Exploring New Physics Frontiers Through Numerical Relativity
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.
Keywords
Gravitation Numerical methods Black holes Extensions of the standard model Alternative theories of gravity Extra dimensions TransPlanckian scatteringAcronyms
 ADM
ArnowittDeserMisner
 (A)dS
(Anti)de Sitter
 AH
Apparent horizon
 BH
Black hole
 BSSN
BaumgarteShapiroShibataNakamura
 CFT
Conformal field theory
 EOB
Effective onebody
 GHG
Generalized Harmonic Gauge
 EM
Electromagnetism, Electromagnetic
 GR
General relativity
 GW
Gravitational wave
 IBVP
Initialboundaryvalue problem
 LHC
Large Hadron Collider
 LIGO
Laser Interferometric GravitationalWave Observatory
 NR
Numerical relativity
 NS
Neutron star
 PDE
Partial differential equation
 PN
PostNewtonian
 PPN
Parametrized PostNewtonian
 QNM
Quasinormal mode
 RN
ReissnerNordström
 SMT
String/M theory
 ZFL
Zerofrequency limit
Notation and conventions
 D
Total number of spacetime dimensions (we always consider one timelike and D − 1 spacelike dimensions).
 L
Curvature radius of (A)dS spacetime, related to the (negative) positive cosmological constant A in the Einstein equations (G_{μν} + Λg_{μν} = 0) through L^{2} = (D − 2)(D − 1)/(2Λ).
 M
BH mass.
 a
BH rotation parameter.
 R_{S}
Radius of the BH’s event horizon in the chosen coordinates.
 ω
Fourier transform variable. The time dependence of any field is ∼ e^{−iωt}. For stable spacetimes, Im(ω) < 0.
 s
Spin of the field.
 l
Integer angular number, related to the eigenvalue A_{lm} = l(l + D − 3 of scalar spherical harmonics in D dimensions.
 a, b, …, h
Index range referred to as “early lower case Latin indices” (likewise for upper case indices).
 i, j, …, v
Index range referred to as “late lower case Latin indices” (likewise for upper case indices).
 g_{αβ}
Spacetime metric; greek indices run from 0 to D − 1.
 \(\Gamma _{\beta \gamma}^\alpha\)
\(= {1 \over 2}{g^{\alpha \mu}}({\partial _\beta}{g_{\gamma \mu}} + {\partial _\gamma}{g_{\mu \beta}}  {\partial _\mu}{g_{\beta \gamma}})\), Christoffel symbol associated with the spacetime metric g_{αβ}.
 \({R^\alpha}_{\beta \gamma \delta}\)
\(= {\partial _\gamma}\Gamma _{\delta \beta}^\alpha  {\partial _\delta}\Gamma _{\gamma \beta}^\alpha + \Gamma _{\gamma \rho}^\alpha \Gamma _{\delta \beta}^\rho  \Gamma _{\delta \rho}^\alpha \Gamma _{\gamma \beta}^\rho\), Riemann curvature tensor of the Ddimensional spacetime.
 ∇_{α}
Ddimensional covariant derivative associated with \(\Gamma _{\beta \gamma}^\alpha\).
 γ_{ij}
Induced metric, also known as first fundamental form, on (D − 1)dimensional spatial hypersurface; latin indices run from 1 to D − 1.
 K_{ij}
Extrinsic curvature, also known as second fundamental form, on (D − 1)dimensional spatial hypersurface.
 \(\Gamma _{jk}^i\)
(D − 1)dimensional Christoffel symbol associated with γ_{ij}.
 \({{\mathcal R}^i}_{jkl}\)
(D − 1)dimensional Riemann curvature tensor of the spatial hypersurface.
 D_{i}
(D − 1)dimensional covariant derivative associated with \(\Gamma _{jk}^i\).
 S^{n}
ndimensional sphere.
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.
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^{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.
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].

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
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 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^{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?
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 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.^{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.
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.
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

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^{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.^{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
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].
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].
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.
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].
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.
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 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.
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.
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

