# Metric tensor: trace

Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 6;  5.

A specific case of the trace of a tensor is the trace of the metric tensor, which is given by ${g_{ij}g^{ij}}$. Since ${g^{ij}}$ is the inverse of the metric tensor ${g_{ij}}$, ${g_{ik}g^{kj}=\delta_{\;\; i}^{j}}$ is the identity matrix, which means it is diagonal with every diagonal element equal to 1. The trace of the identity matrix is simply ${n}$, the dimension of the matrix. Thus in 2-d it would be ${n=2}$ and so on.