Schwarzschild-Tangherlini Metric from Scattering Amplitudes

TL;DR

This paper develops a framework to derive the Schwarzschild-Tangherlini metric in arbitrary dimensions from scattering amplitudes, confirming its validity to second order and revealing a logarithmic radial dependence in five dimensions.

Contribution

It introduces a gauge-independent method to obtain the metric from scattering amplitudes in any dimension, including quantum corrections up to one-loop order.

Findings

The metric is gauge-independent and satisfies the classical gauge condition.

Explicit second-order calculations verify the framework.

A logarithmic radial dependence appears in five dimensions.

Abstract

We present a general framework with which the Schwarzschild-Tangherlini metric of a point particle in arbitrary dimensions can be derived from a scattering amplitude to all orders in the gravitational constant, $G_N$, in covariant gauge (i.e. $R_\xi$-gauge) with a generalized de Donder-type gauge function, $G_\sigma$. The metric is independent of the covariant gauge parameter $\xi$ and obeys the classical gauge condition $G_\sigma=0$. We compute the metric with the generalized gauge choice explicitly to second order in $G_N$ where gravitational self-interactions become important and these results verify the general framework to one-loop order. Interestingly, after generalizing to arbitrary dimension, a logarithmic dependence on the radial coordinate appears in space-time dimension $D=5$.