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.
6.1 Formulations of the Einstein equations
6.1.1 The ADM equations
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.^{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.
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.
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.
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
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.
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].
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].
6.1.7 Einstein’s equations extended to include fundamental fields
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].^{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 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 }.
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.
6.2.2 The cartoon method
The cartoon method has originally been developed in Ref. [25] for evolving axisymmetric fourdimensional spacetimes using an effectively twodimensional spatial grid which employs ghostzones, i.e., a small number of extra gridpoints off the computational plane required for evaluating finite differences in the third spatial direction. Integration in time, however, is performed exclusively on the twodimensional plane whereas the ghostzones are filled in after each timestep by appropriate interpolation of the fields in the plane and subsequent rotation of the solution using the axial spacetime symmetry. A version of this method has been applied to 5dimensional spacetimes in Ref. [820]. For arbitrary spacetime dimensions, however, even the relatively small number of ghostzones required in every extra dimension leads to a substantial increase in the computational resources; for fourthorder finite differencing, for example, four ghostzones are required in each extra dimension resulting in an increase of the computational domain by an overall factor 5D−^{4}. An elegant scheme to avoid this difficulty while preserving all advantages of the cartoon method has been developed in Ref. [630] and is sometimes referred to as the modified cartoon method. This method has been applied to D > 5 dimensions in Refs. [700, 512, 821] and we will discuss it now in more detail.
Let us consider for illustrating this method a Ddimensional spacetime with SO(D − 3) symmetry and Cartesian coordinates x^{ μ } = (t, x, y, z, w^{ a }), where a = 4, …, D − 1. Without loss of generality, the coordinates are chosen such that the SO(D − 3) symmetry implies rotational symmetry in the planes spanned by each choice of two coordinates from^{14} (y, w^{ a }). The goal is to obtain a formulation of the Ddimensional Einstein equations (60)–(69) with SO(D − 3) symmetry that can be evolved exclusively on the xyz hyperplane. The tool employed for this purpose is to use the spacetime symmetries in order to trade derivatives off the hyperplane, i.e., in the w^{ a } directions, for derivatives inside the hyperplane. Furthermore, the symmetry implies relations between the Ddimensional components of the BSSN variables.
6.3 Initial data
In Section 6.1 we have discussed different ways of casting the Einstein equations into a form suitable for numerical simulations. At the start of Section 6, we have listed a number of additional ingredients that need to be included for a complete numerical study and physical analysis of BH spacetimes. We will now discuss the main choices used in practical computations to address these remaining items, starting with the initial conditions.
As we have seen in Section 6.1, initial data to be used in time evolutions of the Einstein equations need to satisfy the Hamiltonian and momentum constraints (54), (55). A comprehensive overview of the approach to generate BH initial data is given by Cook’s Living Reviews article [224]. Here we merely summarize the key concepts used in the construction of vacuum initial data, but discuss in some more detail how solutions to the constraint equations can be generated in the presence of specific matter fields that play an important role in the applications discussed in Section 7.
In spite of its popularity, there remain a few caveats with puncture data that have inspired explorations of alternative initial data. In particular, it has been shown that there exist no maximal, conformally flat spatial slices of the Kerr spacetime [341, 756]. Constructing puncture data of a single BH with nonzero BowenYork parameter S^{ i } will, therefore, inevitably result in a hypersurface containing a BH plus some additional content which typically manifests itself in numerical evolutions as spurious GWs, colloquially referred to as “junk radiation”. For rotation parameters close to the limit of extremal Kerr BHs, the amount of spurious radiation rapidly increases leading to an upper limit of the dimensionless spin parameter J/M^{2} ≈ 0.93 for conformally flat BowenYorktype data [226, 237, 238, 527]; BH initial data of BowenYork type with a spin parameter above this value rapidly relax to rotating BHs with spin χ ≈ 0.93, probably through absorption of some fraction of the spurious radiation. This limit has been overcome [527, 528] by instead constructing initial data with an extended version of the conformal thinsandwich method using superposed KerrSchild BHs [467]. In an alternative approach, most of the above outlined puncture method is applied but using a nonflat conformal metric; see for instance [493, 391].
In practice, puncture data are the methodofchoice for most evolutions performed with the BSSNmovingpuncture technique^{15} whereas GHG evolution schemes commonly start from conformal thinsandwich data using either conformally flat or KerrSchild background data. Alternatively to both these approaches, initial data containing scalar fields which rapidly collapse to one or more BHs has also been employed [629].
6.4 Gauge conditions
We have seen in Section 6.1, that the Einstein equations do not make any predictions about the gauge functions; the ADM equations leave lapse α and shift β^{ i } unspecified and the GHG equations make no predictions about the source functions H^{ α }. Instead, these functions can be freely specified by the user and represent the coordinate or gaugeinvariance of the theory of GR. Whereas the physical properties of a spacetime remain unchanged under gauge transformations, the performance of numerical evolution schemes depends sensitively on the gauge choice. It is wellknown, for example, that evolutions of the Schwarzschild spacetime employing geodesic slicing α = 1 and vanishing shift β^{ i } = 0 inevitably reach a hypersurface containing the BH singularity after a coordinate time interval t = πM [709]; computers respond to singular functions with nonassigned numbers which rapidly swamp the entire computational domain and render further evolution in time practically useless. This problem can be avoided by controlling the lapse function such that the evolution in proper time slows down in the vicinity of singular points in the spacetime [312]. Such slicing conditions are called singularity avoiding and have been studied systematically in the form of the BonaMassó family of slicing conditions [116]; see also [343, 20]. A potential problem arising from the use of singularity avoiding slicing is the different progress in proper time in different regions of the computational domain resulting in a phenomenon often referred to as “grid stretching” or “slice stretching” which can be compensated with suitable nonzero choices for the shift vector [24].
6.5 Discretization of the equations
In the previous sections, we have derived formulations of the Einstein equations in the form of an IBVP. Given an initial snapshot of the physical system under consideration, the evolution equations, as for example in the form of the BSSN equations (60)–(64), then predict the evolution of the system in time. These evolution equations take the form of a set of nonlinear partial differential equations which relate a number of grid variables and their time and spatial derivatives. Computers, on the other hand, exclusively operate with (large sets of) numbers and for a numerical simulation we need to translate the differential equations into expressions relating arrays of numbers.
The common methods to implement this discretization of the equations are finite differencing, the finite element, finite volume and spectral methods. Finite element and volume methods are popular choices in various computational applications, but have as yet not been applied to time evolutions of BH spacetimes. Spectral methods provide a particularly efficient and accurate approach for numerical modelling provided the functions do not develop discontinuities. Even though BH spacetimes contain singularities, the use of singularity excision provides a tool to remove these from the computational domain. This approach has been used with great success in the SpEC code to evolve inspiralling and merging BH binaries with very high accuracy; see, e.g., [122, 220, 526]. Spectral methods have also been used successfully for the modelling of spacetimes with high degrees of symmetry [205, 206, 207] and play an important role in the construction of initial data [39, 38, 836]. An indepth discussion of spectral methods is given in the Living Reviews article [365]. The main advantage of finite differencing methods is their comparative simplicity. Furthermore, they have proved very robust in the modelling of rather extreme BH configurations as for example BHs colliding near the speed of light [719, 587, 716] or binaries with mass ratios up to 1:100 [525, 523, 718].
Mesh refinement and domain decomposition: BH spacetimes often involve lengthscales that differ by orders of magnitude. The BH horizon extends over lengths of the order \({\mathcal O(1)}\) M where M is the mass of the BH. Inspiralling BH binaries, on the other hand, emit GWs with wavelengths of \({\mathcal O}({10^2})\;M\). Furthermore, GWs are rigorously defined only at infinity. In practice, wave extraction is often performed at finite radii but these need to be large enough to ensure that systematic errors are small. In order to accomodate accurate wave extraction, computational domains used for the modelling of asymptotically flat BH spacetimes typically have a size of \({\mathcal O}({10^3})\;M\). With present computational infrastructure it is not possible to evolve such large domains with a uniform, high resolution that is sufficient to accurately model the steep profiles arising near the BH horizon. The solution to this difficulty is the use of mesh refinement, i.e., a grid resolution that depends on the location in space and may also vary in time. The use of mesh refinement in BH modelling is simplified by the remarkably rigid nature of BHs which rarely exhibit complicated structure beyond some mild deformation of a sphere. The requirements of increased resolution are, therefore, simpler to implement than, say, in the modelling of airplanes or helicopters. In BH spacetimes the grid resolution must be highest near the BH horizon and it decreases gradually at larger and larger distances from the BH. In terms of the internal bookkeeping, this allows for a particularly efficient manner to arrange regions of refinement which is often referred to as moving boxes. A set of nested boxes with outwardly decreasing resolution is centered on each BH of the spacetime and follows the BH motion. These sets of boxes are immersed in one or more common boxes which are large enough to accomodate those centered on the BHs. As the BHs approach each other, boxes originally centered on the BHs merge into one and become part of the commonbox hierarchy. A snapshot of such moving boxes is displayed in Figure 4.
Mesh refinement in NR has been pioneered by Choptuik in his seminal study on critical phenomena in the collapse of scalar fields [212]. The first application of mesh refinement to the evolution of BH binaries was performed by Brügmann [140]. There exists a variety of mesh refinement packages available for use in NR including Bam [140], Had [384], Pamr/Amrd [754], Paramesh [534], Samrai [672] and the Carpet [684, 184] package integrated into the Cactus Computational Toolkit [155]. For additional information on Cactus see also the Einstein Toolkit webpage [300] and the lecture notes [840]. A particular meshrefinement algorithm used for many BH applications is the BergerOliger [90] scheme where coarse and fine levels communicate through interpolation in the form of the prolongation and restriction operation; see [684] for details. Alternatively, the different lengthscales can be handled efficiently through the use of multiple domains of different shapes. Communication between the individual subdomains is performed either through overlaps or directly at the boundary for touching domains. Details of this domain decomposition can be found in [618, 146] and references therein.
6.6 Boundary conditions
In NR, we typically encounter two types of physical boundaries, (i) inner boundaries due to the treatment of spacetime singularities in BH solutions and (ii) the outer boundary either at infinite distance from the strongfield sources or, in the form of an approximation to this scenario, at the outer edge of the computational domain at large but finite distances.
Singularity excision: BH spacetimes generically contain singularities, either physical singularities with a divergent Ricci scalar or coordinate singularities where the spacetime curvature is well behaved but some tensor components approach zero or inifinite values. In the case of the Schwarzschild solution in Schwarzschild coordinates, for example, r = 0 corresponds to a physical singularity whereas the singular behaviour of the metric components g_{ tt } and g_{ rr } at r = 2 M merely reflects the unsuitable nature of the coordinates as r →2 M and can be cured, for example, by transforming to KruskalSzekeres coordinates; cf. for example Chapter 7 in Ref. [186]. Both types of singularities give rise to trouble in the numerical modelling of spacetimes because computers only handle finite numbers. Some control is available in the form of gauge conditions as discussed in Section 6.4; the evolution of proper time is slowed down when the evolution gets close to a singularity. In general, however, BH singularities require some special numerical treatment.
Quite remarkably, the moving puncture method for evolving BHs does not employ any such specific numerical treatment near BH singularities, but instead applies the same evolution procedure for points arbitrarily close to singularities as for points far away and appears “to get away with it”. In view of the remarkable success of the moving puncture method, various authors have explored the behaviour of the puncture singularity in the case of a single Schwarzschild BH [392, 136, 137, 390, 138, 264]. Initially, the puncture represents spatial infinity on the other side of the wormhole geometry compactified into a single point. Under numerical evolution using moving puncture gauge conditions, however, the region immediately around this singularity rapidly evolves into a socalled trumpet geometry which is partially covered by the numerical grid to an extent that depends on the numerical resolution; cf. Figure 1 in [138]. In practice, the singularity falls through the inevitably finite resolution of the computational grid which thus facilitates a natural excision of the spacetime singularity without the need of any special numerical treatment.
Outer boundary: Most physical scenarios of interest for NR involve spatial domains of infinite extent and there arises the question how these may be accomodated inside the finite memory of a computer system. Probably the most elegant and rigorous method is to apply a spatial compactification, i.e., a coordinate transformation that maps the entire domain including spatial infinity to a finite coordinate range. Such compactification is best achieved in characteristic formulations of the Einstein equations where the spacetime foliation in terms of ingoing and/or outgoing light cones may ensure adequate resolution of in or outgoing radiation throughout the entire domain. In principle, such a compactification can also be implemented in Cauchytype formulations, but here it typically leads to an increasing blueshift of radiative signals as they propagate towards spatial infinity. As a consequence, any discretization method applied will eventually fail to resolve the propagating features. This approach has been used in Pretorius’ breakthrough [629] and the effective damping of radiative signals at large distances through underresolving them approximates a noingoingradiation boundary condition. An intriguing alternative consists in using instead a spacetime slicing of asymptotically null hypersurfaces which play a key role in the conformal field equations [328]. To our knowledge, this method has not yet been applied successfully to BH simulations in either astrophysical problems or simulations of the type reviewed here, but may well merit more study in the future.
