Perturbations of Schwarzschild black holes in the Lorenz gauge: Formulation and numerical implementation

TL;DR

The paper develops a time-domain, Lorenz-gauge formulation for Schwarzschild perturbations with a tensor-harmonic decomposition into 10 coupled fields per (l,m), incorporating constraint damping to maintain gauge conditions. It provides analytic solutions for the monopole and axially-symmetric odd-parity modes, and a robust 1+1D characteristic evolution code to solve the remaining modes for a particle in circular orbit, with careful reconstruction of the metric perturbation and verification via energy-flux comparisons. The approach enables direct access to the metric perturbation near the particle, facilitating gravitational self-force calculations, and offers a promising path toward Kerr extensions using puncture methods. Overall, the work demonstrates accurate, gauge-consistent Lorenz-gauge perturbations, validated by convergence tests and agreement of radiated flux with established results, and lays groundwork for broader applications in SF and gravitational-wave modeling.

Abstract

We reformulate the theory of Schwarzschild black hole perturbations in terms of the metric perturbation in the Lorenz gauge. In this formulation, each tensor-harmonic mode of the perturbation is constructed algebraically from 10 scalar functions, satisfying a set of 10 wavelike equations, which are decoupled at their principal parts. We solve these equations using numerical evolution in the time domain, for the case of a pointlike test particle set in a circular geodesic orbit around the black hole. Our code uses characteristic coordinates, and incorporates a constraint damping scheme. The axially-symmetric, odd-parity modes of the perturbation are obtained analytically. The approach developed here is especially advantageous in applications requiring knowledge of the local metric perturbation near a point particle; in particular, it offers a useful framework for calculations of the gravitational self force.

Figures (10)

• Figure 1: The Lorenz-gauge monopole solution, Eqs. (\ref{['monopolein']}) and (\ref{['monopoleout']}), plotted here for three values of the orbital radius, $r_0=4M,7M,12M$. By construction (Detweiler and Poisson, Ref. DP) this solution is continuous across the orbit, and well behaved both at the event horizon and at infinity. It is a unique Lorenz-gauge monopole solution with these properties. (The quantity $E$ is the specific energy parameter, denoted elsewhere in this paper by $\cal E$.)
• Figure 2: Numerical grid for characteristic evolution of the field equations. See the text for description.
• Figure 3: Numerical solutions for the (dimensionless) Lorenz-gauge MP functions $\bar{h}^{(i)l=2,m=1,2}$, evaluated at the particle's location, for a particle in a circular orbit at $r=r_0=7M$. In the Lorenz gauge (unlike in the Regge--Wheeler gauge, for example) the mode-decomposed MP is continuous at the particle, and has a definite value there. The early stage of the time evolution is dominated by transient spurious waves associated with the imperfection of the initial data. This part of the evolution (which, of course, is to be discarded in interpreting the physics content of the numerical results) lasts around one orbital period of evolution time, after which the inherent physical behavior is unveiled. Our code evolves separately the real and imaginary parts of the complex functions $\bar{h}^{(i)lm}$, which are both needed to construct the full MP [through formula (\ref{['1-50']})]; for compactness, we show here only the moduli $\left|\bar{h}^{(i)lm}\right| = \left\{[{\rm Re}(\bar{h}^{(i)lm})]^2+[{\rm Im}(\bar{h}^{(i)lm})]^2\right\}^{1/2}$. The mode $l=m=2$ is of even parity, and is constructed from the seven even-parity functions $\bar{h}^{(1,\ldots,7)}$; the odd-parity mode $l=2$, $m=1$ is constructed from the three remaining functions, $\bar{h}^{(8,9,10)}$.
• Figure 4: A different slice cut through the solutions of Fig. \ref{['fig:solutions:fixedr']}, this time showing the behavior of the fields $\bar{h}^{(i)l=2,m=1,2}$ on a $t$=const slice, 2.5 $T_{\rm orb}$ into the evolution. The wavy feature on the left- and right-hand sides (small and large values of $r_*$, respectively) are associated with the spurious initial waves propagating inward (toward the black hole) and outward (to infinity), and are to be discarded. While all functions $\bar{h}^{(i)lm}$ are continuous across the particle (located at $r=7M$, $r_*\simeq 8.83M$), those functions whose corresponding sources $S^{(i)lm}$ are non-zero have discontinues $r$ derivatives there. The fields $\bar{h}^{(2,5,9)lm}$, whose sources $S^{(2,5,9)lm}$ vanish, have continuous derivatives across the particle. The insets expand the peak area of the plots, for better clarity. The code resolves the gradients of the MP fields at the particle with good accuracy (cf. Figs. \ref{['fig:res']} in the next section). These gradients are needed for SF calculations. Note the significant damping in the MP amplitude at $r_*\lesssim 0$. This, presumably, is the effect of the well known potential barrier surrounding the black hole. The high-frequency spurious waves, on the other hand, penetrate this barrier with ease.
• Figure 5: Another slice cut through the solutions of Figs. \ref{['fig:solutions:fixedr']} and \ref{['fig:solutions:fixedt']}, this time showing the behavior along an outgoing ray, $u={\rm const}(=3\;T_{\rm orb})$. Once again, the insets expand the peak areas. All fields $\bar{h}^{(i)lm}$ approach constant values at late advanced times ("null infinity"). The fluxes of energy and angular momentum carried by gravitational waves to infinity are straightforwardly extracted from these values (if fact, only $|\bar{h}^{(7)lm}|$ and $|\bar{h}^{(10)lm}|$ are needed for this purpose), as we discuss in Sec. \ref{['Subsec:flux']}. Note how at large $v$ we seem to have $|\bar{h}^{(3)}|\sim 0$, $|\bar{h}^{(2)}|\sim |\bar{h}^{(1)}|$, $|\bar{h}^{(4)}|\sim |\bar{h}^{(5)}|$, and $|\bar{h}^{(8)}|\sim |\bar{h}^{(9)}|$. In Sec. \ref{['Subsec:flux']} we explain how these empirical asymptotic relations are predicted theoretically.