handle gravitational perturbations of four-dimensional, spherically symmetric asymptotically (A)dS BHs has been developed in Ref. [176], whereas perturbations of rotating AdS BHs were recently tackled [192, 277, 276]. Gravitational perturbations of higher-dimensional BHs can be handled through the elegant approach by Kodama and Ishibashi [479, 480], generalized in Ref. [495] to include perturbations of black strings. Perturbations of higher-dimensional, rotating BHs can be expressed in terms of a single master variable only in few special cases [496]. The generic case has been handled by numerical methods in the linear regime [270, 395].

    Scalar radiation by particles around Schwarzschild-AdS BHs has been studied in Refs. [180, 178, 177]. We are not aware of any studies on gravitational or electromagnetic radiation emitted by particles in orbit about BHs in spacetimes with a cosmological constant.

    The quadrupole formula was generalized to higher-dimensional spacetimes in Ref. [170]. The first fully relativistic calculation of GWs generated by point particles falling from rest into a higher-dimensional asymptotically flat non-rotating BH was done in Ref. [98], and later generalized to arbitrary velocity in Ref. [94]. The mass multipoles induced by an external gravitational field (i.e., the “Love numbers”) to a higher-dimensional BH, have been determined in Ref. [488].

    The close limit approximation was extended to higher-dimensional, asymptotically flat, space-times in Refs. [822, 823].

The zero-frequency limit

Astrophysical systems in general relativity

While conceptually simple, the spacetime perturbation approach does involve solving one or more second-order, non-homogeneous differential equations. A very simple and useful estimate of the energy spectrum and total radiated gravitational energy can be obtained by using what is known as the ZFL or instantaneous collision approach.

The technique was derived by Weinberg in 1964 [773, 774] from quantum arguments, but it is equivalent to a purely classical calculation [707]. The approach is a consequence of the identity

$${\left. {\overline {(\dot h)}} \right\vert _{\omega = 0}} = \underset{\omega \rightarrow 0}{\lim} \int\nolimits_{- \infty}^{+ \infty} {\dot h{e^{- i\omega t}}} {\rm{d}}t = h(t = + \infty) - h(t = - \infty),$$

for the Fourier transform \(\overline {(\dot h)} (\omega)\) of the time derivative of any metric perturbation h(t) (we omitted unimportant constant overall factors in the definition of the transform). Thus, the low-frequency spectrum depends exclusively on the asymptotic state of the colliding particles which can be readily computed from their Coulomb gravitational fields. Because the energy spectrum is related to \(\bar \dot h(\omega)\) via

$${{{{\rm{d}}^2}E} \over {{\rm{d}}\Omega \;{\rm{d}}\omega}} \propto {r^2}{\overline {\left({(\dot h)} \right)} ^2},$$

we immediately conclude that the energy spectrum at low-frequencies depends only on the asymptotic states [774, 14, 707, 93, 489, 513]. Furthermore, if the asymptotic states are an accurate description of the collision at all times, as for instance if the colliding particles are point-like, then one expects the ZFL to be an accurate description of the problem.

For the head-on collision of two equal-mass objects each with mass Mγ/2, Lorentz factor γ and velocity υ in the center-of-mass frame, one finds the ZFL prediction [707, 513]

$${{{{\rm{d}}^2}E} \over {{\rm{d}}\omega \;{\rm{d}}\Omega}} = {{{M^2}{\gamma ^2}{v^4}} \over {4{\pi ^2}}}{{{{\sin}^4}\theta} \over {{{(1 - {v^2}{{\cos}^2}\theta)}^2}}}.$$

The particles collide head-on along the z-axis and we use standard spherical coordinates. The spectrum is flat, i.e., ω-independent, thus the total radiated energy is formally divergent. The approach neglects the details of the interaction and the internal structure of the colliding and final objects, and the price to pay is the absence of a lengthscale, and therefore the appearance of this divergence. The divergence can be cured by introducing a phenomenological cutoff in frequency. If the final object has typical size R, we expect a cutoff ωcutoff ∼ 1/R to be a reasonable assumption. BHs have a more reasonable cutoff in frequency given by their lowest QNMs; because QNMs are defined within a multipole decomposition, one needs first to decompose the ZFL spectrum into multipoles (see Appendix B of Ref. [93] and Appendix B2 of Ref. [513]). Finally, one observes that the high-energy limit υ →1 yields isotropic emission; when translated to a multipole dependence, it means that the energy in each multipole scales as 1/l2 in this limit.

The ZFL has been applied in a variety of contexts, including electromagnetism where it can be used to compute the electromagnetic radiation given away in β-decay [181, 455]; Wheeler used the ZFL to estimate the emission of gravitational and electromagnetic radiation from impulsive events [777]; the original treatment by Smarr considered only head-on collisions and computed only the spectrum and total emitted energy. These results have been generalized to include collisions with finite impact parameter and to a computation of the radiated momentum as well [513, 93]. Finally, recent nonlinear simulations of high-energy BH or star collisions yield impressive agreement with ZFL predictions [719, 93, 288, 134].

State of the art

  • Astrophysical systems. The zero-frequency limit for head-on collisions of particles was used by Smarr [707] to understand gravitational radiation from BH collisions and in Ref. [14] to understand radiation from supernovae-like phenomena. It was later generalized to the nontrivial finite impact parameter case [513], and compared extensively with fully nonlinear numerical simulations [93]. Ref. [181] reports on collisions of an electromagnetic charge with a non-rotating BH in a spacetime perturbation approach and compares the results with a ZFL calculation.

  • Beyond four-dimensional, electrovacuum GR. Recent work has started applying the ZFL to other spacetimes and theories. Brito [134] used the ZFL to understand head-on collisions of scalar charges with four-dimensional BHs. The ZFL has been extended to higher dimensions in Refs. [170, 513] and recently to specific AdS soliton spacetimes in Ref. [173].

Shock wave collisions

An alternative technique to model the dynamics of collisons of two particles (or two BHs) at high energies describes the particles as gravitational shock waves. This method yields a bound on the emitted gravitational radiation using an exact solution, and provides an estimate of the radiation using a perturbative method. In the following we shall review both.

In D = 4 vacuum GR, a point-like particle is described by the Schwarzschild metric of mass M. The gravitational field of a particle moving with velocity υ is then obtained by boosting the Schwarzschild metric. Of particular interest is the limiting case where the velocity approaches the speed of light υc. Taking simultaneously the limit M →0 so that the zeroth component of the 4-momentum, E, is held fixed, \(E = M/\sqrt {1 - {v^2}/{c^2}}\), one observes an infinite Lorentz contraction of the curvature in the spatial direction of the motion. In this limit, the geometry becomes that of an impulsive or shock gravitational pp-wave, i.e., a plane-fronted gravitational wave with parallel rays, sourced by a null particle. This is the Aichelburg-Sexl geometry [16] for which the curvature has support only on a null plane. In Brinkmann coordinates, the line element is:

$${\rm{d}}{s^2} = - {\rm{d}}u\;{\rm{d}}v + \kappa \Phi (\rho)\delta (u)\;{\rm{d}}{u^2} + {\rm{d}}{\rho ^2}{\rm{+}}{\rho ^2}{\rm{d}}{\phi ^2},\quad - \Delta [\kappa \Phi (\rho)] = 4\pi \kappa \delta (\rho){.}$$

Here the shock wave is moving in the positive z-direction, where (u = tz, υ = t+z). This geometry solves the Einstein equations with energy momentum tensor Tuu = (u)δ(ρ) — corresponding to a null particle of energy E = κ/4G, traveling along u = 0 = ρ — provided the equation on the right-hand side of (31) is satisfied, where the Laplacian is in the flat 2-dimensional transverse space. Such a solution is given in closed analytic form by Φ(ρ) = − 2ln(ρ).

The usefulness of shock waves in modelling collisions of particles or BHs at very high energies relies on the following fact. Since the geometry of a single shock wave is flat outside a null plane, one can superimpose two shock wave solutions traveling in opposite directions and still obtain an exact solution of the Einstein equations, valid up to the moment when the two shock waves collide. The explicit metric is obtained by superimposing two copies of (31), one with support at u = 0 and another one with support at υ = 0. But it is more convenient to write down the geometry in coordinates for which test particle trajectories vary continuously as they cross the shock. These are called Rosen coordinates, \((\bar u,\bar v,\bar \rho, \phi)\); their relation with Brinkmann coordinates can be found in [420] and the line element for the superposition becomes

$${\rm{d}}{s^2} = - {\rm{d}}\bar u\;{\rm{d}}\bar v + \left[ {{{\left({1 + {{\kappa \bar u\theta (\bar u)} \over 2}\Phi {\prime\prime}} \right)}^2} + {{\left({1 + {{\kappa \bar v\theta (\bar v)} \over 2}\Phi {\prime\prime}} \right)}^2} - 1} \right]{\rm{d}}{\bar \rho ^2}$$
$$+ {\bar \rho ^2}\left[ {{{\left({1 + {{\kappa \bar u\theta (\bar u)} \over {2\bar \rho}}\Phi {\prime}} \right)}^2} + {{\left({1 + {{\kappa \bar v\theta (\bar v)} \over {2\bar \rho}}\Phi {\prime}} \right)}^2} - 1} \right]{\rm{d}}{\phi ^2}.$$

This metric is a valid description of the spacetime with the two shock waves except in the future light-cone of the collision, which occurs at \(\bar u = 0 = \bar v\). Remarkably, and despite not knowing anything about the future development of the collision, an AH can be found for this geometry within its region of validity, as first pointed out by Penrose. Its existence indicates that a BH forms and moreover its area provides a lower bound for the mass of the BH [766]. This AH is the union of two surfaces,

$$\{{{\mathcal S}_1},\;{\rm{on}}\;\bar u = 0\;{\rm{and}}\;\bar v = - {\psi _1}(\bar \rho) \leq 0\}, \quad {\rm{and}}\;\quad \{{{\mathcal S}_2},\;{\rm{on}}\;\bar v = 0\;{\rm{and}}\;\bar u = - {\psi _2}(\bar \rho) \leq 0\},$$

for some functions ψ1,ψ2 to be determined. The relevant null normals to S1 and S2 are, respectively,

$${l_1} = {\partial _{\bar u}} - {1 \over 2}\psi _1^\prime{g^{\bar \rho \bar \rho}}{\partial _{\bar \rho}} + {1 \over 4}{(\psi _1^\prime)^2}{g^{\bar \rho \bar \rho}}{\partial _{\bar v}}\;,\quad \;{l_2} = {\partial _{\bar v}} - {1 \over 2}\psi _2^\prime{g^{\bar \rho \bar \rho}}{\partial _{\bar \rho}} + {1 \over 4}{(\psi _2^\prime)^2}{g^{\bar \rho \bar \rho}}{\partial _{\bar u}}.$$

One must then guarantee that these normals have zero expansion and are continuous at the intersection \(\bar u = 0 = \bar v\). This yields the solution \({\psi _1}(\bar \rho) = \kappa \Phi (\bar \rho/\kappa) = {\psi _2}(\bar \rho)\). In particular, at the intersection, the AH has a polar radius \(\bar \rho = \kappa\). The area of the AH is straightforwardly computed to be 2π2κ2, and provides a lower bound on the area of a section of the event horizon, and hence a lower bound on the mass of the BH: \(M/\kappa > 1\sqrt 8\). By energy conservation, we then obtain an upper bound on the inelasticity ϵ, i.e., the fraction of the initial centre of mass energy which can be emitted in gravitational radiation:

$${\epsilon _{{\rm{AH}}}} \leq 1 - {1 \over {\sqrt 2}} \simeq 0{.}29{.}$$

Instead of providing a bound on the inelasticity, a more ambitious program is to determine the exact inelasticity by solving the Einstein equations in the future of the collision. Whereas an analytic exact solution seems out of reach, a numerical solution of the fully nonlinear field equations might be achievable, but none has been reported. The approach that has produced the most interesting results, so far, is to solve the Einstein equations perturbatively in the future of the collision.

To justify the use of a perturbative technique and introduce a perturbation expansion parameter, D’Eath and Payne [257, 258, 259] made the following argument. In a boosted frame, say in the negative z direction, one of the shock waves will become blueshifted whereas the other will become redshifted. These are, respectively, the waves with support on u = 0 and υ = 0. The geometry is still given by (33), but with the energy parameter κ multiplying ū terms ( terms) replaced by a new energy parameter ν (parameter λ). For a large boost, λ/ν ≪ 1, or in other words, in the boosted frame there are a strong shock (at u = 0) and a weak shock (at υ = 0). The weak shock is regarded as a perturbation of the spacetime of the strong shock, and λ/ν provides the expansion parameter to study this perturbation. Moreover, to set up initial conditions for the post-collision perturbative expansion, one recasts the exact solution on the immediate future of the strong shock, u = 0+, in a perturbative form, even though it is an exact solution. It so happens that expressing the exact solution in such perturbative fashion only has terms up to second order:

$${g_{\mu \nu}}{\vert _{u = {0^ +}}} = {\nu ^2}\left[ {{\eta _{\mu \nu}} + {\lambda \over \nu}h_{\mu \nu}^{(1)} + {{\left({{\lambda \over \nu}} \right)}^2}h_{\mu \nu}^{(2)}} \right].$$

This perturbative expansion is performed in dimensionless coordinates of Brinkmann type, as in Eq. (31), since the latter are more intuitive than Rosen coordinates. The geometry to the future of the strong shock, on the other hand, will be of the form

$${g_{\mu \nu}}{\vert _{u > 0}} = {\nu ^2}\left[ {{\eta _{\mu \nu}} + \sum\limits_{i = 1}^\infty {{{\left({{\lambda \over \nu}} \right)}^i}} h_{\mu \nu}^{(i)}} \right],$$

where each of the \(h_{\mu \nu}^{(i)}\) will be obtained by solving the Einstein equations to the necessary order. For instance, to obtain \(h_{\mu \nu}^{(1)}\) one solves the linearized Einstein equations. In the de Donder gauge these yield a set of decoupled wave equations of the form \(\square \bar h_{\mu \nu}^{(1)} = 0\), where the \(\bar h_{\mu \nu}^{(1)}\) is the trace reversed metric perturbation. The wave equation must then be subjected to the boundary conditions (36). At higher orders, the problem can also be reduced to solving wave equations for \(h_{\mu \nu}^{(i)}\) but now with sources provided by the perturbations of lower order [221].