The vast majority of Cauchybased NR applications instead resort to an approximative treatment where the infinite spatial domain is truncated and modeled as a compact domain with “suitable” outer boundary conditions. Ideally, the boundary conditions would satisfy the following requirements [651]: They (i) ensure well posedness of the IBVP, (ii) are compatible with the constraint equations, and (iii) correctly represent the physical conditions, which in almost all practical applications means that they control or minimize the ingoing gravitational radiation.
Boundary conditions meeting these requirements at least partially have been studied most extensively for the harmonic or generalized harmonic formulation of the Einstein equations [492, 651, 58, 652, 665].
The boundary treatment inside a numerical modelling of asymptotically AdS spacetimes needs to take care of the singular nature of the metric. In practice, this is achieved through some form of regularization which makes use of the fact that the singular piece of an asymptotically AdS spacetime is known in analytic form, e.g., through Eqs. (130) or (131). In Ref. [70] the spacetime metric is decomposed into an analytically known AdS part plus a deviation which is regular at infinity. In this approach, particular care needs to be taken of the gauge conditions to ensure that the coordinates remain compatible with this decomposition throughout the simulation. An alternative approach consists in factoring out appropriate factors involving the bulk coordinate as for example the term cos ρ in the denominator on the righthand side of Eq. (130). This method is employed in several recent works [207, 415, 108].
We finally note that the boundary plays an active role in AdS spacetimes. The visualization of the AdS spacetime in the form of a Penrose diagram demonstrates that it is not globally hyperbolic, i.e., there exists no Cauchy surface on which initial data can be specified in such a way that the entire future of the spacetime is uniquely determined. This is in marked contrast to the Minkowski spacetime. Put in other words, the outer boundary of asymptotically flat spacetimes is represented in a Penrose diagram by a null surface such that information cannot propagate from infinity into the interior spacetime. In contrast, the outer boundary of asymptotically AdS spacetimes is timelike and, hence, the outer boundary actively influences the evolution of the interior. The specification of boundary conditions in NR applications to the gauge/gravity duality or AdS/CFT correspondence therefore reflects part of the description of the physical system under study; cf. Section 7.8.
6.7 Diagnostics
Once we have numerically generated a spacetime, there still remains the question of how to extract physical information from the large chunk of numbers the computer has written to the hard drive. This analysis of the data faces two main problems in NR applications, (i) the gauge or coordinate dependence of the results and (ii) the fact that many quantities we are familiar with from Newtonian physics are hard or not even possible to define in a rigorous fashion in GR. In spite of these difficulties, a number of valuable diagnostic tools have been developed and the purpose of this section is to review how these are extracted.
The physical information is often most conveniently calculated from the ADM variables and we assume for this discussion that a numerical solution is available in the form of the ADM variables γ_{ IJ }, K_{ IJ }, α and β^{ I }. Even if the time evolution has been performed using other variables as for example the BSSN or GHG variables the conversion between these and their ADM counterparts according to Eq. (58) or (43) is straightforward.
One evident diagnostic directly arises from the structure of the Einstein equations where the number of equations exceeds the number of free variables; cf. the discussion following Eq. (55). Most numerical applications employ “free evolutions” where the evolution equations are used for updating the grid variables. The constraints are thus not directly used in the numerical evolution but need to be satisfied by any solution to the Einstein equations. A convergence analysis of the constraints (see for example Figure 3 in Ref. [714]) then provides an important consistency check of the simulations.
6.7.1 Global quantities and horizons
The event horizon is defined as the boundary between points in the spacetime from which null geodesics can escape to infinity and points from which they cannot. The event horizon is therefore by definition a concept that depends on the entire spacetime. In the context of numerical simulations, this implies that an event horizon can only be computed if information about the entire spacetime is stored which results in large data sets even by contemporary standards. Nevertheless, event horizon finders have been developed in Refs. [278, 223]. For many purposes, however, it is more convenient to determine the existence of a horizon using data from a spatial hypersurface Σ_{ t } only. Such a tool is available in the form of an AH. AHs are one of the most important diagnostic tools in NR and are reviewed in detail in the Living Reviews article [749]. It can be shown under the assumption of cosmic censorship and reasonable energy conditions, that the existence of an AH implies an event horizon whose cross section with Σ_{ t } either lies outside the AH or coincides with it; see [406, 766] for details and proofs.
It is a remarkable feature of BHs that their local properties such as mass and angular momentum can be determined in the way summarized here. In general it is not possible to assign in such a welldefined manner a local energy or momentum content to compact subsets of spacetimes due to the nonlinear nature of GR. For BHs, however, it is possible to derive expressions analogous to the ADM integrals discussed above, but now applied to the apparent horizon. Ultimately, this feature rests on the dynamic and isolated horizon framework; for more details see [281, 52] and the Living Reviews article by Ashtekar & Krishnan [53].
6.7.2 Gravitationalwave extraction
Probably the most important physical quantity to be extracted from dynamical BH spacetimes is the gravitational radiation. It is commonly extracted from numerical simulations in the form of either the NewmanPenrose scalar or a master function obtained through BH perturbation theory (see Section 5.2.1). Simulations using a characteristic formulation also facilitate wave extraction in the form of the Bondi mass loss formula. The LandauLifshitz pseudotensor [500], which has been generalized to D > 4 in [820], has been used for gravitational radiation extraction in Ref. [700] for studies of BH stability in higher dimensions; for applications in D = 4 see, e.g., [529]. Here, we will focus on the former two methods; wave extraction using the Bondi formalism is discussed in detail in Ref. [788].
For reasons already discussed in Section 6.6, extraction of gravitational waves is often performed at finite distance from the sources; but see Refs. [642, 60] for Cauchycharacteristic extraction that facilitates GW calculation at future null infinity. GW extraction at finite distances requires further ingredients which are discussed in more detail in [510]. These include a specific asymptotic behaviour of the Riemann tensor, the socalled peeling property [666, 667, 575], that outgoing null hypersurfaces define sequences of S^{2} spheres which are conformal to unit spheres and a choice of coordinates that ensures appropriate falloff of the metric components in the extraction frame.
Extraction of GWs at finite extraction radii r_{ex} is therefore affected by various potential errors. An attempt to estimate the uncertainty arising from the use of finite r_{ex} consists in measuring the GW signal at different values of the radius and analyzing its behaviour as the distance is increased. Convergence of the signal as 1/r_{ex} →0 may then provide some estimate for the error incurred and improved results may be obtained through extrapolation to infinite r_{ex}; see, e.g., [124, 429]. While such methods appear to work relatively well in practice (applying balance arguments together with measurements of BH horizon masses and the ADM mass or comparison with alternative extraction methods provide useful checks), it is important to bear in mind the possibility of systematic errors arising in the extraction of GWs using this method.
Perturbative wave extraction: The basis of this approach to extract GWs from numerical simulations in D = 4 is the ReggeWheelerZerilliMoncrief formalism developed for the study of perturbations of spherically symmetric BHs. The assumption for applying this formalism to numerically generated spacetimes is that at sufficiently large distances from the GW sources, the spacetime is well approximated by a spherically symmetric background (typically Schwarzschild or Minkowski spacetime) plus nonspherical perturbations. These perturbations naturally divide into odd and even multipoles which obey the ReggeWheeler [641] (odd) and the Zerilli [830] (even) equations respectively (see Section 5.2.1). Moncrief [555] developed a gaugeinvariant formulation for these perturbations in terms of a master function which obeys a wavetype equation with a background dependent scattering potential; for a review and applications of this formalism see for example [566, 721, 643].
An extension of this formalism to higherdimensional spacetimes has been developed by Kodama & Ishibashi [479], and is discussed in Section 5.2.3. This approach has been used to develop wave extraction from NR simulations in D > 4 with SO(D − 2) symmetry [797]. In particular, it has been applied to the extraction of GWs from headon collisions of BHs. As in our discussion of formulations of the Einstein equations in higher dimensions in Section 6.2, it turns out useful to introduce coordinates that are adapted to the rotational symmetry on a S^{D−2} sphere. Here, we choose spherical coordinates for this purpose which we denote by (t, r, ϑ, θ, ϕ^{a}) where a = 4, …, D − 1; we use the same convention for indices as in Section 6.2.
As discussed in Section 5.2.3, the metric perturbations, decomposed in tensor harmonics, can be combined in a gaugeinvariant master function Φ_{ ℓm }. From the master function, we can calculate the GW energy flux and the total radiated energy as discussed in Section 5.2.3.
6.7.3 Diagnostics in asymptotically AdS spacetimes
The gauge/gravity duality, or AdS/CFT correspondence (see Section 3.3.1), relates gravity in asymptotically AdS spacetimes to conformal field theories on the boundary of this spacetime. A key ingredient of the correspondence is the relation between fields interacting gravitationally in the bulk spacetime and expectation values of the field theory on the boundary. Here we restrict our attention to the extraction of the expectation values of the energymomentum tensor 〈T_{ IJ }〉 of the field theory from the falloff behaviour of the AdS metric.
The role of additional (e.g., scalar) fields in the AdS/CFT dictionary is discussed, for example, in Refs. [253, 705].
7 Applications of Numerical Relativity
Numerical relativity was born out of efforts to solve the twobody problem in GR, and aimed mainly at understanding stellar collapse and GW emission from BH and NS binaries. There is therefore a vast amount of important results and literature on NR in astrophysical contexts. Because these results fall outside the scope of this review, we refer the interested reader to Refs. [631, 592, 191, 715, 18, 617, 429] and to the relevant sections of Living Reviews^{18} for (much) more on this subject. Instead, we now focus on applications of NR outside its traditional realm, most of which are relatively recent new directions in the field.
7.1 Critical collapse
The nonlinear stability of Minkowski spacetime was established by Christodoulou and Klainerman, who showed that arbitrarily “small” initial fluctations eventually disperse to infinity [219]. On the other hand, large enough concentrations of matter are expected to collapse to BHs, therefore raising the question of how the threshold for BH formation is approached.
The BH threshold in the space of initial data for GR shows both surprising structure and surprising simplicity. In particular, critical behavior was found at the threshold of BH formation associated with universality, powerlaw scaling of the BH mass, and discrete selfsimilarity, which bear resemblance to more familiar statistical physics systems. Critical phenomena also provide a route to develop arbitrarily large curvatures visible from infinity (starting from smooth initial data) and are therefore likely to be relevant for cosmic censorship (see Section 7.2), quantum gravity, astrophysics, and our general understanding of the dynamics of GR.
Choptuik’s original result was extended in many different directions, to encompass massive scalar fields [125, 586], collapse in higher dimensions [344] or different gravitational theories [265]. Given the difficulty of the problem, most of these studies have focused on 1 + 1 simulations; the first non spherically (but axially) symmetric simulations were performed in Ref. [9], whereas recently the first 3 + 1 simulations of the collapse of minimally coupled scalar fields were reported [411]. The attempt to extend these results to asymptotically AdS spacetimes would uncover a new surprising result, which we discuss below in Section 7.4. A full account of critical collapse along with the relevant references can be found in a Living Reviews article on the subject [381].
7.2 Cosmic censorship
As discussed in Section 3.2.1, an idea behind cosmic censorship is that classical GR is selfconsistent for physical processes. That is, despite the fact that GR predicts the formation of singularities, at which geodesic incompleteness occurs and therefore failure of predictablity, such singularities should be — for physical processes^{19} — causally disconnected from distant observers by virtue of horizon cloaking. In a nutshell: a GR evolution does not lead, generically, to a system GR cannot tackle. To test this idea, one must analyze strong gravity dynamics, which has been done both using numerical evolutions and analytical arguments. Here we shall focus on recent results based on NR methods. The interested reader is referred to some historically relevant numerical [692, 361] and analytical [218, 653] results, as well as to reviews on the subject [768, 88, 649, 461] for further information.
The simplest (and most physically viable) way to violate cosmic censorship would be through the gravitational collapse of very rapidly rotating matter, possibly leading to a Kerr naked singularity with a > M. However, NR simulations of the collapse of a rotating NS to a BH [568, 63, 348] have shown that when the angular momentum of the collapsing matter is too large, part of the matter bounces back, forming an unstable disk that dissipates the excess angular momentum, and eventually collapses to a Kerr BH. Simulations of the coalescence of rapidly rotating BHs [769, 412] and NSs [465] have shown that the a > M bound is preserved by these processes as well. These simulations provide strong evidence supporting the cosmic censorship conjecture. Let us remark that analytical computations and NR simulations show that naked singularities can arise in the collapse of ideal fluids [461] but these processes seem to require finetuned initial conditions, such as in spherically symmetric collapse or in the critical collapse [375] discussed in Section 7.1.
Another suggestion that Planckian scale curvature becomes visible in a classical evolution in D = 5 GR arises in the highenergy scattering of BHs. In Ref. [587], NR simulations of the scattering of two nonspinning boosted BHs with an impact parameter b were reported. For sufficiently small initial velocities (υ ≲ 0.6c) it is possible to find the threshold impact parameter b_{scat} = b_{scat}(υ) such that the BHs merge into a (spinning) BH for b < b_{scat} or scatter off to infinity for b > b_{scat}. For high velocities, however, only a lower bound on the impact parameter for scattering b_{ C } = b_{ C }(υ) and an upper bound on the impact parameter for merger b_{ B } = b_{ B }(υ) could be found, since simulations with b_{ B } < b < b_{ C } crashed before the final outcome could be determined (cf. Figure 16 below). Moreover, an analysis of a scattering configuration with υ = 0.7 and b = b_{ C }, shows that very high curvature develops outside the individual BHs’ AHs, shortly after they have reached their minimum separation — see Figure 7 (right panel). The timing for the creation of the high curvature region, i.e., that it occurs after the scattering, is in agreement with other simulations of high energy collisions. For instance, in Refs. [216, 288] BH formation is seen to occur in the wake of the collision of nonBH objects, which was interpreted as due to focusing effects [288]. In the case of Ref. [587], however, there seems to be no (additional) BH formation. Both the existence and significance of such a high curvature region, seemingly uncovered by any horizon, remains mysterious and deserves further investigation.
