Abstract
The demand to obtain answers to highly complex problems within strongfield gravity has been met with significant progress in the numerical solution of Einstein’s equations — along with some spectacular results — in various setups.
We review techniques for solving Einstein’s equations in generic spacetimes, focusing on fully nonlinear evolutions but also on how to benchmark those results with perturbative approaches. The results address problems in highenergy physics, holography, mathematical physics, fundamental physics, astrophysics and cosmology.
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1 Prologue
“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 weakfield 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 strongfield 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: gravitationalwave (GW) detectors were promising to see the very last stages of BHbinary 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 longterm stable evolutions of BHbinaries in fourdimensional, asymptotically flat spacetimes, starting a phase transition in the field. It is common to refer to such activity — numerically solving Einstein’s equations
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 highenergy 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, timedependent Einsteintype 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 fivedecade 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 Reviews^{Footnote 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 twobody problem in GR for instance, was approached via a slowmotion, large separation postNewtonian 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 twobody 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 twobody 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.
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, highenergy 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.
2 Milestones
Numerical solving is a thousandyearold 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].

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 boundaryvalue and eigenvalue problems, and for the initialvalue 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 — ChoquetBruhat [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 socalled “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 nonminimally 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 (quasistellar radio source) [681]. Quasars are now believed to be supermassive BHs, described by the Kerr solution.

1963 — Tangherlini finds the higherdimensional generalization of the Schwarzschild solution [740].

1964 — Chandrasekhar and Fock develop the postNewtonian 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 Xrays are discovered. One of these Xray sources, Cygnus X1, confirmed in 1971 with the UHURU orbiting Xray observatory, is soon accepted as the first plausible stellarmass BH candidate (see, e.g., [110]). The UHURU orbiting Xray observatory makes the first surveys of the Xray sky discovering over 300 Xray “stars”.

1965 — Penrose and Hawking prove that collapse of ordinary matter leads, under generic conditions, to spacetime singularities (the socalled “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, firstperson 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 ReggeWheeler 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 quasinormal 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. Quasinormal 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 NewmanPenrose formalism [575].

1973 — York [808, 809] introduces a split of the extrinsic curvature leading to the LichnerowiczYork 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 highenergy particle collisions, which were to become important in TeVscale 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 Xray nova A062000 contains a compact object of mass almost certainly larger than 3 M_{⊙}, paving the way for the identification of many more stellarmass BH candidates.

1986 — Myers and Perry construct higherdimensional rotating, topologically spherical, BH solutions [565].

1987 —’ t Hooft [736] argues that the scattering process of two pointlike particles above the fundamental Planck scale is well described and calculable using classical gravity. This idea is behind the application of GR for modeling transPlanckian 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 singularityavoidance 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 headon collision of two BHs at the speed of light. Their second order result will be in good agreement with later numerical simulations of highenergy collisions.

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

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

1993 — Gregory and Laflamme show that black strings, one of the simplest higherdimensional solutions with horizons, are unstable against axisymmetric perturbations [367]. The instability is similar to the RayleighPlateau instability seen in fluids [167, 162]; the endstate 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 socalled 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, highperformance multidimensional componentbased 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 BrillLindquist data to the case of generic BowenYork 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 nonconformal 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 socalled braneworld scenarios, in which we live on a fourdimensional subspace of a higherdimensional 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 ultrahigh energy cosmic ray collisions [315, 33, 304].

1999 — Friedrich & Nagy [335] present the first wellposed formulation of the initialboundaryvalue 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 BHNS binaries [699].

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

2001 — Horowitz and Maeda suggest that black strings do not fragment and that the endstate of the GregoryLaflamme instability may be an inhomogeneous string [440], driving the development of the field. Nonuniform strings are constructed perturbatively by Gubser [371] and numerically by Wiseman, who, however, shows that these cannot be the endstate of the GregoryLaflamme instability [789].

2003 — In a series of papers [479, 452, 480], Kodama and Ishibashi extend the ReggeWheelerZerilli 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 longterm 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 multidomain decomposition [618] and a dual coordinate frame [678].

2008 — The first simulations of highenergy collisions of two BHs are performed [719]. These were later generalized to include spin and finite impact parameter collisions, yielding zoomwhirl 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 YangMills plasma through the gauge/gravity duality [205].

2009 — Dias et al. show that rapidly spinning MyersPerry BHs present zeromodes, 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 MyersPerry BHs nonlinearly and show that a nonaxisymmetric 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 GregoryLaflamme instability in five dimensions, which shows hints of pinchoff and cosmic censorship violation [511].

