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On the Euclidean action of de Sitter black holes and constrained instantons

Morvan, Edward K., van der Schaar, Jan Pieter, Visser, Manus R.·2023·SciPost Physics
DOI 10.21468/scipostphys.14.2.022
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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 independent of the Euclidean time periodicity and 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 constant and a “non-perturbative” 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 as potential non-perturbative corrections to correlators in the de Sitter vacuum.

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Figures

Figure 1 Horizon radii r b , c in units of the de Sitter radius L vs. mass M in units of the Nariai mass MN , where we have set d = 4 (left) and d = 5 (right). The black hole horizon radius (blue curve) becomes larger as M increases, whereas the cosmological horizon (orange curve) shrinks in size as M increases. For the Nariai solution the horizon radii coincide.
Figure 1 Horizon radii r b , c in units of the de Sitter radius L vs. mass M in units of the Nariai mass MN , where we have set d = 4 (left) and d = 5 (right). The black hole horizon radius (blue curve) becomes larger as M increases, whereas the cosmological horizon (orange curve) shrinks in size as M increases. For the Nariai solution the horizon radii coincide.CC-BY · From the PDF
Image 2
CC-BY · From the PDF
Figure 2 Temperatures Tb , c in units of TdS = 1 / 2 π L vs. unitless mass parameter M / MN , where we have set d = 4. The black hole temperature is shown in blue and the cosmological horizon temperature in orange. The left figure shows the temperatures Tb , c = κ b , c 2 π , where the surface gravities are defined with respect to the time translation Killing vector ξ = ∂ t . The right figure shows the Bousso-Hawking temperatures ˜ Tb , c = ˜ κ b , c 2 π , which are normalized with respect to a free-falling static observer, cf. (17). On the left the temperatures go to zero in the Nariai limit, whereas on the right the Nariai temperature is finite TN = π 3 2 π L .
Figure 2 Temperatures Tb , c in units of TdS = 1 / 2 π L vs. unitless mass parameter M / MN , where we have set d = 4. The black hole temperature is shown in blue and the cosmological horizon temperature in orange. The left figure shows the temperatures Tb , c = κ b , c 2 π , where the surface gravities are defined with respect to the time translation Killing vector ξ = ∂ t . The right figure shows the Bousso-Hawking temperatures ˜ Tb , c = ˜ κ b , c 2 π , which are normalized with respect to a free-falling static observer, cf. (17). On the left the temperatures go to zero in the Nariai limit, whereas on the right the Nariai temperature is finite TN = π 3 2 π L .CC-BY · From the PDF
Image 4
CC-BY · From the PDF
Figure 3 Horizon entropies S b , c in units of the de Sitter entropy SdS vs. unitless mass M / MN , for d = 4. The blue curve corresponds to the black hole entropy S b , and the orange curve shows the cosmological horizon entropy Sc . The sum of the entropies SSdS is approximately a straight line (red curve), and is always less than (or equal to) SdS . For the Nariai solution the horizon entropies are both equal to one third of the de Sitter entropy, and their sum is the Nariai entropy SN = 2 3 SdS . A similar plot appeared in [ 58 ] .
Figure 3 Horizon entropies S b , c in units of the de Sitter entropy SdS vs. unitless mass M / MN , for d = 4. The blue curve corresponds to the black hole entropy S b , and the orange curve shows the cosmological horizon entropy Sc . The sum of the entropies SSdS is approximately a straight line (red curve), and is always less than (or equal to) SdS . For the Nariai solution the horizon entropies are both equal to one third of the de Sitter entropy, and their sum is the Nariai entropy SN = 2 3 SdS . A similar plot appeared in [ 58 ] .CC-BY · From the PDF
Figure 4 Euclidean SdS geometry. (i) Euclidean de Sitter space is a sphere S d and the Euclidean time circle has periodicity β dS = 2 π L . (ii) Euclidean Nariai space is S 2 × S d -2 , where the S 2 has a Euclidean time circle with periodicity β N = 2 π L / π d -1 and the S d -2 has radius r N . (iii) Euclidean SdS space is a warped product of S 2 , with one or two conical defects, times S d -2 . We have drawn three possible Euclidean SdS geometries ('ice cream cones'): on the left we removed the conical singularity at the cosmological horizon by fixing the Euclidean time periodicity to be β c = 2 π/κ c , in the middle we removed the conical singularity at the black hole horizon by fixing the periodicity to be β b = 2 π/κ b , and on the right both conical singularities are present since the periodicity β is arbitrary . For these three geometries the radius of the S d -2 varies along spatial sections, with the minimal (maximal) sphere corresponding to the black hole (cosmological) horizon.
Figure 4 Euclidean SdS geometry. (i) Euclidean de Sitter space is a sphere S d and the Euclidean time circle has periodicity β dS = 2 π L . (ii) Euclidean Nariai space is S 2 × S d -2 , where the S 2 has a Euclidean time circle with periodicity β N = 2 π L / π d -1 and the S d -2 has radius r N . (iii) Euclidean SdS space is a warped product of S 2 , with one or two conical defects, times S d -2 . We have drawn three possible Euclidean SdS geometries ('ice cream cones'): on the left we removed the conical singularity at the cosmological horizon by fixing the Euclidean time periodicity to be β c = 2 π/κ c , in the middle we removed the conical singularity at the black hole horizon by fixing the periodicity to be β b = 2 π/κ b , and on the right both conical singularities are present since the periodicity β is arbitrary . For these three geometries the radius of the S d -2 varies along spatial sections, with the minimal (maximal) sphere corresponding to the black hole (cosmological) horizon.CC-BY · From the PDF