In contrast with the two higherdimensional examples above, NR simulations that have tested the cosmic censorship conjecture in different D = 4 setups, found support for the conjecture. We have already mentioned simulations of the gravitational collapse of rotating matter, and of the coalescence of rotating BH and NS binaries. As we discuss below in Section 7.6, the highenergy headon collisions of BHs [719], boson stars [216] or fluid particles [288, 647] in D = 4 result in BH formation but no naked singularities. A different check of the conjecture involves asymptotically dS spacetimes [837]. Here, the cosmological horizon imposes an upper limit on the size of BHs. Thus one may ask what is the outcome of the collision of two BHs with almost the maximum allowed size. In Ref. [837] the authors were able to perform the evolution of two BHs, initially at rest with the cosmological expansion. They observe that for all the (small) initial separations attempted, a cosmological AH, as viewed by an observer at the center of mass of the binary BH system, eventually forms in the evolution, and both BH AHs are outside the cosmological one. In other words, the observer in the center of mass loses causal contact with the two BHs which fly apart rather than merge. This suggests that the background cosmological acceleration dominates over the gravitational attraction between’ large’ BHs. It would be interesting to check if a violation of the conjecture can be produced by introducing opposite charges to the BHs (to increase their mutual attraction) or give them mutually directed initial boosts.
7.3 Hoop conjecture
The hoop conjecture, first proposed by K. Thorne in 1972 [750], states that when the mass M of a system (in D = 4 dimensions) gets compacted into a region whose circumference in every direction has radius R ≲ R_{ s } = 2M, a horizon — and thus a BH — forms (for a generalization in D > 4, see [449]). This conjecture is important in many contexts. In highenergy particle collisions, it implies that a classical BH forms if the centerofmass energy significantly exceeds the Planck energy. This is the key assumption behind the hypothesis — in the TeV gravity scenario — of BH production in particle accelerators (see Section 3.3.2). In the transPlanckian regime, the particles can be treated as classical objects. If two such “classical” particles with equal rest mass m_{0} and radius (corresponding to the de Broglie wavelength of the process) R collide with a boost parameter γ, the massenergy in the centreofmass frame is M = 2γm_{0}. The threshold radius of Thorne’s hoop is then R_{s} = 4γm_{0} and the condition R ≲ R_{ s } = 2M = 4γm_{0} translates into a bound on the boost factor, γ ≳ γ_{h} ≡ R/(4m_{0}).
Even though the hoop conjecture seems plausible, finding a rigorous proof is not an easy task. In the last decades, the conjecture has mainly been supported by studies of the collision of two infinitely boosted point particles [256, 286, 817, 818], but it is questionable that they give an accurate description of an actual particle collision (see, e.g., the discussion in Ref. [216]). In recent years, however, advances in NR have made it possible to model transPlanckian collisions of massive bodies and provided more solid evidence in favor of the validity of the hoop conjecture.
The hoop conjecture has been first addressed in NR by Choptuik & Pretorius [216], who studied headon collisions of boson stars in four dimensions (see Section 4.2). The simulations show that the threshold boost factor for BH formation is ∼ 1/3γ_{h} (where the “hoop” critical boost factor γ_{h} is defined above), well in agreement with the hoop conjecture. These results have been confirmed by NR simulations of fluid star collisions [288], showing that a BH forms when the boost factor is larger than ∼ 0.42ϒ_{h}. Here, the fluid balls are modeled as two superposed TolmanOppenheimerVolkoff “stars” with a Γ = 2 polytropic equation of state.
7.4 Spacetime stability
Understanding the stability of stationary solutions to the Einstein field equations, or generalisations thereof, is central to gauge their physical relevance. If the corresponding spacetime configuration is to play a role in a given dynamical process, it should be stable or, at the very least, its instabilities should have longer time scales than those of that dynamical process. Following the evolution of unstable solutions, on the other hand, may unveil smoking guns for establishing their transient existence. NR provides a unique tool both for testing nonlinear stability and for following the nonlinear development of unstable solutions. We shall now review the latest developments in both these directions, but before doing so let us make a remark. At the linear level, typical studies of spacetime stability are in fact studies of mode stability. A standard example is Whiting’s study of the mode stability for Kerr BHs [780]. For BH spacetimes, however, mode stability does not guarantee linear stability, cf. the discussion in [236]. We refer the reader to this reference for further information on methods to analyse linear stability.
Even if a spacetime does not exhibit unstable modes in a linear analysis it may be unstable when fully nonlinear dynamics are taken into account. A remarkable illustration of this possibility is the turbulent instability of the AdS spacetime reported in Ref. [108]. These authors consider Einstein gravity with a negative cosmological constant Λ and minimally coupled to a massless real scalar field ϕ in D = 4 spacetime dimensions. The AdS metric is obviously a solution of the system together with a constant scalar field. Linear scalarfield perturbations around this solution generate a spectrum of normal modes with real frequencies [150]: ω_{ N }L = 2N + 3+ℓ, where N ∈ ℕ_{0} and L, ℓ are the AdS length scale and total angular momentum harmonic index, respectively. The existence of this discrete spectrum is quite intuitive from the global structure of AdS: a timelike conformal boundary implies that AdS behaves like a cavity. Moreover, the fact that the frequencies are real shows that the system is stable against scalarfield perturbations at linear level.
The central property of AdS to obtain this instability is its global structure, rather than its local geometry. This can be established by noting that a qualitatively similar behaviour is obtained by considering precisely the same dynamical system in Minkowski space enclosed in a cavity [537], see Figure 9 (right panel). Moreover, the mechanism behind the instability seems to rely on nonlinear interactions of the field that tend to shift its energy to higher frequencies and hence smaller wavelenghts. This process stops in GR since the theory has a natural cutoff: BH formation.
It has since been pointed out that collapse to BHs may not be the generic outcome of evolutions in AdS [145, 66, 274, 538, 539]. For example, in Ref. [145] “islands of stability” were discovered for which the initial data, chosen as a small perturbation of a boson star, remain in a nonlinearly stable configuration. In Ref. [586], the authors raised the possibility that some of the features of the AdS instability could also show up in asymptotically flat spacetimes in the presence of some confinement mechanism. They observed that the evolution of minimally coupled, massive scalar wavepackets in asymptotically flat spacetimes can also lead to collapse after a very large number of “bounces” off the massive effective potential barrier in a manner akin to that discovered in AdS. Similarly, in some region of the parameter space the evolution drives the system towards nonlinearly stable, asymptotically flat “oscillatons” [688, 326]. Nevertheless, for sufficiently small initial amplitudes they observe a t^{−3/2} decay of the initial data, characteristic of massive fields, and showing that Minkowski is nonlinearly stable. The “weakly turbulent” instability discovered in AdS is stimulating research on the topic of turbulence in GR. A full understanding of the mechanism(s) will require further studies, including collapse in nonspherically symmetric backgrounds, other forms of matter and boundary conditions, etc.
We now turn to solutions that display an instability at linear level, seen by a mode analysis, and to the use of NR techniques to follow the development of such instabilities into the nonlinear regime. One outstanding example is the GregoryLaflamme instability of black strings already described in Section 7.2. Such black strings exist in higher dimensions, D ≥ 5. It is expected that the same instability mechanism afflicts other higherdimensional BHs, even with a topologically spherical horizon. A notable example are MyersPerry BHs. In D ≥ 6, the subset of these solutions with a single angular momentum parameter have no analogue to the Kerr bound, i.e., a maximum angular momentum for a given mass. As the angular momentum increases, they become ultraspinning BHs and their horizon becomes increasingly flattened and hence resembling the horizon of a black pbrane, which is subject to the GregoryLaflamme instability [305]. It was indeed shown in Ref. [272, 271, 270], by using linear perturbation theory, that rapidly rotating MyersPerry BHs for 7 ≤ D ≤ 9 are unstable against axisymmetric perturbations. The nonlinear growth of this instability is unknown but an educated guess is that it may lead to a deformation of the pancake like horizon towards multiple concentric rings.
A different argument — of entropic nature — for the instability of ultraspinning BHs against nonaxisymmetric perturbations was given by Emparan and Myers [305]. Such type of instability has been tested in 6 ≤ D ≤ 8 [700], but also in D = 5 [701, 700] — for which a slightly different argument for instability was given in [305] — by evolving a MyersPerry BH with a nonaxisymmetric barmode deformation, using a NR code adapted to higher dimensions. In each case sufficiently rapidly rotating BHs are found to be unstable against the barmode deformation. In terms of a dimensionless spin parameter q ≡ a/μ^{1/(D−3)}, where μ, a are the standard mass and angular momentum parameters of the MyersPerry solution, the onset values for the instability were found to be: D = 5, q = 0.87; D = 6, q = 0.74; D = 7, q = 0.73; D = 8, q = 0.77. We remark that the corresponding values found in [272] for the GregoryLaflamme instability in 7 ≤ D ≤ 9 are always larger than unity. Thus, the instability triggered by nonaxisymmetric perturbations sets in for lower angular momenta than the axisymmetric GregoryLaflamme instability. Moreover, in [700], longterm numerical evolutions have been performed to follow the nonlinear development of the instability. The central conclusion is that the unstable BHs relax to stable configurations by radiating away the excess angular momentum. These results have been confirmed for D = 6, 7 by a linear analysis in Ref. [273]; such linear analysis suggests that for D = 5, however, the single spinning MyersPerry BH is linearly stable.^{20}
Another spacetime instability seen at linear level is the superradiant instability of rotating or charged BHs in the presence of massive fields or certain boundary conditions. This instability will be discussed in detail in the next section.
Concerning the nonlinear stability of BH solutions, the only generic statement one can produce at the moment is that hundreds of NR evolutions of binary Kerr or Schwarzschild BHs in vacuum, over the last decade, lend empirical support to the nonlinear stability of these solutions. One must remark, however, on the limitations of testing instabilities with NR simulations. For instance, fully nonlinear dynamical simulations cannot probe — at least at present — extremal Kerr BHs; they are also unable to find instabilities associated with very high harmonic indices ℓ, m (associated with very small scales), as well as instabilities that may grow very slowly. Concerning the first caveat, it was actually recently found by Aretakis that extremal RN and Kerr BHs are linearly unstable against scalar perturbations [42, 41, 43], an observation subsequently generalised to more general linear fields [530] and to a nonlinear analysis [563, 44]. This growth of generic initial data on extremal horizons seems to be a very specific property of extremal BHs, in particular related to the absence of a redshift effect [563], and there is no evidence a similar instability occurs for nonextremal solutions.
To conclude this section let us briefly address the stability of BH interiors already discussed in Section 3.2.2. The picture suggested by Israel and Poisson [621, 622] of mass inflation has been generically confirmed in a variety of toy models — i.e., not Kerr — by numerical evolutions [151, 152, 153, 393, 56, 448, 57] and also analytical arguments [233]. Other numerical/analytical studies also suggest the same holds for the realistic Kerr case [388, 386, 387, 531]. As such, the current picture is that mass inflation will drive the curvature to Planckian values, near or at the Cauchy horizon. The precise nature of the consequent singularity, that is, if it is spacelike or lightlike, is however still under debate (see, e.g., [235]).
7.5 Superradiance and fundamental massive fields
There are several reasons to consider extensions of GR with minimally, or nonminimally coupled massive scalar fields with mass parameter μ_{ s }. As mentioned in Section 3.1.4, ultralight degrees of freedom appear in the axiverse scenario [49, 50] and they play an important role in cosmological models and also in dark matter models. Equally important is the fact that massive scalar fields are a very simple proxy for more complex, realistic matter fields, the understanding of which in full NR might take many years to achieve.
At linearized level, the behavior of fundamental fields in the vicinities of nonrotating BHs has been studied for decades, and the main features can be summarized as follows:
 (i)
A prompt response at early times, whose features depend on the initial conditions. This is the counterpart to lightcone propagation in flat space.
 (ii)
An exponentially decaying “ringdown” phase at intermediate times, where the BH is ringing in its characteristic QNMs. Bosonic fields of mass μ_{ s }ħ introduce both an extra scale in the problem and a potential barrier at distances ∼ 1/μ_{ s }, thus effectively trapping fluctuations. In this case, extra modes appear which are quasibound states, i.e., extremely longlived states effectively turning the BH into a quasihairy BH [76, 794, 280, 656, 604, 135].
 (iii)
At late times, the signal is dominated by a powerlaw falloff, known as “latetime tail” [633, 506, 209, 491]. Tails are caused by backscattering off spacetime curvature (and a potential barrier induced by massive terms) and more generically by a failure of Huygens’ principle. In other words, radiation in curved spacetimes travels not only on, but inside the entire light cone.
Selfinteracting scalars can give rise to stable or very longlived configurations. For example, selfinteracting complex scalar fields can form boson stars for which the scalar field has an oscillatory nature, but the metric is stationary [458, 685, 516, 533]. Realvalued scalars can form oscillating solitons or “oscillatons”, longlived configurations where both the scalar field and the metric are timedependent [688, 689, 594, 586]. Dynamical boson star configurations were studied by several authors [516], with focus on boson star collisions with different velocities and impact parameters [597, 599, 216]. These are important for tests of the hoop and cosmic censorship conjectures, and were reviewed briefly in Sections 7.2 and 7.3. For a thorough discussion and overview of results on dynamical boson stars we refer the reader to the Living Reviews article by Liebling and Palenzuela [516].
Full nonlinear results from Ref. [588] are reproduced in Figure 12, and agree with linearized predictions. Higher multipoles “feel” the centrifugal barrier close to the light ring which, together with the mass barrier at large distances, provides a confining mechanism and gives rise to almost stationary configurations, shown in the right panel of Figure 12. The beating patterns are a consequence of the excitation of different overtones with similar ringing frequency [794, 588]. These “scalar condensates” are extremely longlived and can, under some circumstances, be considered as adding hair to the BH. They are not however really stationary: the changing quadrupole moment of the “scalar cloud” triggers the simultaneous release of gravitational radiation [588, 816, 585]. In fact, gravitational radiation is one of the most important effects not captured by linearized calculations. These nontrivial results extend to higher multipoles, which display an even more complex behavior [588, 585].