After obtaining the metric perturbations to a given order, one must still compute the emitted gravitational radiation, in order to obtain the inelasticity. In the original work [256, 257, 258, 259], the metric perturbations were computed to second order and the gravitational radiation was extracted using Bondi’s formalism and the Bondi mass loss formula. The first-order results can equivalently be obtained using the Landau-Lifshitz pseudo-tensor for GW extraction [420]. The results in first and second order are, respectively:

$${\epsilon ^{(1)}} = 0.25,\quad {\epsilon ^{(2)}} = 0{.}164{.}$$

Let us close this subsection with three remarks on these results. Firstly, the results (38) are below the AH bound (35), as they should. Secondly, and as we shall see in Section 7.6, the second-order result is in excellent agreement with results from NR simulations. Finally, as we comment in the next subsection, the generalisation to higher dimensions of the first-order result reveals a remarkably simple pattern.

State of the art

The technique of superimposing two Aichelburg-Sexl shock waves [16] was first used by Penrose in unpublished work but quoted, for instance, in Ref. [257]. Penrose showed the existence of an AH for the case of a head-on collision, thus suggesting BH formation. Computing the area of the AH yields an upper bound on the fraction of the overall energy radiated away in GWs, i.e., the inelasticity. In the early 2000s, the method of superimposing shock waves and finding an AH was generalized to D ≥ 5 and non-zero impact parameter in Refs. [286, 818] and refined in Ref. [819] providing, in addition to a measure of the inelasticity, an estimate of the cross section for BH formation in a high-energy particle collision. A potential improvement to the AH based estimates was carried out in a series of papers by D’Eath and Payne [256, 257, 258, 259]. They computed the metric in the future of the collision perturbatively to second order in the head-on case. This method was generalized to D ≥ 5 in first-order perturbation theory [420, 222] yielding a very simple result: ϵ(1) = 1/2 − 1/D. A formalism for higher order and the caveats of the method in the presence of electric charge were exhibited in [221]. AH formation in shock wave collisions with generalized profiles and asymptotics has been studied in [19, 739, 31, 282].

Numerical Relativity

Generating time-dependent solutions to the Einstein equations using numerical methods involves an extended list of ingredients which can be loosely summarized as follows.

  • Cast the field equations as an IBVP.

  • Choose a specific formulation that admits a well-posed IBVP, i.e., there exist suitable choices for the following ingredients that ensure well posedness.

  • Choose numerically suitable coordinate or gauge conditions.

  • Discretize the resulting set of equations.

  • Handle singularities such that they do not result in the generation of non-assigned numbers which rapidly swamp the computational domain.

  • Construct initial data that solve the Einstein constraint equations and represent a realistic snapshot of the physical system under consideration.

  • Specify suitable outer boundary conditions.

  • Fix technical aspects: mesh refinement and/or multi-domains as well as use of multiple computer processors through parallelization.

  • Apply diagnostic tools that measure GWs, BH horizons, momenta and masses, and other fields.

In this section, we will discuss state-of-the-art choices for these ingredients.

Formulations of the Einstein equations

The ADM equations

The Einstein equations in D dimensions describing a spacetime with cosmological constant Λ and energy-matter content Tαβ are given by

$${R_{\alpha \beta}} - {1 \over 2}R{g_{\alpha \beta}} + \Lambda {g_{\alpha \beta}} = 8\pi {T_{\alpha \beta}}\quad \Leftrightarrow \quad {R_{\alpha \beta}} = 8\pi \left({{T_{\alpha \beta}} - {1 \over {D - 2}}{g_{\alpha \beta}}T} \right) + {2 \over {D - 2}}\Lambda {g_{\alpha \beta}}.$$

Elegant though this tensorial form of the equations is from a mathematical point of view, it is not immediately suitable for a numerical implementation. For one thing, the character of the equations as a hyperbolic, parabolic or elliptic system is not evident. In other words, are we dealing with an initial-value or a boundary-value problem? In fact, the Einstein equations are of mixed character in this regard and represent an IBVP. Well-posedness of the IBVP then requires a suitable formulation of the evolution equations, boundary conditions and initial data. We shall discuss this particular aspect in more detail further below, but first consider the general structure of the equations. The multitude of possible ways of writing the Einstein equations are commonly referred to as formulations of the equations and a good starting point for their discussion is the canonical “3 + 1” or “(D − 1) + 1” split originally developed by Arnowitt, Deser & Misner [47] and later reformulated by York [810, 812].

The tensorial form of the Einstein equations (39) fully reflects the unified viewpoint of space and time; it is only through the Lorentzian signature (−, +, …, +) of the metric that the timelike character of one of the coordinates manifests itself.Footnote 9 It turns out crucial for understanding the character of Einstein’s equations to make the distinction between spacelike and timelike coordinates more explicit.

Let us consider for this purpose a spacetime described by a manifold equipped with a metric gαβ of Lorentzian signature. We shall further assume that there exists a foliation of the spacetime in the sense that there exists a scalar function t : →ℝ with the following properties. (i) The 1-form dt associated with the function t is timelike everywhere; (ii) The hypersurfaces Σt defined by t = const are non-intersecting and ∪t∈ℝΣt = . Points inside each hypersurface Σt are labelled by spatial coordinates xI, I = 1, …, D − 1, and we refer to the coordinate system (t, xI) as adapted to the spacetime split.

Next, we define the lapse function α and shift vector β through

$$\alpha \equiv {1 \over {\vert \vert {\bf{d}}t\vert \vert}},\quad \;\;{\beta ^\mu} \equiv {({\partial _t})^\mu} - \alpha {n^\mu},$$

where n ≡ − αdt is the timelike unit normal field. The geometrical interpretation of these quantities in terms of the timelike unit normal field nα and the coordinate basis vector t is illustrated in Figure 2. Using the relation 〈dt, t〉 = 1 and the definition of α and β, one directly finds 〈dt, β〉 = 0, so that the shift β is tangent to the hypersurfaces Σt. It measures the deviation of the coordinate vector t from the normal direction n. The lapse function relates the proper time measured by an observer moving with four velocity nα to the coordinate time t: Δτ = αΔt.

Figure 2
figure 2

Illustration of two hypersurf aces of a foliation Σt. Lapse α and shift βμ are defined by the relation of the timelike unit normal field nμ and the basis vector t associated with the coordinate t. Note that 〈dt, αn〉 = 1 and, hence, the shift vector β is tangent to Σt.

A key ingredient for the spacetime split of the equations is the projection of tensors onto time and space directions. For this purpose, the space projection operator is defined as \({\bot ^\alpha}_\mu \equiv {\delta ^\alpha}_\mu + {n^\alpha}{n_\mu}\). For a generic tensor \({T^{{\alpha _1}{\alpha _2} \ldots}}_{{\beta _1}{\beta _2} \ldots}\), its spatial projection is given by projecting each index speparately

$${(\bot T)^{{\alpha _1}{\alpha _2} \ldots}}_{{\beta _1}{\beta _2} \ldots} \equiv {\bot ^{{\alpha _1}}}_{{\mu _1}}{\bot ^{{\alpha _2}}}_{{\mu _2}} \ldots {\bot ^{{\nu _1}}}_{{\beta _1}}{\bot ^{{\nu _2}}}_{{\beta _2}} \ldots {T^{{\mu _1}{\mu _2} \ldots}}_{{\nu _1}{\nu _2} \ldots}.$$

A tensor S is called tangent to Σt if it is invariant under projection, i.e., ⊥S = S. In adapted coordinates, we can ignore the time components of such spatial tensors and it is common practice to denote their components with Latin indices I, J, … = 1, …, (D − 1). We similarly obtain time projections of a tensor by contracting its indices with nα. Mixed projections are obtained by contracting any combination of tensor indices with nα and projecting the remaining ones with \({\bot ^\alpha}_\mu\). A particularly important tensor is obtained from the spatial projection of the spacetime metric

$${\gamma _{\alpha \beta}} \equiv {(\bot g)_{\alpha \beta}} = {\bot ^\mu}_\alpha {\bot ^\nu}_\beta {g_{\mu \nu}} = ({\delta ^\mu}_\alpha + {n^\mu}{n_\alpha})({\delta ^\nu}_\beta + {n^\nu}{n_\beta}){g_{\mu \nu}} = {g_{\alpha \beta}} + {n_\alpha}{n_\beta} = {\bot _{\alpha \beta}}.$$

γαβ is known as the first fundamental form or spatial metric and describes the intrinsic geometry of the spatial hypersurfaces Σt. As we see from Eq. (42), it is identical to the projection operator. In the remainder, we will use both the ⊥ and γ symbols to denote this tensor depending on whether the emphasis is on the projection or the hypersurface geometry.

With our definitions, it is straightforward to show that the spacetime metric in adapted coordinates (t, xI) can be written as ds2 = −α2 dt2 + γIJ(dxI + βI dt)(dxJ + βJ dt) or, equivalently,


It can be shown [364] that the spatial metric γIJ defines a unique, torsion-free and metric-compatible connection \(\Gamma _{JK}^I = {1 \over 2}{\gamma ^{IM}}({\partial _J}{\gamma _{KM}} + {\partial _K}{\gamma _{MJ}} - {\partial _M}{\gamma _{JK}})\) whose covariant derivative for an arbitrary spatial tensor is given by

$${D_\gamma}{S^{{\alpha _1}{\alpha _2} \ldots}}_{{\beta _1}{\beta _2} \ldots} = {\bot ^\lambda}_\gamma {\bot ^{{\alpha _1}}}_{{\mu _1}}{\bot ^{{\alpha _2}}}_{{\mu _2}} \ldots {\bot ^{{\nu _1}}}_{{\beta _1}}{\bot ^{{\nu _2}}}_{{\beta _2}} \ldots {\nabla _\lambda}{S^{{\mu _1}{\mu _2} \ldots}}_{{\nu _1}{\nu _2} \ldots}.$$

The final ingredient required for the spacetime split of the Einstein equations is the extrinsic curvature or second fundamental form defined as

$${K_{\alpha \beta}} \equiv - \bot {\nabla _\beta}{n_\alpha}.$$

The sign convention employed here is common in NR but the “−” is sometimes omitted in other studies of GR. The definition (45) provides an intuitive geometric interpretation of the extrinsic curvature as the change in direction of the timelike unit normal field n as we move across the hypersurface Σt. As indicated by its name, the extrinsic curvature thus describes the embedding of Σt inside the higher-dimensional spacetime manifold. The projection ⊥βnα is symmetric under exchange of its indices in contrast to its non-projected counterpart ∇βnα. For the formulation of the Einstein equations in the spacetime split, it is helpful to introduce the vector field mμαnμ = (∂t)μβμ. A straightforward calculation shows that the extrinsic curvature can be expressed in terms of the Lie derivative of the spatial metric along either n or m according to

$${K_{\alpha \beta}} = - {1 \over 2}{{\mathcal L}_n}{\gamma _{\alpha \beta}} = - {1 \over {2\alpha}}{{\mathcal L}_m}{\gamma _{\alpha \beta}}.$$

We have now assembled all tools to calculate the spacetime projections of the Riemann tensor. In the following order, these are known as the Gauss, the contracted Gauss, the scalar Gauss, the Codazzi, the contracted Codazzi equation, as well as the final projection of the Riemann tensor and its contractions:

$$\begin{array}{*{20}c} {{\bot ^\mu}_\alpha {\bot ^\nu}_\beta {\bot ^\gamma}_\rho {\bot ^\sigma}_\delta {R^\rho}_{\sigma \mu \nu} = {{\mathcal R}^\gamma}_{\delta \alpha \beta} + {K^\gamma}_\alpha {K_{\delta \beta}} - {K^\gamma}_\beta {K_{\delta \alpha}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{\bot ^\mu}_\alpha {\bot ^\nu}_\beta {R_{\mu \nu}} + {\bot _{\mu \alpha}}{\bot ^\nu}_\beta {n^\rho}{n^\sigma}{R^\mu}_{\rho \nu \sigma} = {{\mathcal R}_{\alpha \beta}} + K{K_{\alpha \beta}} - {K^\mu}_\beta {K_{\alpha \mu}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {R + 2{R_{\mu \nu}}{n^\mu}{n^\nu} = {\mathcal R} + {K^2} - {K^{\mu \nu}}{K_{\mu \nu}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,\quad \quad \quad} \\ {{\bot ^\gamma}_\rho {n^\sigma}{\bot ^\mu}_\alpha {\bot ^\nu}_\beta {R^\rho}_{\sigma \mu \nu} = {D_\beta}{K^\gamma}_\alpha - {D_\alpha}{K^\gamma}_\beta ,\,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;} \\ {{n^\sigma}{\bot ^\nu}_{\;\beta}{R_{\sigma \nu}} = {D_\beta}K - {D_\mu}{K^\mu}_\beta ,\,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,\,\,\quad \quad \quad} \\ {{\bot _{\alpha \mu}}{\bot ^\nu}_\beta {n^\sigma}{n^\rho}{R^\mu}_{\rho \nu \sigma} = {1 \over \alpha}{{\mathcal L}_m}{K_{\alpha \beta}} + {K_{\alpha \mu}}{K^\mu}_\beta + {1 \over \alpha}{D_\alpha}{D_\beta}\alpha ,\,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{\bot ^\mu}_\alpha {\bot ^\nu}_\beta {R_{\mu \nu}} = - {1 \over \alpha}{{\mathcal L}_m}{K_{\alpha \beta}} - 2{K_{\alpha \mu}}{K^\mu}_\beta - {1 \over \alpha}{D_\alpha}{D_\beta}\alpha + {{\mathcal R}_{\alpha \beta}} + K{K_{\alpha \beta}},\,\,\quad \quad \quad} \\ {R = - {2 \over \alpha}{{\mathcal L}_m}K - {2 \over \alpha}{\gamma ^{\mu \nu}}{D_\mu}{D_\nu}\alpha + {\mathcal R} + {K^2} + {K^{\mu \nu}}{K_{\mu \nu}}.\,\,} \\ \end{array}$$

Here, denotes the Riemann tensor and its contractions as defined in standard fashion from the spatial metric γIJ. For simplicity, we have kept all spacetime indices here even for spatial tensors. As mentioned above, the time components can and will be discarded eventually.

