Lovelock gravity and Weyl's tube formula

TL;DR

This paper explores the connection between Lovelock gravity in higher dimensions and Weyl's tube formula, proposing that certain curvature inequalities naturally arise from geometric considerations, with implications for classical gravity theories.

Contribution

It demonstrates how Weyl's tube formula can be used to derive curvature inequalities that ensure well-posed Lovelock gravity in higher dimensions.

Findings

Curvature inequalities emerge naturally from Weyl's tube formula.

Weyl's formula is generalized for tubes in Minkowski space.

Connections between geometric volume formulas and gravitational theories are established.

Abstract

In four space-time dimensions, there are good theoretical reasons for believing that General Relativity is the correct geometrical theory of gravity, at least at the classical level. If one admits the possibility of extra space-time dimensions, what would we expect classical gravity to be like? It is often stated that the most natural generalisation is Lovelock's theory, which shares many physical properties with GR. But there are also key differences and problems. A potentially serious problem is the breakdown of determinism, which can occur when the matrix of coefficients of second time derivatives of the metric degenerates. This can be avoided by imposing inequalities on the curvature. Here it is argued that such inequalities occur naturally if the Lovelock action is obtained from Weyl's formulae for the volume and surface area of a tube. Part of the purpose of this article is to give a treatment of the Weyl tube formula in terminology familiar to relativists and to give an appropriate (straightforward) generalisation to a tube embedded in Minkowski space.