[Paper Review] de Sitter Black Holes as Constrained States in the Euclidean Path Integral

Patrick Draper, Szilárd Farkas|arXiv (Cornell University)|Mar 4, 2022

Black Holes and Theoretical PhysicsPhysics and Astronomy21 references29 citations

TL;DR

This paper shows that Schwarzschild-de Sitter black holes, despite having two horizons at different temperatures and thus a non-smooth Euclidean continuation, can be treated as genuine saddle points in a constrained Euclidean path integral. By fixing boundary data (metric, extrinsic curvature, lapse) on a sphere enclosing the black hole, the path integral factorizes, and the semiclassical action of the black hole solution is found to be −(S_bh + S_c) = −S_SdS, leading to a probability weight e^{−(S_SdS − S_dS)} for finding a black hole in de Sitter space.

ABSTRACT

Schwarzschild-de Sitter black holes have two horizons that are at different temperatures for generic values of the black hole mass. Since the horizons are out of equilibrium the solutions do not admit a smooth Euclidean continuation and it is not immediately clear what role they play in the gravitational path integral. We show that Euclidean SdS is a genuine saddle point of a certain constrained path integral, providing a consistent Euclidean computation of the probability $\sim e^{-(S_{dS}-S_{SdS})}$ to find a black hole in the de Sitter bath.

Motivation & Objective

• To resolve the ambiguity in the role of de Sitter black holes in the gravitational path integral, which are not smooth saddle points due to differing horizon temperatures.
• To show that a constrained path integral—fixing boundary data on a sphere enclosing the black hole—can yield a well-defined saddle point for SdS solutions.
• To provide a semiclassical derivation of the probability e^{−(S_SdS − S_dS)} for black hole formation in de Sitter space.
• To clarify the physical interpretation of the Euclidean SdS geometry as a constrained state, where conical singularities are replaced by discontinuities in local temperature.

Proposed method

• Use of the Hamiltonian (ADM) formalism with a foliation of spacetime into regions separated by a static surface T at fixed ρ.
• Imposition of a constraint via delta functionals on boundary data (sab, k, Na) at T to fix the black hole mass and extrinsic curvature.
• Adoption of microcanonical boundary terms (BN and BNa) to ensure a well-posed variational problem and consistent factorization of the path integral.
• Computation of the action using a hybrid action: ADM form in bulk regions and EH+GHY in infinitesimal neighborhoods around horizons.
• Evaluation of the total action as solely due to Gibbons-Hawking-York (GHY) boundary terms at the black hole and cosmological horizons, yielding Itot = −(Ab + Ac)/4.
• Use of the resulting action to compute the path integral weight e^{−I_total} = e^{−(S_SdS − S_dS)} for the constrained black hole state.

Experimental results

Research questions

• RQ1 Can Schwarzschild-de Sitter black holes, which are not smooth in Euclidean signature due to differing horizon temperatures, still serve as valid saddle points in the gravitational path integral?
• RQ2 How can a constrained path integral be formulated such that the SdS solution emerges as a genuine saddle point despite its conical singularities?
• RQ3 What is the semiclassical action of the SdS solution under such a constrained path integral, and how does it relate to the entropy of the black hole and cosmological horizons?
• RQ4 How does the probability of finding a black hole in de Sitter space emerge from the constrained path integral, and what is its dependence on the entropy deficit?
• RQ5 Can the discontinuity in the lapse function (and thus temperature) at the constraint surface be interpreted as a physical encoding of thermal non-equilibrium?

Key findings

• The total classical action of the Euclidean SdS solution in the constrained path integral is I_total = −(A_b + A_c)/4 = −S_SdS, independent of the radial position of the constraint surface.
• The probability to find a black hole of mass M in the de Sitter ensemble is given by P ∼ e^{−(S_SdS − S_dS)}, where S_dS is the entropy of empty de Sitter space.
• The constrained path integral formulation resolves the issue of non-smoothness in the Euclidean SdS geometry by replacing conical singularities with a discontinuity in the lapse function at the constraint surface.
• The action is computed via GHY boundary terms at the two horizons, with bulk contributions vanishing in the limit of infinitesimal boundary neighborhoods.
• The result is independent of the specific location of the constraint surface T as long as it encloses the black hole, confirming the physical consistency of the state.
• The formal path integral continuation to Lorentzian signature is achieved by decompactifying time in each region after making the lapse continuous, yielding the unique Lorentzian SdS solution with the given mass.