Decomposition of Schwarzschild Green's Function

TL;DR

This work addresses the problem of decomposing the Schwarzschild Green's function for gravitational perturbations into physically interpretable components: direct waves, quasinormal-mode (QNM) contributions, and late-time tails. It develops a frequency-domain framework that splits $\tilde{G}$ into $\tilde{G}^{+} + \tilde{G}^{-}$, where QNMs arise from zeros of $A^{\mathrm{inc}}_{\mathrm{in}}(\omega)$ and the direct part and tail originate from branch cuts on the imaginary axis, avoiding the problematic large-arc direct term. The authors implement Mano–Suzuki–Takasugi (MST) based calculations in Julia to evaluate the Regge–Wheeler radial functions and assemble the three Green's-function components, validating the results against a time-domain Regge–Wheeler solver and aligning with Schwarzschild–de Sitter limits. The resulting framework provides a practical, self-consistent tool for analyzing early-time direct waves, ringdown, and nonlinear interactions in Schwarzschild spacetimes, with potential extensions to Kerr black holes.

Abstract

In this work, we present a full description of the spherically decomposed Green's function of a non-rotating black hole, which naturally splits into three components: the direct part, the quasinormal modes, and the tail. Both the direct part and the tail are contributed by branch cut integrals on the complex-frequency domain, and the quasinormal modes correspond to poles of the Green's function. We show that these different components match the Green's function numerically obtained by solving a time-domain Regge-Wheeler code. In addition, the components of the Green's function also agree with earlier studies in Schwarzschild spacetime with small cosmological constant. The identification of all the various parts of the Schwarzschild Green's function represents an important step towards analyzing direct waves and quasinormal modes in the ringdown stage of binary black hole coalescence, as well as their nonlinear interaction near the merger.

Figures (6)

• Figure 1: The decomposition of Schwarzschild black hole Green's function.
• Figure 2: The frequency-domain Green’s function evaluated along the right-hand side of the BC on the imaginary-frequency axis. For positive (negative) imaginary frequencies, the curves show the real and imaginary parts of $G^{+}$ ($G^{-}$).
• Figure 3: $G^{-}$ evaluated along the right-hand side of the small circular contour. The radius of the circle is $\delta=0.05$. The definition of the angular parameter $\phi$ is shown in Fig. \ref{['fig:Contour']}. Here $r=30$, $r^\prime=10$, and $\ell=2$.
• Figure 4: Early-time behavior of the time-domain Green's function. The blue dashed curve shows the full Green's function obtained from the time-domain simulation. The orange curve represents the BC direct part, while the green curve corresponds to the sum of QNMs with overtones $n=0$--$3$. The black dashed line indicates the time $t = r_* - r^\prime_*$, and the black dotted line indicates $t = r_* + r^\prime_*$. This convention is adopted for all subsequent figures. In this figure, $r=30$, $r^\prime = 10$ and $\ell = 2$.
• Figure 5: Comparison between the BC tail contribution and the numerically simulated waveform after applying the QNM filter described in PhysRevD.106.084036. In this figure, $r=30$, $r^\prime = 10$ and $\ell = 2$.