2010, 2011 — First nonlinear simulations describing collisions of higherdimensional 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].
3 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 endstate of collapse and the twobody 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 stronggravity simulations are required will hopefully become clear.
3.1 Astrophysics
3.1.1 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 ultralow 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 spacebased 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 Xrays 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, templatematching techniques — frequently used in data analysis — can be helpful to extract the signal from the detector noise, but they require an apriori 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 twobody 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.
Groundbased 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. Spacebased 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 BHBH 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.
BHNS and NSNS binary coalescences pose a different sort of problems than those posed by BHBH 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 BHNS and NSNS 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 nonperturbative 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 groundbased interferometers. When perturbed by an external or internal event, a NS can be set into nonradial 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 nonrotating 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].
3.1.2 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 m_{F}. The momentum of each fermion is p_{F} ∼ ħn^{1/3}, with n = N/R^{3} the number density of fermions. In the relativistic regime, the Fermi energy per particle then reads E_{F} = p_{F}c = ħcN^{1/3}/R. The gravitational energy per fermion is approximately \({E_G} \sim  Gm_F^2/R\), and the star’s total energy is thus,
For small N, the total energy is positive, and we can decrease it by increasing R. At some point the fermion becomes nonrelativistic and \({E_F} \sim p_F^2 \sim 1/{R^2}\). In this regime, the gravitational binding energy E_{G} dominates over E_{F}, 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
for neutrons, stars with masses above ∼ 3 M_{⊙} cannot attain equilibrium.
What is the fate of massive stars whose pressure cannot counterbalance 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.
3.1.3 Kicks
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 10^{5} M_{⊙} to 10^{10} 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, postNewtonian techniques and the closelimit 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 nonspinning 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.
3.1.4 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 strongfield 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 strongfield 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 strongfield gravity regime [635, 826]). However, our present theoretical knowledge of strongfield astrophysical processes is based, in most cases, on the apriori 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 wellposed initialvalue 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.
Scalartensor 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 BransDicke gravity [128], the metric tensor is nonminimally coupled with one or more scalar fields. In the case of a single scalar field (which can be generalized to multiscalartensor theories [242]), the action can be written as
where R is the Ricci scalar associated to the metric g_{μν}, F, Z, U are arbitrary functions of the scalar field ϕ, and S_{m} is the action describing the dynamics of the other fields (which we call “matter fields”, ψm). A more general formulation of scalartensor theories yielding second order equations of motion has been proposed by Horndeski [435] (see also Ref. [260]).
Scalartensor theories can be obtained as lowenergy 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 ultralight axion fields (pseudoscalar fields, behaving under many respects as scalar fields) arise from the dimensional reduction of SMT, and play a role in cosmological models.
Scalartensor 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 (ℓ =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 scalarfield 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 scalartensor gravity.
These theories, whose wellposedness has been proved [669, 670], are a perfect arena for NR. Recovering some of the above smokinggun 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 R^{2}, R_{μν}R^{μν}, \({R_{\mu \nu \alpha \beta}}{R^{\mu \nu \alpha \beta}},{\epsilon _{\mu \nu \alpha \beta}}{R^{\mu \nu \rho \sigma}}{R^{\alpha \beta}}_{\rho \sigma}\)), possibly coupled with scalar fields, or theories which explicitly break Lorentz invariance. In particular, EinsteinDilatonGaussBonnet gravity and Dynamical ChernSimons gravity [602, 27] can arise from SMT compactifications, and Dynamical ChernSimons gravity also arises in Loop Quantum Gravity; theories such as EinsteinAether [456] and “HoravaLifshitz” 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.
3.2 Fundamental and mathematical issues
3.2.1 Cosmic censorship
Spacetime singularities signal the breakdown of the geometric description of the spacetime, and can be diagnosed by either the blowup of observerinvariant quantities or by the impossibility to continue timelike or null geodesics past the singular point. For example, the Schwarzschild geometry has a curvature invariant R^{abcd}R_{abcd} = 48 G^{2}M^{2}/(c^{4}r^{6}) 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 E^{2} − B^{2}) 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 extraGR 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 cosmiccensorship 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.
3.2.2 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 2parameter 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 pointlike, spacelike singularity creates a boundary for spacetime. Inside the 0 < J < M^{2} Kerr event horizon, by contrast, there is a ringlike, timelike singularity, beyond which another asymptotically flat spacetime region, with r < 0 in BoyerLindquist 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 counterstreaming^{Footnote 2} between those streams leads to exponential growth of gaugeinvariant quantities such as the interior (MisnerSharp [552]) mass, the centerofmass 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 spacelike nature [234]. Fully nonlinear numerical simulations will certainly be important for understanding this process.
3.2.3 Most luminous events in the universe
The most advanced laser units on the planet can output luminosities as high as ∼ 10^{18} W [301], while at ∼ 10^{26} W the Tsar Bomba remains the most powerful artificial explosion ever [732]. These numbers pale in comparison with strongly dynamical astrophysical events: a γray burst, for instance, reaches luminosities of approximately ∼ 10^{45} 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 10^{23} [443]. If all of them have a luminosity equal to our Sun, we get a total luminosity of approximately ∼ 10^{49} 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 \({\mathcal L_G}\),
The quantity \({\mathcal L_G}\) should control gravitydominated 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 R ∼ GM/c^{2}. 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 Mc^{2}. 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 c^{4}/(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 strongfield 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].
3.2.4 Higher dimensions
Higherdimensional 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 fourdimensional 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 higherdimensional theories as part of the fundamental picture [463, 476] (for a historical view, see [283]).
Consider first for simplicity the Ddimensional KleinGordon equation □ϕ(x^{μ},z^{i}) = 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 fourdimensional d’Alembertian operator. As a consequence,

