Abstract
We show that the Schwarzschild-(A)dS black hole mechanics possesses a hidden symmetry under the three-dimensional Poincaré group. This symmetry shows up after having gauge-fixed the diffeomorphism invariance in the symmetry-reduced homogeneous Einstein-Λ model and stands as a physical symmetry of the system. It dictates the geometry both in the black hole interior and exterior regions, as well as beyond the cosmological horizon in the Schwarzschild-dS case. It follows that one can associate a set of non-trivial conserved charges to the Schwarzschild-(A)dS black hole which act in each causally disconnected regions. In T-region, they act on fields living on spacelike hypersurface of constant time, while in R-regions, they act on time-like hypersurface of constant radius. We find that while the expression of the charges depend explicitly on the location of the hypersurface, the charge algebra remains the same at any radius in R-regions (or time in T-regions). Finally, the analysis of the Casimirs of the charge algebra reveals a new solution-generating map. The \( \mathfrak{sl}\left(2,\mathrm{\mathbb{R}}\right) \) Casimir is shown to generate a one-parameter family of deformation of the black hole geometry labelled by the cosmological constant. This gives rise to a new conformal bridge allowing one to continuously deform the Schwarzschild-AdS geometry to the Schwarzschild and the Schwarzschild-dS solutions.
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R. M. Wald and A. Zoupas, A General definition of ‘conserved quantities’ in general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
G. Barnich and G. Compere, Surface charge algebra in gauge theories and thermodynamic integrability, J. Math. Phys. 49 (2008) 042901 [arXiv:0708.2378] [INSPIRE].
W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].
G. Compère and A. Fiorucci, Advanced Lectures on General Relativity, arXiv:1801.07064 [INSPIRE].
D. Harlow and J.-Q. Wu, Covariant phase space with boundaries, JHEP 10 (2020) 146 [arXiv:1906.08616] [INSPIRE].
L. Freidel, M. Geiller and D. Pranzetti, Edge modes of gravity. Part I. Corner potentials and charges, JHEP 11 (2020) 026 [arXiv:2006.12527] [INSPIRE].
G. Odak and S. Speziale, Brown-York charges with mixed boundary conditions, arXiv:2109.02883 [INSPIRE].
L. Freidel, R. Oliveri, D. Pranzetti and S. Speziale, Extended corner symmetry, charge bracket and Einstein’s equations, JHEP 09 (2021) 083 [arXiv:2104.12881] [INSPIRE].
H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
A. Ashtekar and R. O. Hansen, A unified treatment of null and spatial infinity in general relativity. I — Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity, J. Math. Phys. 19 (1978) 1542 [INSPIRE].
A. Ashtekar and M. Streubel, Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity, Proc. Roy. Soc. Lond. A 376 (1981) 585 [INSPIRE].
G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
G. Compère, R. Oliveri and A. Seraj, The Poincaré and BMS flux-balance laws with application to binary systems, JHEP 10 (2020) 116 [arXiv:1912.03164] [INSPIRE].
L. Freidel, R. Oliveri, D. Pranzetti and S. Speziale, The Weyl BMS group and Einstein’s equations, JHEP 07 (2021) 170 [arXiv:2104.05793] [INSPIRE].
J. Ben Achour and E. R. Livine, Cosmology as a CFT1, JHEP 12 (2019) 031 [arXiv:1909.13390] [INSPIRE].
J. Ben Achour and E. R. Livine, The cosmological constant from conformal transformations: Möbius invariance and Schwarzian action, Class. Quant. Grav. 37 (2020) 215001 [arXiv:2004.05841] [INSPIRE].
J. Ben Achour, Proper time reparametrization in cosmology: Möbius symmetry and Kodama charges, arXiv:2103.10700 [INSPIRE].
J. Ben Achour and E. R. Livine, Cosmological spinor, Phys. Rev. D 101 (2020) 103523 [arXiv:2004.06387] [INSPIRE].
J. Ben Achour and E. R. Livine, Conformal structure of FLRW cosmology: spinorial representation and the so(2, 3) algebra of observables, JHEP 03 (2020) 067 [arXiv:2001.11807] [INSPIRE].
B. Pioline and A. Waldron, Quantum cosmology and conformal invariance, Phys. Rev. Lett. 90 (2003) 031302 [hep-th/0209044] [INSPIRE].
V. de Alfaro, S. Fubini and G. Furlan, Conformal Invariance in Quantum Mechanics, Nuovo Cim. A 34 (1976) 569 [INSPIRE].
A. Galajinsky, Conformal mechanics in Newton-Hooke spacetime, Nucl. Phys. B 832 (2010) 586 [arXiv:1002.2290] [INSPIRE].
G. W. Gibbons, Dark Energy and the Schwarzian Derivative, arXiv:1403.5431 [INSPIRE].
L. Inzunza, M. S. Plyushchay and A. Wipf, Conformal bridge between asymptotic freedom and confinement, Phys. Rev. D 101 (2020) 105019 [arXiv:1912.11752] [INSPIRE].
L. Inzunza and M. S. Plyushchay, Conformal bridge transformation and PT symmetry, arXiv:2104.08351 [INSPIRE].
M. Geiller, E. R. Livine and F. Sartini, Symmetries of the black hole interior and singularity regularization, SciPost Phys. 10 (2021) 022 [arXiv:2010.07059] [INSPIRE].
M. Geiller, E. R. Livine and F. Sartini, BMS3 Mechanics and the Black Hole Interior, arXiv:2107.03878 [INSPIRE].
J. Ben Achour and E. R. Livine, Polymer Quantum Cosmology: Lifting quantization ambiguities using a SL(2, ℝ) conformal symmetry, Phys. Rev. D 99 (2019) 126013 [arXiv:1806.09290] [INSPIRE].
J. Ben Achour and E. R. Livine, Protected SL(2, ℝ) Symmetry in Quantum Cosmology, JCAP 09 (2019) 012 [arXiv:1904.06149] [INSPIRE].
J. Ben Achour and E. R. Livine, Thiemann complexifier in classical and quantum FLRW cosmology, Phys. Rev. D 96 (2017) 066025 [arXiv:1705.03772] [INSPIRE].
J. D. Bekenstein and V. F. Mukhanov, Spectroscopy of the quantum black hole, Phys. Lett. B 360 (1995) 7 [gr-qc/9505012] [INSPIRE].
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Achour, J.B., Livine, E.R. Symmetries and conformal bridge in Schwarschild-(A)dS black hole mechanics. J. High Energ. Phys. 2021, 152 (2021). https://doi.org/10.1007/JHEP12(2021)152
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DOI: https://doi.org/10.1007/JHEP12(2021)152