Time-Domain Master-Equation Evolution

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Time-domain master equations reveal prompt response, quasinormal ringing, and late-time decay in one causal evolution. Their apparent simplicity is deceptive: a stable waveform can still contain boundary reflections, inconsistent characteristic data, or a tail manufactured by finite precision. The numerical claim is finite and testable only when the evolution scheme, extraction location, convergence interval, and decomposition diagnostics are recorded together. Here the accepted perturbation equation is an input; the focus is its trustworthy evolution.

Required background. Ordinary Differential Equations supplies method-of-lines time integration, stability, and convergence tests.

Helpful background. Formulations of General Relativity provides the geometric setting, Black-Hole Perturbation Theory derives the master variables and potentials, and Gravitational Radiation and Waveforms supplies waveform and flux interpretation.

The causal master equation

For a Schwarzschild perturbation, let $M_{\rm geom}=GM_{\rm phys}$ denote the mass length; all times and radii quoted below in units of $M$ mean $M_{\rm geom}$ . A representative equation is

\left(-\partial_t^2+\partial_{r_*}^2-V_\ell(r)\right)\Psi_\ell(t,r_*)=S_\ell(t,r_*).

The domain, potential, source, initial data, and boundary data completely define the computational problem. For the reduced scalar field defined by $\phi=\Psi_\ell(t,r)Y_{\ell m}(\theta,\varphi)/r$ , the massless scalar potential is

V_\ell^{\rm scalar}=\left(1-\frac{2M_{\rm geom}}r\right) \left[\frac{\ell(\ell+1)}{r^2}+\frac{2M_{\rm geom}}{r^3}\right].

The axial gravitational potential instead has $-6M_{\rm geom}/r^3$ in the second bracket and requires $\ell\ge2$ , as stated on the frequency-domain page. Thus the gravitational ringing test and scalar-monopole tail test below are two distinct equations; do not fit the gravitational frequency to the scalar waveform.

In null coordinates $u=t-r_*$ and $v=t+r_*$ , the homogeneous equation becomes $4\partial_u\partial_v\Psi+V\Psi=0$ . A characteristic grid respects the causal geometry and avoids a separate time integrator. Define the diamond corners by $S=(u,v)$ , $E=(u,v+h)$ , $W=(u+h,v)$ , and $N=(u+h,v+h)$ , and evaluate $V_C$ at its center. A second-order update is

\Psi_N=\Psi_E+\Psi_W-\Psi_S-\frac{h^2}{8}V_C(\Psi_E+\Psi_W)+O(h^4).

Integrating $4\Psi_{uv}=-V\Psi$ over the cell gives $\Psi_N-\Psi_E-\Psi_W+\Psi_S=-\tfrac14\int\!\int V\Psi\,du\,dv$ . Centered quadrature and $\Psi_C=(\Psi_E+\Psi_W)/2+O(h^2)$ give the displayed coefficient $h^2/8$ . The local defect is fourth order but the accumulated global error is second order. This distinction is derived and tested in Gundlach, Price, and Pullin 1994, §III, Eqs. (26)–(27); reporting fourth-order convergence for the full waveform would confuse local and global accuracy.

A method-of-lines implementation on $r_*$ is equally legitimate. It should state the spatial stencil, dissipation, Runge–Kutta scheme, CFL factor, and boundary closure. Smooth compact initial data provide a clean vacuum benchmark. Characteristic initial data on intersecting null rays must satisfy the corner compatibility condition; independently prescribing all four edges overdetermines the diamond.

Separate physical epochs from numerical artifacts

At a fixed timelike extraction radius, a compact pulse typically produces direct propagation, damped ringing, and a tail. Those labels are interpretations, not fit instructions. First remove causal intervals contaminated by startup data or the outer boundary. Then estimate an instantaneous complex frequency or fit multiple damped sinusoids over a family of windows; a plateau under window movement is stronger evidence than a single low residual. At the precision supported by Leaver’s table, the recovered Schwarzschild gravitational fundamental should approach $GM_{\rm phys}\omega=M_{\rm geom}\omega\simeq0.3736715-0.0889625i$ Leaver 1985, Table 1, p. 293.

