On the Euclidean Action of de Sitter Black Holes and Constrained Instantons

Abstract

We compute the on-shell Euclidean action of Schwarzschild-de Sitter black holes, and take their contributions in the gravitational path integral into account using the formalism of constrained instantons. Although Euclidean de Sitter black hole geometries have conical singularities for generic masses, their on-shell action is finite and is shown to be equal to minus the sum of the black hole and cosmological horizon entropy. We apply this result to compute the probability for a nonrotating, neutral arbitrary mass black hole to nucleate spontaneously in empty de Sitter space, which separates into a "perturbative" and a "nonperturbative" contribution, the latter corresponding to the proper saddle-point instanton in the Nariai limit. We also speculate on some further applications of our results, most notably the possible non-perturbative corrections to anti-podal correlators in the de Sitter vacuum.

Black holes in dS3

“…In figure 7 we plot the temperatures T c , T h , and T as a function of mass M . The behavior of the temperatures are essentially identical to that of classical four-dimensional Schwarzschild-de Sitter black holes (see, for instance figure 2 of [31]). In particular, notice for small mass M black holes, the black hole temperature diverges, such that it may be approximated by the temperature of a (quantum) Schwarzschild black hole,…”

“…We observe that the sum of the black hole and cosmological horizon entropies S (3) gen,h and S (3) gen,c produces an approximately linear curve always equal to or less than the entropy of the quantum de Sitter solution (5.28), see figure 9. This is reminiscent of the observation in [31,38] for the classical SdS solution that the sum of the horizon entropies is approximately a linear function of the mass. We will return to this point in section 6, as it will prove useful when computing the nucleation rate of quantum dS black holes.…”

“…Therefore, when = 0 the Nariai limit places an upper bound on µ. Note that the relation (4.16) may be inverted to find closed form expressions of r h,c in terms of µ and r N (see, e.g., [30,31])…”

“…Importantly, we will see the entropy deficit of a quantum SdS black hole takes on a similar form as its classical four-dimensional counterpart, however, where the classical Bekenstein-Hawking entropy is replaced by the generalized entropy. Moreover, using the fact that the generalized entropy is a linear function of the mass, we compute the nucleation rate using the method of constrained instantons, extending [31] to the case when the backreaction of quantum fields is included.…”

Comments on the Entanglement Spectrum of de Sitter Space

“…It was also noted in [3] that the Schwarzschild de Sitter black hole entropy formula implies that maximal entropy corresponds to empty de Sitter space, so that localized objects in a causal patch of de Sitter space correspond to constrained, low entropy states. In terms of the gravitational path integral, this means that Euclidean SdS is a semiclassical solution subject to a constraint on the quasilocal energy [6] [7]. The migration of a typical localized object to the cosmological horizon, where it becomes absorbed in the featureless background, is then understood as equilibration of this low entropy state.…”