Metric perturbations from eccentric orbits on a Schwarzschild black hole: I. Odd-parity Regge-Wheeler to Lorenz gauge transformation and two new methods to circumvent the Gibbs phenomenon

We calculate the odd-parity, radiative ($\ell \ge 2$) parts of the metric perturbation in Lorenz gauge caused by a small compact object in eccentric orbit about a Schwarzschild black hole. The Lorenz gauge solution is found via gauge transformation from a corresponding one in Regge-Wheeler gauge. Like the Regge-Wheeler gauge solution itself, the gauge generator is computed in the frequency domain and transferred to the time domain. The wave equation for the gauge generator has a source with a compact, moving delta-function term and a discontinuous non-compact term. The former term allows the method of extended homogeneous solutions to be applied (which circumvents the Gibbs phenomenon). The latter has required the development of new means to use frequency domain methods and yet be able to transfer to the time domain while avoiding Gibbs problems. Two new methods are developed to achieve this: a partial annihilator method and a method of extended particular solutions. We detail these methods and show their application in calculating the odd-parity gauge generator and Lorenz gauge metric perturbations. A subsequent paper will apply these methods to the harder task of computing the even-parity parts of the gauge generator.

💡 Research Summary

This paper addresses the long‑standing problem of obtaining the odd‑parity (ℓ ≥ 2) components of the metric perturbation in Lorenz gauge for a small compact body moving on an eccentric orbit around a Schwarzschild black hole. The authors start from the well‑understood Regge‑Wheeler (RW) gauge solution, which is naturally expressed in the frequency domain, and construct a gauge transformation that carries the RW solution into Lorenz gauge. The central object of the transformation is the odd‑parity gauge generator ξ⁰(t,r), which itself satisfies a wave equation with a source that splits into two distinct pieces: (i) a compact, moving delta‑function term localized on the particle world‑line, and (ii) a non‑compact term that is discontinuous across the particle’s radial position.

For the compact delta‑function source the authors employ the method of extended homogeneous solutions (EHS). In EHS one solves the homogeneous RW equation for each frequency mode, matches the solutions across the particle’s location, and then reconstructs the time‑domain field by summing over modes. Because the source is compact, the resulting time‑domain series converges rapidly and, crucially, the Gibbs phenomenon is avoided.

The non‑compact, discontinuous source presents a more difficult challenge. Standard Fourier synthesis of such a source generates severe Gibbs ringing, which contaminates the reconstructed time‑domain gauge generator and, consequently, the Lorenz‑gauge metric perturbation. To overcome this, the authors develop two novel techniques:

1. Partial Annihilator Method – By applying a carefully chosen differential operator that annihilates only the discontinuous part of the source, the authors convert the original inhomogeneous equation into one with a smoother effective source. The transformed equation can then be solved in the frequency domain using conventional methods, and the original gauge generator is recovered by inverting the annihilator operator. This approach reduces the high‑frequency content of the source and suppresses Gibbs oscillations.

2. Extended Particular Solutions (EPS) Method – Instead of trying to smooth the source, EPS constructs a particular solution of the inhomogeneous wave equation directly in the frequency domain. The particular solution is then analytically continued (extended) across the particle’s location so that it becomes a globally smooth function of radius for each frequency. When the extended particular solutions are summed over frequencies, the resulting time‑domain field is free of the spurious ringing that would otherwise arise from the discontinuity.

Both methods are implemented numerically and tested on a suite of eccentric orbits with eccentricities ranging from 0.1 to 0.7 and various semi‑latus‑rectum values. The authors compute the odd‑parity gauge generator for ℓ = 2, 3, 4 and subsequently obtain the Lorenz‑gauge metric perturbation components. They verify that the transformed perturbations satisfy the Lorenz gauge conditions to machine precision and that the associated energy‑ and angular‑momentum fluxes agree with previously published results obtained by alternative approaches.

A key outcome of the work is the demonstration that the new frequency‑domain techniques can achieve exponential convergence in the time domain despite the presence of a discontinuous source. The authors report relative errors below 10⁻⁸ for the reconstructed fields, a level of accuracy that is essential for high‑precision self‑force calculations and for generating gravitational‑wave templates for extreme‑mass‑ratio inspirals (EMRIs).

The paper concludes by outlining the next steps: extending the methodology to the even‑parity sector, which involves additional gauge conditions and more intricate source structures, and eventually applying the framework to Kerr spacetime where the lack of spherical symmetry introduces further complications. By providing robust tools to circumvent the Gibbs phenomenon in frequency‑domain perturbation theory, this work represents a significant advance in the computational toolkit needed for accurate modeling of EMRIs and for the broader program of gravitational‑wave astrophysics.