Late-time decay supplies a different benchmark. For a massless scalar monopole on Schwarzschild, generic compact data give $\Psi\propto t^{-3}$ at fixed radius and $\Psi\propto u^{-2}$ at future null infinity. Gundlach, Price, and Pullin 1994, §IV and Figs. 2–6 distinguish the observation regions and demonstrate why a finite-time slope must not be mistaken for the asymptotic exponent. Compute the local power index $p(t)=-d\ln|\Psi|/d\ln t$ and show its approach rather than quoting one fitted number.

Run at least three grids, such as $h/M_{\rm geom}=0.10,0.05,0.025$ , with the same physical initial data and extraction event. For a second-order scheme the differences should satisfy

\frac{\lVert\Psi_h-\Psi_{h/2}\rVert_2} {\lVert\Psi_{h/2}-\Psi_{h/4}\rVert_2}\longrightarrow 4

before roundoff or interpolation dominates. Align samples by physical coordinate time only; maximizing over arbitrary time and phase shifts can manufacture convergence. Monitor the discrete equation residual, energy or flux balance, QNM frequency, and tail index. Move the outer boundary while holding the extraction radius fixed. A reflected pulse arrives after a predictable light-travel time, so this variation separates boundary error from truncation error.

The most useful independent check changes representation: compare the fitted ringdown with a frequency-domain solver, or compare characteristic evolution with a method-of-lines code. A waveform extracted twice from the same grid is not independent. The domain of validity is linear perturbation theory on the declared background; agreement does not validate nonlinear merger dynamics.

Environment and rerun specification

The experiment record pins the background and harmonic labels, potential definition, source revision, compiler or interpreter, numerical libraries, floating-point precision, stencil, time step, dissipation, boundary locations, initial-data file checksum, and extraction procedure. A future executable should accept a single configuration through an interface such as master-evolve --config scalar-l0.toml --output run/. Outputs should include the waveform, flux, residuals, refinement differences, fitted intervals, boundary-arrival estimate, and environment metadata in machine-readable form. This is a rerun specification, not evidence that code shipped with the prose was executed.

Failure criteria are explicit: reject growing high-frequency noise, a convergence factor inconsistent with the designed order, flux imbalance outside the error estimate, a tail exponent that changes under boundary movement, or a QNM fit stable only after arbitrary alignment. At very late times cancellation and underflow can imitate a steeper tail; increased precision and an independent extraction variable test that breakdown.

Diagnostic

A characteristic solver gives convergence factors near four until $t=300M_{\rm geom}$ . At $t=420M_{\rm geom}$ the fitted monopole slope is $-3.0$ , but moving the outer boundary from $500M_{\rm geom}$ to $700M_{\rm geom}$ shifts the onset of that slope by $200M_{\rm geom}$ . Should the slope be accepted as a physical tail or identified specifically as a reflected outgoing pulse?

Solution

The slope should not be accepted, but the observed shift does not by itself identify the causal path. A startup disturbance injected at the boundary shifts its arrival by approximately $\Delta r_*=200M_{\rm geom}$ , whereas reflection of the same outgoing pulse shifts its return by approximately $2\Delta r_*=400M_{\rm geom}$ . Predict both arrival times from the initial-data support, boundary, and extraction worldtube, then identify the contaminant from the matching path. The early second-order convergence verifies the discretization only before contamination. Enlarge or compactify the domain and recover a slope stable under both grid refinement and boundary movement.

References

Gundlach, C., Price, R. H., and Pullin, J. (1994). “Late-time behavior of stellar collapse and explosions: I. Linearized perturbations.” Physical Review D 49, 883–889. arXiv.
Leaver, E. W. (1985). “An analytic representation for the quasi-normal modes of Kerr black holes.” Proceedings of the Royal Society A 402, 285–298. DOI.