We wish to understand black hole dynamics in gravitational theories that go beyond Einstein’s General Relativity. At present, our research focuses mainly in studying such dynamics, as dictated by (generalised) Einstein’s equations, in a generic background and number of dimensions.

*Why is this interesting? *

The main reason is that 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. Over the last five 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. Here is an animation (made by Ulrich Sperhake), constructed from the numerical simulations^{[2]} of non-head on black hole collisions at high energies:

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.

**The TeV gravity scenario and lowering the Planck scale**

A class of models that has been studied since the work of Kaluza and Klein (in 1919 and 1926) includes extra space-time dimensions. The basic idea is that introducing extra dimensions makes observable degrees of freedom, like the electromagnetic or other the charges that couple to the other gauge interactions, geometric. In the last 30 years, ideas emerging from high energy physics, such as supersymmetry and super string theory, further motivated the study of gravitational theories with more than four space-time dimensions. To be consistent with the fact that we do not seem to observe them (at least as geometric degrees of freedom) these dimensions have been considered to be compact and very small (with a length scale of the order of the Planck length - around 10^-35 m). But at the end of the 1990s it was realised that taking extra dimensions much larger than the Planck length could not only be compatible with observations but also desirable to solve a long standing problem in high energy physics.

The long standing problem is the hierarchy problem. There is a huge hierarchy between two very important scales in nature. The electroweak scale, at which there is a partial unification of the nuclear and electromagnetic forces, is of the order of 200 GeV. The four dimensional Planck scale, at which it is believed that there is a unification of the electromagnetic, nuclear and gravitational forces is 10^19 GeV. There are 17 orders of magnitude between the two unification scales, and that does not sound natural. One other way to pose the problem is: why is gravity so much weaker than the electromagnetic and nuclear forces? It is this apparent weakness that requires going to extremely high energies, in order for gravity to become as relevant as the gauge interactions.

The key ingredient to legitimate large extra dimensions and address the hierarchy problem is the idea of “brane”. A p-brane is a hyper-surface with p-spatial dimensions. So a membrane is a 2-brane. A particular type of branes that appear in string theory, called D-branes, offer a very natural mechanism to confine gauge interactions to the brane, independently of the number of space-time dimensions the brane lives in. By contrast gravity is the * geometrodynamics of space-time itself *. Thus one may consider a model of a D-dimensional space-time with a 3-brane in it. Gravity will see the full D-dimensional geometry whereas the gauge interactions (electromagnetic and nuclear forces) will propagate solely in the 3+1 dimensional brane.

The point now is that the gauge interactions are well tested down to scales of around 10^-19 m, and exhibit no signatures of extra-dimensions. But gravity is only tested, for short distances, down to fractions of a millimeter! Thus extra dimensions that only gravity can see (i.e propagate in them) are very weakly bounded by observations/experiments. They could be many orders of magnitude larger than the four dimensional Planck scale or even than the scales tested in particle accelerators for standard model physics. This would also explain why gravity is so weak: roughly, because it is “diluted” propagating into a number of extra dimensions. The fundamental Planck scale could then be much smaller than the four dimensional Planck scale. Current bounds are that it could be as small as 1 TeV. That is very exciting because the Large Hadron Collider (LHC) at CERN will make hadron collisions with centre of mass energies of up to 14 TeV; therefore at the LHC there may be particle colliding with energies above the fundamental Planck scale.

** Colliding particles with energies above the fundamental Planck scale **

Well above the fundamental Planck scale gravity becomes the dominant interaction. It was conjectured by Kip Thorne that a (head on) collision of two particles with energies above the fundamental Planck scale should form a black hole: the hoop conjecture. In 2009 evidence for this was presented by M. Choptuik and F. Pretorius. They performed numerical simulations for the collision of two particle like objects (not black holes) in General Relativity and showed that a black hole indeed formed, actually at smaller energies than those suggested by Thorne’s conjecture. Moreover it seems likely that that is a generic phenomenon. That is, a black hole will form irrespectively of the types of particles that are being collided. This is the idea that “matter does not matter” for scattering at energies above the fundamental Planck scale; it only matters how much gravitational energy there is. Again, this is because gravity is the dominating interaction.

