The imaginary part of the gravity action and black hole entropy

As observed recently in arXiv:1212.2922, the action of General Relativity (GR) in finite spacetime regions has an imaginary part that resembles the Bekenstein entropy. In this paper, we expand on that argument, with attention to different causal types of boundaries. This property of the GR action may open a new approach to the puzzles of gravitational entropy. In particular, the imaginary action can be evaluated for non-stationary finite regions, where the notion of entropy is not fully understood. As a first step in constructing the precise relationship between the imaginary action and entropy, we focus on stationary black hole spacetimes. There, we identify a class of bounded regions for which the action's imaginary part precisely equals the black hole entropy. As a powerful test on the validity of the approach, we also calculate the imaginary action for Lovelock gravity. The result is related to the corresponding entropy formula in the same way as in GR.