Gravitational perturbations and metric reconstruction: Method of extended homogeneous solutions applied to eccentric orbits on a Schwarzschild black hole

We calculate the gravitational perturbations produced by a small mass in eccentric orbit about a much more massive Schwarzschild black hole and use the numerically computed perturbations to solve for the metric. The calculations are initially made in the frequency domain and provide Fourier-harmonic modes for the gauge-invariant master functions that satisfy inhomogeneous versions of the Regge-Wheeler and Zerilli equations. These gravitational master equations have specific singular sources containing both delta function and derivative-of-delta function terms. We demonstrate in this paper successful application of the method of extended homogeneous solutions, developed recently by Barack, Ori, and Sago, to handle source terms of this type. The method allows transformation back to the time domain, with exponential convergence of the partial mode sums that represent the field. This rapid convergence holds even in the region of $r$ traversed by the point mass and includes the time-dependent location of the point mass itself. We present numerical results of mode calculations for certain orbital parameters, including highly accurate energy and angular momentum fluxes at infinity and at the black hole event horizon. We then address the issue of reconstructing the metric perturbation amplitudes from the master functions, the latter being weak solutions of a particular form to the wave equations. The spherical harmonic amplitudes that represent the metric in Regge-Wheeler gauge can themselves be viewed as weak solutions. They are in general a combination of (1) two differentiable solutions that adjoin at the instantaneous location of the point mass (a result that has order of continuity $C^{-1}$ typically) and (2) (in some cases) a delta function distribution term with a computable time-dependent amplitude.

Figure 1

The standard FD approach to reconstructing the TD master function and its $r$ derivative. The left panel shows ${\Psi }_{22}^{\mathrm{std}}$ and the right shows ${\partial }_r{\Psi }_{22}^{\mathrm{std}}$ at $t=51.78M$ for a particle orbiting with $p=7.50478$ and $e=0.188917$. This figure is analogous to Fig. 1 of BOS [38]. Partial sums are computed with Eq. (3.3) and shown for different $N$. For contrast we plot the converged solution from the new method with a solid curve (see Fig. 3). The arrow in the right panel gives a notional representation of the delta function singularity present in ${\partial }_r{\Psi }_{22}$; the amplitude of this singular term is related to the jump in ${\Psi }_{22}$ seen in the left panel.

Figure 2

An alternate view of the behavior presented in Fig. 1. A change in the scale in the left panel emphasizes the Gibbs overshoots in ${\Psi }_{22}$. On the right, a zoom-out of the vertical scale more clearly indicates the attempt of the Fourier synthesis to capture the delta function at $r_p(t)$.

Figure 3

The EHS approach to reconstructing the TD master function and its radial derivative. As in Fig. 1, we give ${\Psi }_{22}^{\mathrm{EHS}}$ and ${\partial }_r{\Psi }_{22}^{\mathrm{EHS}}$ at $t=51.78M$ for a particle orbiting with $p=7.50478$ and $e=0.188917$. Partial sums of ${\Psi }_{22}^{\mathrm{EHS}}$ are computed from Eq. (3.5), with a range of $-N\le n\le N$. The full ${\Psi }_{22}^{\mathrm{EHS}}$ and its $r$ derivative result from $N=10$, which gives agreement in the jumps in ${\Psi }_{22}^{\mathrm{EHS}}$ and ${\partial }_r{\Psi }_{22}^{\mathrm{EHS}}$ to a relative error of ${10}^{-10}$. On the right, the presence of a delta function singularity is notionally depicted with an arrow. The time-dependent amplitude of this singularity is separately computable from the jump in ${\Psi }_{22}$.

Figure 4

A plot of the convergence of the master function using the two methods. For a particle orbiting with $p=7.50478$ and $e=0.188917$ at $t=51.78M$ we compute the master function ${\Psi }_{22}(n_{\max })$ by summing over modes ranging from $-n_{\max }\le n\le n_{\max }$ for $n_{\max }=15$. We plot the log of the difference between ${\Psi }_{22}(n_{\max })$ and the partial sum ${\Psi }_{22}(N)$, for different $N<n_{\max }$. For the standard approach (left panel), we see exponential convergence in the homogeneous region, but only algebraic convergence in the region of the source. The method of extended homogeneous solutions (right panel) yields exponentially converging results at all points outside and inside the region of the source. The method of extended homogeneous solutions gives exponential convergence for ${\partial }_r{\Psi }_{\ell m}^{\mathrm{EHS}}$ as well.

Figure 5

The EHS approach to reconstructing the TD MP amplitudes. We consider a particle orbiting with $p=7.50478$ and $e=0.188917$ at $t=80.62M$. The left plot shows the odd-parity MP amplitudes $h_r^{21}$ and $h_t^{21}$. The right shows the even-parity $h_{tt}^{22}$, $h_{rr}^{22}$, $h_{tr}^{22}$, and $K^{22}$. Note that the amplitudes $h_{tt}^{22}$, $h_{rr}^{22}$, and $h_{tr}^{22}$ are singular along the particle’s worldline, as indicated by arrows in the plot on the right. The magnitudes of these singularities are given in Eqs. (5.10, 5.13, 5.14). The remaining three MP amplitudes approach the particle location smoothly, and have only a finite jump at that point.