Entropy of Null Surfaces and Dynamics of Spacetime

Abstract

The null surfaces of a spacetime act as one-way membranes and can block information for a corresponding family of observers (time-like curves). Since lack of information can be related to entropy, this suggests the possibility of assigning an entropy to the null surfaces of a spacetime. We motivate and introduce such an entropy functional for any vector field in terms of a fourth-rank divergence free tensor P_{ab}^{cd} with the symmetries of the curvature tensor. Extremising this entropy for all the null surfaces then leads to equations for the background metric of the spacetime. When P_{ab}^{cd} is constructed from the metric alone, these equations are identical to Einstein's equations with an undetermined cosmological constant (which arises as an integration constant). More generally, if P_{ab}^{cd} is allowed to depend on both metric and curvature in a polynomial form, one recovers the Lanczos-Lovelock gravity. In all these cases: (a) We only need to extremise the entropy associated with the null surfaces; the metric is not a dynamical variable in this approach. (b) The extremal value of the entropy agrees with standard results, when evaluated on-shell for a solution admitting a horizon. The role of full quantum theory of gravity will be to provide the specific form of P_{ab}^{cd} which should be used in the entropy functional. With such an interpretation, it seems reasonable to interpret the Lanczos-Lovelock type terms as quantum corrections to classical gravity.