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Register a new userUsing curvature invariants for wave extraction in numerical relativity
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We present a new expression for the Weyl scalar Psi_4 that can be used in numerical relativity to extract the gravitational wave content of a spacetime. The formula relies upon the identification of transverse tetrads, namely the ones in which Psi_1=Psi_3=0. It is well known that tetrads with this property always exist in a general Petrov type I spacetime. A sub-class of these tetrads naturally converges to the Kinnersley tetrad in the limit of Petrov type D spacetime. However, the transverse condition fixes only four of the six parameters coming from the Lorentz group of transformations applied to tetrads. Here we fix the tetrad completely, in particular by giving the expression for the spin-boost transformation that was still unclear. The value of Psi_4 in this optimal tetrad is given as a function of the two curvature invariants I and J.
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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.
We present the recent results of a research project aimed at constructing a robust wave extraction technique for numerical relativity. Our procedure makes use of Weyl scalars to achieve wave extraction. It is well known that, with a correct choice of null tetrad, Weyl scalars are directly associated to physical properties of the space-time under analysis in some well understood way. In particular it is possible to associate $Psi_4$ with the outgoing gravitational radiation degrees of freedom, thus making it a promising tool for numerical wave--extraction. The right choice of the tetrad is, however, the problem to be addressed. We have made progress towards identifying a general procedure for choosing this tetrad, by looking at transverse tetrads where $Psi_1=Psi_3=0$. As a direct application of these concepts, we present a numerical study of the evolution of a non-linearly disturbed black hole described by the Bondi--Sachs metric. This particular scenario allows us to compare the results coming from Weyl scalars with the results coming from the news function which, in this particular case, is directly associated with the radiative degrees of freedom. We show that, if we did not take particular care in choosing the right tetrad, we would end up with incorrect results.
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.
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.
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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