By using Eq. (47), we can express the space and time projections of the Einstein equations (39) exclusively in terms of the first and second fundamental forms and their derivatives. It turns out helpful for this purpose to introduce the corresponding projections of the energy-momentum tensor which are given by

$$\rho = {T_{\mu \nu}}{n^\mu}{n^\nu},\quad \;{j_\alpha} = - {\bot ^\nu}_\alpha {T_{\mu \nu}}{n^\mu},$$
$${S_{\alpha \beta}} = {\bot ^\mu}_\alpha {\bot ^\nu}_\beta {T_{\mu \nu}},\quad S = {\gamma ^{\mu \nu}}{S_{\mu \nu}}.$$

then, the energy-momentum tensor is reconstructed according to Tαβ = Sαβ+nαjβ+nβjα+ρnαnβ. Using the explicit expressions for the Lie derivatives

$${{\mathcal L}_m}{K_{IJ}} = {{\mathcal L}_{{\partial _t} - \beta}}{K_{IJ}} = {\partial _t}{K_{IJ}} - {\beta ^M}{\partial _M}{K_{IJ}} - {K_{MJ}}{\partial _I}{\beta ^M} - {K_{IM}}{\partial _J}{\beta ^M},$$
$${{\mathcal L}_m}{\gamma _{IJ}} = {{\mathcal L}_{{\partial _t} - \beta}}{\gamma _{IJ}} = {\partial _t}{\gamma _{IJ}} - {\beta ^M}{\partial _M}{\gamma _{IJ}} - {\gamma _{MJ}}{\partial _I}{\beta ^M} - {\gamma _{IM}}{\partial _J}{\beta ^M},$$

we obtain the spacetime split of the Einstein equations

$${\partial _t}{\gamma _{IJ}} = {\beta ^M}{\partial _M}{\gamma _{IJ}} + {\gamma _{MJ}}{\partial _I}{\beta ^M} + {\gamma _{IM}}{\partial _J}{\beta ^M} - 2\alpha {K_{IJ}},$$
$$\begin{array}{*{20}c} {{\partial _t}{K_{IJ}} = {\beta ^M}{\partial _M}{K_{IJ}} + {K_{MJ}}{\partial _I}{\beta ^M} + {K_{IM}}{\partial _J}{\beta ^M} - {D_I}{D_J}\alpha + \alpha ({{\mathcal R}_{IJ}} + K{K_{IJ}} - 2{K_{IM}}{K^M}_J)}\\ {+ 8\pi \alpha \left({{{S - \rho} \over {D - 2}}{\gamma _{IJ}} - {S_{IJ}}} \right) - {2 \over {D - 2}}\alpha \Lambda {\gamma _{IJ}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ \end{array}$$
$$0 = {\mathcal R} + {K^2} - {K^{MN}}{K_{MN}} - 2\Lambda - 16\pi \rho,$$
$$0 = {D_I}K - {D_M}{K^M}_I + 8\pi {j_I}.$$

By virtue of the Bianchi identities, the constraints (54) and (55) are preserved under the evolution equations. Furthermore, we can see that D(D − 1)/2 second-order-in-time evolution equations for the γIJ are written as a first-order-in-time system through introduction of the extrinsic curvature. Additionally, we have obtained D constraint equations, the Hamiltonian and momentum constraints, which relate data within a hypersurface Σt. We note that the Einstein equations do not determine the lapse α and shift βI. For the case of D = 4, these equations are often referred to as the ADM equations, although we note that Arnowitt, Deser & Misner used the canonical momentum in place of the extrinsic curvature in their original work [47]. Counting the degrees of freedom, we start with D(D + 1)/2 components of the spacetime metric. The Hamiltonian and momentum constraints determine D of these while D gauge functions represent the gauge freedom, leaving D(D − 3)/2 physical degrees of freedom as expected.


The suitability of a given system of differential equations for a numerical time evolution critically depends on a continuous dependency of the solution on the initial data. This aspect is referred to as well posedness of the IBVP and is discussed in great detail in Living Reviews articles and other works [645, 674, 383, 427]. Here, we merely list the basic concepts and refer the interested reader to these articles.

Consider for simplicity an initial-value problem in one space and one time dimension for a single variable u(t, x) on an unbounded domain. Well-posedness requires a norm ∥ · ∥, i.e., a map from the space of functions f(x) to the real numbers ℝ, and a function F(t) independent of the initial data such that

$$\vert \vert \delta u(t, \cdot)\vert \vert \leq F(t)\vert \vert \delta u(0, \cdot)\vert \vert,$$

where δu denotes a linear perturbation relative to a solution u0(t, x) [380]. We note that F(t) may be a rapidly growing function, for example an exponential, so that well posedness represents a necessary but not sufficient criterion for suitability of a numerical scheme.

Well posedness of formulations of the Einstein equations is typically studied in terms of the hyperbolicity properties of the system in question. Hyperbolicity of a system of PDEs is often defined in terms of the principal part, that is, the terms of the PDE which contain the highest-order derivatives. We consider for simplicity a quasilinear first-order system for a set of variables u(t, x)

$${\partial _t}u = P(t,x,u,{\partial _x})u.$$

The system is called strongly hyperbolic if P is a smooth differential operator and its associated principal symbol is symmetrizeable [567]. For the special case of constant coefficient systems this definition simplifies to the requirement that the principal symbol has only imaginary eigenvalues and a complete set of linearly independent eigenvectors. If linear independence of the eigenvectors is not satisfied, the system is called weakly hyperbolic. For more complex systems of equations, strong and weak hyperbolicity can be defined in a more general fashion [645, 567, 646, 674].

In our context, it is of particular importance that strong hyperbolicity is a necessary condition for a well posed IBVP [741, 742]. The ADM equations (52)(53), in contrast, have been shown to be weakly but not strongly hyperbolic for fixed gauge [567]; likewise, a first-order reduction of the ADM equations has been shown to be weakly hyperbolic [468]. These results strongly indicate that the ADM formulation is not suitable for numerical evolutions of generic spacetimes.

A modification of the ADM equations which has been used with great success in NR is the BSSN system [78, 695] which is the subject of the next section.

The BSSN equations

It is interesting to note that the BSSN formulation had been developed in the 1990s before a comprehensive understanding of the hyperbolicity properties of the Einstein equations had been obtained; it was only about a decade after its first numerical application that strong hyperbolicity of the BSSN system [380] was demonstrated. We see here an example of how powerful a largely empirical approach can be in the derivation of successful numerical methods. And yet, our understanding of the mathematical properties is of more than academic interest as we shall see in Section 6.1.5 below when we discuss recent investigations of potential improvements of the BSSN system.

The modification of the ADM equations which results in the BSSN formulation consists of a trace split of the extrinsic curvature, a conformal decomposition of the spatial metric and of the traceless part of the extrinsic curvature and the introduction of the contracted Christoffel symbols as independent variables. For generality, we will again write the definitions of the variables and the equations for the case of an arbitrary number D of spacetime dimensions. We define

$$\begin{array}{*{20}c} {\chi = {\gamma ^{- 1/(D - 1)}},\quad \quad K = {\gamma ^{MN}}{K_{MN}},\quad \quad \quad}\\ {{{\tilde \gamma}_{IJ}} = \chi {\gamma _{IJ}}\quad \quad \quad \Leftrightarrow {{\tilde \gamma}^{IJ}} = {1 \over \chi}{\gamma ^{IJ}},\quad \quad \quad}\\ {{{\tilde A}_{IJ}} = \chi \left({{K_{IJ}} - {1 \over {D - 1}}{\gamma _{IJ}}K} \right)\quad \Leftrightarrow {K_{IJ}} = {1 \over \chi}\left({{{\tilde A}_{IJ}} + {1 \over {D - 1}}{{\tilde \gamma}_{IJ}}K} \right),\quad \quad}\\ {{{\tilde \Gamma}^I} = {{\tilde \gamma}^{MN}}\tilde \Gamma _{MN}^I,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ \end{array}$$

where γ ≡ detγIJ and \(\tilde \Gamma _{MN}^I\) is the Christoffel symbol defined in the usual manner in terms of the conformal metric \({\tilde \gamma _{IJ}}\). Note that the definition (58) implies two algebraic and one differential constraints

$$\tilde \gamma = 1,\quad {\tilde \gamma ^{MN}}{\tilde A_{MN}} = 0,\quad \;{{\mathcal G}^I} = {\tilde \Gamma ^I} - {\tilde \gamma ^{MN}}\tilde \Gamma _{MN}^I = 0.$$

Inserting the definition (58) into the ADM equations (52)(53) and using the Hamiltonian and momentum constraints respectively in the evolution equations for K and \({\tilde \Gamma ^I}\) results in the BSSN evolution system

$${\partial _t}\chi = {\beta ^M}{\partial _M}\chi + {2 \over {D - 1}}\chi (\alpha K - {\partial _M}{\beta ^M}),$$
$${\partial _t}{\tilde \gamma _{IJ}} = {\beta ^M}{\partial _M}{\tilde \gamma _{IJ}} + 2{\tilde \gamma _{M(I}}{\partial _{J)}}{\beta ^M} - {2 \over {D - 1}}{\tilde \gamma _{IJ}}{\partial _M}{\beta ^M} - 2\alpha {\tilde A_{IJ}},$$
$$\begin{array}{*{20}c} {{\partial _t}K = {\beta ^M}{\partial _M}K - \chi {{\tilde \gamma}^{MN}}{D_M}{D_N}\alpha + \alpha {{\tilde A}^{MN}}{{\tilde A}_{MN}} + {1 \over {D - 1}}\alpha {K^2}}\\ {+ {{8\pi} \over {D - 2}}\alpha [S + (D - 3)\rho ] - {2 \over {D - 2}}\alpha \Lambda, \quad \quad \quad}\\ \end{array}$$
$$\begin{array}{*{20}c} {{\partial _t}{{\tilde A}_{IJ}} = {\beta ^M}{\partial _M}{{\tilde A}_{IJ}} + 2{{\tilde A}_{M(I}}{\partial _{J)}}{\beta ^M} - {2 \over {D - 1}}{{\tilde A}_{IJ}}{\partial _M}{\beta ^M} + \alpha K{{\tilde A}_{IJ}} - 2\alpha {{\tilde A}_{IM}}{{\tilde A}^M}_{\;\;\;\;J},}\\ {+ \chi {{(\alpha {{\mathcal R}_{IJ}} - {D_I}{D_J}\alpha - 8\pi \alpha {S_{IJ}})}^{{\rm{TF}}}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ \end{array}$$
$$\begin{array}{*{20}c} {{\partial _t}{{\tilde \Gamma}^I} = {\beta ^M}{\partial _M}{{\tilde \Gamma}^I} + {2 \over {D - 1}}{{\tilde \Gamma}^I}{\partial _M}{\beta ^M} - {{\tilde \Gamma}^M}{\partial _M}{\beta ^I} + {{\tilde \gamma}^{MN}}{\partial _M}{\partial _N}{\beta ^I} + {{D - 3} \over {D - 1}}{{\tilde \gamma}^{IM}}{\partial _M}{\partial _N}{\beta ^N}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ {- {{\tilde A}^{IM}}\left[ {(D - 1)\alpha {{{\partial _M}\chi} \over \chi} + 2{\partial _M}\alpha} \right] + 2\alpha \tilde \Gamma _{MN}^I{{\tilde A}^{MN}} - 2{{D - 2} \over {D - 1}}\alpha {{\tilde \gamma}^{IM}}{\partial _M}K - 16\pi {\alpha \over \chi}{j^I}.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ \end{array}$$

Here, the superscript “TF” denotes the trace-free part and we further use the following expressions that relate physical to conformal variables:

$$\Gamma _{JK}^I = \tilde \Gamma _{JK}^I - {1 \over {2\chi}}({\delta ^I}_K{\partial _J}\chi + {\delta ^I}_J{\partial _K}\chi - {\tilde \gamma _{JK}}{\tilde \gamma ^{IM}}{\partial _M}\chi),$$
$${{\mathcal R}_{IJ}} = {\tilde {\mathcal R}_{IJ}} + {\mathcal R}_{IJ}^\chi,$$
$${\mathcal R}_{IJ}^\chi = {{{{\tilde \gamma}_{IJ}}} \over {2\chi}}\left[ {{{\tilde \gamma}^{MN}}{{\tilde D}_M}{{\tilde D}_N}\chi - {{D - 1} \over {2\chi}}{{\tilde \gamma}^{MN}}{\partial _M}\chi \;{\partial _N}\chi} \right] + {{D - 3} \over {2\chi}}\left({{{\tilde D}_I}{{\tilde D}_J}\chi - {1 \over {2\chi}}{\partial _I}\chi \;{\partial _J}\chi} \right),$$
$${\tilde {\mathcal R}_{IJ}} = - {1 \over 2}{\tilde \gamma ^{MN}}{\partial _N}\;{\partial _N}{\tilde \gamma _{IJ}} + {\tilde \gamma _{M(I}}{\partial _{J)}}{\tilde \Gamma ^M} + {\tilde \Gamma ^M}{\tilde \Gamma _{(IJ)M}} + {\tilde \gamma ^{MN}}\left[ {2\tilde \Gamma _{M(I}^K{{\tilde \Gamma}_{J)KN}} + \tilde \Gamma _{IM}^K{{\tilde \Gamma}_{KJN}}} \right],$$
$${D_I}{D_J}\alpha = {\tilde D_I}{\tilde D_J}\alpha + {1 \over \chi}{\partial _{(I}}\chi {\partial _{J)}}\alpha - {1 \over {2\chi}}{\tilde \gamma _{IJ}}{\tilde \gamma ^{MN}}{\partial _M}\chi {\partial _N}\alpha.$$

In practical applications, it turns out necessary for numerical stability to enforce the algebraic constraint \({\tilde \gamma ^{MN}}{\tilde A_{MN}} = 0\) whereas enforcement of the unit determinant \(\tilde \gamma = 1\) appears to be optional. A further subtlety is concerned with the presence of the conformal connection functions \({\tilde \Gamma ^I}\) on the right-hand side of the BSSN equations. Two recipes have been identified that provide long-term stable numerical evolutions. (i) The independently evolved \({\tilde \Gamma ^I}\) are only used when they appear in differentiated form but are replaced by their definition in terms of the conformal metric \({\tilde \gamma _{IJ}}\) everywhere else [23]. (ii) Alternatively, one can add to the right-hand side of Eq. (64) a term \(- \sigma {\mathcal G^I}{\partial _M}{\beta ^M}\), where σ is a positive constant [803].

