6 AcknowledgmentsCharacteristic Evolution and Matching4.8 The Binary Black Hole

5 Numerical Hydrodynamics on Null Cones 

Numerical evolution of relativistic hydrodynamics has been traditionally carried out on spacelike Cauchy hypersurfaces. Although the Bondi-Sachs evolution algorithm can easily be extended to include matter [92], the advantage of a light cone approach for treating fluids is not as apparent as for a massless field whose physical characteristics lie on the light cone. However, preliminary results from recent studies of a fluid moving in the vicinity of a black hole indicate that this approach can provide accurate simulations of mass accretion and the gravitational radiation from an infalling body.

In a three-dimensional study, a naive hydrodynamic code for a perfect fluid was incorporated into the PITT null code [29]. The fully nonlinear three-dimensional matter-gravity null code was tested for stability and accuracy to verify that nothing breaks down as long as the fluid remains well behaved, e.g. hydrodynamical shocks do not form. The code was used to simulate a localized blob of matter falling into a black hole, verifying that the motion of the center of the blob approximates a geodesic and determining the waveform of the emitted gravitational radiation at tex2html_wrap_inline1655 . The results establish the feasibility of a characteristic matter plus gravity evolution.

This simulation was a prototype of a neutron star orbiting a black hole. It would be unrealistic to expect that a naive fluid code would allow evolution of a compact star for several orbits as it spiraled into a black hole. However, a refined characteristic hydrodynamic code would open the way to explore this important astrophysical problem. Recently, J. A. Font and P. Papadopoulos [109] successfully implemented such a code in the case of spherical symmetry, using the Bondi-Sachs formalism as adapted to describe gravity coupled to matter inside a worldtube boundary [139].

The approach is based upon a high resolution shock-capturing (HRSC) version of relativistic hydrodynamics developed by the Valencia group (for a review see Ref. [57]). In the HRSC scheme, the hydrodynamic equations are written in flux conservative, hyperbolic form. In each computational cell, the system of equations is diagonalized to determine the characteristic fields and velocities, and the local Riemann problem is solved to obtain a solution consistent with physical discontinuities. This allows a finite differencing scheme along the fluid characteristics that preserves the physically conserved quantities and leads to a stable and accurate treatment of shocks. Because the general relativistic system of hydrodynamical equations is formulated in covariant form, it can equally well be applied to spacelike or null foliations of the spacetime. The null formulation gave remarkable performance in the standard Riemann shock tube test carried out in a Minkowski background and gave second order convergence in curved space tests based upon Tolman-Oppenheimer-Volkoff equilibrium solutions. In the dynamic self-gravitating case, simulations of spherical accretion of a fluid onto a black hole were stable and free of any numerical problems. Accretion was successfully carried out in the regime where the mass of the black hole doubled. In more recent work, they have used the code to study how accretion modulates both decay rates and oscillation frequencies of the quasi-normal modes of the interior black hole [110].

Plans are being made to combine the characteristic evolution codes for vacuum spacetimes and HRSC relativistic hydrodynamics.

6 AcknowledgmentsCharacteristic Evolution and Matching4.8 The Binary Black Hole

image Characteristic Evolution and Matching
Jeffrey Winicour
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