Short term issues were explored with the code in subsequent work [55]. 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:

- a Newtonian correspondence method, which guarantees that the Einstein quadrupole formula is satisfied in the Newtonian limit [303], and
- 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, independent 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.

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 orbiting a black hole of much larger mass, say [51]. 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 that the assumption of Newtonian equilibrium is valid but small enough 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 10^{3} 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 [237].

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 [271] described in Section 7.2. In the three-dimensional characteristic simulations of a star orbiting a black hole [55, 51], the lack of resolution introduced by a localized star makes an accurate calculation of the news highly problematic. 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 [47] 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 the computation of mass loss or radiation reaction effects.

Living Rev. Relativity 15, (2012), 2
http://www.livingreviews.org/lrr-2012-2 |
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