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arXiv · 1202.5287

The Riemann-Lovelock Curvature Tensor

David Kastor
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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.

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David Kastor. 2012-02-23. The Riemann-Lovelock Curvature Tensor. https://doi.org/10.1088/0264-9381%2F29%2F15%2F155007

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