On the naturalness of Einstein's equation

TL;DR

This paper characterizes the Einstein tensor as the unique divergence-free 2-covariant tensor derived from a metric that is scale-invariant, providing insights into the foundations of Einstein's field equations in General Relativity.

Contribution

It provides a mathematical characterization of the Einstein tensor based on natural tensor constructions and divergence-free conditions, clarifying its uniqueness in the context of General Relativity.

Findings

Einstein tensor is uniquely characterized among natural tensors by divergence-free and scale-invariance properties.

Theorems connect the Einstein tensor to the energy-momentum tensor, supporting Einstein's field equations.

Results clarify the geometric foundations underlying Einstein's equations.

Abstract

We compute all 2-covariant tensors naturally constructed from a semiriemannian metric which are divergence-free and have weight greater than -2. As a consequence, it follows a characterization of the Einstein tensor as the only, up to a constant factor, 2-covariant tensor naturally constructed from a semiriemannian metric which is divergence-free and has weight 0 (i.e., is independent of the unit of scale). Since these two conditions are also satisfied by the energy-momentum tensor of a relativistic space-time, we discuss in detail how these theorems lead to the field equation of General Relativity.