Laws of black hole mechanics from the Holst action

The formulation of weak isolated horizons (WIH) based on the standard isolated horizon formulation of black hole horizons has been reconsidered in the context of the Holst action. Both the zeroth and the first laws of WIH mechanics have been derived. While the zeroth law follows directly from the WIH boundary conditions, first law depends on the action chosen. We construct the covariant phase space of the Holst action for a space-time having WIH as the inner boundary. This requires the introduction of new potential functions so that the symplectic structure is foliation independent. We show that a precise cancellation among various terms leads to the usual first law. Subsequently, we show from the same covariant phase space that for spherically symmetric horizons, the effective theory on the inner boundary is a $U(1)$ Chern-Simons theory.

Figure 1

$M_{\pm }$ are two partial Cauchy surfaces enclosing a region of space-time and intersecting $\Delta$ in the two-spheres $S_{\pm }$, respectively, and extend to spatial infinity $i^o$. Another Cauchy slice $M$ is drawn which intersects $\Delta$ in $S_{\Delta }$.