Two Theorems on Flat Space-Time Gravitational Theories

TL;DR

This paper presents two theorems constraining flat space-time gravitational theories, showing they must have specific Lagrangian forms and cannot satisfy the strong equivalence principle simultaneously.

Contribution

It proves that all flat space-time gravitational theories have a particular Lagrangian structure and cannot meet the strong equivalence principle, highlighting fundamental limitations.

Findings

Flat space-time theories have a Lagrangian with a homogeneous first term.

Theories satisfying the strong equivalence principle have a specific Lagrangian form.

Such theories cannot be both flat space-time and satisfy the strong equivalence principle.

Abstract

The first theorem states that all flat space-time gravitational theories must have a Lagrangian with a first term that is an homogeneous (degree-I) function of the 4-velocity $u^i$, plus a functional of $\eta_{ij}u^i u^j$. The second theorem states that all gravitational theories that satisfy the strong equivalence principle have a Lagrangian with a first term $g_{ij}(x)u^i u^j$ plus an irrelevant term. In both cases the theories must issue from a unique variational principle. Therefore, under this condition it is impossible to find a flat space-time theory that satisfies the strong equivalence principle.