Green functions of the Regge-Wheeler and Teukolsky equations in Schwarzschild spacetime

We present a calculation of the full retarded Green functions of the Regge-Wheeler and Teukolsky equations obeyed by gravitational field perturbations of Schwarzschild spacetime. We perform the calculations for spacetime points along: (i) a timelike circular geodesic (where null-separated points are not at caustics); and (ii) a static worldline (where null-separated points are at caustics). These Green functions show a 4-fold singularity structure away from caustics, and 2-fold at caustics (similarly to the case of scalar field perturbations, which we also reproduce). Physical oscillations near the singularities appear in the gravitational case, which were not present in the scalar case. We obtain our results by developing various numerical and analytical methods.

💡 Research Summary

This paper presents the first complete calculation of the retarded Green functions (GFs) for the Regge‑Wheeler (RW) and Teukolsky (BPT) equations governing gravitational perturbations of a Schwarzschild black hole. The authors focus on two specific worldlines: (i) a timelike circular geodesic of radius (r_0=6M) and (ii) a static worldline at the same radius. These choices allow them to probe two distinct singularity regimes of the GF: points connected by null geodesics that have not crossed any caustics (the circular orbit case) and points where the null connection always passes through a caustic (the static case).

The paper begins with a review of the global singularity structure of Green functions in curved spacetime. Using Synge’s world function (\sigma), the authors recall that a direct null geodesic yields a Dirac‑delta singularity (\delta(\sigma)). When the null geodesic has crossed one caustic, the singularity cycles through a four‑fold pattern: principal‑value (1/\sigma), (-\delta(\sigma)), (-)PV(1/\sigma), (\delta(\sigma)), and repeats. At caustics the pattern collapses to a two‑fold cycle involving a (|\sigma|^{-3/2}) divergence that is asymmetric on either side of (\sigma=0). These structures were previously demonstrated for scalar fields; the present work shows that they persist for spin‑2 RW and BPT Green functions, but with an additional physical oscillation near each singularity that is absent in the scalar case.

To obtain the full GF, the authors combine several analytical and numerical techniques. In the quasi‑local (QL) region, where the two spacetime points are within a normal neighbourhood, they employ the Hadamard form of the Green function. They derive explicit coordinate‑distance expansions for the tail bitensor of the RW Green function, providing the coefficients in a downloadable code. For the BPT equation they do not yet have a full Hadamard expansion, limiting their ability to reach the coincidence limit for that case.

In the distant‑past (DP) region, they use two complementary numerical approaches. First, they evolve characteristic initial data directly in the time domain, which captures the early‑time behaviour and validates the singularity structure. Second, they compute inverse Fourier integrals (inverse‑F) over complex frequencies. To ensure convergence they deform the integration contour, treat the high‑frequency tail analytically, and verify the results against the time‑domain evolution. Additionally, they perform a spectroscopic decomposition of each (\ell)-mode in the complex‑frequency plane, isolating contributions from quasinormal modes (QNMs) and the branch cut. This decomposition is used as a cross‑check for a selected (\ell)-mode and to illustrate the dominant physical content of the Green function at late times.

A key technical insight is the analysis of resonances between (\ell)-modes, which explains how the singularities arise from the infinite sum over angular modes. The authors also present asymptotic expansions for large real frequencies, confirming the expected power‑law tails and providing analytic control over the numerical inversions.

The results are displayed in a series of figures: the scalar Green function (Fig. 6), the spin‑2 RW Green function (Fig. 8) and its radial derivative (Figs. 9–10), and the spin‑2 BPT Green function and its radial derivative (Figs. 15–18). In the circular‑geodesic setting, the four‑fold singularity pattern is clearly visible, while in the static setting the two‑fold pattern emerges. Near each singularity, the gravitational Green functions exhibit oscillatory behaviour with frequencies matching the dominant QNM spectrum, a feature not present in the scalar case.

The paper concludes with a discussion of implications. The full RW and BPT Green functions are essential ingredients in the MiSaTaQuWa equation for the gravitational self‑force, and the demonstrated “matched‑expansions” method can now be applied to spin‑2 fields. The oscillatory singularities may affect the convergence of mode‑sum regularization schemes and could provide new diagnostics for gravitational wave memory and tail effects. Moreover, the availability of the complete Green functions opens the door to precise calculations of quantum detector response, entanglement harvesting, and communication protocols in black‑hole spacetimes.

Overall, this work delivers a comprehensive, validated set of Green functions for gravitational perturbations of Schwarzschild spacetime, bridging a long‑standing gap between scalar field results and the more physically relevant spin‑2 case, and establishing a robust toolbox for future analytical and numerical studies in black‑hole physics.