BLACK-HOLE ENTROPY IS THE NOETHER CHARGE

We consider a general, classical theory of gravity in n dimensions, ar ising from a diffeomorphism-invariant Lagrangian. In any such theory, to each vector field xi(a) on spacetime one can associate a local symm etry and, hence, a Noether current (n - 1)-form j and (for solutions t o the field equations) a Noether charge (n - 2)-form Q, both of which are locally constructed from xi(a) and the fields appearing in the Lag rangian. Assuming only that the theory admits stationary black hole so lutions with a bifurcate Killing horizon (with bifurcation surface SIG MA), and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole me chanics always holds for perturbations to nearby stationary black hole solutions. The quantity playing the role of black hole entropy in thi s formula is simply 2pi times the integral over SIGMA of the Noether c harge (n - 2)-form associated with the horizon Killing field. Furtherm ore, we show that this black hole entropy always is given by a local g eometrical expression on the horizon of the black hole. We thereby obt ain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity. Our results show that the validity of the ''second law'' of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equival ent to the positivity of a total Noether flux, and thus may be intimat ely related to the positive energy properties of the theory. The relat ionship between the derivation of our formula for black hole entropy a nd the derivation via 'Euclidean methods'' also is explained.