We show that retarded Green’s functions of black hole spacetimes can be expressed as a convergent mode sum everywhere in spacetime. At late times a quasinormal mode sum converges, while at early times a Matsubara (or Euclidean) mode sum converges. The two regions are separated by a light cone that scatters from the black hole potential. The Matsubara sum is a Fourier series on the Euclidean thermal circle associated with the early time region. We illustrate our results for the Pöschl-Teller model, the Bañados-Teitelboim-Zanelli black hole, and the Schwarzschild black hole. In the case of the Schwarzschild black hole, we express the branch cut contribution as a convergent sum of de Sitter quasinormal modes as $\mathrm{\Lambda }\to 0^{+}$ and exploit recent exact solutions to the Heun connection problem. In each case, we analytically show convergence by studying the asymptotics of residue sums and also provide numerical demonstrations.

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• Classical black holes
• Gravitational wave sources