What is the meaning of non-uniqueness of FRW and Schwarzschild metrics?

TL;DR

The paper discusses how the non-uniqueness of FRW and Schwarzschild metrics arises from invariance under geodesic mappings, implying that these metrics do not uniquely determine the gravitational field in general relativity.

Contribution

It demonstrates that geodesic invariance leads to multiple metrics representing the same gravitational field, challenging the uniqueness of standard cosmological and black hole solutions.

Findings

FRW and Schwarzschild metrics are invariant under geodesic mappings.

Standard metrics do not uniquely determine the gravitational field.

Geodesic invariance affects the interpretation of space-time geometries.

Abstract

It is shown that any theory of gravitation, based on the hypothesis of the geodesic motion of test particles must be invariant under geodesic (projecive) mappings of the used space-time. The reason is that due to invariance of the equations of geodesic lines under a continuous group of transformations of the coefficients of affine connection, there is a wide class of transformations of the geometrical objects of Riemannian space-time which leaves invariant the equations of motion of test particles. The FRW metric in cosmology and the Schwarzschild metric are a good example to make sure that the standard space-time metrics does not determine the gravitational field unequivocally.