It has been known since the early s 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. 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.

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 [ ]. The first analysis of GW momentum flux generated by binary systems was performed by Fitchett [ ] in for two masses in Keplerian orbit.

The following two decades saw various semi- analytic calculations for inspiraling compact binary systems using the particle approximation, post-Newtonian techniques and the close-limit approach see Section 5 for a description of these techniques and main results. 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 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. Although GR is widely accepted as the standard theory of gravity and has survived all experimental and observational weak field scrutiny, there is convincing evidence that it is not the ultimate theory of gravity: since GR is incompatible with quantum field theory, it should be considered as the low energy limit of some, still elusive, more fundamental theory.

In addition, GR itself breaks down at small length scales, since it predicts singularities. For large scales, on the other hand, cosmological observations show that our universe is filled with dark matter and dark energy, of as yet unknown nature. This suggests that the strong-field regime of gravity — which has barely been tested so far — could be described by some modification or extension of GR.

In the next few years both GW detectors [ , ] and astrophysical observations [ ] will provide an unprecedented opportunity to probe the strong-field regime of the gravitational interaction, characterized by large values of the gravitational field or of the spacetime curvature it is a matter of debate which of the two parameters is the most appropriate for characterizing the strong-field gravity regime [ , ]. However, our present theoretical knowledge of strong-field astrophysical processes is based, in most cases, on the a-priori assumption that GR is the correct theory of gravity.

This sort of theoretical bias [ ] 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. 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 [ , ] and, to some extent, Loop Quantum Gravity [ ]. In addition, such theories should allow a well-posed initial-value formulation of the field equations. Various arguments suggest that the modifications to GR could involve [ ] i additional degrees of freedom scalar fields, vector fields ; ii corrections to the action at higher order in the spacetime curvature; iii additional dimensions.

Scalar-tensor theories for example see, e. In these theories, which include for instance Brans-Dicke gravity [ ], the metric tensor is non-minimally coupled with one or more scalar fields. In the case of a single scalar field which can be generalized to multi-scalar-tensor theories [ ] , the action can be written as. A more general formulation of scalar-tensor theories yielding second order equations of motion has been proposed by Horndeski [ ] see also Ref.

Scalar-tensor theories can be obtained as low-energy limits of SMT [ ]; this provides motivation for studying these theories on the grounds of fundamental physics. Scalar-tensor theories are also appealing alternatives to GR because they predict new phenomena, which are not allowed in GR. A detection of one of these phenomena would be a smoking gun of scalar-tensor gravity. These theories, whose well-posedness has been proved [ , ], are a perfect arena for NR. Recovering some of the above smoking-gun effects is extremely challenging, as the required timescales are typically very large when compared to any other timescales in the problem.

Spacetime singularities signal the breakdown of the geometric description of the spacetime, and can be diagnosed by either the blow-up of observer-invariant quantities or by the impossibility to continue timelike or null geodesics past the singular point. 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. 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.

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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 [ , , , ]. Does this always occur, i. Loosely speaking, the conjecture states that physically reasonable matter under generic initial conditions only forms singularities hidden behind horizons [ ].

### History Topics

The cosmic-censorship conjecture and the possible existence of naked singularities in our universe has triggered interest in complex problems which can only be addressed by NR. As discussed in Section 3. Thus, if no other presently unkown physical effect can prevent it, according to GR, a BH forms. From the uniqueness theorems cf. Section 4. Outside the event horizon, the Kerr family — a 2-parameter family described by mass M and angular momentum J — varies smoothly with its parameters.

But inside the event horizon a puzzling feature occurs. Indeed, inside the Schwarzschild event horizon a point-like, spacelike singularity creates a boundary for spacetime. 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.

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Indeed, there are arguments, of both physical [ ] and mathematical nature [ ], indicating that the Cauchy horizon i. 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 [ , ]: if ingoing and outgoing streams of matter are simultaneously present near the inner horizon, then relativistic counter-streaming 2 between those streams leads to exponential growth of gauge-invariant quantities such as the interior Misner-Sharp [ ] mass, the center-of-mass energy density, or curvature scalar invariants.

Since this effect is causally disconnected from any external observers, the mass of the BH measured by an outside observer remains unchanged by the mass inflation going on in the interior. But this inflation phenomenon causes the spacetime curvature to grow to Planckian values in the neighbourhood of the Cauchy horizon.

The precise nature of this evolution for the Kerr case is still under study. For the simpler RN case, it has been argued by Dafermos, using analytical methods, that the singularity that forms is not of space-like nature [ ]. Fully nonlinear numerical simulations will certainly be important for understanding this process.

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 [ ]. Can one possibly surpass this astronomical number? In four spacetime dimensions, there is only one constant with dimensions of energy per second that can be built out of the classical universal constants. This is the Planck luminosity ,. The quantity should control gravity-dominated dynamical processes; as such it is no wonder that these events release huge luminosities. During a collapse time of the order of the infall time, , the star can release an energy of up to Mc 2.

The process can therefore yield a power as large as. It was conjectured by Thorne [ ] that the Planck luminosity is in fact an upper limit for the luminosity of any process in the universe. Are such luminosities ever attained in practice, is there any process that can reach the Planck luminosity and outshine the entire universe? The answer to this issue requires once again a peek at gravity in strongly dynamical collisions with full control of strong-field regions.

## Exploring New Physics Frontiers Through Numerical Relativity

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 [ , , , ]. Higher-dimensional spacetimes are a natural framework for mathematicians and have been of general interest in physics, most notably as a tool to unify gravity with the other fundamental interactions. The quest for a unified theory of all known fundamental interactions is old, and seems hopeless in four-dimensional arenas.

In a daring proposal however, Kaluza and Klein, already in and showed that such a programme might be attainable if one is willing to accept higher-dimensional theories as part of the fundamental picture [ , ] for a historical view, see [ ]. As a consequence,. One important, generic conclusion is that the higher-dimensional fundamental theory imparts mass terms as imprints of the extra dimensions.

As such, the effects of extra dimensions are in principle testable. The attempts by Kaluza and Klein to unify gravity and electromagnetism considered five-dimensional Einstein field equations with the metric appropriately decomposed as,. Assuming all the fields are independent of the extra dimension z , one finds a set of four-dimensional Einstein-Maxwell-scalar equations, thereby almost recovering both GR and EM [ ]. This is the basic idea behind the Kaluza-Klein procedure, which unfortunately failed due to the presence of the undetected scalar field. The idea of using higher dimensions was to be revived decades later in a more sophisticated model, eventually leading to SMT.

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 [ ]. The uniqueness theorems, for example, are known to break down in higher dimensions, at least in the sense that solutions are uniquely characterized by asymptotic charges. BHs of spherical topology — the extension of the Kerr solution to higher dimensions — can co-exist with black rings [ ].

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.