Kerr Metric - Gradient Operator

Gradient Operator

Since even a direct check on the Kerr metric involves cumbersome calculations, the contravariant components of the metric tensor are shown below in the expression for the square of the four-gradient operator:

$g^{\mu\nu}\frac{\partial}{\partial{x^{\mu}}}\frac{\partial}{\partial{x^{\nu}}} = \frac{1}{c^{2}\Delta}\left(r^{2} + \alpha^{2} + \frac{r_{s}r\alpha^{2}}{\rho^{2}}\sin^{2}\theta\right)\left(\frac{\partial}{\partial{t}}\right)^{2} + \frac{2r_{s}r\alpha}{c\rho^{2}\Delta}\frac{\partial}{\partial{\phi}}\frac{\partial}{\partial{t}} \,\, -$

$\frac{1}{\Delta\sin^{2}\theta}\left(1 - \frac{r_{s}r}{\rho^{2}}\right)\left(\frac{\partial}{\partial{\phi}}\right)^{2} - \frac{\Delta}{\rho^{2}}\left(\frac{\partial}{\partial{r}}\right)^{2} - \frac{1}{\rho^{2}}\left(\frac{\partial}{\partial{\theta}}\right)^{2}$