i)
the fundamental, homogenous mode n = 0 is a massless fourdimensional field obeying the same KleinGordon equation, whereas

ii)
even though we started with a higherdimensional massless theory, we end up with a tower of massive modes described by the fourdimensional massive KleinGordon equation, with mass terms proportional to n/L. One important, generic conclusion is that the higherdimensional (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 highenergy 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 nonobservation of extra dimensions in everyday laboratory experiments.
The attempts by Kaluza and Klein to unify gravity and electromagnetism considered fivedimensional Einstein field equations with the metric appropriately decomposed as,
Here, ds^{2} = g_{μν} dx^{μ} dx^{ν} is a fourdimensional 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 fourdimensional theory in the Einstein frame. Assuming all the fields are independent of the extra dimension z, one finds a set of fourdimensional EinsteinMaxwellscalar equations, thereby almost recovering both GR and EM [283]. This is the basic idea behind the KaluzaKlein 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 TeVscale scenarios in highenergy 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 Ddimensional 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 coexist 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 fourdimensional 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 GregoryLaflamme instability [367]. The instability is similar in many aspects to the RayleighPlateau 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 pinchoff would be accompanied by (naked) regions of unbounded curvature.^{Footnote 4} Evidence that the GregoryLaflamme does lead to disruption of strings was recently put forward [511].
3.3 Highenergy physics
3.3.1 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 Ddimensional 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 HawkingPage phase transition for AdS BHs [799]. Away from thermal equilibrium, the quasinormal 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 nonAbelian quantum field theories at nonzero 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 latticeregularized thermodynamical calculations of quantum chromodynamics, 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 lowenergy 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 quarkgluon plasma. Holography uses \({\mathcal N} = 4\) Super YangMills theory as a learning ground for the real quarkgluon plasma. Then, the formation of such plasma in the collision of highenergy ions has been modeled, in its gravity dual, by colliding gravitational shock waves in fivedimensional 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 EinsteinMaxwellcharged scalar theory with negative cosmological constant. The RNAdS BH solution of this theory, for which the scalar field vanishes, is unstable for temperatures T below a critical temperature T_{c}. If triggered, the instability leads the scalar field to condense into a nonvanishing 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 RNAdS BH at low temperature was identified with a superconducting phase transition, and the RNAdS 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, nonequilibrium 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 fivedimensional 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.
3.3.2 Theories with lower fundamental Planck scale
As discussed in Section 3.2.4, higherdimensional 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 + 1dimensional subspace (which is called the “brane”, while the higherdimensional 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 Dirichletbranes, 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 RandallSundrum 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 (∼ 10^{19} 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 + 1dimensional application of Gauss’s law then yields an incomplete account of the total gravitational flux. Thus, the apparent (3 + 1dimensional) gravitational coupling appears smaller than the real (Ddimensional) one. Or, equivalently, the real fundamental Planck energy scale becomes much smaller than the apparent one. An estimate is obtained considering the Ddimensional gravitational action and integrating the compact dimensions by assuming the metric is independent of them:
thus the fourdimensional Newton’s constant is related to the Ddimensional one by the volume of the compact dimensions G_{4} = G_{D}/V_{D−4}.
In units such that c = ħ =1 (different from the units G = c =1 used in the rest of this paper), the massenergy 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 Ddimensional Planck energy \(E_{{\rm{Planck}}}^{(D)}\) is related to the fourdimensional one by
where we have defined the fourdimensional 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 transPlanckian 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 highenergy cosmic rays [69, 279, 353]. Well into the transPlanckian regime, i.e., for energies significantly larger than the Planck scale, classical gravity described by GR in Ddimensions 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 centerofmass 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 Ddimensions, 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.
4 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].
4.1 Exact solutions
4.1.1 Fourdimensional, 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 3dimensional 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 higherdimensional 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 GoldbergSachs 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 KerrNewman solution [576] — and to include a cosmological constant by Carter [187]. In BoyerLindquist coordinates, the KerrNewman(A)dS metric reads:
where
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 KerrNewman 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 timedependent 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.
4.1.2 Beyond fourdimensional, electrovacuum general relativity with Λ
As discussed in Section 3 there are various motivations to consider generalizations of (or alternative theories to) fourdimensional 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 righthand side — i.e., theories with different matter fields, including nonminimally coupled ones —, or by changing the lefthand 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 higherdimensional 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 Ddimensional 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 fourdimensional 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 pbranes, where p stands for the spatial dimensionality of the object [441, 285]. Thus, a pointlike BH is a 0brane, a string is a 1brane 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 ‘RamondRamond’ fields, in which case they are called Dpbranes or simply Dbranes [284]. Here we wish to emphasize that the GregoryLaflamme instability discussed in Section 3.2.4 was unveiled in the context of pbranes, 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 is^{Footnote 6} \(SO{(2)^{[{{D  1} \over 2}]}}\) — symmetric solution to the Ddimensional 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 KerrSchild 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 nonvanishing, i.e., for the singly spinning MyersPerry solution, in dimensions D ≥ 6 there is no bound on the angular momentum J in terms of the BH mass M. Ultraspinning MyersPerry BHs are then possible and their horizon appears highly deformed, becoming locally analogous to that of a pbrane. This similarity suggests that ultraspinning BHs should suffer from the GregoryLaflamme 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 nonsimply connected horizon, having spatial sections that are topologically S^{2} × S^{1}. Its discovery raised questions about how the D = 4 results on uniqueness and stability of vacuum solutions generalized to higherdimensional gravity. Moreover, using the generalization to higher dimensions of Weyl solutions [306] and of the inverse scattering technique [394], geometries with a nonconnected event horizon — i.e., multiobject 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 nonspherical topology; regular multiobject solutions with only MyersPerry BHs do not seem to exist [425], just as regular multiobject 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 3dimensional Newton’s constant. The necessary extra ingredient is a negative cosmological constant; considering it leads to the celebrated BañadosTeitelboimZanelli (BTZ) BH [68]. In D = 2 a BH spacetime was obtained by Callan, Giddings, Harvey and Strominger (the CGHS BH), by considering GR nonminimally 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 fourdimensional 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 “nohair 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 nohair 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 scalartensor 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 lefthand 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 GaussBonnet [844] (in D ≥ 5), EinsteinDilatonGaussBonnet gravity and Dynamical ChernSimons 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 GaussBonnet 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, ChernSimons 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 GaussBonnet theory), illdefined.
4.1.3 State of the art