Superradiant instability and its saturation. The timescales probed in current nonlinear simulations are still not sufficient to unequivocally observe superradiance with test scalar fields. The main reason for this is the feebleness of such instabilities: for scalar fields they have, at best, an instability timescale of order 10^{7} M for carefully tuned scalar field mass. However, current longterm simulations are able to extract GWs induced by the scalar cloud [588].
The biggest challenge ahead is to perform simulations which are accurate enough and last long enough to observe the scalarinstability growth and its subsequent saturation by GW emission. This will allow GW templates for this mechanism to finally be released.
Due to their simplicity, scalar fields are a natural candidate to carry on this program, but they are not the only one. Massive vector fields, which are known to have amplification factors one order of magnitude larger, give rise to stronger superradiant instabilities, and might also be a good candidate to finally observe superradiant instabilities at the nonlinear level. We note that the development of the superradiant instability may, in some special cases, lead to a truly asymptotically flat, hairy BH solution of the type recently discussed in [422].

Turbulence of massive fields in strong gravity. Linearized results indicate that the development of superradiant instabilities leaves behind a scalar cloud with scalar particles of frequency ω ∼ mΩ, in a nearly stationary state. This system may therefore be prone to turbulent effects, where nonlinear terms may play an important role. One intriguing aspect of these setups is the possibility of having gravitational turbulence or collapse on sufficiently large timescales. Such effects were recently observed in “closed” systems where scalar fields are forced to interact gravitationally for long times [108, 537, 145]. It is plausible that quasibound states are also prone to such effects, but in asymptotically flat spacetimes.