We finally note that in place of the variable χ, alternative choices for evolving the conformal factor are in use in some NR codes, namely ϕ≡ − (ln χ)/4 [65] or \(W \equiv \sqrt \chi\) [540]. An overview of the specific choices of variables and treatment of the BSSN constraints for the present generation of codes is given in Section 4 of [429].

The generalized harmonic gauge formulation

It has been realized a long time ago that the Einstein equations have a mathematically appealing form if one imposes the harmonic gauge condition \({\square x^\alpha} = - {g^{\mu \nu}}\Gamma _{\mu \nu}^\alpha = 0\) [294]. Taking the derivative of this condition eliminates a specific combination of second derivatives from the Ricci tensor such that its principal part is that of the scalar wave operator

$${R_{\alpha \beta}} = - {1 \over 2}{g^{\mu \nu}}{\partial _\mu}{\partial _\nu}{g_{\alpha \beta}} + \ldots,$$

where the dots denote terms involving at most the first derivative of the metric. In consequence of this simplification of the principal part, the Einstein equations in harmonic gauge can straightforwardly be written as a strongly hyperbolic system. This formulation even satisfies the stronger condition of symmetric hyperbolicity which is defined in terms of the existence of a conserved, positive energy [674], and harmonic coordinates have played a key part in establishing local uniqueness of the solution to the Cauchy problem in GR [327, 141, 321].

This particularly appealing property of the Ricci tensor can be maintained for arbitrary coordinates by introducing the functions [333, 343]

$${H^\alpha} \equiv \square{x^\alpha} = - {g^{\mu \nu}}\Gamma _{\mu \nu}^\alpha,$$

and promoting them to the status of independently evolved variables; see also [630, 519]. This is called the Generalized Harmonic Gauge formulation.

With this definition, it turns out convenient to consider the generalized class of equations

$${R_{\alpha \beta}} - {\nabla _{(\alpha}}{{\mathcal C}_{\beta)}} = 8\pi \left({{T_{\alpha \beta}} - {1 \over {D - 2}}T{g_{\alpha \beta}}} \right) + {2 \over {D - 2}}\Lambda {g_{\alpha \beta}},$$

where \({\mathcal C^\alpha} \equiv {H^\alpha} - {\square x^\alpha}\). The addition of the term \({\nabla _{(\alpha}}{\mathcal C_{\beta)}}\) replaces the contribution of \({\nabla _{(\alpha}}{\square x_{\beta)}}\) to the Ricci tensor in terms of \({\nabla _{(\alpha}}{H_{\beta)}}\) and thus changes the principal part to that of the scalar wave operator. A solution to the Einstein equations is now obtained by solving Eq. (72) subject to the constraint \({\mathcal C_\alpha} = 0\).

The starting point for a Cauchy evolution are initial data gαβ and tgαβ which satisfy the constraints \({\mathcal C^\alpha} = 0 = {\partial _t}{\mathcal C^\alpha}\). A convenient manner to construct such initial data is to compute the initial Hα directly from Eq. (71) so that \({{\mathcal C}^\alpha} = 0\) by construction. It can then be shown [519] that the ADM constraints (54), (55) imply \({\partial _t}{\mathcal C^\mu} = 0\). By virtue of the contracted Bianchi identities, the evolution of the constraint system obeys the equation

$$\square{{\mathcal C}_\alpha} = - {{\mathcal C}^\mu}{\nabla _{(\mu}}{{\mathcal C}_{\alpha)}} - {{\mathcal C}^\mu}\left[ {8\pi \left({{T_{\mu \alpha}} - {1 \over {D - 2}}T{g_{\mu \alpha}}} \right) + {2 \over {D - 2}}\Lambda {g_{\mu \alpha}}} \right],$$

and the constraint \({{\mathcal C}^\alpha} = 0\) is preserved under time evolution in the continuum limit.

A key addition to the GHG formalism has been devised by Gundlach et al. [377] in the form of damping terms which prevent growth of numerical violations of the constraints \({{\mathcal C}^\alpha} = 0\) due to discretization or roundoff errors.

Including these damping terms and using the definition (71) to substitute higher derivatives in the Ricci tensor, the generalized Einstein equations (72) can be written as

$$\begin{array}{*{20}c} {{g^{\mu \nu}}{\partial _\mu}{\partial _\nu}{g_{\alpha \beta}} = - 2{\partial _\nu}{g_{\mu (\alpha}}\,{\partial _{\beta)}}{g^{\mu \nu}} - 2{\partial _{(\alpha}}{H_{\beta)}} + 2{H_\mu}\Gamma _{\alpha \beta}^\mu - 2\Gamma _{\nu \alpha}^\mu \Gamma _{\mu \beta}^\nu \quad \quad \quad \quad \quad \quad \quad} \\ {- 8\pi {T_{\alpha \beta}} + {{8\pi T - 2\Lambda} \over {D - 2}}{g_{\alpha \beta}} - 2\kappa \;[2{n_{(\alpha}}{{\mathcal C}_{\beta)}} - \lambda {g_{\alpha \beta}}{n^\mu}{{\mathcal C}_\mu}]\;\;,} \\ \end{array}$$

where κ, λ are user-specified constraint-damping parameters. An alternative first-order system of the GHG formulation has been presented in Ref. [519].

Beyond BSSN: Improvements for future applications

The vast majority of BH evolutions in generic 4-dimensional spacetimes have been performed with the GHG and the BSSN formulations. It is interesting to note in this context the complementary nature of the two formulations’ respective strengths and weaknesses. In particular, the constraint subsystem of the BSSN equations contains a zero-speed mode [100, 379, 378] which may lead to large Hamiltonian constraint violations. The GHG system does not contain such modes and furthermore admits a simple way of controlling constraint violations in the form of damping terms [377]. Finally, the wave-equation-type principal part of the GHG system allows for the straightforward construction of constraint-preserving boundary conditions [650, 492, 665]. On the other hand, the BSSN formulation is remarkably robust and allows for the simulation of BH binaries over a wide range of the parameter space with little if any modifications of the gauge conditions; cf. Section 6.4. Combination of these advantages in a single system has motivated the exploration of improvements to the BSSN system and in recent years resulted in the identification of a conformal version of the Z4 system, originally developed in Refs. [113, 112, 114], as a highly promising candidate [28, 163, 775, 428].

The key idea behind the Z4 system is to replace the Einstein equations with a generalized class of equations given by

$${G_{\alpha \beta}} = 8\pi {T_{\alpha \beta}} - {\nabla _\alpha}{Z_\beta} - {\nabla _\beta}{Z_\alpha} + {g_{\alpha \beta}}{\nabla _\mu}{Z^\mu} + {\kappa _1}[{n_\alpha}{Z_\beta} + {n_\beta}{Z_\alpha} + {\kappa _2}{g_{\alpha \beta}}{n_M}{Z^M}],$$

where Zα is a vector field of constraints which is decomposed into space and time components according to \(\Theta \equiv - {n^\mu}{Z_\mu}\;{\rm{and}}\;{Z_I} = \;{\bot ^\mu}_I{Z_\mu}\). Clearly, a solution to the Einstein equations is recovered provided the constraint Zμ = 0 is satisfied. The conformal version of the Z4 system is obtained in the same manner as for the BSSN system and leads to time evolution equations for a set of variables nearly identical to the BSSN variables but augmented by the constraint variable Θ. The resulting evolution equations given in the literature vary in details, but clearly represent relatively minor modifications for existing BSSN codes [28, 163, 428]. Investigations have shown that the conformal Z4 system is indeed suitable for implementation of constraint preserving boundary conditions [664] and that constraint violations in simulations of gauge waves and BH and NS spacetimes are indeed smaller than those obtained for the BSSN system, in particular when constraint damping is actively enforced [28, 428]. This behaviour also manifests itself in more accurate results for the gravitational radiation in binary inspirals [428]. In summary, the conformal Z4 formulation is a very promising candidate for future numerical studies of BH spacetimes, including in particular the asymptotically AdS case where a rigorous control of the outer boundary is of utmost importance; cf. Section 6.6 below.

Another modification of the BSSN equations is based on the use of densitized versions of the trace of the extrinsic curvature and the lapse function as well as the traceless part of the extrinsic curvature with mixed indices [497, 795]. Some improvements in simulations of colliding BHs in higher-dimensional spacetimes have been found by careful exploration of the densitization parameter space [791].

Alternative formulations

The formulations discussed in the previous subsections are based on a spacetime split of the Einstein equations. A natural alternative to such a split is given by the characteristic approach pioneered by Bondi et al. and Sachs [118, 667]. Here, at least one coordinate is null and thus adapted to the characteristics of the vacuum Einstein equations. For generic four-dimensional spacetimes with no symmetry assumptions, the characteristic formalism results in a natural hierarchy of two evolution equations, four hypersurface equations relating variables on hypersurfaces of constant retarded (or advanced) time, as well as three supplementary and one trivial equations. A comprehensive overview of characteristic methods in NR is given in the Living Reviews article [788]. Although characteristic codes have been developed with great success in spacetimes with additional symmetry assumptions, evolutions of generic BH spacetimes face the problem of formation of caustics, resulting in a breakdown of the coordinate system; see [59] for a recent investigation. One possibility to avoid the problem of caustic formation is Cauchy-characteristic matching, the combination of a (D − 1) + 1 or Cauchy-type numerical scheme in the interior strong-field region with a characteristic scheme in the outer parts. In the form of Cauchy-characteristic extraction, i.e., ignoring the injection of information from the characteristic evolution into the inner Cauchy region, this approach has been used to extract GWs with high accuracy from numerical simulations of compact objects [642, 60].

All the Cauchy and characteristic or combined approaches we have discussed so far, evolve the physical spacetime, i.e., a manifold with metric (ℳ, gαβ). An alternative approach for asymptotically flat spacetimes dating back to Hübner [444] instead considers the numerical construction of a conformal spacetime \((\tilde {\mathcal M},{\tilde g_{\alpha \beta}})\) where \({\tilde g_{\alpha \beta}} = {\Omega ^2}{g_{\alpha \beta}}\) subject to the condition that gαβ satisfies the Einstein equations on ℳ. The conformal factor Ω vanishes at null infinity ℐ = ℐ+ ∪ ℐ− of the physical spacetime which is thus conformally related to an interior of the unphysical manifold \(\tilde{\mathcal M},{\tilde g_{\alpha \beta}}\) which extends beyond the physical manifold. A version of these conformal field equations that overcomes the singular nature of the transformed Einstein equations at has been developed by Friedrich [332, 331]. This formulation is suitable for a 3+1 decomposition into a symmetric hyperbolic systemFootnote 10 of evolution equations for an enhanced (relative to the ADM decomposition) set of variables. The additional cost resulting from the larger set of variables, however, is mitigated by the fact that these include projections of the Weyl tensor that directly encode the GW content. Even though the conformal field equations have as yet not resulted in simulations of BH systems analogous to those achieved in BSSN or GHG, their elegance in handling the entire spacetime without truncation merits further investigation. For more details about the formulation and numerical applications, we refer the reader to the above articles, Lehner’s review [509], Frauendiener’s Living Reviews article [328] as well as [329, 26] and references therein. A brief historic overview of many formulations of the Einstein equations (including systems not discussed in this work) is given in Ref. [702]; see in particular Figures 3 and 4 therein.

Figure 3
figure 3

D-dimensional representation of head-on collisons for spinless BHs, with isometry group SO(D−2) (left), and non-head-on collisons for BHs spinning in the orbital plane, with isometry group SO(D − 3) (right). Image reproduced with permission from [841], copyright by APS.

Figure 4
figure 4

Illustration of mesh refinement for a BH binary with one spatial dimension suppressed. Around each BH (marked by the spherical AH), two nested boxes are visible. These are immersed within one large, common grid or refinement level.