D ≠ 4: The essential results in higherdimensional vacuum gravity are the Tangherlini [740] and MyersPerry [565] BHs, the (vacuum) black pbranes [441, 285] and the EmparanReall black ring [307]. Solutions with multiobjects 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 MyersPerry 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 pbranes 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 BransDicke gravity the only stationary BH solutions are the same as in GR. This result was recently extended by Sotiriou and Faraoni to more general scalartensor theories [712]. Such type of nohair 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 scalartensor gravity [713].
4.2 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 solitonlike 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 BartnikMcKinnon particlelike solutions in EinsteinYangMills 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 EinsteinYangMills 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 selfinteractions. Boson stars are horizonless gravitating solitons kept in equilibrium by a balance between their selfgenerated gravity and the dispersion effect of the scalar field’s wavelike 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 MyersPerry 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 MyersPerryAdS BHs precisely at the threshold of the superradiant instability for a given scalarfield mode [423], and display new physical properties, e.g., new shapes of ergoregions [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 GregoryLaflamme instability [367] (see Section 3.2.4 and Section 7.2) into a family of nonuniform 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 nonuniform 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) KaluzaKlein solutions. Also in higher dimensions, a number of other numerical solutions have been reported in recent years, most notably generalizations of the EmparanReall 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 EinsteinGaussBonnet 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 AdSCondensed 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 RandallSundrum 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 Ricciflat metrics on CalabiYau manifolds [409].
5 Approximation Schemes
The exact and numericallyconstructed 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 orderofmagnitude 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 fourdimensional, asymptotically flat spacetimes, see Ref. [502].
5.1 PostNewtonian schemes
5.1.1 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:
which uses the fact that, for bound systems, the virial theorem implies υ^{2} ∼ GM/r. In this approach, called “postNewtonian”, 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 bookkeeping parameter it is customary to consider ϵ = υ/c. The spacetime metric and the stressenergy tensor are expanded in powers of ϵ and terms of order ϵ^{n} 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 λ_{Gw} of the emitted GWs. Outside this region, and in particular in the wave zone (e.g., at a distance ≫ λ_{gw} from the source), radiative processes make the PN expansion illdefined, and different approaches have to be employed, such as the postMinkowskian expansion, which assumes weak fields but not slow motion. In the postMinkowskian 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, postMinkowskian expansion in the wave zone, multipolar expansions, regularization of pointlike 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 postMinkowskian expansion can itself be PNexpanded, 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 Nbody 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 twobody 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 twobody problem, finding the equations of motion of the source and the emitted gravitational waveform: the multipolar postMinkowskian 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 postMinkowskian 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
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 halfinteger 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 Q_{ab} is the (traceless) quadrupole moment of the source. The leading term in C is then of 2.5PN order (i.e., ∼ 1/c^{5}), but since Q_{ab} is computed in the Newtonian limit, it is often considered as a “Newtonian” term. A remarkable result of the multipolar postMinkowskian approach and of the direct integration of relaxed Einstein’s equations, is that once the equations are solved at nth PN order both in the near zone and in the wave zone, E is known at nPN order, and \({\mathcal L}\) is known at nPN 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 nPN order.
Presently, PN schemes determine the motion of a compact binary, and the emitted gravitational waveform, up to 3.5PN order for nonspinning binaries in circular orbits [109], but up to lower PNorders for eccentric orbits and for spinning binaries [48, 148]. It is estimated that Advanced LIGO/Virgo data analysis requires 3.5PN 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 stateoftheart 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 ADMHamiltonian approach [676], in which using the ADM formulation of GR, the source is described as a canonical system in terms of its Hamiltonian. The ADMHamiltonian approach is equivalent to the multipolar postMinkowskian 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 postNewtonian 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]. ADMHamiltonian and effective field theory are probably the most promising approaches to extend the accuracy of PN computations for spinning binaries.
The effective onebody (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 twobody system and the “dual” onebody system is not unique, the EOB Hamiltonian depends on a number of free parameters, which are fixed using results of PN schemes, of gravitational selfforce 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 pointlike, described by delta functions in the stressenergy 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 5PN 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].
