Assuming a Lagrangian proportional to the following terms:
##L \sim ( \partial_\mu \sigma) (\partial^\mu \sigma) - g^{m\bar{n}} g^{r\bar{p}} (\partial_\mu g_{mr} ) ( \partial^\mu g_{\bar{n}\bar{p}} ) ~~~~~ \to (1) ##
Where ##\mu =0,1,2,3,4## and m, n,r, p and ##\bar{n}, \bar{p}, \bar{m}## and...