We finally note that for simulations of spacetimes with high degrees of symmetry, it often turns out convenient to directly impose the symmetries on the shape of the line element rather than use one of the general formalisms discussed so far. As an example, we consider the classic study by May and White [544, 545] of the dynamics of spherically symmetric perfect fluid stars. A four-dimensional spherically symmetric spacetime can be described in terms of the simple line element

$${\rm{d}}{s^2} = - {a^2}(x,t)\;{\rm{d}}{t^2} + {b^2}(x,t)\;{\rm{d}}{x^2} + {R^2}(x,t)\;{\rm{d}}\Omega _2^2,$$

where \({\rm{d}}\Omega _2^2\) is the line element of the 2-sphere. May and White employ Lagrangian coordinates co-moving with the fluid shells which is imposed through the form of the energy-momentum tensor \({T^0}_0 = - \rho (1 + \epsilon),\;{T^1}_1 = {T^2}_2 = {T^3}_3 = P\). Here, the rest-mass density ρ, internal energy ϵ, and pressure P are functions of the radial and time coordinates. Plugging the line element (76) into the Einstein equations (39) with D = 4, Λ = 0 and the equations of conservation of energy-momentum \({\nabla _\mu}{T^\mu}_\alpha = 0\), result in a set of equations for the spatial and time derivatives of the metric and matter functions amenable for a numerical treatment; cf. Section II in Ref. [544] for details.

Einstein’s equations extended to include fundamental fields

The addition of matter to the spacetime can, in principle, be done using the formalism just laid downFootnote 11. The simplest extension of the field equations to include matter is described by the Einstein-Hilbert action (in 4-dimensional asymptotically flat spacetimes) minimally coupled to a complex, massive scalar field Φ with mass parameter μs = ms/ħ,

$$S = \int {{{\rm{d}}^4}} x\sqrt {- g} \left({{R \over {16\pi}} - {1 \over 2}{g^{\mu \nu}}{\partial _\mu}{\Phi ^{\ast}}{\partial _\nu}\Phi - {1 \over 2}\mu _S^2{\Phi ^{\ast}}\Phi} \right).$$

if we introduce a time reduction variable defined as

$$\Pi = - {1 \over \alpha}({\partial _t} - {{\mathcal L}_\beta})\Phi,$$

we recover the equations of motion and constraints (52)(55) with D = 4, Λ = 0 and with energy density ρ, energy-momentum flux ji and spatial components Sij of the energy-momentum tensor given by

$$\rho = {1 \over 2}\Pi ^{\ast}\Pi + {1 \over 2}\mu _S^2\Phi ^{\ast}\Phi + {1 \over 2}{D^i}\Phi ^{\ast}{D_i}\Phi,$$
$${j_i} = {1 \over 2}(\Pi ^{\ast}{D_i}\Phi + \Pi {D_i}\Phi ^{\ast}),$$
$${S_{ij}} = {1 \over 2}({D_i}\Phi ^{\ast}{D_j}\Phi + {D_i}\Phi {D_j}\Phi ^{\ast}) + {1 \over 2}{\gamma _{ij}}(\Pi ^{\ast}\Pi - \mu _S^2\Phi ^{\ast}\Phi - {D^k}\Phi ^{\ast}{D_k}\Phi).$$

Vector fields can be handled in similar fashion, we refer the reader to Ref. [794] for linear studies and to Refs. [595, 598, 838, 839] for full nonlinear evolutions.

In summary, a great deal of progress has been made in recent years concerning the well-posedness of the numerical methods used for the construction of spacetimes. We note, however, that the well-posedness of many problems beyond electrovacuum GR remains unknown at present. This includes, in particular, a wide class of alternative theories of gravity where it is not clear whether they admit well-posed IBVPs.

Higher-dimensional NR in effective “3 + 1” form

Performing numerical simulations in generic higher-dimensional spacetimes represents a major challenge for simple computational reasons. Contemporary simulations of compact objects in four spacetime dimensions require \({\mathcal O(100)}\) cores and \({\mathcal O(100)}\) Gb of memory for storage of the fields on the computational domain. In the absence of spacetime symmetries, any extra spatial dimension needs to be resolved by \({\mathcal O(100)}\) grid points resulting in an increase by about two orders of magnitude in both memory requirement and computation time. In spite of the rapid advance in computer technology, present computational power is pushed to its limits with D = 5 or, at best, D = 6 spacetime dimensions. For these reasons, as well as the fact that the community already has robust codes available in D = 4 dimensions, NR applications to higher-dimensional spacetimes have so far focussed on symmetric spacetimes that allow for a reduction to an effectively four-dimensional formalism. Even though this implies a reduced class of spacetimes available for numerical study, many of the most important questions in higher-dimensional gravity actually fall into this class of spacetimes. In the following two subsections we will describe two different approaches to achieve such a dimensional reduction, for the cases of spacetimes with SO(D − 2) or SO(D − 3) isometry, i.e., the rotational symmetry leaving invariant SD−3. or SD−4, respectively (we denote with Sn the n-dimensional sphere). The group SO(D − 2) is the isometry of, for instance, head-on collisions of non-rotating BHs, while the group SO(D − 3) is the isometry of non-head-on collisions of non-rotating BHs; SO(D − 3) is also the isometry of non-head-on collisions of rotating BHs with one nonvanishing angular momentum, generating rotations on the orbital plane (see Figure 3). Furthermore, the SO(D − 3) group is the isometry of a single rotating BH, with one non-vanishing angular momentum. We remark that, in order to implement the higher-dimensional system in (modified) four-dimensional evolution codes, it is necessary to perform a 4 + (D − 4) splitting of the spacetime dimensions. With such splitting, the equations have a manifest SO(D − 3) symmetry, even when the actual isometry is larger.

We shall use the following conventions for indices. As before, Greek indices a, β, … cover all spacetime dimensions and late upper case capital Latin indices I, J, … = 1, … D − 1 cover the D − 1 spatial dimensions, whereas late lower case Latin indices i, j, … = 1, 2, 3 cover the three spatial dimensions of the eventual computational domain. In addition, we introduce barred Greek indices \(\bar \alpha, \bar \beta, \ldots = 0, \ldots, 3\) which also include time, and early lower case Latin indices a, b, … = 4, …, D − 1 describing the D − 4 spatial directions associated with the rotational symmetry. Under the 4 + (D − 4) splitting of spacetime dimensions, then, the coordinates xμ decompose as \({x^\mu} \rightarrow ({x^{\bar \mu}},{x^a})\). When explicitly stated, we shall consider instead a 3 + (D − 3) splitting, e.g., with barred Greek indices running from 0 to 2, and early lower case Latin indices running from 3 to D − 1.

Dimensional reduction by isometry

The idea of dimensional reduction had originally been developed by Geroch [347] for four-dimensional spacetimes possessing one Killing field as for example in the case of axisymmetry; for numerical applications see for example Refs. [535, 704, 722, 214]. The case of arbitrary spacetime dimensions and number of Killing vectors has been discussed in Refs. [210, 211].Footnote 12 More recently, this idea has been used to develop a convenient formalism to perform NR simulations of BH dynamical systems in higher dimensions, with SO(D − 2) or SO(D − 3) isometry [841, 797]. Comprehensive summaries of this approach are given in Refs. [835, 791, 792].

The starting point is the general D-dimensional spacetime metric written in coordinates adapted to the symmetry

$${\rm{d}}{s^2} = {g_{\alpha \beta}}\;{\rm{d}}{x^\alpha}\;{\rm{d}}{x^\beta} = \left({{g_{\bar \mu \bar \nu}} + {e^2}{\kappa ^2}{g_{ab}}{B^a}_{\bar \mu}{B^b}_{\bar \nu}} \right)\;{\rm{d}}{x^{\bar \mu}}\;{\rm{d}}{x^{\bar \nu}} + 2e\kappa {B^a}_{\bar \mu}{g_{ab}}\;{\rm{d}}{x^{\bar \mu}}\;{\rm{d}}{x^b} + {g_{ab}}\;{\rm{d}}{x^a}\;{\rm{d}}{x^b}.$$

Here, κ and e represent a scale parameter and a coupling constant that will soon drop out and play no role in the eventual spacetime reduction. We note that the metric (82) is fully general in the same sense as the spacetime metric in the ADM split discussed in Section 6.1.1.

The special case of a SO(D − 2) (SO(D − 3)) isometry admits (n+1)n/2 Killing fields ξ(i) where nD − 3 (n ≡ D − 4) stands for the number of extra dimensions. For n = 2, for instance, there exist three Killing fields given in spherical coordinates by ξ(1) = ϕ, ξ(2) = sin ϕ θ +cot θ cos ϕ ϕ, ξ(3) = cos ϕθ − cot θ sin ϕ ϕ.

Killing’s equation ξ(i)gAB = 0 implies that

$${{\mathcal L}_{{\xi _{(i)}}}}{g_{ab}} = 0,\quad \quad {{\mathcal L}_{{\xi _{(i)}}}}{B^a}_{\bar \mu} = 0,\quad \quad {{\mathcal L}_{{\xi _{(i)}}}}{g_{\bar \mu \bar \nu}} = 0,$$

where, as discussed above, the decomposition \({x^\mu} \rightarrow ({x^{\bar \mu}},{x^a})\) describes a 4 + (D − 4) splitting in the case of SO(D − 3) isometry, and a 3 + (D − 3) splitting in the case of SO(D − 2) isometry.

From these conditions, we draw the following conclusions: (i) \({g_{ab}} = {e^{2\psi ({x^{\bar \mu}})}}{\Omega _{ab}}\), where Ωab is the metric on the Sn sphere with unit radius and ψ is a free field; (ii) \({g_{\bar \mu \bar \nu}} = {g_{\bar \mu \bar \nu}}({x^{\bar \sigma}})\) in adapted coordinates; (iii) \([{\xi _{(i)}},{B_{\bar \mu}}] = 0\). We here remark an interesting consequence of the last property. Since, for n ≥ 2, there exist no nontrivial vector fields on Sn that commute with all Killing fields, all vector fields \({B^a}_{\bar \mu}\) vanish; when, instead, n = 0,1 (i.e., when D = 4, or D = 5 for SO(D − 3) isometry), this conclusion can not be made. In this approach, as it has been developed up to now [841, 797, 796], one restricts to the n ≥ 2 case, and it is then possible to assume \({B^a}_{\bar \mu} \equiv 0\). Eq. (82) then reduces to the formFootnote 13

$${\rm{d}}{s^2} = {g_{\bar \mu \bar \nu}}\;{\rm{d}}{x^{\bar \mu}}\;{\rm{d}}{x^{\bar \nu}} + {e^{2\psi ({x^{\bar \mu}})}}\;{\Omega _{ab}}\;{\rm{d}}{x^a}\;{\rm{d}}{x^b}.$$

for this reason, this approach can only be applied when D ≥ 5 in the case of SO(D − 2) isometry, and D ≥ 6 in the case of SO(D − 3) isometry.

As mentioned above, since the Einstein equations have to be implemented in a four-dimensional NR code, we eventually have to perform a 4 + (D − 4) splitting, even when the spacetime isometry is SO(D − 2). This means that the line element is (84), with \(\bar \alpha, \bar \beta, \ldots = 0, \ldots, 3\) and a, b, … = 4, …, D − 1. In this case, only a subset SO(D − 3) ⊂ SO(D − 2) of the isometry is manifest in the line element; the residual symmetry yields an extra relation among the components \({g_{\bar \mu \bar \nu}}\). If the isometry group is SO(D − 3), the line element is the same, but there is no extra relation.

A tedious but straightforward calculation [835] shows that the components of the D-dimensional Ricci tensor can then be written as

$$\begin{array}{*{20}c} {{R_{ab}} = \{(D - 5) - {e^{2\psi}}[(D - 4){\partial ^{\bar \mu}}\psi {\partial _{\bar \mu}}\psi + {{\bar \nabla}^{\bar \mu}}{\partial _{\bar \mu}}\psi ]\} {\Omega _{ab}},\quad \quad \quad \;}\\ {{R_{\bar \mu a}} = 0,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ {{R_{\bar \mu \bar \nu}} = {{\bar R}_{\bar \mu \bar \nu}} - (D - 4)({{\bar \nabla}_{\bar \nu}}{\partial _{\bar \mu}}\psi + {\partial _{\bar \mu}}\psi {\partial _{\bar \nu}}\psi),\quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ {R = \bar R + (D - 4)[(D - 5){e^{- 2\psi}} - 2{{\bar \nabla}^{\bar \mu}}{\partial _{\bar \mu}}\psi - (D - 3){\partial ^{\bar \mu}}\psi {\partial ^{\bar \mu}}\psi ],}\\ \end{array}$$

where \({\bar R_{\bar \mu \bar \nu}},\;\bar R\;{\rm{and}}\;\bar \nabla\) respectively denote the 3 + 1-dimensional Ricci tensor, Ricci scalar and covariant derivative associated with the 3 + 1 metric \({\bar g_{\bar \mu \bar \nu}} \equiv {g_{\bar \mu \bar \nu}}\). The D-dimensional vacuum Einstein equations with cosmological constant Λ can then be formulated in terms of fields on a 3 + 1-dimensional manifold

$${\bar R_{\bar \mu \bar \nu}} = (D - 4)({\bar \nabla _{\bar \nu}}{\partial _{\bar \mu}}\psi - {\partial _{\bar \mu}}\psi {\partial _{\bar \nu}}\psi) - \Lambda {\bar g_{\bar \mu \bar \nu}},$$
$${e^{2\psi}}[(D - 4){\partial ^{\bar \mu}}\psi {\partial _{\bar \mu}}\psi + {\bar \nabla ^{\bar \mu}}{\partial _{\bar \mu}}\psi - \Lambda ] = (D - 5).$$