These ideas suggest that the LHC may therefore be a black hole factory! Moreover it suggests that one may model parton scattering at energies above the fundamental Planck scale by the simplest possible objects (with the same gravitational energy as the colliding particles): black holes. This is exactly what we are doing. Modelling black hole production in a particle accelerator by studying black hole collisions in D dimensions at high energies. Check our first results.^{[3]}^{[4]}

The AdS/CFT correspondence

The same D-branes that were mentioned above also gave rise to a remarkable correspondence which is known and the Anti-de-Sitter (AdS)/Conformal Field Theory (CFT) correspondence. By analysing the physics of these objects in two different limits, it was observed by J. Maldacena in 1997 that they were described by two very different models: a gravitational model in AdS space or by a non-abelian gauge theory with conformal invariance in the absence of gravity. Moreover, the correspondence is such that the classical gravitational description maps into strongly coupled physics in the gauge theory. Thus the correspondence offers a calculational tool for probing the strongly coupled regime of non-abelian gauge theories, which cannot be studied by perturbation theory.

The most relevant non-abelian gauge theory in our world is Quantum Chromodynamics (QCD). This theory describes the physics of quarks and gluons, i.e. the strong nuclear force. Although the non-abelian gauge theories that are directly described by the AdS/CFT correspondence are different from QCD, there has been a growing hope that some features of QCD may be described by the correspondence. This hope has been reinforced by experimental data found at the Relativistic Heavy Ion Collider (RHIC).

At RHIC heavy ions (like gold nuclei) were collided near the speed of light. The state that formed in the instants following the collision was dubbed “quark-gluon plasma”. Quite remarkably it shares some properties with a fluid; in particular it may be given a viscosity and an entropy density. Using AdS/CFT, P. Kovtun, D. Son and A. Starinets showed in 2004 that the correspondence predicts, for any gauge theory with a gravity dual, a very similar ratio between viscosity and entropy density to that observed at RHIC. Their calculation was made by analysing physical properties of black holes in the gravitational description. Therefore analysing black holes in AdS spaces renders information about real world QCD! This is a good reason to study more thoroughly the dynamics of black holes in these spaces.

The numerical evolution of black holes in AdS has one extra twist as compared to that in asymptotically flat space. AdS space is a natural gravitational “box”, in the sense that a light ray shot from any point inside AdS will arrive at infinity in finite proper time as measured by the observer that shot it. This means that AdS has a time-like conformal boundary. And, in turn, this means that evolution in AdS requires boundary conditions to be specified, because the * boundary has an active role for the bulk evolution *. In order to understand this active role we have started by building a toy model for AdS: model black holes in AdS by considering black holes in a boxed, asymptotically flat space.^{[5]} Here is a movie (made by Helvi Witek) for the evolution of a black hole binary in the case of a spherical box at which reflecting boundary conditions are imposed:

** References: **

- ^
*High-Energy Collision of Two Black Holes*,

U. Sperhake, V. Cardoso, F. Pretorius, E. Berti, J. González, Phys. Rev. Lett. 101, 161101 (2008) - ^
*Cross Section, Final Spin, and Zoom-Whirl Behavior in High-Energy Black-Hole Collisions*,

U. Sperhake, V. Cardoso, F. Pretorius, E. Berti, T. Hinderer, N. Yunes, Phys. Rev. Lett. 103, 131102 (2009) - ^
*Numerical relativity for D dimensional axially symmetric space-times: formalism and code tests*,

M. Zilhao, H. Witek, U. Sperhake, V. Cardoso, L. Gualtieri, C. Herdeiro, A. Nerozzi, Phys. Rev. D 81, 084052 (2010) - ^
*Numerical relativity for D dimensional space-times: head-on collisions of black holes and gravitational wave extraction*,

H. Witek, M. Zilhao, L. Gualtieri, V. Cardoso, C. Herdeiro, A. Nerozzi, U. Sperhake, arXiv:1006.3081 - ^
*Black holes in a box: towards the numerical evolution of black holes in AdS*,

H. Witek, V. Cardoso, C. Herdeiro, A. Nerozzi, U. Sperhake, M. Zilhao, arXiv:1004.4633