5.1.2 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 topdown and bottomup, respectively [636]).
Let us discuss topdown approaches first. Nearly fifty years ago, Will and Nordtvedt developed the PPN formalism [784, 581], in which the PN metric of an Nbody 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 weakfield regime of GR. It is particularly well suited to perform tests in the solar system. All solarsystem 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 timedelay 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 BransDicke 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 socalled “parametrized postEinsteinian” 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 bottomup, 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 scalartensor 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 scalartensor 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.
5.1.3 State of the art
The postNewtonian approach has mainly been used to study the relativistic twobody 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 stateoftheart computations give the gravitational waveform emitted by a compact binary system, up to 3.5PN order for nonspinning binaries in circular orbits [109], up to 3PN order for eccentric orbits [48], and up to 2PN 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 solarsystem tests (see [782, 783] for extensive reviews on the subject), and in the parametrized postEinsteinian 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 scalartensor theories [244, 787, 30].
5.2 Spacetime perturbation approach
5.2.1 Astrophysical systems in general relativity
The PN expansion is less successful at describing strongfield, 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 stellarmass BH of mass μ into a supermassive BH of mass M ≫ μ [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
where h_{μν} ≪ 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 firstorder terms in h_{μν} (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 h_{μν}. 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 “ReggeWheeler gauge”, which allows us to set to zero some of the perturbation functions), the harmonic expansion of the metric perturbation is
with
where f(r) = 1 − 2M/r, and Y_{lm}(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 \(H_{0,1,2}^{lm},\;{K^{lm}}\), K^{lm} which describe completely the propagation of GWs. These functions, called the ReggeWheeler and the Zerilli function, satisfy Schrödingerlike wave equations of the form
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 Ψ_{RW, Zer}.
This approach was soon extended to general spherically symmetric BH backgrounds and a gaugeinvariant 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 NewmanPenrose formalism [575], in which the spacetime perturbation is not described by the metric perturbation h_{μν}, but by a set of gaugeinvariant complex scalars, the Weyl scalars, obtained by projecting the Weyl tensor C_{αβγδ} 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, Ψ_{4}, describes the (outgoing) gravitational radiation; it is defined as
In the literature one may also find Ψ_{4} defined without the minus sign, but all physical results derived from Ψ_{4} 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 C_{αβγδ} in Eq. (20) with R_{αβγδ}.
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 BardeenPressTeukolsky 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 higherdimensional spacetimes. It is not clear whether such generalizations are possible with the BardeenPressTeukolsky approach.
The sources \({\mathcal S_{{\rm{RW,Zer}}}}\) describe the objects that excite the spacetime perturbations, and can arise either directly from a nonvanishing stressenergy 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 stresstensor, 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 μ ≪ M, 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 backreaction 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 “selfforce”. For a general account on the selfforce 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 M_{1}M_{2}/(M_{1} + M_{2}) describes surprisingly well the dynamics of generic BHs with masses M_{1}, M_{2}.
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, equalmass 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 ReggeWheeler 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 quasinormal 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 ReggeWheelerZerilli 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].
5.2.2 Beyond electrovacuum GR
The ReggeWheelerZerilli 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 scalartensor theories [668, 165, 824], Dynamical ChernSimons theory [175, 554, 603], EinsteinDilatonGaussBonnet [602], Horndeski gravity [477, 478], and massive theories of gravity [135].
5.2.3 Beyond four dimensions
Spacetime perturbation theory is a powerful tool to study BHs in higherdimensional spacetimes. The tensor harmonic approach has been successfully extended by Kodama and Ishibashi [479, 452] to GR in higherdimensional spacetimes, with or without cosmological constant. Their approach generalizes the gaugeinvariant formulation of the ReggeWheelerZerilli construction to perturbations of Tangherlini’s solution describing spherically symmetric BHs.
Since many dynamical processes involving higherdimensional 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 Ddimensional 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 h_{μν} represents a small perturbation. Decomposing the Ddimensional 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 h_{μν} can be expanded in spherical harmonics, as in the fourdimensional case (see Section 5.2.1). However, the expansion in D > 4 is more complex than its fourdimensional 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 S^{D−2} and their gradients, as follows:
where ll′… denote harmonic indices on S^{D−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 x^{A} = (t,r), the metric perturbations can be written as