One important comment is in order at this stage. If we describe the three spatial dimensions in terms of Cartesian coordinates (x, y, z), one of these is now a quasi-radial coordinate. Without loss of generality, we choose y and the computational domain is given by x, z ∈ ℝ, y ≥ 0. In consequence of the radial nature of the y direction, e2ψ = 0 at y = 0. Numerical problems arising from this coordinate singularity can be avoided by working instead with a rescaled version of the variable e2ψ. More specifically, we also include the BSSN conformal factor e4ϕ in the redefinition and write

$$\zeta \equiv {{{e^{- 4\phi}}} \over {{y^2}}}{e^{2\psi}}.$$

The BSSN version of the D-dimensional vacuum Einstein equations (86), (87) with Λ = 0 in its dimensionally reduced form on a 3 + 1 manifold is then given by Eqs. (60)(64) with the following modifications, (i) Upper-case capital indices I, J, … are replaced with their lower case counterparts i, j, … = 1, 2, 3. (ii) The (D − 1) dimensional metric γIJ, Christoffel symbols \(\Gamma _{JK}^I\), covariant derivative D, conformal factor χ and extrinsic curvature variables K and ÃIJ are replaced by the 3 dimensional metric γij, the 3 dimensional Christoffel symbols \(\Gamma _{jk}^i\), the covariant derivative D, as well as the conformal factor χ, K and Aij defined in analogy to Eq. (58) with D = 4, i.e.

$$\begin{array}{*{20}c} {\chi = {\gamma ^{- 1/3}},\quad \quad K = {\gamma ^{nm}}{K_{mn}},\quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ {{{\tilde \gamma}_{ij}} = \chi {\gamma _{ij}}\quad \quad \quad \Leftrightarrow {{\tilde \gamma}^{ij}} = {1 \over \chi}{\gamma ^{ij}},\quad \quad \quad \quad \quad \quad}\\ {{{\tilde A}_{ij}} = \chi \left({{K_{ij}} - {1 \over 3}{\gamma _{ij}}K} \right)\quad \Leftrightarrow {K_{ij}} = {1 \over \chi}\left({{{\tilde A}_{ij}} + {1 \over 3}{{\tilde \gamma}_{ij}}K} \right),\quad \quad \quad \quad}\\ {{{\tilde \Gamma}^i} = {{\tilde \gamma}^{mn}}\tilde \Gamma _{mn}^i.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ \end{array}$$

(iii) The extra dimensions manifest themselves as quasi-matter terms given by

$$\begin{array}{*{20}c} {{{4\pi (\rho + S)} \over {D - 4}} = (D - 5){\chi \over \zeta}{{{{\tilde \gamma}^{yy}}\zeta - 1} \over {{y^2}}} - {{2D - 7} \over {4\zeta}}{{\tilde \gamma}^{mn}}{\partial _m}\eta \;{\partial _n}\chi - \chi {{{{\tilde \Gamma}^y}} \over y} + {{D - 6} \over 4}{\chi \over {{\zeta ^2}}}{{\tilde \gamma}^{mn}}{\partial _m}\zeta \;{\partial _n}\zeta \quad \quad \quad \quad \quad \;\quad}\\ {+ {1 \over {2\zeta}}{{\tilde \gamma}^{mn}}(\chi {{\tilde D}_m}{\partial _n}\zeta - \zeta {{\tilde D}_m}{\partial _n}\chi) + (D - 4){{{{\tilde \gamma}^{ym}}} \over y}\left({{\chi \over \zeta}{\partial _m}\zeta - {\partial _m}\chi} \right) - {{K{K_\zeta}} \over \zeta} - {{{K^2}} \over 3}}\\ {- {1 \over 2}{{{{\tilde \gamma}^{ym}}} \over y}{\partial _m}\chi + {{D - 1} \over 4}{{\tilde \gamma}^{ym}}{{{\partial _m}\chi \;{\partial _n}\chi} \over \chi} - (D - 5){{\left({{{{K_\zeta}} \over \zeta} + {K \over 3}} \right)}^2},\quad \quad \quad \quad \quad \quad \;}\\ \end{array}$$
$$\begin{array}{*{20}c} {{{8\pi \chi S_{ij}^{{\rm{TF}}}} \over {D - 4}} = - \left({{{{K_\zeta}} \over \zeta} + {K \over 3}} \right){{\tilde A}_{ij}} + {1 \over 2}\left[ {{{2\chi} \over {y\zeta}}({\delta ^y}_{(j}{\partial _{i)}}\zeta - \zeta \tilde \Gamma _{ij}^y) + {1 \over {2\chi}}{\partial _i}\chi \;{\partial _j}\chi - {{\tilde D}_i}{\partial _j}\chi} \right. + {\chi \over \zeta}{{\tilde D}_i}{\partial _j}\zeta}\\ {{{\left. {+ {1 \over {2\chi}}{{\tilde \gamma}_{ij}}{{\tilde \gamma}^{mn}}{\partial _n}\chi \left({{\partial _m}\chi - {\chi \over \zeta}{\partial _m}\zeta} \right) - {{\tilde \gamma}_{ij}}{{{{\tilde \gamma}^{ym}}} \over y}{\partial _m}\chi - {\chi \over {2{\zeta ^2}}}{\partial _i}\zeta \;{\partial _j}\zeta} \right]}^{{\rm{TF}}}}}\\ \end{array}$$
$$\begin{array}{*{20}c} {{{16\pi {j_i}} \over {D - 4}} = {2 \over y}\left({{\delta ^y}_i{{{K_\zeta}} \over \zeta} - {{\tilde \gamma}^{ym}}{{\tilde A}_{mi}}} \right) + {2 \over \zeta}{\partial _i}{K_\zeta} - {{{K_\zeta}} \over \zeta}\left({{1 \over \chi}{\partial _i}\chi + {1 \over \zeta}{\partial _i}\zeta} \right) + {2 \over 3}{\partial _i}K}\\ {- {{\tilde \gamma}^{nm}}{{\tilde A}_{mi}}\left({{1 \over \zeta}{\partial _n}\zeta - {1 \over \chi}{\partial _n}\chi} \right).\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ \end{array}$$

Here, Kζ ≡ −(2αy2)−1(∂tℒβ)(ζy2). The evolution of the field ζ is determined by Eq. (87) which in terms of the BSSN variables becomes

$${\partial _t}\zeta = {\beta ^m}{\partial _m}\zeta - 2\alpha {K_\zeta} - {2 \over 3}\zeta {\partial _m}{\beta ^m} + 2\zeta {{{\beta ^y}} \over y},$$
$$\begin{array}{*{20}c} {{\partial _t}{K_\zeta} = {\beta ^m}{\partial _m}{K_\zeta} - {2 \over 3}{K_\zeta}{\partial _m}{\beta ^m} + 2{{{\beta ^y}} \over y}{K_\zeta} - {1 \over 3}\zeta ({\partial _t} - {{\mathcal L}_\beta})K - \chi \zeta {{{{\tilde \gamma}^{ym}}} \over y}{\partial _m}\alpha \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ {- {1 \over 2}{{\tilde \gamma}^{ym}}{\partial _m}\alpha \;(\chi {\partial _n}\zeta - \zeta {\partial _n}\chi) + \alpha \left[ {(5 - D)\chi {{\zeta {{\tilde \gamma}^{yy}} - 1} \over {{y^2}}} + (4 - D)\chi {{{{\tilde \gamma}^{ym}}} \over y}{\partial _m}\zeta \quad \quad \quad \quad} \right.}\\ {+ {{2D - 7} \over 2}\zeta {{{{\tilde \gamma}^{ym}}} \over y}{\partial _m}\chi + {{6 - D} \over 4}{\chi \over \zeta}{{\tilde \gamma}^{mn}}{\partial _m}\zeta \;{\partial _n}\zeta + {{2D - 7} \over 4}{{\tilde \gamma}^{mn}}{\partial _m}\zeta \;{\partial _n}\chi \quad \quad \quad \quad \quad \quad}\\ {+ {{1 - D} \over 4}{\zeta \over \chi}{{\tilde \gamma}^{mn}}{\partial _m}\chi \;{\partial _n}\chi + (D - 6){{K_\zeta ^2} \over \zeta} + {{2D - 5} \over 3}K{K_\zeta} + {{D - 1} \over 9}\zeta {K^2}\quad \quad \quad \quad \quad \quad}\\ {\left. {+ {1 \over 2}{{\tilde \gamma}^{mn}}(\zeta {{\tilde D}_m}{\partial _n}\chi - \chi {{\tilde D}_m}{\partial _n}\zeta) + \chi \zeta {{{{\tilde \Gamma}^y}} \over y}} \right].\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ \end{array}$$

It has been demonstrated in Ref. [841] how all terms containing factors of y in the denominator can be regularized using the symmetry properties of tensors and their derivatives across y = 0 and assuming that the spacetime does not contain a conical singularity.

The cartoon method

The cartoon method has originally been developed in Ref. [25] for evolving axisymmetric four-dimensional spacetimes using an effectively two-dimensional spatial grid which employs ghostzones, i.e., a small number of extra gridpoints off the computational plane required for evaluating finite differences in the third spatial direction. Integration in time, however, is performed exclusively on the two-dimensional plane whereas the ghostzones are filled in after each timestep by appropriate interpolation of the fields in the plane and subsequent rotation of the solution using the axial spacetime symmetry. A version of this method has been applied to 5-dimensional spacetimes in Ref. [820]. For arbitrary spacetime dimensions, however, even the relatively small number of ghostzones required in every extra dimension leads to a substantial increase in the computational resources; for fourth-order finite differencing, for example, four ghostzones are required in each extra dimension resulting in an increase of the computational domain by an overall factor 5D−4. An elegant scheme to avoid this difficulty while preserving all advantages of the cartoon method has been developed in Ref. [630] and is sometimes referred to as the modified cartoon method. This method has been applied to D > 5 dimensions in Refs. [700, 512, 821] and we will discuss it now in more detail.

Let us consider for illustrating this method a D-dimensional spacetime with SO(D − 3) symmetry and Cartesian coordinates xμ = (t, x, y, z, wa), where a = 4, …, D − 1. Without loss of generality, the coordinates are chosen such that the SO(D − 3) symmetry implies rotational symmetry in the planes spanned by each choice of two coordinates fromFootnote 14 (y, wa). The goal is to obtain a formulation of the D-dimensional Einstein equations (60)(69) with SO(D − 3) symmetry that can be evolved exclusively on the xyz hyperplane. The tool employed for this purpose is to use the spacetime symmetries in order to trade derivatives off the hyperplane, i.e., in the wa directions, for derivatives inside the hyperplane. Furthermore, the symmetry implies relations between the D-dimensional components of the BSSN variables.

These relations are obtained by applying a coordinate transformation from Cartesian to polar coordinates in any of the two-dimensional planes spanned by y and w, where wwa for any particular choice of a ∈ {4, …, D − 1}

$$\begin{array}{*{20}c} {\rho = \sqrt {{y^2} + {w^2}}, y = \rho \cos \varphi,}\\ {\varphi = \arctan \;{w \over y},\;w = \rho \sin \varphi.}\\ \end{array}$$

Spherical symmetry in nD − 4 dimensions implies the existence of n(n + 1)/2 Killing vectors, one for each plane with rotational symmetry. For each Killing vector ξ, the Lie derivative of the spacetime metric vanishes. For the yw plane, in particular, the Killing vector field is ξ = ϕ and the Killing condition is given by the simple relation

$${\partial _\varphi}{g_{\mu \nu}} = 0.$$

All ADM and BSSN variables are constructed from the spacetime metric and a straightforward calculation demonstrates that the Lie derivatives along ϕ of all these variables vanish. For D ≥ 6, we can always choose the coordinates such that for µ, ≠ ϕ, gµϕ = 0 which implies the vanishing of the BSSN variables \({\beta ^\varphi} = {\tilde \gamma ^{\mu \varphi}} = {\tilde \Gamma ^\varphi} = 0\) The case of SO(D − 3) symmetry in D = 5 dimensions is special in the same sense as already discussed in Section 6.2.1 and the vanishing of \({\tilde \Gamma ^\varphi}\) does not in general hold. As before, we therefore consider in D = 5 the more restricted class of SO(D − 2) isometry which implies \({\tilde \Gamma ^\varphi} = 0\). Finally, the Cartesian coordinates wa can always be chosen such that the diagonal metric components are equal,

$${\gamma _{{w^1}{w^1}}} = {\gamma _{{w^2}{w^2}}} = \ldots \equiv {\gamma _{ww}}.$$

We can now exploit these properties in order to trade derivatives in the desired manner. We shall illustrate this for the second w derivative of the ww component of a symmetric \((_2^0)\) tensor density S of weight \({\mathcal W}\) which transforms under change of coordinates \({x^\mu} \leftrightarrow {x^{\hat \alpha}}\) according to

$${S_{\hat \alpha \hat \beta}} = {{\mathcal J}^{\mathcal W}}{{\partial {x^\mu}} \over {\partial {x^{\hat \alpha}}}}{{\partial {x^\nu}} \over {\partial {x^{\hat \beta}}}}{S_{\mu \nu}},\quad {\mathcal J} \equiv \det \left({{{\partial {x^\mu}} \over {\partial {x^{\hat \alpha}}}}} \right).$$

Specifically, we consider the coordinate transformation (95) where ℑ = ρ. In particular, this transformation implies

$${\partial _w}{S_{ww}} = {{\partial \rho} \over {\partial w}}{\partial _\rho}{S_{ww}} + {{\partial \varphi} \over {\partial w}}{\partial _\varphi}{S_{ww}},$$

and we can substitute

$${S_{ww}} = {{\mathcal J}^{- {\mathcal W}}}\left({{{\partial \rho} \over {\partial w}}{{\partial \rho} \over {\partial w}}{S_{\rho \rho}} + 2{{\partial \rho} \over {\partial w}}{{\partial \varphi} \over {\partial w}}{S_{\rho \varphi}} + {{\partial \varphi} \over {\partial w}}{{\partial \varphi} \over {\partial w}}{S_{\varphi \varphi}}} \right).$$

