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Register a new userNumerical relativity for D dimensional space-times: head-on collisions of black holes and gravitational wave extraction
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Black objects in higher dimensional space-times have a remarkably richer structure than their four dimensional counterparts. They appear in a variety of configurations (e.g. black holes, black branes, black rings, black Saturns), and display complex stability phase diagrams. They might also play a key role in high energy physics: for energies above the fundamental Planck scale, gravity is the dominant interaction which, together with the hoop-conjecture, implies that the trans-Planckian scattering of point particles should be well described by black hole scattering. Higher dimensional scenarios with a fundamental Planck scale of the order of TeV predict, therefore, black hole production at the LHC, as well as in future colliders with yet higher energies. In this setting, accurate predictions for the production cross-section and energy loss (through gravitational radiation) in the formation of black holes in parton-parton collisions is crucial for accurate phenomenological modelling in Monte Carlo event generators. In this paper, we use the formalism and numerical code reported in arXiv:1001.2302 to study the head-on collision of two black holes. For this purpose we provide a detailed treatment of gravitational wave extraction in generic D-dimensional space-times, which uses the Kodama-Ishibashi formalism. For the first time, we present the results of numerical simulations of the head-on collision in five space-time dimensions, together with the relevant physical quantities. We show that the total radiated energy, when two black holes collide from rest at infinity, is approximately (0.089pm 0.006)% of the centre of mass energy, slightly larger than the 0.055% obtained in the four dimensional case, and that the ringdown signal at late time is in very good agreement with perturbative calculations.
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The numerical evolution of Einsteins field equations in a generic background has the potential to answer a variety of important questions in physics: from applications to the gauge-gravity duality, to modelling black hole production in TeV gravity scenarios, analysis of the stability of exact solutions and tests of Cosmic Censorship. In order to investigate these questions, we extend numerical relativity to more general space-times than those investigated hitherto, by developing a framework to study the numerical evolution of D dimensional vacuum space-times with an SO(D-2) isometry group for Dge 5, or SO(D-3) for Dge 6. Performing a dimensional reduction on a (D-4)-sphere, the D dimensional vacuum Einstein equations are rewritten as a 3+1 dimensional system with source terms, and presented in the Baumgarte, Shapiro, Shibata and Nakamura (BSSN) formulation. This allows the use of existing 3+1 dimensional numerical codes with small adaptations. Brill-Lindquist initial data are constructed in D dimensions and a procedure to match them to our 3+1 dimensional evolution equations is given. We have implemented our framework by adapting the LEAN code and perform a variety of simulations of non-spinning black hole space-times. Specifically, we present a modified moving puncture gauge which facilitates long term stable simulations in D=5. We further demonstrate the internal consistency of the code by studying convergence and comparing numerical versus analytic results in the case of geodesic slicing for D=5,6.
Gravitational waves are one of the most important diagnostic tools in the analysis of strong-gravity dynamics and have been turned into an observational channel with LIGOs detection of GW150914. Aside from their importance in astrophysics, black holes and compact matter distributions have also assumed a central role in many other branches of physics. These applications often involve spacetimes with $D>4$ dimensions where the calculation of gravitational waves is more involved than in the four dimensional case, but has now become possible thanks to substantial progress in the theoretical study of general relativity in $D>4$. Here, we develop a numerical implementation of the formalism by Godazgar and Reall (Ref.[1]) -- based on projections of the Weyl tensor analogous to the Newman-Penrose scalars -- that allows for the calculation of gravitational waves in higher dimensional spacetimes with rotational symmetry. We apply and test this method in black-hole head-on collisions from rest in $D=6$ spacetime dimensions and find that a fraction $(8.19pm 0.05)times 10^{-4}$ of the Arnowitt-Deser-Misner mass is radiated away from the system, in excellent agreement with literature results based on the Kodama-Ishibashi perturbation technique. The method presented here complements the perturbative approach by automatically including contributions from all multipoles rather than computing the energy content of individual multipoles.
Binary black holes with spins that are aligned with the orbital angular momentum do not precess. However, post-Newtonian calculations predict that up-down binaries, in which the spin of the heavier (lighter) black hole is aligned (antialigned) with the orbital angular momentum, are unstable when the spins are slightly perturbed from perfect alignment. This instability provides a possible mechanism for the formation of precessing binaries in environments where sources are preferentially formed with (anti) aligned spins. In this paper, we present the first full numerical relativity simulations capturing this instability. These simulations span $sim 100$ orbits and $sim 3$-$5$ precession cycles before merger, making them some of the longest numerical relativity simulations to date. Initialized with a small perturbation of $1^{circ}$-$10^{circ}$, the instability causes a dramatic growth of the spin misalignments, which can reach $sim 90^{circ}$ near merger. We show that this leaves a strong imprint on the subdominant modes of the gravitational wave signal, which can potentially be used to distinguish up-down binaries from other sources. Finally, we show that post-Newtonian and effective-one-body approximants are able to reproduce the unstable dynamics of up-down binaries extracted from numerical relativity.
A numerical-relativity calculation yields in general a solution of the Einstein equations including also a radiative part, which is in practice computed in a region of finite extent. Since gravitational radiation is properly defined only at null infinity and in an appropriate coordinate system, the accurate estimation of the emitted gravitational waves represents an old and non-trivial problem in numerical relativity. A number of methods have been developed over the years to extract the radiative part of the solution from a numerical simulation and these include: quadrupole formulas, gauge-invariant metric perturbations, Weyl scalars, and characteristic extraction. We review and discuss each method, in terms of both its theoretical background as well as its implementation. Finally, we provide a brief comparison of the various methods in terms of their inherent advantages and disadvantages.
Wave extraction plays a fundamental role in the binary black hole simulations currently performed in numerical relativity. Having a well defined procedure for wave extraction, which matches simplicity with efficiency, is critical especially when comparing waveforms from different simulations. Recently, progress has been made in defining a general technique which uses Weyl scalars to extract the gravitational wave signal, through the introduction of the {it quasi-Kinnersley tetrad}. This procedure has been used successfully in current numerical simulations; however, it involves complicated calculations. The work in this paper simplifies the procedure by showing that the choice of the {it quasi-Kinnersley tetrad} is reduced to the choice of the time-like vector used to create it. The space-like vectors needed to complete the tetrad are then easily identified, and it is possible to write the expression for the Weyl scalars in the right tetrad, as simple functions of the electric and magnetic parts of the Weyl tensor.
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