Figure 5 On-shell action I E , SdS of Euclidean SdS in units of SdS vs. the dimensionless parameter ε = ± 1 -r c / L , where we have set d = 4. At ε = 0 the action attains a local minimum, which corresponds to Euclidean de Sitter space S 4 , and at ε = ± 1 -1 / π 3 (dashed lines) the action attains a local maximum, corresponding to Euclidean Nariai space S 2 × S 2 . At | ε | = π 2 the action has a global minimum which resembles pure de Sitter space, see Appendix A. There is a negative mode between the de Sitter and Nariai stationary points.
Figure 5 On-shell action I E , SdS of Euclidean SdS in units of SdS vs. the dimensionless parameter ε = ± 1 -r c / L , where we have set d = 4. At ε = 0 the action attains a local minimum, which corresponds to Euclidean de Sitter space S 4 , and at ε = ± 1 -1 / π 3 (dashed lines) the action attains a local maximum, corresponding to Euclidean Nariai space S 2 × S 2 . At | ε | = π 2 the action has a global minimum which resembles pure de Sitter space, see Appendix A. There is a negative mode between the de Sitter and Nariai stationary points.CC-BY · From the PDF
Image 8
CC-BY · From the PDF
Figure 6 Total gravitational entropy SSdS = S b + Sc in units of the de Sitter entropy SdS vs. the dimensionless mass M / MN , where we have set d = 4. The red curve corresponds to the actual sum of the horizon entropies, whereas the cyan curve is a linear fit. The total entropy is approximately a straight line, which also holds in higher dimensions d > 4.
Figure 6 Total gravitational entropy SSdS = S b + Sc in units of the de Sitter entropy SdS vs. the dimensionless mass M / MN , where we have set d = 4. The red curve corresponds to the actual sum of the horizon entropies, whereas the cyan curve is a linear fit. The total entropy is approximately a straight line, which also holds in higher dimensions d > 4.CC-BY · From the PDF
Figure 7 The difference between the linear fit Slin and the actual total entropy SSdS , in units of SdS vs. the mass M in units of MN . The different plots correspond to d = 4 (red), d = 5 (orange), d = 6 (green), d = 7 (cyan), d = 8 (blue).
Figure 7 The difference between the linear fit Slin and the actual total entropy SSdS , in units of SdS vs. the mass M in units of MN . The different plots correspond to d = 4 (red), d = 5 (orange), d = 6 (green), d = 7 (cyan), d = 8 (blue).CC-BY · From the PDF
Image 11
CC-BY · From the PDF
Figure 8 The roots of f ( r ) in units of L as a function of ε for d = 4. The black hole horizon radius r b is shown in blue, the cosmological horizon radius r c in orange and the third root r a = -( r b + r c ) in green.
Figure 8 The roots of f ( r ) in units of L as a function of ε for d = 4. The black hole horizon radius r b is shown in blue, the cosmological horizon radius r c in orange and the third root r a = -( r b + r c ) in green.CC-BY · From the PDF
Image 13
CC-BY · From the PDF
Figure 9 Mass M in units of MN as a function of ε for d = 4. For the values 1 < | ε | < π 2 the mass M becomes negative, and for | ε | > 1 + 2 / π 3 the mass exceeds MN .
Figure 9 Mass M in units of MN as a function of ε for d = 4. For the values 1 < | ε | < π 2 the mass M becomes negative, and for | ε | > 1 + 2 / π 3 the mass exceeds MN .CC-BY · From the PDF
Image 15
CC-BY · From the PDF
Figure 10 Euclidean action I E of SdS in units of SdS as a function of ε for d = 4. Empty de Sitter is a minimum and Nariai a maximum of the action, and there is an additional global minimum at | ε | = π 2 for which M = 0.
Figure 10 Euclidean action I E of SdS in units of SdS as a function of ε for d = 4. Empty de Sitter is a minimum and Nariai a maximum of the action, and there is an additional global minimum at | ε | = π 2 for which M = 0.CC-BY · From the PDF
Figure 11 The roots of f ( r ) in units of L as a function of ε for d = 5. The black hole horizon radius r b is shown in blue, the cosmological horizon radius r c in orange, the root r a = -r c in green and the fourth root r d = -r b in purple.
Figure 11 The roots of f ( r ) in units of L as a function of ε for d = 5. The black hole horizon radius r b is shown in blue, the cosmological horizon radius r c in orange, the root r a = -r c in green and the fourth root r d = -r b in purple.CC-BY · From the PDF
Image 18
CC-BY · From the PDF
Figure 12 Mass in units of MN as a function of ε for d = 5.
Figure 12 Mass in units of MN as a function of ε for d = 5.CC-BY · From the PDF
Figure 13 Euclidean action in units of SdS as a function of ε for d = 5.
Figure 13 Euclidean action in units of SdS as a function of ε for d = 5.CC-BY · From the PDF
Image 21
CC-BY · From the PDF

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Citations

Cites

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Metadata

Authors
Morvan, E. K., van der Schaar, J. P., & Visser, M. R.
Affiliations
  • Delta Institute for Theoretical Physics
  • University of Geneva
ORCID
Venue
SciPost Physics
Publisher
Stichting SciPost
Published
2023-02-23
Volume
14
Issue
2
ISSN
2542-4653
DOI
10.21468/scipostphys.14.2.022
Cited by
32
References
7
Language
en
Licence
Funders
  • Ministerie van Onderwijs, Cultuur en Wetenschap
  • Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (200020-182513, NCCR 51NF40-141869)

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