Inserting (100) into (99) and setting Sρϕ = 0 yields a lengthy expression involving derivatives of Sρρ and Sϕϕ with respect to ρ and ϕ. The latter vanish due to symmetry and we substitute for the ρ derivatives using

$$\begin{array}{*{20}c} {{\partial _\rho}{S_{\rho \rho}} = \left({{{\partial y} \over {\partial \rho}}{\partial _y} + {{\partial w} \over {\partial \rho}}{\partial _w}} \right)\left[ {{{\mathcal J}^{\mathcal W}}\left({{{\partial y} \over {\partial \rho}}{{\partial y} \over {\partial \rho}}{S_{yy}} + 2{{\partial y} \over {\partial \rho}}{{\partial w} \over {\partial \rho}}{S_{yw}} + {{\partial w} \over {\partial \rho}}{{\partial w} \over {\partial \rho}}{S_{ww}}} \right)} \right],}\\ {{\partial _\rho}{S_{\varphi \varphi}} = \left({{{\partial y} \over {\partial \rho}}{\partial _y} + {{\partial w} \over {\partial \rho}}{\partial _w}} \right)\left[ {{{\mathcal J}^{\mathcal W}}\left({{{\partial y} \over {\partial \varphi}}{{\partial y} \over {\partial \varphi}}{S_{yy}} + 2{{\partial y} \over {\partial \varphi}}{{\partial w} \over {\partial \varphi}}{S_{yw}} + {{\partial w} \over {\partial \varphi}}{{\partial w} \over {\partial \varphi}}{S_{ww}}} \right)} \right].}\\ \end{array}$$

This gives a lengthy expression relating the y and w derivatives of Sww. Finally, we recall that we need these derivatives in the xyz hyperplane and therefore set w = 0. In order to obtain an expression for the second w derivative of Sww, we first differentiate the expression with respect to w and then set w = 0. The final result is given by

$${\partial _w}{S_{ww}} = 0,\quad \;{\partial _w}{\partial _w}{S_{ww}} = {{{\partial _y}{S_{ww}}} \over y} + 2{{{S_{yy}} - {S_{ww}}} \over {{y^2}}}.$$

Note that the density weight dropped out of this calculation, so that Eq. (102) is valid for the BSSN variables Ãμν and \({\tilde \gamma _{\mu \nu}}\) as well.

Applying a similar procedure to all components of scalar, vector and symmetric tensor densities gives all expressions necessary to trade derivatives off the xyz hyperplane for those inside it. We summarize the expressions recalling our notation: a late Latin index, i = 1,…, 3 stands for either x, y or z whereas an early Latin index, a = 4,…, D − 1 represents any of the wa directions. For scalar, vector and tensor fields Ψ, V and T we obtain

$$\begin{array}{*{20}c} {0 = {\partial _a}\Psi = {\partial _i}{\partial _a}\Psi = {V^a} = {\partial _i}{V^a} = {\partial _a}{\partial _b}{V^c} = {\partial _a}{V^i} = {\partial _a}{S_{bc}} = {\partial _i}{\partial _a}{S_{bc}} = {S_{ia}}} \\ {= {\partial _a}{\partial _b}{S_{ic}} = {\partial _a}{S_{ij}} = {\partial _i}{\partial _a}{S_{jk}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;} \\ {{\partial _a}{\partial _b}\Psi = {\delta _{ab}}{{{\partial _y}\Psi} \over y},\;\;\,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;} \\ {{\partial _a}{V^b} = {\delta ^b}_a{{{V^y}} \over y},\quad \;\;\,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;} \\ {{\partial _i}{\partial _a}{V^b} = {\delta ^b}_a\left({{{{\partial _i}{V^y}} \over y} - {\delta _{iy}}{{{V^y}} \over {{y^2}}}} \right)\;\;,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;} \\ {{\partial _a}{\partial _b}{V^i} = {\delta _{ab}}\;\left({{{{\partial _y}{V^i}} \over y} - \delta _y^i{{{V^y}} \over {{y^2}}}} \right)\;\;,\quad \;\,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{S_{ab}} = {\delta _{ab}}{S_{ww}},\quad \;\quad \,\;\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{\partial _a}{\partial _b}{S_{cd}} = \left({{\delta _{ac}}{\delta _{bd}} + {\delta _{ad}}{\delta _{bc}}} \right){{{S_{yy}} - {S_{ww}}} \over {{y^2}}} + {\delta _{ab}}{\delta _{cd}}{{{\partial _y}{S_{ww}}} \over y},\;\;\;\;\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,} \\ {{\partial _a}{S_{ib}} = {\delta _{ab}}{{{S_{iy}} - {\delta _{iy}}{S_{ww}}} \over y},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{\partial _i}{\partial _a}{S_{jb}} = {\delta _{ab}}\left({{{{\partial _i}{S_{jy}} - {\delta _{jy}}{\partial _i}{S_{ww}}} \over y} - {\delta _{iy}}{{{S_{jy}} - {\delta _{jy}}{S_{ww}}} \over {{y^2}}}} \right)\;\;,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\quad \quad \;} \\ {{\partial _a}{\partial _b}{S_{ij}} = {\delta _{ab}}\left({{{{\partial _y}{S_{ij}}} \over y} - {{{\delta _{iy}}{S_{jy}} + {\delta _{iy}}{S_{iy}} - 2{\delta _{iy}}{\delta _{jy}}{S_{ww}}} \over {{y^2}}}} \right)\;\;.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\quad} \\ \end{array}$$

By trading or eliminating derivatives using these relations, a numerical code can be written to evolve D-dimensional spacetimes with SO(D − 3) symmetry on a strictly three-dimensional computational grid. We finally note that y is a quasi-radial variable so that y ≥ 0.

Initial data

In Section 6.1 we have discussed different ways of casting the Einstein equations into a form suitable for numerical simulations. At the start of Section 6, we have listed a number of additional ingredients that need to be included for a complete numerical study and physical analysis of BH spacetimes. We will now discuss the main choices used in practical computations to address these remaining items, starting with the initial conditions.

As we have seen in Section 6.1, initial data to be used in time evolutions of the Einstein equations need to satisfy the Hamiltonian and momentum constraints (54), (55). A comprehensive overview of the approach to generate BH initial data is given by Cook’s Living Reviews article [224]. Here we merely summarize the key concepts used in the construction of vacuum initial data, but discuss in some more detail how solutions to the constraint equations can be generated in the presence of specific matter fields that play an important role in the applications discussed in Section 7.

One obvious way to obtain constraint-satisfying initial data is to directly use analytical solutions to the Einstein equations as for example the Schwarzschild solution in D = 4 in isotropic coordinates

$${\rm{d}}{s^2} = - {\left({{{M - 2r} \over {M + 2r}}} \right)^2}\;{\rm{d}}{t^2} + {\left({1 + {M \over {2r}}} \right)^4}[{\rm{d}}{r^2} + {r^2}({\rm{d}}{\theta ^2} + {\sin ^2}\theta \;{\rm{d}}{\phi ^2})].$$

Naturally, the numerical evolution of an analytically known spacetime solution does not generate new physical insight. It still serves as an important way to test numerical codes and, more importantly, analytically known solutions often form the starting point to construct generalized classes of initial data whose time evolution is not known without numerical study. Classic examples of such analytic initial data are the Misner [550] and Brill-Lindquist [133] solutions describing n non-spinning BHs at the moment of time symmetry. In Cartesian coordinates, the Brill-Lindquist data generalized to arbitrary D are given by

$${K_{IJ}} = 0,\quad {\gamma _{IJ}} = {\psi ^{4/(D - 3)}}{\delta _{IJ}},\quad \psi = 1 + \sum\limits_A {{{{\mu _A}} \over {4{{\left[ {\sum\nolimits_{K = 1}^{D - 1} {{{({x^K} - x_0^K)}^2}}} \right]}^{(D - 3)/2}}}},}$$

where the summations over A and κ run over the number of BHs and the spatial coordinates, respectively, and μA are parameters related to the mass of the A-th BH through the surface area ΩD−2 of the (D − 2)-dimensional sphere by μA = 16πM/[(D − 2)ΩD−2]. We remark that in the case of a single BH, the Brill-Lindquist initial data (105) reduce to the Schwarzschild spacetime in Cartesian, isotropic coordinates (see Eq. (137) in Section 6.7.1).

A systematic way to generate solutions to the constraints describing BHs in D = 4 dimensions is based on the York-Lichnerowicz split [515, 806, 807]. This split employs a conformal spatial metric defined by \({\gamma _i}_j = {\psi ^{4 -}}{\gamma _{ij}}\); note that in contrast to the BSSN variable \({\tilde \gamma _{ij}}\), in general det \({\bar \gamma _{ij}} \neq 1\). Applying a conformal traceless split to the extrinsic curvature according to

$${K_{ij}} = {A_{ij}} + {1 \over 3}{\gamma _{ij}}K,\quad {A^{ij}} = {\psi ^{- 10}}{\bar A^{ij}}\quad \Leftrightarrow \quad {A_{ij}} = {\psi ^{- 2}}{\bar A_{ij}},$$

and further decomposing Āij into a longitudinal and a transverse traceless part, the momentum constraints simplify significantly; see [224] for details as well as a discussion of the alternative physical transverse-traceless split and conformal thin-sandwich decomposition [813]. The conformal thin-sandwich approach, in particular, provides a method to generate initial data for the lapse and shift which minimize the initial rate of change of the spatial metric, i.e., data in a quasi-equilibrium configuration [225, 190].

Under the further assumption of vanishing trace of the extrinsic curvature K = 0, a flat conformal metric \({\bar \gamma _{ij}} = {f_{ij}}\), where fij describes a flat Euclidean space, and asymptotic flatness limr→∞ ψ = 1, the momentum constraint admits an analytic solution known as Bowen-York data [121]

$${\bar A_{ij}} = {3 \over {2{r^2}}}\left[ {{P_i}{n_j} + {P_j}{n_i} - ({f_{ij}} - {n_i}{n_j}){P^k}{n_k}} \right] + {3 \over {{r^3}}}\left({{\epsilon_{kil}}{S^l}{n^k}{n_j} + {\epsilon_{kjl}}{S^l}{n^k}{n_i}} \right),$$

with \(r = \sqrt {{x^2} + {y^2} + {z^2}}, \;{n^i} = {x^i}/r\) the unit radial vector and user-specified parameters Pi, Si. By calculating the momentum associated with the asymptotic translational and rotational Killing vectors \(\xi _{(k)}^i\) [811], one can show that Pi and Si represent the components of the total linear and angular momentum of the initial hypersurface. The linearity of the momentum constraint further allows us to superpose solutions \(\bar A_{ij}^{(a)}\) of the type (107) and the total linear momentum is merely obtained by summing the individual \(P_{(a)}^i\). The total angular momentum is given by the sum of the individual spins \(S_{(a)}^i\) plus an additional contribution representing the orbital angular momentum. For the generalization of Misner data, it is necessary to construct inversion-symmetric solutions of the type (107) using the method of images [121, 224]. Such a procedure is not required for generalizing Brill-Lindquist data where a superposition of solutions \(\bar A_{ij}^{(a)}\) of the type (107) can be used directly to calculate the extrinsic curvature from Eq. (106) and insert the resulting expressions into the vacuum Hamiltonian constraint given with the above listed simplifications by

$${\bar \nabla ^2}\psi + {1 \over 8}{K^{mn}}{K_{mn}}{\psi ^{- 7}} = 0,$$

where \({\bar \nabla ^2}\) is the Laplace operator associated with the flat metric fij. This elliptic equation is commonly solved by decomposing ψ into a Brill-Lindquist piece \({\psi _{{\rm{BL}}}} = \sum\nolimits_{a = 1}^N {{m_a}/\vert \vec r - {{\vec r}_a}\vert}\) plus a regular piece u = ψ − ψBL, where \({\vec r_a}\) denotes the location of the a-th BH and ma a parameter that determines the BH mass and is sometimes referred to as the bare mass. Brandt & Brügmann [126] have proven existence and uniqueness of C2 regular solutions u to Eq. (108) and the resulting puncture data are the starting point of the majority of numerical BH evolutions using the BSSN moving puncture technique. The simplest example of this type of initial data is given by Schwarzschild’s solution in isotropic coordinates where

$${K_{mn}} = 0\,,\qquad \psi = 1 + {m \over {2r}}.$$

In particular, this solution admits the isometry rm2 /(4r) which leaves the coordinate sphere r = m/2 invariant, but maps the entire asymptotically flat spacetime r > m/2 into the interior and vice versa. The solution, therefore, consists of 2 asymptotically flat regions connected by a “throat” and spatial infinity of the far region is compactified into the single point r = 0 which is commonly referred to as the puncture. Originally, time evolutions of puncture initial data split the conformal factor, in analogy to the initial-data construction, into a singular Brill-Lindquist contribution given by the ψ in Eq. (109) plus a deviation u that is regular everywhere; cf. Section IV B in [24]. In this approach, the puncture locations remain fixed on the computational domain. The simulations through inspiral and merger by [159, 65], in contrast, evolve the entire conformal factor using gauge conditions that allow for the puncture to move across the domain and are, therefore, often referred to as “moving puncture evolutions”.

In spite of its popularity, there remain a few caveats with puncture data that have inspired explorations of alternative initial data. In particular, it has been shown that there exist no maximal, conformally flat spatial slices of the Kerr spacetime [341, 756]. Constructing puncture data of a single BH with non-zero Bowen-York parameter Si will, therefore, inevitably result in a hyper-surface containing a BH plus some additional content which typically manifests itself in numerical evolutions as spurious GWs, colloquially referred to as “junk radiation”. For rotation parameters close to the limit of extremal Kerr BHs, the amount of spurious radiation rapidly increases leading to an upper limit of the dimensionless spin parameter J/M2 ≈ 0.93 for conformally flat Bowen-York-type data [226, 237, 238, 527]; BH initial data of Bowen-York type with a spin parameter above this value rapidly relax to rotating BHs with spin χ ≈ 0.93, probably through absorption of some fraction of the spurious radiation. This limit has been overcome [527, 528] by instead constructing initial data with an extended version of the conformal thin-sandwich method using superposed Kerr-Schild BHs [467]. In an alternative approach, most of the above outlined puncture method is applied but using a non-flat conformal metric; see for instance [493, 391].

