Entropy of null surfaces and dynamics of spacetime

Abstract

The null surfaces of a spacetime act as oneway membranes and can block information for a corresponding family of observers (timelike 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 Pcd ab with the symmetries of the curvature tensor. Extremizing this entropy for all the null surfaces then leads to equations for the background metric of the spacetime. When Pcd ab 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 Pcd ab 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 extremize 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 the full quantum theory of gravity will be to provide the specific form of Pcd ab 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.