# Reduction of a family of metric gravities

## TL;DR

This paper reviews Shuler's family of metric theories of gravity, highlighting that only the Schwarzschild metric aligns with a relativistic Newtonian law, and discusses the limitations of the postulates used to derive these metrics.

## Contribution

It critically analyzes Shuler's derivation of a family of metrics, demonstrating only the Schwarzschild metric's physical consistency with relativistic gravity laws.

## Key findings

- All Shuler's metrics agree with Schwarzschild in weak fields.
- Only the Schwarzschild metric satisfies a relativistic Newtonian law.
- Other metrics in the family are likely unphysical.

## Abstract

A recent proposal by Shuler regarding a postulate-based derivation of a family of metrics describing the gravitational field outside a static spherically symmetric mass distribution is reviewed. All of Shuler's gravities agree with the Schwarzschild solution in the weak-field limit, but they differ in the strong-field domain, i.e., close enough to a sufficiently compact source of the field. It is found that the evoked postulates of i) momentum conservation and ii) consistency of field strength measurement are satisfied in all metric theories of gravity compatible with the Einstein equivalence principle, no matter what the form of the metric. Therefore, they cannot be used, within any correct deduction, to derive a particular metric. Shuler's derivations are based on an inconsistent set of correspondences between local and distant quantities. Furthermore, it is shown here that out of the family of possible metrics given by Shuler only one member, the Schwarzschild metric, satisfies a standard relativistic generalization of Newton's law of gravitation, suggesting the others to be unphysical.