# What is the action for a rotating blackhole with angular momentum $J$?

The action for an electromagnetic theory is given by $$S_{EM}[g^{\mu\nu},A^{\mu}]=-\frac{1}{4\mu_0}\int F_{\alpha\beta}F^{\alpha\beta}\sqrt{-g}d^4x$$ and the action for charged relativistic dust is $$S_q[x^{\mu},A^{\mu}]=-\int\rho_{EM}v^{\mu}A_{\mu}\sqrt{-g}d^4x$$ where $A^{\mu}$ is the electromagnetic field (potential), $g^{\mu\nu}$ is the gravitational field, $\rho_{EM}$ is the charge density of dust, $v^{\mu}=\frac{dx^{\mu}}{dt}$ is the $4$-velocity of dust, and $F_{\alpha\beta}=\nabla_{\alpha}A_{\beta}-\nabla_{\beta}A_{\alpha}$ is the electromagnetic field tensor. Using these as general references for the form of an action, is it then possible to write down an action for a rotating blackhole with angular momentum $J$?