Floating orbits. Our discussion until now has focused on minimally coupled fundamental fields. If couplings to matter exist, new effects are possible: a small object (for example, a star), orbiting a rotating, supermassive BH might be able to extract energy and angular momentum from the BH and convert it to gravitational radiation. For this to happen, the object would effectively stall at a superradiant orbit, with a Newtonian frequency Ω = 2μ_{ s } for the dominant quadrupolar emission. These are called floating orbits, and were verified at linearized level [165, 164].^{21} Nonlinear evolutions of systems on floating orbits are extremely challenging on account of all the different extreme scales involved.

Superradiant instabilities in AdS. The mechanism behind superradiant instabilities relies on amplification close to the horizon and reflection by a barrier at large distances. Asymptotically AdS spacetimes provide an infiniteheight barrier, ideal for the instability to develop [171, 166, 172, 755, 169].
Because in these backgrounds there is no dissipation at infinity, it is both possible and likely that new, nonsymmetric final states arise as a consequence of the superradiant instability [172, 168, 514, 275]. Following the instability growth and its final state remains a challenge for NR in asymptotically AdS spacetimes.
 Superradiant instabilities of charged BHs. Superradiant amplification of charged bosonic fields can occur in the background of charged BHs, in quite a similar fashion to the rotating case above, as long as the frequency of the impinging wave ω obeyswhere q is the charge of the field and Φ_{ H } is the electric potential on the BH horizon. In this case both charge and Coulomb energy are extracted from the BH in a way compatible with the first and second law of BH thermodynamics [83]. In order to have a recurrent scattering, and hence, an instability, it is not enough, however, to add a mass term to the field [339, 432, 430, 261, 671]; but an instability occurs either by imposing a mirror like boundary condition at some distance from the BH (i.e., a boxed BH) or by considering an asymptotically AdS spacetime. In Refs. [421, 262] it has been established, through both a frequency and a time domain analysis, that the time scales for the development of the instability for boxed BHs can be made much smaller than for rotating BHs (in fact, arbitrarily small [431]). Together with the fact that even swaves, i.e., ℓ = 0 modes, can trigger the instability in charged BHs, makes the numerical study of the nonlinear development of this type of superradiant instabilities particularly promising. One should be aware, however, that there may be qualitative differences in both the development and endpoint of superradiant instabilities in different setups. For instance, for AdS and boxed BHs, the endpoint is likely a hairy BH, such as those constructed in Ref. [275] (for rotating BHs), since the scalar field cannot be dissipated anywhere. This applies to both charged and rotating backgrounds. By contrast, this does not apply to asymptotically flat spacetimes, wherein rotating (but not charged) superradiance instability may occur. It is then an open question if the system approaches a hairy BH — of the type constructed in [422] — or if the field is completely radiated/absorbed by the BH. Concerning the development of the instability, an important difference between the charged and the rotating cases may arise from the fact that a similar role, in Eqs. (173) and (174), is played by the field’s azimuthal quantum number m and the field charge q; but whereas the former may change in a nonlinear evolution, the latter is conserved [169].$$\omega < q{\Phi _H},$$(174)
7.6 Highenergy collisions

Does cosmic censorship still apply under the extreme conditions of collisions near the speed of light? As has already been discussed in Section 7.2, numerical simulations of these collisions in four dimensions have so far identified horizon formation in agreement with the censorship conjecture. The results of higherdimensional simulations are still not fully understood, cf. Section 7.2.

Do NR simulations of highenergy particle collisions provide evidence supporting the validity of the hoop conjecture? As discussed in Section 7.3, NR results have so far confirmed the hoop conjecture.

In collisions near the speed of light, the energy mostly consists of the kinetic energy of the colliding particles such that their internal structure should be negligible for the collision dynamics. Furthermore, the gravitational field of a particle moving at the speed of light is nonvanishing only near the particle’s worldline [16], suggesting that the gravitational interaction in highenergy collisions should be dominant at the instant of collision and engulfed inside the horizon that forms. This conjecture has sometimes been summarized by the statement that “matter does not matter” [216], and is related to the hoop conjecture discussed above. Do NR simulations of generic highenergy collisions of compact objects support this argument in the classical regime, i.e., does the modelling of the colliding objects as point particles (and, in particular, as BHs) provide an accurate description of the dynamics?

Assuming that the previous question is answered in the affirmative, what is the scattering threshold for BH formation? This corresponds to determining the threshold impact parameter b_{scat} that separates collisions resulting in the formation of a single BH (b < b_{scat}) from scattering encounters (b > b_{scat}), as a function of the number of spacetime dimensions D and the collision velocity v in the centerofmass frame or boost parameter \(\gamma = 1/\sqrt {1  {v^2}}\).

How much energy and momentum is lost in the form of GWs during the collision? By conversion of energy and momentum, the GW emission determines the mass and spin of the BH (if formed) as a function of the spacetime dimension D, scattering parameter b, and boost factor γ of the collision. Collisions near the speed of light are also intriguing events to probe the extremes of GR; in particular what is the maximum radiation that can be extracted from any collision and does the luminosity approach Dyson’s limit dE/ dt ≲ 1 [157]? (See discussion in Section 3.2.3 about this limit.)
The relevance of the internal structure of the colliding bodies has been studied in Ref. [716], comparing the GW emission and scattering threshold in highenergy collisions of rotating and nonrotating BHs in D = 4. The BH spins of the rotating configurations are either aligned or antialigned with the orbital angular momentum corresponding to the socalled hangup and antihangup cases which were found to have particularly strong effects on the dynamics in quasicircular BH binary inspirals [160]. In highenergy collisions, however, this (anti)hangup effect disappears; the GW emission as well as the scattering threshold are essentially independent of the BH spin at large collision velocities (cf. Figure 15 which will be discussed in more detail further below). These findings suggest that ultrarelativistic collisions are indeed well modelled by colliding pointparticles or BHs in GR. In the centerofmass frame, and assuming that the two particles have equal mass, the collisions are characterized by three parameters. (i) The number D of spacetime dimensions, (ii) the Lorentz factor γ or, equivalently, the collision velocity υ, and (iii) the impact parameter b = L/P, where L and P are the initial orbital angular momentum and the linear momentum of either BH in the centerofmass frame.
Collisions of BHs with electric charge have been simulated by Zilhão et al. [838, 839]. For the special case of BHs with equal chargetomass ratio Q/M and initially at rest, constraintsatisfying initial data are available in closed analytic form. The electromagnetic wave signal generated in these headon collisions reveals three regimes similar to the pattern known for the GW signal, (i) an infall phase prior to formation of a common horizon, (ii) the nonlinear merger phase where the wave emission reaches its maximum and (iii) the quasinormal ringdown. As the chargetomass ratio is increased towards Q/M ≲ 1, the emitted GW energy decreases by about 3 orders of magnitude while the electromagnetic wave energy reaches a maximum at Q/M ≈ 0.6, and drops towards 0 in both the uncharged and the extreme limit. This behaviour of the radiated energies is expected because of the decelerating effect of the repulsive electric force between equally charged BHs. For opposite electric charges, on the other hand, the larger collision velocity results in an increased amount of GWs and electromagnetic radiation [839].
An extended study of BH collisions using various analytic approximation techniques including geodesic calculations and the ZFL has been presented in Berti et al. [93]; see also [94] for a first exploration in higher dimensions. Weak scattering of BHs in D = 4, which means large scattering parameters b/M ∼ 10, and for velocities υ ≈ 0.2, has been studied by Damour et al. [245] using NR as well as PN and EOB calculations. Whereas PN calculations start deviating significantly from the NR results for b/M ≲ 10, the NR calibrated EOB model yields good agreement in the scattering angle throughout the weak scattering regime.
BH collisions in D ≥ 5 spacetime dimensions are not as well understood as their fourdimensional counterparts. This is largely a consequence of the fact that NR in higher dimensions is not yet that robust and suffers more strongly from numerical instabilities. Such complications in the higherdimensional numerics do not appear to cause similar problems in the construction of constraint satisfying initial data. The spectral elliptic solver originally developed by Ansorg et al. [39] for D = 4 has been successfully generalized to higher D in Ref. [836] and provides solutions with comparable accuracy as in D = 4.
The main challenges for future numerical work in the field of highenergy collisions are rather evident. For applications in the analysis of experimental data in the context of TeV gravity scenarios (cf. Section 3.3.2), it will be vital to generalize the results obtained in four dimensions to D ≥ 5. Furthermore, it is currently not known whether the impact of electric charge on the collision dynamics becomes negligible at high velocities, as suggested by the matterdoesnotmatter conjecture, and as is the case for the BH spin.
7.7 Alternative theories
As discussed in Section 6.1.7, one of the most straightforward extensions of Einstein’s theory is obtained by the addition of minimally coupled scalar fields. When the scalar couples to the Ricci scalar however, one gets a modification of Einstein gravity, called scalartensor theory. In vacuum, scalartensor theories are described by the generic action in Eq. (5), where R is the Ricci scalar associated to the metric g_{ μν }, and F(ϕ), Z(ϕ) and U(ϕ) are arbitrary functions (see e.g., [92, 824] and references therein). The matter fields minimally coupled to g_{ μν } are collectively denoted by Ψ_{ m }. This form of the action corresponds to the choice of the socalled “Jordan frame”, where the matter fields Ψ_{ m } obey the equivalence principle. For F = ϕ, Z = ω_{BD}/ϕ, U = 0, the action (5) reduces to the standard BransDicke theory.
The equations of motion derived from the action (5) are secondorder and the theory admits a wellposed initialvalue problem [670]. These facts turn scalartheories into an attractive alternative to Einstein’s equations, embodying at least some of the physics one expects from an ultimate theory of gravity, and have been a major driving force behind the efforts to understand scalartensor theories from a NR point of view [696, 679, 680, 582, 410, 92, 73, 670]. In fact, scalartensor theories remain the only alternative theory to date where full nonlinear dynamical evolutions of BH spacetimes have been performed.
In summary, the study of scalartensor theories of gravity can be directly translated, in vacuum, to the study of minimally coupled scalar fields. For a trivial potential V = const, the equations of motion in the Einstein frame (180) — (181) admit GR (with _{ ϕ } = const) as a solution. Because stationary BH spacetimes in GR are stable, i.e., any scalar fluctuations die away rather quickly, the dynamical evolution of vacuum BHs is expected to be the same as in GR. This conclusion relies on handwaving stability arguments, but was verified to be true to first PN order by Will and Zaglauer [787], at 2.5 PN order by Mirshekari and Will [549] and to all orders in the point particle limit in Ref. [824].

