Riemannian general relativity

General relativity has as one of its fundamental principle the signature of the metric tensor. Spacetime has a metric tensor with a mixed signature, usually $(1, n-1)$ or $(n-1, 1)$, for a spacetime of dimension $n$, so that the metric tensor has $1$ eigenvector of positive norm and $(n-1)$ eigenvectors of negative norms, or vice-versa (I usually use the convention of $1$ negative norm eigenvector).

There are some benefits to treating spacetime as a Riemannian manifold, where every eigenvector has a positive norm, usually for mathematical simplicity, as used in Wick rotations, Euclidian gravity and other such processes. But here we will not consider this as a mathematical simplification, but as a physical theory. This is inspired by Greg Egan's treatment of the idea in his series of books Orthogonal, as well as the notes on his websites, as I thought this was an interesting topic.

Topology and causality

Despite the relatively simple change of just switching the metric signature, the changes can be quite widespread. Fortunately, most of the theorems relating to this already exist, as this is simply Riemannian geometry.

Unlike the Lorentzian case, every (paracompact) manifold admits a Riemannian metric, unlike say the case of the sphere in even dimensions for the Lorentz signature.

There are no differentiated curves in the Riemannian case, and therefore no causal structure as such. This makes for a very different

The Einstein field equations

There is no fundamental difference in the Riemannian case as far as the dynamics go. The Einstein-Hilbert action can remain

\begin{equation} S[g] = \int_{M} (R[g] + \Lambda + \mathcal{L}_M[g]) d\mu[g] \end{equation}

Leading to the Einstein field equations

\begin{equation} R_{mn} - \frac{1}{2} R g_{mn} = T_{mn} \end{equation}

Matter fields

An important case to consider is as usual the point particle, using for instance the Nambu-Goto action.

\begin{equation} S[g, X] = \int d\lambda \sqrt{g_{mn}(X(\lambda)) \dot{X}^m(\lambda)\dot{X}^n(\lambda)} d\mu[g] \end{equation}

or, in the

Last updated : 2019-10-15 11:40:43
Tags : physics , general-relativity