Conservation of energy and Gauss Bonnet gravity

TL;DR

This paper classifies tensors derived from the Riemann tensor that conserve gravitational energy, revealing a unique tensor structure linked to Euler gravity in even dimensions, with detailed focus on the case n=2.

Contribution

It identifies a unique tensor of degree n from the Riemann tensor that ensures energy conservation, linking it to Euler gravity in all dimensions, specifically detailed for n=2.

Findings

Existence of a unique energy-conserving tensor from the Riemann tensor.

This tensor characterizes Euler gravity in dimension 2n.

The classification applies broadly across dimensions, with detailed analysis for n=2.

Abstract

It is shown how can be made the classification of all tensors constructed from the Riemann tensor that verify the conservation of gravitational energy momentum. More precisely we explain that there exists a unique tensor of degree n in the Riemann tensor and its contractions that verifies the conservation of energy. We show that this tensor, only because it obeys this degree n structure as well as energy conservation, two facts which are true in all dimensions, verifies in dimension 2n this striking particularity of being Euler gravity. We stick here to the case n=2 but explain briefly why the general case is similar.