Nontrivial potential V and initial conditions. Healy et al. studied an equalmass BH binary in an inflationinspired potential \(V = \lambda {({\varphi ^2}  \varphi _0^2)^2}/8\) with nontrivial initial conditions on the scalar given by φ = φ_{0} tanh (r − r_{0})/σ [410]. This setup is expected to cause deviations in the dynamics of the inspiralling binary, because the binary is now accreting scalar field energy. The larger the initial amplitude of the field, the larger those deviations are expected to be. This is summarized in Figure 18, where the BH positions are shown as a function of time for varying initial scalar amplitude.

Nontrivial boundary conditions. As discussed, GR is recovered for constant scalar fields. For nontrivial timedependent boundary conditions or background scalar fields, however, nontrivial results show up. These boundary conditions could mimic cosmological scenarios or dark matter profiles in galaxies [434, 92]. Reference [92] modelled a BH binary evolving nonlinearly in a constantgradient scalar field. The scalarfield gradient induces scalar charge on the BHs, and the accelerated motion of each BH in the binary generates scalar radiation at large distances, as summarized in Figure 19.
The scalarsignal at large distances, shown in the right panel of Figure 19, mimics the inspiral, merger and ringdown stages in the GW signal of an inspiralling BH binary.

Matter. When matter is present, new effects (due to the coupling of matter to the effective metric \({F^{ 1}}g_{\mu \nu}^E\) can dominate the dynamics and wave emission. For example, it has been shown that, for \(\beta \equiv \partial _\varphi ^2(\ln F(\varphi))\underset{\sim}{<}  4\), NSs can “spontaneously scalarize,” i.e., for sufficiently large compactnesses the GR solution is unstable. The stable branch has a nonzero expectation value for the scalar field [242].
Scalarized matter offers a rich new phenomenology. For example, the dynamics and GW emission of scalarized NSs can be appreciably different (for given coupling function F^{−1}(ϕ)) from the corresponding GR quantities, as shown by Palenzuela et al. [73, 596] and summarized in Figure 20. Strongfield gravity can even induce dynamical scalarization of otherwise GR stars during inspiral, offering new ways to constrain such theories [73, 596].

Understanding the well posedness of some theory, in particular those having some motivation from fundamental physics, as for example EinsteinDilatonGaussBonnet and Dynamical ChernSimons gravity [602, 27]. A study on the well posedness of the latter has recently been presented in Ref. [263].

