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5.3 Three-dimensional characteristic hydrodynamic simulations

The PITT code has been coupled with a rudimentary matter source to carry out three-dimensional characteristic simulations of a relativistic star orbiting a black hole. Animations can be viewed at [148]. A naive numerical treatment of the Einstein-hydrodynamic system for a perfect fluid was incorporated into the code [40], but a more accurate HRSC hydrodynamic algorithm has not yet been implemented. The fully nonlinear 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., hydrodynamic 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 + I. This simulation was a prototype of a neutron star orbiting a black hole, although it would be unrealistic to expect that this naive fluid code would reliably evolve a compact star for several orbits as it spiraled into a black hole. A 3D HRSC characteristic hydrodynamic code would open the way to explore this important astrophysical problem.

Short term issues were explored with the code in subsequent work [41Jump To The Next Citation Point]. The code was applied to the problem of determining realistic initial data for a star in circular orbit about a black hole. In either a Cauchy or characteristic approach to this initial data problem, a serious source of physical ambiguity is the presence of spurious gravitational radiation in the gravitational data. Because the characteristic approach is based upon a retarded time foliation, the resulting spurious outgoing waves can be computed by carrying out a short time evolution. Two very different methods were used to prescribe initial gravitational null data:

  1. a Newtonian correspondence method, which guarantees that the Einstein quadrupole formula is satisfied in the Newtonian limit [222], and
  2. setting the shear of the initial null hypersurface to zero.

Both methods are mathematically consistent but suffer from physical shortcomings. Method 1 has only approximate validity in the relativistic regime of a star in close orbit about a black hole while Method 2 completely ignores the gravitational lensing effect of the star. It was found that, independently of the choice of initial gravitational data, the spurious waves quickly radiate away, and that the system relaxes to a quasi-equilibrium state with an approximate helical symmetry corresponding to the circular orbit of the star. The results provide justification of recent approaches for initializing the Cauchy problem which are based on imposing an initial helical symmetry, as well as providing a relaxation scheme for obtaining realistic characteristic data.

5.3.1 Massive particle orbiting a black hole

One attractive way to avoid the computational expense of hydrodynamics in treating a star orbiting a massive black hole is to treat the star as a particle. This has been attempted using the PITT code to model a star of mass m orbiting a black hole of much larger mass, say 1000 m [38Jump To The Next Citation Point]. The particle was described by the perfect fluid energy-momentum tensor of a rigid Newtonian polytrope in spherical equilibrium of a fixed size in its local proper rest frame, with its center following a geodesic. The validity of the model requires that the radius of the polytrope be large enough so that the assumption of Newtonian equilibrium is valid but small enough so that the assumption of rigidity is consistent with the tidal forces produced by the black hole. Characteristic initial gravitational data for a double null initial value problem were taken to be Schwarzschild data for the black hole. The system was then evolved using a fully nonlinear characteristic code. The evolution equations for the particle were arranged to take computational advantage of the energy and angular momentum conservation laws which would hold in the test body approximation.

The evolution was robust and could track the particle for two orbits as it spiraled into the black hole. Unfortunately, the computed rate of inspiral was much too large to be physically realistic: the energy loss was ~~ 103 greater than the value expected from perturbation theory. This discrepancy might have a physical origin, due to the choice of initial gravitational data that ignores the particle or due to a breakdown of the rigidity assumption, or a numerical origin due to improper resolution of the particle. It is a problem whose resolution would require the characteristic AMR techniques being developed [174].

5.3.2 Computing the radiation field

The Bondi news function, which represents the gravitational radiation field at I+, is computed by post-processing the output data at + I for the primary evolution variables. This is a delicate numerical procedure involving large terms which partially cancel to give a small result. It is somewhat analogous to the experimental task of isolating a transverse radiation field from the longitudinal fields representing the total mass, while in a very non-inertial laboratory.

In the original algorithm [31Jump To The Next Citation Point], the procedure was carried out in the coordinate system of the code by giving a geometric procedure for computing the news. This approach has been tested to be second order convergent in a wide number of testbeds [31228229]. Alternatively, a coordinate transformation may be carried out to inertial Bondi coordinates (as originally formulated by Bondi [46]), in which the news calculation is quite clean. This approach was recently implemented in [35] and shown to be second order convergent in Robinson-Trautman and Schwarzschild testbeds. A direct comparison of the two approaches was not made, although it is clear both face the same delicate numerical problem of extracting a small radiation field in the presence of large gauge effects in the primary output data.

The procedure is further complicated by sources of numerical error, such as

These sources of error can be further aggravated by the introduction of matter fields, as encountered in trying to make definitive comparisons between the Bondi news and the Einstein quadrupole formula in the axisymmetric studies of supernova collapse [193] described in Section 5.2. In the three-dimensional characteristic simulations of a star orbiting a black hole [4138], the lack of resolution introduced by a localized star makes an accurate calculation of the news highly problematical. There exists no good testbed for validating the news calculation in the presence of a fluid source. A perturbation analysis in Bondi coordinates of the oscillations of an infinitesimal fluid shell in a Schwarzschild background [34] might prove useful for testing constraint propagation in the presence of a fluid. However, the underlying Fourier mode decomposition requires the gravitational field to be periodic so that the solution cannot be used to test computation of mass loss or radiation reaction effects.


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