We wish to understand Gravitation. Are the dark, massive compact objects that we `see' black holes? Are they described by Einstein's theory? We wish to understand black hole dynamics in gravitational theories that go beyond Einstein’s General Relativity. In parallel, we want to understand how much information about these objects we can extract from gravitational-wave data.

*This project is funded by the European Research Council through H2020 ERC Consolidator Grant “Matter and strong- field gravity: New frontiers in Einstein’s theory” grant agreement no. MaGRaTh–646597.*

*Why is this interesting? *

There are several reasons. One is that we can finally test these theories and the black hole paradigm. Gravitational waves were finally detected and will be routinely analyzed over the next few years. The confirmation that dark objects are indeed black holes described by Einstein's theory is equivalent to stating that we understand how gravity works even in extreme conditions...it is important to test this!

In addition, scenarios have been suggested, by high energy physics models, in which the dynamics of black holes in higher dimensional space-times and/or space-times with more general geometries than flat Minkowski space, play an important role for making phenomenological predictions of events at particle accelerators (such as the Relativistic Heavy Ion Collider - RHIC - or the Large Hadron Collider - LHC).

*How can we study such highly dynamical systems?*

Using numerical relativity techniques, or other approaches, such as perturbation theory. Over the last years the field of numerical relativity has seen major breakthroughs which have allowed the successful evolution of black hole binaries (and even more complex systems) in four dimensional asymptotically flat space-times. As the black holes inspiral and merge they produce a strong gravitational wave signal. The numerical simulations have allowed the construction of theoretical templates that may be used for filtering the signal that is expected to be observed in current and future gravitational wave observatories. It is these successful techniques (extended to more general space-times) that we use.

** Numerical relativity - an old story of recent success**

The Einstein equations of General Relativity are manifestly hard to solve. If one considers a dynamical system, like two black holes interacting and evolving in time, either one uses approximate analytical methods or one attempts an exact (within numerical error) numerical solution. In the evolution of a black hole binary one expects the non-linearities of the field equations to be very important during the very last orbits before the two black holes merge. Thus approximate methods should be unreliable and one must use a numerical solution.

The first published attempt to solve the field equations for a binary black hole space-time was performed by S. G. Hahn and R. W. Lindquist in 1964. Their simulation took about 4 hours to complete 50 time steps, after which it was concluded that errors had grown too large to enable further evolution. In fact, with the wisdom of hindsight, one may now appreciate that neither the computational resources nor the understanding of the problem itself were sufficiently developed for this attempt to succed. In fact it took another 40 years (!) for the first successful numerical simulations of the problem to be reported. Around 2005, two independent evolution schemes, the “generalised harmonic evolution” and the “Baumgarte-Shibata-Shapiro-Nakamura (BSSN) formulation with moving punctures”, successfully evolved the last orbits and merger of a black hole binary space-time. These breakthroughs initiated a golden era for numerical relativity, which is still ongoing.

*What have we learned about the physics of a black hole binary?*

Quite a lot, although this basic problem - the two-body problem in General Relativity - is not yet completely solved. A complete solution must explore the full parameter space of initial conditions.

The most basic conceivable binary system consists of two equal mass black holes, each without intrinsic spin, in circular orbits around a common centre of mass (energy). The simulations show that the final result of the evolution is a spinning (Kerr) black hole with spin parameter 0.69 (angular momentum per unit mass). Moreover, in the last orbit and merger, around 3.5% of the system’s energy is radiated away in the form of gravitational radiation. In the inspiralling phase prior to this, analytical (approximation) methods suggest that another 1.5% of the energy is radiated away.

If the two black holes have different masses, a new phenomenon is found. There is an asymmetric beaming of the gravitational radiation produced in the inspiral and merger. Thus, more linear momentum is radiated away in a given direction, and therefore the remnant black hole gets a recoil velocity or *kick* in the opposite direction. This recoil velocity is along the orbital plane and can be, at most, of the order of 175 Km/s.

But if the two black holes also have spin, there can be *superkicks*. The configuration that was found to maximise this recoil velocity is that of two (equal mass) spinning black holes, with opposite spins along the (initial) orbital plane. Due to the spin-orbit coupling, the orbital plane in the simulations is seen oscillating up and down. The kick velocity will then depend on the orbital phase at the merger and it can be as large as 4000 Km/s !

*How about colliding black holes at high velocities? *

The binary black hole systems we have been discussing are obviously of astrophysical interest. One may also consider a *Gedanken* experiment wherein black holes collide with relative speeds close to that of light. This could be one of the most energetic phenomena allowed by the laws of physics. Although such events are not expected to occur in astrophysical scenarios, they are important to probe a high energy regime of General Relativity and also for modelling events at particle accelerators, due to Thorne’s hoop conjecture. These types of simulations were reported starting in 2008. For a head on collision of non-spinning equal mass black holes it was found that, in the ultra-relativistic regime, the radiated energy could be about 14% of the total centre of mass energy of the system, and the luminosity could get to 1% of the Dyson luminosity limit.^{[1]} For non-head on collisions, for certain impact parameters, gravitational radiation can carry up to 35% of the energy of the system and the luminosity can be above 10% of the Dyson limit. ^{}

A most important conclusion is that these simulations *can* be done in four space-time dimensions. So, if there is a strong enough motivation one could attempt them in higher dimensions as well. Is there such motivation?

**Motivation to go beyond General Relativity**

There are strong reasons to believe that Einstein’s theory of gravity, albeit extremely successful, will not be the last word in our understanding of the gravitational interaction. The main reason is that it does not seem to be compatible with Quantum Field Theory (QFT). General Relativity is typically relevant in large-scale processes, while QFT is typically relevant in small-scale processes. However, in physical processes at energies larger than the Planck scale (10^19 GeV) both theories should become relevant, and a quantum theory of gravity should be the correct framework to describe processes at these energies. Such theory is unknown. What we know is that, in the low energy limit, it is well described by an effective classical field theory: General Relativity. But it is quite natural that the effective field theory can be improved by considering more degrees of freedom or more complex dynamics. In other words, General Relativity is an incomplete theory of gravity.

This incompleteness also manifests itself in the fact that the theory predicts space-time singularies, such as the ones occuring inside black holes, at the Big Bang or when certain gravitational waves collide. Thus, the theory is predicting its own demise: there is a regime (corresponding to very high energies/short distance scales) when General Relativity should no longer be a good description of the gravitational interaction. Thus, there is a theoretical motivation to study more general theories of gravity which, in the low energy limit, reduce to General Relativity, even if at the present moment there is no experimental/observational evidence to support them (or to falsify them!…). The best

probe of these theories are of course processes involving black holes, like black hole coalescences, since they are (second only to the Big Bang itself) the most violent gravitational processes in the Universe.

** References: **

1. V. Cardoso, L. Gualtieri, C. Herdeiro and U. Sperhake, Exploring New Physics Frontiers Through Numerical Relativity, Living Rev. Relativity 18, 1 (2015) [arXiv:1409.0014 [gr-qc]].

2. V. Cardoso and L. Gualtieri, Testing the black hole ‘no-hair’ hypothesis,

Class. Quant. Grav.33, no. 17, 174001 (2016) [arXiv:1607.03133 [gr-qc]].

3. V. Cardoso, E. Franzin and P. Pani, Is the gravitational-wave ringdown a probe of the event horizon?,

Phys. Rev. Lett.11, no. 17, 171101 (2016) [arXiv:1602.07309 [gr-qc]].