Black Hole Mass Formula Is a Vanishing Noether Charge

The Noether current and its variation relation with respect to diffeomorphism invariance of gravitational theories have been derived from the horizontal variation and vertical-horizontal bi-variation of the Lagrangian, respectively. For Einstein’s GR in the stationary, axisymmetric black holes, the mass formula in vacuum can be derived from this Noether current although it definitely vanishes. This indicates that the mass formula of black holes is a vanishing Noether charge in this case. The first law of black hole thermodynamics can also be derived from the variation relation of this vanishing Noether current. PACS No. 04.20.Cv, 97.60.Lf 1. It is well known that Noether’s theorem links the conservation law of certain quantity with certain symmetry. From 1970’s, one of the most exciting discoveries in GR is the black hole mechanics that relates among others the geometric properties of black holes with the thermodynamic laws. In thermodynamics, the first law exhibits the relation among the changes of energy, entropy and other macroscopical quantities. On the other hand, in either classical mechanics or field theories the energy conservation law is directly related with the time translation invariance. Furthermore, its covariant form is closely related with the reparameterization invariance of spacetime coordinates, i.e. the diffeomorphism invariance of the theory . Therefore, it is significant to explore whether the diffeomorphism invariance of the gravitational theories should be behind the mass formula and its differential one, i.e. the first law of black hole thermodynamics. During last decade, Wald and his collaborators as well as other authors (see, for example, [1]-[5]) have studied this problem and found certain link between the first law of black hole Generally speaking, diffeomorphism is not a group but a pseudo-group. In some literatures, the term of diffeomorphism invariance is restricted to the re-parameterization invariance of spacetime coordinates that keeps the line-element ds being invariant. This restriction is adopted here.