Metric tensor at second perturbation order for spherically symmetric space-times

TL;DR

This paper explores the conditions under which the metric components in spherically symmetric space-times relate, revealing that they are generally unequal unless the potential follows an inverse radial law, with implications for gravitational theories.

Contribution

It derives a general formula for the metric components at second perturbation order and applies it to specific cases in spherically symmetric space-times.

Findings

Time metric component differs from the negative inverse radial component unless potential is inverse proportional to radius.

A general perturbation formula is established for spherically symmetric metrics.

Applications demonstrate the conditions for metric component relations in weak field limits.

Abstract

It is shown in this article that if the Einstein Equivalence Principle is valid on a particular metric theory of gravitation in a spherically symmetric space-time, then the time metric component is not equal to the negative of the inverse radial one unless the underlying potential is inversely proportional to the radial coordinate. At the weak field limit of approximation, a general formula is calculated and applied to some useful cases.