Abstract
We consider four-dimensional general relativity with vanishing cosmological constant defined on a manifold with a boundary. In Lorentzian signature, the timelike boundary is of the form σ × ℝ, with σ a spatial two-manifold that we take to be either flat or S2. In Euclidean signature we take the boundary to be S2 × S1. We consider conformal boundary conditions, whereby the conformal class of the induced metric and trace K of the extrinsic curvature are fixed at the timelike boundary. The problem of linearised gravity is analysed using the Kodama-Ishibashi formalism. It is shown that for a round metric on S2 with constant K, there are modes that grow exponentially in time. We discuss a method to control the growing modes by varying K. The growing modes are absent for a conformally flat induced metric on the timelike boundary. We provide evidence that the Dirichlet problem for a spherical boundary does not suffer from non-uniqueness issues at the linearised level. We consider the extension of black hole thermodynamics to the case of conformal boundary conditions, and show that the form of the Bekenstein-Hawking entropy is retained.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Fefferman, Conformal invariants, Élie Cartan et les Mathématiques d’Aujourd’hui, Lyon, 25–29 Juin 1984, Astérisque, no. S131 (1985), p. 95–116.
S. Kichenassamy, On a conjecture of fefferman and graham, Adv. Math. 184 (2004) 268.
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
M.T. Anderson, Geometric aspects of the AdS/CFT correspondence, IRMA Lect. Math. Theor. Phys. 8 (2005) 1 [hep-th/0403087] [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
K. Prabhu, G. Satishchandran and R.M. Wald, Infrared finite scattering theory in quantum field theory and quantum gravity, Phys. Rev. D 106 (2022) 066005 [arXiv:2203.14334] [INSPIRE].
D. Anninos, S.A. Hartnoll and D.M. Hofman, Static Patch Solipsism: Conformal Symmetry of the de Sitter Worldline, Class. Quant. Grav. 29 (2012) 075002 [arXiv:1109.4942] [INSPIRE].
E. Coleman et al., De Sitter microstates from \( T\overline{T} \) + Λ2 and the Hawking-Page transition, JHEP 07 (2022) 140 [arXiv:2110.14670] [INSPIRE].
B. Banihashemi and T. Jacobson, Thermodynamic ensembles with cosmological horizons, JHEP 07 (2022) 042 [arXiv:2204.05324] [INSPIRE].
E. Witten, Algebras, Regions, and Observers, arXiv:2303.02837 [INSPIRE].
M.J. Blacker and S.A. Hartnoll, Cosmological quantum states of de Sitter-Schwarzschild are static patch partition functions, arXiv:2304.06865 [INSPIRE].
R. Loganayagam and O. Shetye, Influence Phase of a dS Observer I: Scalar Exchange, arXiv:2309.07290 [INSPIRE].
T. Damour, Black Hole Eddy Currents, Phys. Rev. D 18 (1978) 3598 [INSPIRE].
R.L. Znajek, The electric and magnetic conductivity of a Kerr hole, Mon. Not. Roy. Astron. Soc. 185 (1978) 833.
G. ’t Hooft, On the Quantum Structure of a Black Hole, Nucl. Phys. B 256 (1985) 727 [INSPIRE].
R.H. Price and K.S. Thorne, Membrane Viewpoint on Black Holes: Properties and Evolution of the Stretched Horizon, Phys. Rev. D 33 (1986) 915 [INSPIRE].
L. Susskind, L. Thorlacius and J. Uglum, The Stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].
I. Bredberg and A. Strominger, Black Holes as Incompressible Fluids on the Sphere, JHEP 05 (2012) 043 [arXiv:1106.3084] [INSPIRE].
L. Freidel, M. Geiller and W. Wieland, Corner symmetry and quantum geometry, arXiv:2302.12799 [INSPIRE].
T. Banks, Some thoughts on the quantum theory of de sitter space, in the proceedings of the The Davis Meeting on Cosmic Inflation, Davis, U.S.A., 22–23 March 2003 (2003) [astro-ph/0305037] [INSPIRE].
T. Banks, B. Fiol and A. Morisse, Towards a quantum theory of de Sitter space, JHEP 12 (2006) 004 [hep-th/0609062] [INSPIRE].
D. Anninos, T. Anous, I. Bredberg and G.S. Ng, Incompressible Fluids of the de Sitter Horizon and Beyond, JHEP 05 (2012) 107 [arXiv:1110.3792] [INSPIRE].
L. Susskind, De Sitter Holography: Fluctuations, Anomalous Symmetry, and Wormholes, Universe 7 (2021) 464 [arXiv:2106.03964] [INSPIRE].