Building initial data describing interesting setups for such theories. Unless the theory admits particularly simple analytic solutions, it is likely that initial data construction will also have to be done numerically. Apart from noteworthy exceptions, such as GaussBonnet gravity in higher dimensions [814], initial data have hardly been considered in the literature.
Once wellposedness is established and initial data are constructed, NR evolutions will help us understanding how these theories behave in the nonlinear regime.
7.8 Holography
Holography provides a fascinating new source of problems for NR. As such, in recent years, a number of numerical frameworks have been explored in asymptotically AdS spacetimes, as to face the various pressing questions raised within the holographic correspondence, cf. Section 3.3.1. At the moment of writing, no general purpose code has been reported, comparable to existing codes in asymptotically flat spacetime, which can evolve, say, BH binaries with essentially arbitrary masses, spins and momenta. Progress has occured in specific directions to address specific issues. We shall now review some of these developments emphasizing the gravity side of the problems.
Timeplusspace decompositions have also been initiated, both based on a generalized harmonic evolution scheme [70] and in an ADM formulation [416]. In particular the latter formulation seems very suited for extracting relevant physical quantities for holography, such as the boundary time for the thermalization process discussed in Section 3.3.1.
Evolutions of BHs deformed by a scalar field in AdS_{5} have been presented in Ref. [70]. The evolution leads the system to oscillate in a (expected) superposition of quasinormal modes, some of which are nonlinearly driven. On the boundary, the dual CFT stress tensor behaves like that of a thermalized \({\mathcal N = 4}\) superYangMills fluid, with an equation of state consistent with conformal invariance and transport coefficients that match holographic calculations at all times. Similar conclusions were reached in Ref. [417], where the numerical scheme of Ref. [207, 208] was used to study the isotropization of a homogeneous, strongly coupled, nonAbelian plasma by means of its gravity dual, comparing the time evolution of a large number of initially anisotropic states. They find that the linear approximation seems to work well even for initial states with large anisotropies. This unreasonable effectiveness of linearized predictions hints at something more fundamental at work, perhaps a washing out of nonlinearities close to the horizon. Such effects were observed before in asymptotically flat spacetimes, for example the already mentioned agreement between ZFL (see Section 5.3) or close limit approximation predictions (see Section 5.2) and full nonlinear results.
Also of interest for accelerator physics, and the subject of intense work in recent years, are holographic descriptions of jetquenching, i.e., the loss of energy of partons as they cross strongly coupled plasmas produced in heavy ion collisions [2, 199, 200]. Numerical work using schemes similar to that of Refs. [207, 208] have been used to evolve dual geometries describing the quenches [143, 144, 204]; see also [319] for numerical stationary solutions in this context.
Another development within the gauge/gravity duality that gained much attention, also discussed in Section 3.3.1, is related to condensed matter physics. In asymptotically AdS spacetimes, a simple theory, say, with a scalar field minimally coupled to the Maxwell field and to gravity admits RNAdS as a solution. Below a critical temperature, however, this solution is unstable against perturbations of the scalar field, which develops a tachyonic mode. Since the theory admits another set of charged BH solutions, which have scalar hair, it was suggested that the development of the instability of the RNAdS BHs leads the system to a hairy solution. From the dual field theory viewpoint, this corresponds to a phase transition between a normal and a superconducting phase. A numerical simulation showing that indeed the spacetime evolution of the unstable RNAdS BHs leads to a hairy BH was reported in [562]. Therein, the authors performed a numerical evolution of a planar RNAdS BH perturbed by the scalar field and using EddingtonFinkelstein coordinates. A particular numerical scheme was developed, adapted to this problem. The development of the scalar field density is shown in Figure 21. The initial exponential growth of the scalar field is eventually replaced by an approach to a fixed value, corresponding to the value of the scalar condensate on the hairy BH.
Finally, the gauge/gravity duality itself may provide insight into turbulence. Turbulent flows of CFTs are dual to dynamical BH solutions in asymptotically AdS spacetimes. Thus, urgent questions begging for answers include how and when do turbulent BHs arise, and what is the (gravitational) origin of Kolmogorov scaling observed in turbulent fluid flows. These problems are just now starting to be addressed [185, 12, 13].
7.9 Applications in cosmological settings
Some initial applications of NR methods addressing specific issues in cosmology have been reviewed in the Living Reviews article by Anninos [36], ranging from the Big Bang singularity dynamics to the interactions of GWs and the largescale structure of the universe. The first of these problems — the understanding of cosmological singularities — actually motivated the earlier applications of NR to cosmological settings, cf. the Living Reviews article [88]. The set of homogeneous but anisotropic universes was classified by Bianchi in 1898 into nine different types (corresponding to different independent groups of isometries for the 3dimensional space). Belinskii, Khalatnikov and Lifshitz (BKL) proposed that the singularity of a generic inhomogeneous cosmology is a “chaotic” spacelike curvature singularity, and that it would behave asymptotically like a Bianchi IX or VIII homogeneous cosmological model. This is called BKL dynamics or mixmaster universe. The accuracy of the BKL dynamics has been investigated using numerical evolutions in Refs. [87, 558, 662], and the BKL sensitivity to initial conditions in various references (see for instance Ref. [89]). For further details, we refer the reader to Ref. [88].
More recently, NR methods have been applied to the study of bouncing cosmologies, by studying the evolution of adiabatic perturbations in a nonsingular bounce [801]. The results of Ref. [801] show that the bounce is disrupted in regions of the universe with significant inhomogeneity and anisotropy over the background energy density, but is achieved in regions that are relatively homogeneous and isotropic. Sufficiently small perturbations, consistent with observational constraints, can pass through the nonsingular bounce with negligible alteration from nonlinearity.
In parallel, studies of “bubble universes”, in which our universe is one of many nucleating and growing inside an everexpanding false vacuum, have also been made with NR tools. In particular, Refs. [765, 764] investigated the collisions between bubbles, by computing the cosmological observables arising from bubble collisions directly from the Lagrangian of a single scalar field.
Applications of NR in more standard cosmological settings are still in their infancy, but remarkable progress has been achieved. One of these concerns the impact of cosmic inhomogeneities on the value of the cosmological constant and the acceleration of the universe. In other words, how good are models of homogeneous and isotropic universes — the paradigmatic FriedmannLemaîtreRobertsonWalker (FLRW) geometry — when we know that our universe has structure and is inhomogeneous?
The effects of local inhomogeneities have been investigated in Ref. [804] using different initial data, describing an expanding inhomogeneous universe model composed of regularly aligned BHs of identical mass. The evolution of these initial data also indicates that local inhomogeneities do not significantly affect the global expansion law of the universe, despite the fact that the inhomogeneities themselves are extremely nonlinear [804, 805]. Similar conclusions were reached in Ref. [834], where the ADM formalism is used to develop a practical scheme to calculate a proposed domain averaging effect in an inhomogeneous cosmology within the context of numerical largescale structure simulations. This study finds that in the weakfield, slowmotion limit, the proposed effect implies a small correction to the global expansion rate of the universe. In this limit, their simulations are always dominated by the expanding underdense regions, hence the correction to the energy density is negative and the effective pressure is positive. The effects of strong gravity in more general scenarios are yet to be understood [834]. For an earlier NR code developed to address inhomogeneous cosmologies see Ref. [426].
More complex NR codes aimed at understanding cosmological evolutions are currently being developed. NR simulations of large scale dynamical processes in the early universe have recently been reported [345]. These take into account interactions of dark matter, scalar perturbations, GWs, magnetic fields and turbulent plasma. Finally, Ref. [837] considers the effect of (extreme) cosmological expansion on the headon collision and merger of two BHs, by modelling the collision of BHs in asymptotically dS spacetimes.
8 Conclusions
“Somewhere, something incredible is waiting to be known.” ^{22}
Einstein’s theory of general relativity celebrates its 100th anniversary in 2015 as perhaps the most elegant and successful attempt by humankind to capture the laws of physics.
Until recently, this theory has been studied mostly in the weakfield regime, where it passed all experimental and observational tests with flying colors. Studies in the strongfield regime, in contrast, largely concerned the mathematical structure of the theory but made few and indirect connections with observation and experiment. Then, a few years ago, a phase transition in the field of strong gravity occurred: on one hand, new experimental efforts are promising to test gravity for the first time in the strong field regime; on the other hand, a new tool — numerical relativity — has made key breakthroughs opening up the regime of strongfield gravity phenomena for accurate modelling. Driven by these advances, gravitation in the strongfield regime has proven to have remarkable connections to other branches of physics.
With the rise of numerical relativity as a major tool to model and study physical processes involving strong gravity, decadeold problems — brushed aside for their complexity — are now tackled with the use of personal or highperformance computers. Together with analytic methods, old and new, the new numerical tools are pushing forward one of the greatest human endeavours: understanding the universe.
Footnotes
 1.
 2.
That is, the existence of mass currents in opposite spatial directions (in between the horizons) and at relativistic velocities in the centre of energy frame.
 3.
This bound has an interesting story. Kip Thorne, and others after him, attribute the conjecture to Freeman Dyson; Freeman Dyson denies he ever made such a conjecture, and instead attributes such notion to his 1962 paper [157], where he works out the power emitted by a binary of compact objects. (We thank Gary Gibbons and Christoph Schiller for correspondence on this matter.)
 4.
Such fragmentation may however not be a counterexample to the spirit of the cosmic censorhip conjecture, if black strings do not form in generic collapse situations. One hint that this may indeed be the case comes from the DysonChandrasekharFermi instability of higherdimensional cylindrical matter configurations [174]: if cylindrical matter configurations are themselves unstable it is unlikely that their collapse leads to black strings.
 5.
In the context of quantum gravity, it has ben shown that including a fundamental minimal length, a solution exists in which an interior regular solution is matched to the exterior Kerr metric. Such configuration, however, is a “regularized” BH rather than a description of stars [706].
 6.
Here, \([{{D  1} \over 2}]\) denotes the integer part of \({{D  1} \over 2}\).
 7.
Numerical solutions of axially symmetric, rotating NSs in GR have been derived by several groups (see [330] and the Living Reviews article [728], and references therein), and in some cases their codes have been made publically available [729, 117]. These solutions are used to build initial data for NR simulations of NSNS and BHNS binary inspiral and merger.
 8.
If matter or energy is present, there is a stressenergy tensor which is also perturbed, \({T_{\mu \nu}} = T_{\mu \nu}^{(0)} + \delta {T_{\mu \nu}}\). If \({T_{\mu \nu}}\) describes a fluid, its perturbation can be described in terms of the perturbations of the thermodynamic quantities
 9.
Strictly speaking, the signature represents the signs of the eigenvalues of the metric: g_{ αβ } has 1 negative and D − 1 positive eigenvalues even when the timelike coordinate is replaced in terms of one or two null coordinates.
 10.
Decompositions in terms of null foliations have to our knowledge not been studied yet, although there is no evident reason that speaks against such an approach.
 11.
… but beware! For many realistic types of matter, novel effects — such as shocks — can hamper an efficient evolution. These have to be handled with care and would require a review of its own.
 12.
We remark that the dimensional reduction here discussed is different from KaluzaKlein dimensional reduction [463, 283], an idea first proposed about one century ago, which in recent decades has attracted a lot of interest in the context of SMT. Indeed, a crucial feature of KaluzaKlein dimensional reduction is spacetime compactification, which does not occur in our case.
 13.
There is a lengthsquared factor multiplying the exponential which we set to unity.
 14.
Note that Ref. [700] chooses z instead of y.
 15.
 16.
“Petrov type D” is a class of algebraically special spacetimes, which includes in particular the Schwarzschild and Kerr solutions.
 17.
 18.
 19.
Cosmic censorship does not apply to cosmological singularities, i.e., Big Bang or Big Crunch.
 20.
 21.
Floating orbits would manifest themselves in observations by depleting the inner part of accretion disks of stars and matter, and modifying the emitted gravitational waveform.
 22.
This quote, usually attributed to Carl Sagan, was published in Seeking Other Worlds in Newsweek magazine (April 15, 1977), a tribute to Carl Sagan by D. Gelman, S. Begley, D. Gram and E. Clark.
Notes
Acknowledgements
This work benefited greatly from discussions over the years with many colleagues. In particular, we thank E. Berti, R. Brito, J. C. Degollado, R. Emparan, P. Figueras, D. Hilditch, P. Laguna, L. Lehner, D. Mateos, A. Nerozzi, H. Okawa, P. Pani, F. Pretorius, E. Radu, H. Reall, H. Rúnarsson, M. Sampaio, J. Santos, M. Shibata, C. F. Sopuerta, H. Witek and M. Zilhão for very useful comments and suggestions. This work was partially funded by the NRHEP 295189 FP7PEOPLE2011IRSES and PTDC/FIS/116625/2010 grants and the CIDMA strategic project UID/MAT/04106/2013. V.C. acknowledges partial financial support provided under the European Union’s FP7 ERC Starting Grant “The dynamics of black holes: testing the limits of Einstein’s theory” grant agreement no. DyBHo–256667. L.G. acknowledges partial financial support from NewCompStar (COST Action MP1304). C.H. is funded by the FCTIF programme. U.S. acknowledges support from FP7PEOPLE2011CIG Grant No. 293412 CBHEO, STFC GR Roller Grant No. ST/L000636/1, NSF XSEDE Grant No. PHY090003 and support by the Cosmos system, part of DiRAC, funded by STFC and BIS under Grant Nos. ST/K00333X/1 and ST/J005673/1. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development.
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