In practice, puncture data are the method-of-choice for most evolutions performed with the BSSN-moving-puncture techniqueFootnote 15 whereas GHG evolution schemes commonly start from conformal thin-sandwich data using either conformally flat or Kerr-Schild background data. Alternatively to both these approaches, initial data containing scalar fields which rapidly collapse to one or more BHs has also been employed [629].

The constraint equations in the presence of matter become more complex. A simple procedure can however be used to yield analytic solutions to the initial data problem in the presence of minimally coupled scalar fields [588, 586]. Although in general the constraints (54)(55) have to be solved numerically, there is a large class of analytic or semi-analytic initial data for the Einstein equations extended to include scalar fields. The construction of constraint-satisfying initial data starts from a conformal transformation of the ADM variables [224]

$${\gamma _{ij}} = {\psi ^4}{\bar \gamma _{ij}},\quad \bar \gamma = \det {\bar \gamma _{ij}} = 1,$$
$${K_{ij}} = {A_{ij}} + {1 \over 3}{\gamma _{ij}}K,\quad {A_{ij}} = {\psi ^{- 2}}{\bar A_{ij}},$$

which can be used to re-write the constraints as

$${\mathcal H} = \bar \Delta \psi - {1 \over 8}\bar R\psi - {1 \over {12}}{K^2}{\psi ^5} + {1 \over 8}{\bar A^{ij}}{\bar A_{ij}}{\psi ^{- 7}} + \pi \psi [{\bar D^i}{\Phi ^{\ast}}{\bar D_i}\Phi + {\psi ^4}({\Pi ^{\ast}}\Pi + \mu _S^2{\Phi ^{\ast}}\Phi)],$$
$${{\mathcal M}_i} = {\bar D_j}\bar A_i^j - {2 \over 3}{\psi ^6}{\bar D_i}K - 4\pi {\psi ^6}({\Pi ^{\ast}}{\bar D_i}\Phi + \Pi {\bar D_i}{\Phi ^{\ast}}).$$

Here, \(\bar \Delta = {\bar \gamma ^{ij}}{\bar D_i}{\bar D_j},\;\bar D\) and \(\bar R\) denote the conformal covariant derivative and Ricci scalar and Π is a time reduction variable defined in (78).

Take for simplicity a single, non-rotating BH surrounded by a scalar field (more general cases are studied in Ref. [588, 586]). If we adopt the maximal slicing condition K = 0 and set Āij = 0, Φ = 0, then the momentum constraint is immediately satisfied, and one is left with the the Hamiltonian constraint, which for conformal flatness, i.e., \({\bar \gamma _{ij}} = {f_{ij}}\) reads

$${\Delta _{{\rm{flat}}}}\psi = \left[ {{1 \over {{r^2}}}{\partial \over {\partial r}}{r^2}{\partial \over {\partial r}} + {1 \over {{r^2}\sin \theta}}{\partial \over {\partial \theta}}\sin \theta {\partial \over {\partial \theta}} + {1 \over {{r^2}{{\sin}^2}\theta}}{{{\partial ^2}} \over {\partial {\Phi ^2}}}} \right]\psi = - \pi {\psi ^5}\Pi {\Pi ^{\ast}}.$$

The ansatz

$$\Pi = {{{\psi ^{- 5/2}}} \over {\sqrt {r\pi}}}F(r)Z(\theta, \phi),$$
$$\psi = 1 + {M \over {2r}} + \sum\limits_{lm} {{{{u_{lm}}(r)} \over r}{Y_{lm}}(\theta, \phi),}$$

reduces the Hamiltonian constraint to

$$\sum\limits_{lm} {\left({u_{lm}^{\prime\prime} - {{l(l + 1)} \over {{r^2}}}{u_{lm}}} \right)\;} {Y_{lm}} = - F{(r)^2}Z{(\theta, \phi)^2}.$$

By a judicious choice of the angular function Z(θ, ϕ), or in other words, by projecting Z(θ, ϕ) onto spherical harmonics Ylm, the above equation reduces to a single second-order, ordinary differential equation. Thus, the complex problem of finding appropriate initial data for massive scalar fields was reduced to an almost trivial problem, which admits some interesting analytical solutions [588, 586]. Let us focus for defmiteness on spherically symmetric solutions (we refer the reader to Ref. [588, 586] for the general case), by taking a Gaussian-type solution ansatz,

$$Z(\theta, \phi) = {1 \over {\sqrt {4\pi}}}\,,\quad F(r) = {A_{00}} \times \sqrt r {e^{- {{{{{(r - {r_0})}^2}} \over {{w^2}}}}}},$$

where A00 is the scalar field amplitude and r0 and w are the location of the center of the Gaussian and its width. By solving Eq. (117), we obtain the only non-vanishing component of ulm(r)

$${u_{00}} = A_{00}^2{{w[{w^2} - 4{r_0}(r - {r_0})]} \over {16\sqrt 2}}\left[ {{\rm{erf}}\left({{{\sqrt 2 (r - {r_0})} \over w}} \right) - 1} \right] - A_{00}^2{{{r_0}{w^2}} \over {8\sqrt \pi}}{e^{- 2{{(r - {r_0})}^2}/{w^2}}}\,,$$

where we have imposed that ulm → 0 at infinity. Other solutions can be obtained by adding a constant to (119).

Gauge conditions

We have seen in Section 6.1, that the Einstein equations do not make any predictions about the gauge functions; the ADM equations leave lapse α and shift βi unspecified and the GHG equations make no predictions about the source functions Hα. Instead, these functions can be freely specified by the user and represent the coordinate or gauge-invariance of the theory of GR. Whereas the physical properties of a spacetime remain unchanged under gauge transformations, the performance of numerical evolution schemes depends sensitively on the gauge choice. It is well-known, for example, that evolutions of the Schwarzschild spacetime employing geodesic slicing α = 1 and vanishing shift βi = 0 inevitably reach a hypersurface containing the BH singularity after a coordinate time interval t = πM [709]; computers respond to singular functions with non-assigned numbers which rapidly swamp the entire computational domain and render further evolution in time practically useless. This problem can be avoided by controlling the lapse function such that the evolution in proper time slows down in the vicinity of singular points in the spacetime [312]. Such slicing conditions are called singularity avoiding and have been studied systematically in the form of the Bona-Massó family of slicing conditions [116]; see also [343, 20]. A potential problem arising from the use of singularity avoiding slicing is the different progress in proper time in different regions of the computational domain resulting in a phenomenon often referred to as “grid stretching” or “slice stretching” which can be compensated with suitable non-zero choices for the shift vector [24].

The particular coordinate conditions used with great success in the BSSN-based moving puncture approach [159, 65] in D = 4 dimensions are variants of the “1+log” slicing and “Γ-driver” shift condition [24]

$${\partial _t}\alpha = {\beta ^m}{\partial _m}\alpha - 2\alpha K,$$
$${\partial _t}{\beta ^i} = {\beta ^m}{\partial _m}{\beta ^i} + {3 \over 4}{B^i},$$
$${\partial _t}{B^i} = {\beta ^m}{\partial _m}{B^i} + {\partial _t}{\tilde \Gamma ^i} - \eta {B^i}\,.$$

We note that the variable Bi introduced here is an auxiliary variable to write the second-order-in-time equation for the shift vector as a first-order system and has no relation with the variable of the same name introduced in Eq. (82). The “damping” factor η in Eq. (122) is specified either as a constant, a function depending on the coordinates xi and BH parameters [683], a function of the BSSN variables [559, 560], or evolved as an independent variable [29]. A first-order-in-time evolution equation for βi has been suggested in [758] which results from integration of Eqs. (121), (122)

$${\partial _t}{\beta ^i} = {\beta ^m}{\partial _m}{\beta ^i} + {3 \over 4}{\tilde \Gamma ^i} - \eta {\beta ^i}.$$

Some NR codes omit the advection derivatives of the form βmm in Eqs. (120)(123). Long-term stable numerical simulations of BHs in higher dimensions require modifications in the coefficients in Eqs. (120)(123) [700] and/or the addition of extra terms [841]. Reference [313] recently suggested a modification of Eq. (120) for the lapse function α that significantly reduces noise generated by a sharp initial gauge wave pulse as it crosses mesh refinement boundaries.

BH simulations with the GHG formulation employ a wider range of coordinate conditions. For example, Pretorius’ breakthrough evolutions [629] set Hi = 0 and

$${\square H_t} = - {\xi _1}{{\alpha - 1} \over {{\alpha ^\eta}}} + {\xi _2}{n^\mu}{\partial _\mu}{H_t},$$

with parameters ξ1 = 19/m, ξ2 = 2.5/m, η = 5 where m denotes the mass of a single BH. An alternative choice used with great success in long binary BH inspiral simulations [735] sets Hα such that the dynamics are minimized at early stages of the evolution, gradually changes to harmonic gauge Hα = 0 during the binary inspiral and uses a damped harmonic gauge near merger

$${H_\alpha} = {\mu _0}{\left[ {\ln \left({{{\sqrt \gamma} \over \alpha}} \right)} \right]^2}\left[ {\ln \left({{{\sqrt \gamma} \over \alpha}} \right){n_\alpha} - {\alpha ^{- 1}}{g_{\alpha m}}{\beta ^m}} \right],$$

where μ0 is a free parameter. We note in this context that for D = 4, the GHG source functions Hα are related to the ADM lapse and shift functions through [630]

$${n^\mu}{H_\mu} = - K - {n^\mu}{\partial _\mu}\ln \alpha,$$
$${\gamma ^{\mu i}}{H_\mu} = - {\gamma ^{mn}}\Gamma _{mn}^i + {\gamma ^{im}}{\partial _m}\ln \alpha + {1 \over \alpha}{n^\mu}{\partial _\mu}{\beta ^i}.$$

Discretization of the equations

In the previous sections, we have derived formulations of the Einstein equations in the form of an IBVP. Given an initial snapshot of the physical system under consideration, the evolution equations, as for example in the form of the BSSN equations (60)(64), then predict the evolution of the system in time. These evolution equations take the form of a set of nonlinear partial differential equations which relate a number of grid variables and their time and spatial derivatives. Computers, on the other hand, exclusively operate with (large sets of) numbers and for a numerical simulation we need to translate the differential equations into expressions relating arrays of numbers.

The common methods to implement this discretization of the equations are finite differencing, the finite element, finite volume and spectral methods. Finite element and volume methods are popular choices in various computational applications, but have as yet not been applied to time evolutions of BH spacetimes. Spectral methods provide a particularly efficient and accurate approach for numerical modelling provided the functions do not develop discontinuities. Even though BH spacetimes contain singularities, the use of singularity excision provides a tool to remove these from the computational domain. This approach has been used with great success in the SpEC code to evolve inspiralling and merging BH binaries with very high accuracy; see, e.g., [122, 220, 526]. Spectral methods have also been used successfully for the modelling of spacetimes with high degrees of symmetry [205, 206, 207] and play an important role in the construction of initial data [39, 38, 836]. An indepth discussion of spectral methods is given in the Living Reviews article [365]. The main advantage of finite differencing methods is their comparative simplicity. Furthermore, they have proved very robust in the modelling of rather extreme BH configurations as for example BHs colliding near the speed of light [719, 587, 716] or binaries with mass ratios up to 1:100 [525, 523, 718].

Mesh refinement and domain decomposition: BH spacetimes often involve lengthscales that differ by orders of magnitude. The BH horizon extends over lengths of the order \({\mathcal O(1)}\) M where M is the mass of the BH. Inspiralling BH binaries, on the other hand, emit GWs with wavelengths of \({\mathcal O}({10^2})\;M\). Furthermore, GWs are rigorously defined only at infinity. In practice, wave extraction is often performed at finite radii but these need to be large enough to ensure that systematic errors are small. In order to accomodate accurate wave extraction, computational domains used for the modelling of asymptotically flat BH spacetimes typically have a size of \({\mathcal O}({10^3})\;M\). With present computational infrastructure it is not possible to evolve such large domains with a uniform, high resolution that is sufficient to accurately model the steep profiles arising near the BH horizon. The solution to this difficulty is the use of mesh refinement, i.e., a grid resolution that depends on the location in space and may also vary in time. The use of mesh refinement in BH modelling is simplified by the remarkably rigid nature of BHs which rarely exhibit complicated structure beyond some mild deformation of a sphere. The requirements of increased resolution are, therefore, simpler to implement than, say, in the modelling of airplanes or helicopters. In BH spacetimes the grid resolution must be highest near the BH horizon and it decreases gradually at larger and larger distances from the BH. In terms of the internal book-keeping, this allows for a particularly efficient manner to arrange regions of refinement which is often referred to as moving boxes. A set of nested boxes with outwardly decreasing resolution is centered on each BH of the spacetime and follows the BH motion. These sets of boxes are immersed in one or more common boxes which are large enough to accomodate those centered on the BHs. As the BHs approach each other, boxes originally centered on the BHs merge into one and become part of the common-box hierarchy. A snapshot of such moving boxes is displayed in Figure 4.

Mesh refinement in NR has been pioneered by Choptuik in his seminal study on critical phenomena in the collapse of scalar fields [212]. The first application of mesh refinement to the evolution of BH binaries was performed by Brügmann [140]. There exists a variety of mesh refinement packages available for use in NR including Bam [140], Had [384], Pamr/Amrd [