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, Ω_{ab} is the metric on S^{D−2}, S_{a} = −S,_{a}/k, S_{ab} = S_{:ab}/k^{2} minus trace terms, where κ^{2} = l(l + D − 3) is the eigenvalue of the scalar harmonics, and the “:” denotes the covariant derivative on S^{D−2}; the traceless \({\mathcal V_{ab}}\) is defined in a similar way.
A set of gaugeinvariant variables and the socalled “master functions”, generalizations of the ReggeWheeler and Zerilli functions, can be constructed out of the metric perturbation functions and satisfy wavelike 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 gaugeinvariant quantities
where we have dropped harmonic indices,
and \({\hat D_A}\) denotes the covariant derivative associated with (t,r) subsector of the background metric. A master function Φ can be conveniently defined in terms of its time derivative according to
From the master function, we can calculate the GW energy flux
The total radiated energy is obtained from integration in time and summation over all multipoles
In summary, this approach can be used, in analogy with the ReggeWheelerZerilli formalism in four dimensions, to determine the quasinormal mode spectrum (see, e.g., the review [95] and references therein), to determine the gravitationalwave 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 higherdimensional rotating (MyersPerry [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 higherdimensional 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 KodamaIshibashi approach.
5.2.4 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 quasinormal 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 KerrNewman 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 quasicircular motion, and headons or scatters.
Circular and quasicircular motion. Gravitational radiation from point particles in circular geodesics was studied in Refs. [551, 252, 130] for nonrotating 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 selfgravity of the objects implies that particles do not follow geodesics of the background spacetime. Inclusion of dissipative effects is usually done by balancetype arguments [445, 446, 733, 338] but it can also be properly accounted for by computing the selfforce 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].
Headon or finite impact parameter collisions: nonrotating 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 headon collisions at nonrelativistic velocities [660, 317, 524, 93], at exactly the speed of light [179, 93], and to nonheadon collisions at nonrelativistic 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 highenergy collisions of charged particles with uncharged BHs was studied in Ref. [181] including a comparison with zerofrequency limit (ZFL) predictions. Gravitational and EM radiation generated in collisions of charged BHs has been considered in Refs. [459, 460].
Headon 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 highenergy collisions with nearextremal Kerr BHs, have recently inspired further scrutiny of these scenarios [71, 72] as well as the investigation of enhanced absorption effects in the ultrarelativistic regime [376].
Close Limit approximation. The close limit approximation was first compared against nonlinear simulations of equalmass, nonrotating BHs starting from rest [634]. It has since been generalized to unequalmass [35] or even the point particle limit [524], rotating BHs [494] and boosted BHs at secondorder 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 scalartensor 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 scalarcharged 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 fourdimensions and asymptotic flatness. The gauge/gravity duality and related frameworks highlight the importance of (A)dS and higherdimensional background spacetimes. The formalism to handle gravitational perturbations of fourdimensional, 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 higherdimensional 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 higherdimensional, 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 SchwarzschildAdS 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 higherdimensional spacetimes in Ref. [170]. The first fully relativistic calculation of GWs generated by point particles falling from rest into a higherdimensional asymptotically flat nonrotating 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 higherdimensional BH, have been determined in Ref. [488].
The close limit approximation was extended to higherdimensional, asymptotically flat, spacetimes in Refs. [822, 823].
5.3 The zerofrequency limit
5.3.1 Astrophysical systems in general relativity
While conceptually simple, the spacetime perturbation approach does involve solving one or more secondorder, nonhomogeneous 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
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 lowfrequency 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
we immediately conclude that the energy spectrum at lowfrequencies 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 pointlike, then one expects the ZFL to be an accurate description of the problem.
For the headon collision of two equalmass objects each with mass M_{γ}/2, Lorentz factor γ and velocity υ in the centerofmass frame, one finds the ZFL prediction [707, 513]
The particles collide headon along the zaxis and we use standard spherical coordinates. The spectrum is flat, i.e., ωindependent, 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 ω_{cutoff} ∼ 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 highenergy limit υ →1 yields isotropic emission; when translated to a multipole dependence, it means that the energy in each multipole scales as 1/l^{2} 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 βdecay [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 headon 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 highenergy BH or star collisions yield impressive agreement with ZFL predictions [719, 93, 288, 134].
5.3.2 State of the art

