where the index 5 indicates the fifth coordinate by convention even though the first four coordinates are indexed with 0, 1, 2, and 3.
\widetilde{g}^{ab} \equiv \begin{bmatrix}g^{\mu\nu} & -A^\mu \\ -A^\nu & g_{\alpha\beta}A^\alpha A^\beta + {1 \over \phi^2} \end{bmatrix}.