Lovelock gravity and Weyl’s tube formula

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 generalization is Lovelock’s theory, which shares many physical properties with general relativity. 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 formulas 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) generalization to a tube embedded in Minkowski space.

Figure 1

The tube of a section of de Sitter space $d{S}_r^{N-1}$ embedded in Minkowski space ${\mathbb{M}}^N$. A slice through the $(X,T)$ plane is shown. The tube is the shaded region between concentric $dS$ spaces of curvature radii $r-l$ and $r+l$. The normal vectors are aligned with rays through the origin.