Imaginary part of the gravitational action at asymptotic boundaries and horizons

We study the imaginary part of the Lorentzian gravitational action for bounded regions, as described in previous work [Y. Neiman, arXiv:1301.7041]. By comparison to a Euclidean calculation, we explain the agreement between the formula for this imaginary part and the formula for black hole entropy. We also clarify the topological structure of the imaginary part in Lovelock gravity. We then evaluate the action’s imaginary part for some special regions. These include cylindrical slabs spanning the exterior of a stationary black hole spacetime and “maximal diamonds” in various symmetric spacetimes, as well as local near-horizon regions. In the first setup, the black hole’s entropy and conserved charges contribute to the action’s imaginary and real parts, respectively. In the other two setups, the imaginary part coincides with the relevant entropy.

Figure 1

A smooth closed boundary in Lorentzian spacetime. The arrows indicate the normal direction at various points. 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.

Figure 2

A Lorentzian region in a stationary black hole spacetime, analogous to the “entire-spacetime” regions considered in Euclidean calculations. The Lorentzian region is bounded by two time slices and two radial slices. One radial slice is just outside the bifurcation surface, while the other is at a very large radius $r_{\infty }$.

Figure 3

Integration contours in the exponentiated time coordinate $\zeta$. The $\Delta t$-dependent piece of the action for the Lorentzian region in Fig. 2 is given by a line integral along the positive axis. The corresponding Euclidean action is given by a circular integral around the pole.

Figure 4

Some terminology for null-bounded regions. The lines represent codimension-1 light sheets; the dots represent codimension-2 spacelike surfaces. The arrows indicate the “outgoing” null normal that corresponds to the outgoing covector, for a mostly plus metric signature. In $d>2$ dimensions with trivial topology, the left and right sides of the figure are understood to be connected.

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. A continuous boost between these two normals involves two signature flips. As a result, the corner angle has an imaginary part equal to $\pi$.

Figure 6

Regions with null asymptotic boundaries in Minkowski-like spacetimes. (a) The region covering all of Minkowski space ${\mathbb{R}}^{d-1,1}$ for $d>2$. (b) The region covering an entire spacetime of the form ${\mathbb{R}}^{1,1}\times {\mathcal{M}}_{d-2}$, where ${\mathcal{M}}_{d-2}$ is some compact manifold with area $A$. (c) The region covering the Rindler wedge around the origin of ${\mathbb{R}}^{1,1}$ in the same spacetime. We have $\mathrm{Im}S=0$ for case (a) and $\mathrm{Im}S=A/4G$ for cases (b, c).

Figure 7

Regions in the external spacetime of stationary black holes. In (a–b), the spacetime is asymptotically flat, while in (c–f) it is asymptotically AdS. Figures (a, c, e) depict finite regions, which through a limiting procedure become the asymptotic regions in (b, d, f), respectively. In every case, the action’s imaginary part equals the black hole’s entropy.

Figure 8

Two kinds of “maximal diamonds” in anti–de Sitter space: (a) A causal diamond with tip points in the bulk and an equator at spatial infinity. (b) A Poincaré patch. In both cases, the action’s imaginary part vanishes.

Figure 9

“Maximal diamonds” (shaded) in (a) standard de Sitter space $d{S}_d$ and (b) elliptic de Sitter space $d{S}_d/{\mathbb{Z}}_2$. Every point in the Penrose diagrams is a sphere $S_{d-2}$ in spacetime. In (b), each sphere is identified with the sphere that lies opposite from it across the diagram’s center. Each point on one sphere is identified with the antipodal point on the other. The numbered arrows trace a continuous null path along the diamond’s boundary.

Figure 10

A local near-horizon region in a noneternal, nonstationary black hole spacetime. The imaginary part of this region’s action reproduces the instantaneous entropy of the black hole. The relevant horizon for the discussion in the text is the event horizon.