Astrophysical systems. The zerofrequency limit for headon collisions of particles was used by Smarr [707] to understand gravitational radiation from BH collisions and in Ref. [14] to understand radiation from supernovaelike 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 nonrotating BH in a spacetime perturbation approach and compares the results with a ZFL calculation.

Beyond fourdimensional, electrovacuum GR. Recent work has started applying the ZFL to other spacetimes and theories. Brito [134] used the ZFL to understand headon collisions of scalar charges with fourdimensional BHs. The ZFL has been extended to higher dimensions in Refs. [170, 513] and recently to specific AdS soliton spacetimes in Ref. [173].
5.4 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 pointlike 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 υ → c. Taking simultaneously the limit M →0 so that the zeroth component of the 4momentum, 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 ppwave, i.e., a planefronted gravitational wave with parallel rays, sourced by a null particle. This is the AichelburgSexl geometry [16] for which the curvature has support only on a null plane. In Brinkmann coordinates, the line element is:
Here the shock wave is moving in the positive zdirection, where (u = t−z, υ = t+z). This geometry solves the Einstein equations with energy momentum tensor T_{uu} = Eδ(u)δ(ρ) — corresponding to a null particle of energy E = κ/4G, traveling along u = 0 = ρ — provided the equation on the righthand side of (31) is satisfied, where the Laplacian is in the flat 2dimensional 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
This metric is a valid description of the spacetime with the two shock waves except in the future lightcone 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,
for some functions ψ_{1},ψ_{2} to be determined. The relevant null normals to S_{1} and S_{2} are, respectively,
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 2π^{2}κ^{2}, 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:
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 (ῡ 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 postcollision 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:
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
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 firstorder results can equivalently be obtained using the LandauLifshitz pseudotensor for GW extraction [420]. The results in first and second order are, respectively:
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 secondorder 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 firstorder result reveals a remarkably simple pattern.
5.4.1 State of the art
The technique of superimposing two AichelburgSexl 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 headon 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 nonzero 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 highenergy 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 headon case. This method was generalized to D ≥ 5 in firstorder 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].
6 Numerical Relativity
Generating timedependent 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 wellposed 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 nonassigned 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 multidomains 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 stateoftheart choices for these ingredients.
6.1 Formulations of the Einstein equations
6.1.1 The ADM equations
The Einstein equations in D dimensions describing a spacetime with cosmological constant Λ and energymatter content T_{αβ} are given by
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 initialvalue or a boundaryvalue problem? In fact, the Einstein equations are of mixed character in this regard and represent an IBVP. Wellposedness 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 ℳ equipped with a metric g_{αβ} 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 : ℳ →ℝ with the following properties. (i) The 1form dt associated with the function t is timelike everywhere; (ii) The hypersurfaces Σ_{t} defined by t = const are nonintersecting and ∪_{t∈ℝ}Σ_{t} = ℳ. Points inside each hypersurface Σ_{t} are labelled by spatial coordinates x^{I}, I = 1, …, D − 1, and we refer to the coordinate system (t, x^{I}) as adapted to the spacetime split.
Next, we define the lapse function α and shift vector β through
where n ≡ − αdt is the timelike unit normal field. The geometrical interpretation of these quantities in terms of the timelike unit normal field n^{α} and the coordinate basis vector ∂_{t} is illustrated in Figure 2. Using the relation 〈d_{t}, ∂_{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 Σ_{t}. 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 n^{α} to the coordinate time t: Δτ = αΔt.
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
A tensor S is called tangent to Σ_{t} 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 n^{α}. Mixed projections are obtained by contracting any combination of tensor indices with n^{α} and projecting the remaining ones with \({\bot ^\alpha}_\mu\). A particularly important tensor is obtained from the spatial projection of the spacetime metric
γ_{αβ} is known as the first fundamental form or spatial metric and describes the intrinsic geometry of the spatial hypersurfaces Σ_{t}. 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, x^{I}) can be written as ds^{2} = −α^{2} dt^{2} + γ_{IJ}(dx^{I} + β^{I} dt)(dx^{J} + β^{J} dt) or, equivalently,
It can be shown [364] that the spatial metric γ_{IJ} defines a unique, torsionfree and metriccompatible 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
The final ingredient required for the spacetime split of the Einstein equations is the extrinsic curvature or second fundamental form defined as
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 Σ_{t}. As indicated by its name, the extrinsic curvature thus describes the embedding of Σ_{t} inside the higherdimensional spacetime manifold. The projection ⊥_{β}n_{α} is symmetric under exchange of its indices in contrast to its nonprojected counterpart ∇_{β}n_{α}. For the formulation of the Einstein equations in the spacetime split, it is helpful to introduce the vector field m^{μ} ≡ αn^{μ} = (∂t)^{μ} − β^{μ}. 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
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:
Here, ℛ denotes the Riemann tensor and its contractions as defined in standard fashion from the spatial metric γ_{IJ}. 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 energymomentum tensor which are given by
then, the energymomentum tensor is reconstructed according to T_{αβ} = S_{αβ}+n_{α}j_{β}+n_{β}j_{α}+ρn_{α}n_{β}. Using the explicit expressions for the Lie derivatives
