Alternatives to Schwarzschild in the weak field limit of General Relativity

TL;DR

This paper investigates alternative static spherically symmetric solutions in General Relativity with power-law asymptotic behavior, exploring conditions on matter sources and implications for astrophysical observations.

Contribution

It derives conditions for constructing non-Schwarzschild metrics with arbitrary power-law limits and shows scalar fields can support such solutions.

Findings

Derived expressions for gravitational redshift and force in these metrics

Identified matter conditions for arbitrary power-law asymptotic behavior

Demonstrated scalar fields can support non-Schwarzschild solutions

Abstract

The metric outside an isolated object made up of ordinary matter is bound to be the classical Schwarzschild vacuum solution of General Relativity. Nevertheless, some solutions are known (e.g. Morris-Thorne wormholes) that do not match Schwarzschild asymptotically. On a phenomenological point of view, gravitational lensing in metrics falling as $1/r^q$ has recently attracted great interest. In this work, we explore the conditions on the source matter for constructing static spherically symmetric metrics exhibiting an arbitrary power-law as Newtonian limit. For such space-times we also derive the expressions of gravitational redshift and force on probe masses, which, together with light deflection, can be used in astrophysical searches of non-Schwarzschild objects made up of exotic matter. Interestingly, we prove that even a minimally coupled scalar field with a power-law potential can support non-Schwarzschild metrics with arbitrary asymptotic behaviour.