D. Anninos and E. Harris, Three-dimensional de Sitter horizon thermodynamics, JHEP 10 (2021) 091 [arXiv:2106.13832] [INSPIRE].
E. Shaghoulian, The central dogma and cosmological horizons, JHEP 01 (2022) 132 [arXiv:2110.13210] [INSPIRE].
E. Shaghoulian and L. Susskind, Entanglement in De Sitter space, JHEP 08 (2022) 198 [arXiv:2201.03603] [INSPIRE].
B. Banihashemi, T. Jacobson, A. Svesko and M. Visser, The minus sign in the first law of de Sitter horizons, JHEP 01 (2023) 054 [arXiv:2208.11706] [INSPIRE].
D. Anninos and D.M. Hofman, Infrared Realization of dS2 in AdS2, Class. Quant. Grav. 35 (2018) 085003 [arXiv:1703.04622] [INSPIRE].
D. Anninos, D.A. Galante and D.M. Hofman, De Sitter horizons & holographic liquids, JHEP 07 (2019) 038 [arXiv:1811.08153] [INSPIRE].
A. Svesko, E. Verheijden, E.P. Verlinde and M.R. Visser, Quasi-local energy and microcanonical entropy in two-dimensional nearly de Sitter gravity, JHEP 08 (2022) 075 [arXiv:2203.00700] [INSPIRE].
D. Anninos and E. Harris, Interpolating geometries and the stretched dS2 horizon, JHEP 11 (2022) 166 [arXiv:2209.06144] [INSPIRE].
V. Shyam, \( T\overline{T} \) + Λ2 deformed CFT on the stretched dS3 horizon, JHEP 04 (2022) 052 [arXiv:2106.10227] [INSPIRE].
D.A. Galante, Modave lectures on de Sitter space & holography, PoS Modave2022 (2023) 003 [arXiv:2306.10141] [INSPIRE].
V. Chandrasekaran, R. Longo, G. Penington and E. Witten, An algebra of observables for de Sitter space, JHEP 02 (2023) 082 [arXiv:2206.10780] [INSPIRE].
O. Sarbach and M. Tiglio, Continuum and Discrete Initial-Boundary-Value Problems and Einstein’s Field Equations, Living Rev. Rel. 15 (2012) 9 [arXiv:1203.6443] [INSPIRE].
Y. Fourès-Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math. 88 (1952) 141.
Y. Choquet-Bruhat and R.P. Geroch, Global aspects of the Cauchy problem in general relativity, Commun. Math. Phys. 14 (1969) 329 [INSPIRE].
M.T. Anderson, On boundary value problems for Einstein metrics, Geom. Topol. 12 (2008) 2009 [math/0612647] [INSPIRE].
E. Witten, A note on boundary conditions in Euclidean gravity, Rev. Math. Phys. 33 (2021) 2140004 [arXiv:1805.11559] [INSPIRE].
H. Friedrich and G. Nagy, The Initial boundary value problem for Einstein’s vacuum field equations, Commun. Math. Phys. 201 (1999) 619 [INSPIRE].
G. Fournodavlos and J. Smulevici, The Initial Boundary Value Problem for the Einstein Equations with Totally Geodesic Timelike Boundary, Commun. Math. Phys. 385 (2021) 1615 [arXiv:2006.01498] [INSPIRE].
G. Fournodavlos and J. Smulevici, The Initial Boundary Value Problem in General Relativity: The Umbilic Case, Int. Math. Res. Not. 2023 (2023) 3790 [arXiv:2104.08851] [INSPIRE].
Z. An and M.T. Anderson, The initial boundary value problem and quasi-local Hamiltonians in General Relativity, arXiv:2103.15673 [https://doi.org/10.1088/1361-6382/ac0a86] [INSPIRE].
D. Anninos, D.A. Galante and B. Mühlmann, Finite features of quantum de Sitter space, Class. Quant. Grav. 40 (2023) 025009 [arXiv:2206.14146] [INSPIRE].
H. Friedrich, Einstein equations and conformal structure — Existence of anti de Sitter type space-times, J. Geom. Phys. 17 (1995) 125 [INSPIRE].
P. Figueras, J. Lucietti and T. Wiseman, Ricci solitons, Ricci flow, and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua, Class. Quant. Grav. 28 (2011) 215018 [arXiv:1104.4489] [INSPIRE].
H. Kodama, A. Ishibashi and O. Seto, Brane world cosmology: Gauge invariant formalism for perturbation, Phys. Rev. D 62 (2000) 064022 [hep-th/0004160] [INSPIRE].
