Lagrangian perfect uids and black hole mechanics

The rst law of black hole mechanics (in the form derived by Wald), is expressed in terms of integrals over surfaces, at the horizon and spatial in nity, of a stationary, axisymmetric black hole, in a di eomorphism invariant Lagrangian theory of gravity. The original statement of the rst law given by Bardeen, Carter and Hawking for an Einstein-perfect uid system contained, in addition, volume integrals of the uid elds, over a spacelike slice stretching between these two surfaces. One would expect that Wald's methods, applied to a Lagrangian Einstein-perfect uid formulation, would convert these terms to surface integrals. However, because the elds appearing in the Lagrangian of a gravitating perfect uid are typically nonstationary, (even in a stationary black hole-perfect uid spacetime) a direct application of these methods generally yields restricted results. We therefore rst approach the problem of incorporating general nonstationary matter elds into Wald's analysis, and derive a rst law-like relation for an arbitrary Lagrangian metric theory of gravity coupled to arbitrary Lagrangian matter elds, requiring only that the metric eld be stationary. This relation includes a volume integral of matter elds over a spacelike slice between the black hole horizon and spatial in nity, and reduces to the rst law originally derived by Bardeen, Carter and Hawking when the theory is general relativity coupled to a perfect uid. We then turn to consider a speci c Lagrangian formulation for an isentropic perfect uid given by Carter, and directly apply Wald's analysis, assuming that both the metric and uid elds are stationary and axisymmetric in the black hole spacetime. The rst law we derive contains only surface integrals at the black hole horizon and spatial in nity, but the assumptions of stationarity and axisymmetry of the uid elds make this relation much more restrictive in its allowed uid congurations and perturbations than that given by Bardeen, Carter and Hawking. In the Appendix, we use the symplectic structure of the Einstein-perfect uid system to derive a conserved current for perturbations of this system: this current reduces to one derived ab initio for this system by Chandrasekhar and Ferrari.