we obtain the spacetime split of the Einstein equations
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 secondorderintime evolution equations for the γ_{IJ} are written as a firstorderintime 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 Σ_{t}. We note that the Einstein equations do not determine the lapse α and shift β^{I}. 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.
6.1.2 Wellposedness
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 initialvalue problem in one space and one time dimension for a single variable u(t, x) on an unbounded domain. Wellposedness 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
where δu denotes a linear perturbation relative to a solution u_{0}(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 highestorder derivatives. We consider for simplicity a quasilinear firstorder system for a set of variables u(t, x)
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 firstorder 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.
6.1.3 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
where γ ≡ detγ_{IJ} 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
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
Here, the superscript “TF” denotes the tracefree part and we further use the following expressions that relate physical to conformal variables:
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 righthand side of the BSSN equations. Two recipes have been identified that provide longterm 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 righthand 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].
6.1.4 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
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]
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
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 g_{α}β and ∂_{t}g_{α}β 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 H^{α} 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
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
where κ, λ are userspecified constraintdamping parameters. An alternative firstorder system of the GHG formulation has been presented in Ref. [519].
6.1.5 Beyond BSSN: Improvements for future applications
The vast majority of BH evolutions in generic 4dimensional 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 zerospeed 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 waveequationtype principal part of the GHG system allows for the straightforward construction of constraintpreserving 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
where Z_{α} 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 Z_{μ} = 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 higherdimensional spacetimes have been found by careful exploration of the densitization parameter space [791].
6.1.6 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 fourdimensional 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 Cauchycharacteristic matching, the combination of a (D − 1) + 1 or Cauchytype numerical scheme in the interior strongfield region with a characteristic scheme in the outer parts. In the form of Cauchycharacteristic 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, g_{α}β). 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 g_{α}β 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 ℐ has been developed by Friedrich [332, 331]. This formulation is suitable for a 3+1 decomposition into a symmetric hyperbolic system^{Footnote 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.
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 fourdimensional spherically symmetric spacetime can be described in terms of the simple line element
where \({\rm{d}}\Omega _2^2\) is the line element of the 2sphere. May and White employ Lagrangian coordinates comoving with the fluid shells which is imposed through the form of the energymomentum tensor \({T^0}_0 =  \rho (1 + \epsilon),\;{T^1}_1 = {T^2}_2 = {T^3}_3 = P\). Here, the restmass 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 energymomentum \({\nabla _\mu}{T^\mu}_\alpha = 0\), 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.
6.1.7 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 down^{Footnote 11}. The simplest extension of the field equations to include matter is described by the EinsteinHilbert action (in 4dimensional asymptotically flat spacetimes) minimally coupled to a complex, massive scalar field Φ with mass parameter μ_{s} = m_{s}/ħ,
if we introduce a time reduction variable defined as
we recover the equations of motion and constraints (52)–(55) with D = 4, Λ = 0 and with energy density ρ, energymomentum flux j_{i} and spatial components S_{ij} of the energymomentum tensor given by
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 wellposedness of the numerical methods used for the construction of spacetimes. We note, however, that the wellposedness 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 wellposed IBVPs.
6.2 Higherdimensional NR in effective “3 + 1” form
Performing numerical simulations in generic higherdimensional 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 higherdimensional spacetimes have so far focussed on symmetric spacetimes that allow for a reduction to an effectively fourdimensional formalism. Even though this implies a reduced class of spacetimes available for numerical study, many of the most important questions in higherdimensional 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 S^{D−3}. or S^{D−4}, respectively (we denote with S^{n} the ndimensional sphere). The group SO(D − 2) is the isometry of, for instance, headon collisions of nonrotating BHs, while the group SO(D − 3) is the isometry of nonheadon collisions of nonrotating BHs; SO(D − 3) is also the isometry of nonheadon 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 nonvanishing angular momentum. We remark that, in order to implement the higherdimensional system in (modified) fourdimensional 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 x^{μ} 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.
6.2.1 Dimensional reduction by isometry
The idea of dimensional reduction had originally been developed by Geroch [347] for fourdimensional 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 Ddimensional spacetime metric written in coordinates adapted to the symmetry
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 n ≡ D − 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 ϕ∂_{θ} − cot θ sin ϕ ∂_{ϕ}.
Killing’s equation ℒ_{ξ}_{(i)}gAB = 0 implies that
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 Ω_{ab} is the metric on the S^{n} 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 S^{n} 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 form^{Footnote 13}
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 fourdimensional 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 [