The imaginary part of the gravity action and black hole entropy

TL;DR

The paper investigates the imaginary part of the gravitational action in finite Lorentzian regions, showing it arises from signature flips of the boundary and is tied to an area-like entropy density. In GR, the imaginary action is expressed through flip-surface areas and reproduces black-hole entropy for certain boundary choices in stationary spacetimes; this extends to Lovelock gravity, where the imaginary part matches the corresponding entropy functional on flip surfaces. The results provide a Lorentzian, finite-region perspective on gravitational entropy, offering a potential non-stationary generalization and highlighting deep connections between boundary causality, topology, and thermodynamics. Collectively, the work suggests $ ext{Im} S$ may play a foundational role akin to entropy in gravity, with implications for holography and non-stationary gravitational thermodynamics.

Abstract

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.

Figures (8)

• Figure 1: An assignment of boost angles $\eta$ in the Lorentzian plane, according to eq. \ref{['eq:eta']}. This is one of two complex-conjugate choices, distinguished by the sign convention \ref{['eq:sign']}. The horizontal and vertical axes describe the spacelike and timelike components of a vector $n^\mu$. The angles are defined up to integer multiples of $2\pi i$.
• Figure 2: The boost angles $\eta$ from figure \ref{['fig:angles_plane']}, as applied to the normal vectors of a smooth closed boundary in 1+1d spacetime. The arrows indicate the normal direction. The normal's sign is chosen so that it has a positive scalar product with outgoing vectors. Empty circles denote "signature flips", where the normal becomes momentarily null. Contributions to the Gibbons-Hawking boundary term can be read off by traveling counter-clockwise and picking up angle differences, multiplied by $1/(8\pi G)$. The angles are defined up to integer multiples of $2\pi i - \theta$, where $\theta$ is a real finite deficit angle arising from spacetime curvature.
• Figure 3: A smooth closed boundary in spacetime, with flip surfaces indicated by empty circles. The dashed lines represent the two lightsheets passing through one of the flip surfaces. One lightsheet is tangent to the boundary and shares its normal vector $n^\mu \equiv L^\mu$. The other lightsheet is transverse, and has a different null normal $\ell^\mu$. The extents of the two null vectors are arbitrary.
• Figure 4: A boundary of mixed signature with the flip surfaces "hidden" in topological corners (designated by full circles). The imaginary part of the corner angles is $\pi/2$. The angles' real part can be arbitrary. For the particular boundary depicted here, the spacelike and timelike patches are orthogonal to each other. In this case, the angles' real part is zero.
• Figure 5: A purely spacelike closed boundary, composed of two intersecting hypersurfaces. The full circles denote the intersection surface. The arrows indicate the two boundary normals at each intersection point (due to the normals' timelike signature, the future-pointing "outgoing" normal belongs to the initial hypersurface, and vice versa). A continuous boost between these two normals involves two signature flips. As a result, the corner angle has an imaginary part equal to $\pi$.