5d Schwarzschild-Tangherlini spacetime: MST-like formalism for a Reduced Confluent Heun Equation

Abstract

We study the five-dimensional Schwarzschild-Tangherlini solution, with particular attention to its geodesic structure and massless scalar perturbations. In the probe limit, we present two applications. First, we compute the scattering angle for unbound geodesics showing both post-Newtonian and post-Minkowskian type expansions, and succeeding in resumming the resulting series in terms of hypergeometric functions. Second, we derive the Lyapunov exponent for deviations from a critical circular orbit, which is relevant to the eikonal estimation of quasinormal modes. We then investigate the dynamics of massless scalar $(s=0)$ perturbations, for which the radial equation becomes a Reduced Confluent Heun equation. In this $d=5$ Schwarzschild-Tangherlini case we develop an original extension of the standard Mano-Suzuki-Takasugi (MST) formalism and validate the construction by computing the renormalized angular-momentum parameter $ν$, whose value agrees with an independent determination based on the quantum Seiberg-Witten formalism. Finally, we analyze the energy flux from circular orbits, obtaining post-Newtonian results through 2.5PN order.