LAGRANGIAN PERFECT FLUIDS AND BLACK-HOLE MECHANICS

The first law of black hole mechanics (in the form derived by Wald) is expressed in terms of integrals over surfaces, at the horizon and spa tial infinity, of a stationary, axisymmetric black hole, in a diffeomo rphism-invariant Lagrangian theory of gravity. The original statement of the first law given by Bardeen, Carter, and Hawking for an Einstein -perfect fluid system contained, in addition, volume integrals of the fluid fields, over a spacelike slice stretching between these two surf aces. One would expect that Wald's methods, applied to a Lagrangian Ei nstein-perfect fluid formulation, would convert these terms to surface integrals. However, because the fields appearing in the Lagrangian of a gravitating perfect fluid are typically nonstationary (even in a st ationary black-hole-perfect-fluid spacetime) a direct application of t hese methods generally yields restricted results. We therefore first a pproach the problem of incorporating general nonstationary matter fiel ds into Wald's analysis, and derive a first-law-like relation for an a rbitrary Lagrangian metric theory of gravity coupled to arbitrary Lagr angian matter fields, requiring only that the metric field be stationa ry. This relation includes a volume integral of matter fields over a s pacelike slice between the black hole horizon and spatial infinity, an d reduces to the first law originally derived by Bardeen, Carter, and Hawking when the theory is general relativity coupled to a perfect flu id. We then turn to consider a specific Lagrangian formulation for an isentropic perfect fluid given by Carter, and directly apply Wald's an alysis, assuming that both the metric and fluid fields are stationary and axisymmetric in the black hole spacetime. The first law we derive contains only surface integrals at the black hole horizon and spatial infinity, but the assumptions of stationarity and axisymmetry of the f luid fields make this relation restrictive in its allowed fluid config urations and perturbations than that given by Bardeen, Carter, and Haw king. In the Appendix, we use the symplectic structure of the Einstein -perfect fluid 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.