H. Kodama and A. Ishibashi, A Master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions, Prog. Theor. Phys. 110 (2003) 701 [hep-th/0305147] [INSPIRE].
T. Andrade, W.R. Kelly, D. Marolf and J.E. Santos, On the stability of gravity with Dirichlet walls, Class. Quant. Grav. 32 (2015) 235006 [arXiv:1504.07580] [INSPIRE].
J.W. York Jr., Black hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D 33 (1986) 2092 [INSPIRE].
J.W. York Jr., Role of conformal three geometry in the dynamics of gravitation, Phys. Rev. Lett. 28 (1972) 1082 [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
P. Bizon and A. Rostworowski, On weakly turbulent instability of anti-de Sitter space, Phys. Rev. Lett. 107 (2011) 031102 [arXiv:1104.3702] [INSPIRE].
T. Andrade and D. Marolf, Asymptotic Symmetries from finite boxes, Class. Quant. Grav. 33 (2016) 015013 [arXiv:1508.02515] [INSPIRE].
G. Odak and S. Speziale, Brown-York charges with mixed boundary conditions, JHEP 11 (2021) 224 [arXiv:2109.02883] [INSPIRE].
J.D. Brown and J.W. York Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].
S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-De Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
F. Alessio, G. Barnich and M. Bonte, Gravitons in a Casimir box, JHEP 02 (2021) 216 [Erratum ibid. 03 (2021) 228] [arXiv:2011.14432] [INSPIRE].
S.K. Asante and B. Dittrich, Perfect discretizations as a gateway to one-loop partition functions for 4D gravity, JHEP 05 (2022) 172 [arXiv:2112.03307] [INSPIRE].
L. Susskind, Entanglement and Chaos in De Sitter Space Holography: An SYK Example, JHAP 1 (2021) 1 [arXiv:2109.14104] [INSPIRE].
S. Chapman et al., Complex geodesics in de Sitter space, JHEP 03 (2023) 006 [arXiv:2212.01398] [INSPIRE].
L. Aalsma et al., Late-time correlators and complex geodesics in de Sitter space, SciPost Phys. 15 (2023) 031 [arXiv:2212.01394] [INSPIRE].
S. Gao and R.M. Wald, Theorems on gravitational time delay and related issues, Class. Quant. Grav. 17 (2000) 4999 [gr-qc/0007021] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with \( T\overline{T} \), JHEP 04 (2018) 010 [arXiv:1611.03470] [INSPIRE].
M. Taylor, TT deformations in general dimensions, arXiv:1805.10287 [INSPIRE].
T. Hartman, J. Kruthoff, E. Shaghoulian and A. Tajdini, Holography at finite cutoff with a T2 deformation, JHEP 03 (2019) 004 [arXiv:1807.11401] [INSPIRE].
E. Coleman and V. Shyam, Conformal boundary conditions from cutoff AdS3, JHEP 09 (2021) 079 [arXiv:2010.08504] [INSPIRE].
D. Anninos, F. Denef, Y.T.A. Law and Z. Sun, Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions, JHEP 01 (2022) 088 [arXiv:2009.12464] [INSPIRE].
M. Alishahiha, A. Karch, E. Silverstein and D. Tong, The dS/dS correspondence, AIP Conf. Proc. 743 (2004) 393 [hep-th/0407125] [INSPIRE].
D. Anninos, D.M. Hofman and S. Vitouladitis, One-dimensional Quantum Gravity and the Schwarzian theory, JHEP 03 (2022) 121 [arXiv:2112.03793] [INSPIRE].
https://people.brandeis.edu/~headrick/HeadrickCompendium.pdf.
Acknowledgments
It is a pleasure to acknowledge Vijay Balasubramanian, Bianca Dittrich, Pau Figueras, Chris Herzog, Diego Hofman, Luis Lehner, Hong Liu, and Beatrix Mühlmann for useful discussions. D.A. is funded by the Royal Society under the grant “The Atoms of a deSitter Universe”. The work of D.A.G. is funded by UKRI Stephen Hawking Fellowship “Quantum Emergence of an Expanding Universe”. D.A. and D.A.G. are further funded by STFC Consolidated grant ST/X000753/1. C.M. is funded by STFC under grant number ST/X508470/1.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2310.08648
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Anninos, D., Galante, D.A. & Maneerat, C. Gravitational observatories. J. High Energ. Phys. 2023, 24 (2023). https://doi.org/10.1007/JHEP12(2023)024
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2023)024