The Riemann-Lovelock Curvature Tensor

TL;DR

This paper introduces the Riemann-Lovelock tensor, a higher-order curvature tensor in Lovelock gravity, and explores its properties and implications for solutions in various dimensions, generalizing Einstein gravity results.

Contribution

It defines the kth-order Riemann-Lovelock tensor, analyzes its properties, and shows its implications for solutions in low-dimensional Lovelock gravity theories.

Findings

In D=2k+1, all pure kth-order Lovelock solutions are Riemann-Lovelock flat.

The tensor is determined by its traces in dimensions 2k <4k.

Static, spherically symmetric solutions satisfy the flatness property.

Abstract

In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann-Lovelock tensor as a certain quantity having a total 4k-indices, which is kth-order in the Riemann curvature tensor and shares its basic algebraic and differential properties. We show that the kth-order Riemann-Lovelock tensor is determined by its traces in dimensions 2k \le D <4k. In D=2k+1 this identity implies that all solutions of pure kth-order Lovelock gravity are `Riemann-Lovelock' flat. It is verified that the static, spherically symmetric solutions of these theories, which are missing solid angle space times, indeed satisfy this flatness property. This generalizes results from Einstein gravity in D=3, which corresponds to the k=1 case. We speculate about some possible further consequences of Riemann-Lovelock curvature.