Solutions of the Spherically Symmetric Wave Equation in $p+q$ Dimensions

TL;DR

This paper explores solutions to the spherically symmetric wave and Klein-Gordon equations across various dimensions, introducing methods to generate new solutions, analyze dimensional shifts, and extend to curved spacetimes, with implications for black hole physics.

Contribution

It presents novel procedures for generating solutions in different dimensions, including fractional derivatives, and discusses analytic continuation and transformations related to black holes.

Findings

Methods for generating solutions across dimensions

Analysis of pole structures in fractional operators

Operators transforming time into space coordinates

Abstract

We discuss solutions of the spherically symmetric wave equation and Klein Gordon equation in an arbitrary number of spatial and temporal dimensions. Starting from a given solution, we present various procedures to generate futher solutions in the same or in different dimensions. The transition from odd to even or non integer dimensions can be performed by fractional derivation or integration. The dimensional shift, however, can also be interpreted simply as a modification of the dynamics. We also discuss the analytic continuation to arbitrary real powers of the D'Alembert operator. There, particular peculiarities in the pole structure show up when $p$ and $q$ are both even. Finally we give operators which transform a time into a space coordinate and v.v. and comment on their possible relation to black holes. In this context, we describe a few aspects of the extension of our discussion to a curved metrics.