The initial conditions for the exterior solution were fixed by the requirement that the interior sources be stationary prior to some fixed time, As a result, the exterior analytic solution is causal in the sense that it is stationary in the past of some null cone. This effectively introduces a condition that eliminates extraneous incoming radiation from the system in a physically plausible way and determines the exterior solution uniquely. Their strategy was to introduce an overlap region between the numerical interior and the analytic exterior. In the overlap, the numerical solution is matched to the causal analytic solution, resulting in an evolution that is everywhere causally meaningful.
This is the physically correct approach to a system which is stationary prior to a fixed time but is nontrivial to generalize, say, to the problem of radiation from an orbiting binary. Anderson and Hobill first tackled the 1D model problem of an oscillator coupled to a spherically symmetric, flat space, massless scalar field. The numerical results were in excellent agreement with the exact analytic solution (which could be obtained globally for this problem).
They extended the model to include spherical scalar waves propagating in a spherically symmetric curved background space-time. This introduces backscattering which obscures the concept of a purely outgoing wave. Also, no exact solution exists to this problem so that an approximation method is necessary to determine the exterior analytic solution. The approximation was based upon an expansion in terms of a scale parameter that controls the amount of backscatter. In the 0th approximation, the scale parameter vanishes and the problem reduces to the flat space case which can be solved exactly. The flat space Green function is then used to generate higher order corrections.
This is a standard perturbative approach, but a key ingredient of the scheme is that the wave equation is solved in retarded null coordinates (u, r) for the curved space metric, so that causality is built into the Green function at each order of approximation. The transformation from null coordinates (u, r) to Cauchy coordinates (t, r) is known analytically for this problem. This allows simple matching between the null and Cauchy solutions at the boundary of the Cauchy grid. Their scheme is efficient and leads to consistent results in the region that the numerical and analytic solutions overlap. It is capable of handling both strong fields and fast motions.
Later, a global, characteristic, numerical study of the self-gravitating version of this problem confirmed that the use of the true null cones is essential in getting the correct radiated waveform . For quasi-periodic radiation, the phase of the waveform is particular sensitive to the truncation of the outer region at a finite boundary. Although a perturbative estimate would indicate an O (M / R) error, this error accumulates over many cycles to produce an error of order in the phase.
Anderson and Hobill proposed that their method be extended to general relativity by matching a numerical solution to an analytic 1/ r expansion in null coordinates. However, the only analytic-numerical matching schemes that have been implemented in general relativity have been based upon perturbations of a Schwarzschild background using the standard Schwarzschild time slicing [2, 3, 4, 5]. It would be interesting to compare results with an analytic-numeric matching scheme based upon the true null cones. However the original proposal by Anderson and Hobill has not been carried out.
|Characteristic Evolution and Matching
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