Abstract
We show that a holographic description of four-dimensional asymptotically locally flat spacetimes is reached smoothly from the zero-cosmological-constant limit of anti-de Sitter holography. To this end, we use the derivative expansion of fluid/gravity correspondence. From the boundary perspective, the vanishing of the bulk cosmological constant appears as the zero velocity of light limit. This sets how Carrollian geometry emerges in flat holography. The new boundary data are a two-dimensional spatial surface, identified with the null infinity of the bulk Ricci-flat spacetime, accompanied with a Carrollian time and equipped with a Carrollian structure, plus the dynamical observables of a conformal Carrollian fluid. These are the energy, the viscous stress tensors and the heat currents, whereas the Carrollian geometry is gathered by a two-dimensional spatial metric, a frame connection and a scale factor. The reconstruction of Ricci-flat spacetimes from Carrollian boundary data is conducted with a flat derivative expansion, resummed in a closed form in Eddington-Finkelstein gauge under further integrability conditions inherited from the ancestor anti-de Sitter set-up. These conditions are hinged on a duality relationship among fluid friction tensors and Cotton-like geometric data. We illustrate these results in the case of conformal Carrollian perfect fluids and Robinson-Trautman viscous hydrodynamics. The former are dual to the asymptotically flat Kerr-Taub-NUT family, while the latter leads to the homonymous class of algebraically special Ricci-flat spacetimes.
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References
C. Fefferman and C.R. Graham, Conformal invariants, in Elie Cartan et les mathématiques d’aujourd’hui, Astérisque Hors série Soc. Math. (1985) 95.
C. Fefferman and C.R. Graham, The ambient metric, Ann. Math. Stud. 178 (2011) 1 [arXiv:0710.0919] [INSPIRE].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
V.E. Hubeny, S. Minwalla and M. Rangamani, The fluid/gravity correspondence, in Black holes in higher dimensions, G. Horowitz ed., Cambridge University Press, Cambridge U.K. (2012), pp. 348-383 [arXiv:1107.5780] [INSPIRE].
M. Haack and A. Yarom, Nonlinear viscous hydrodynamics in various dimensions using AdS/CFT, JHEP 10 (2008) 063 [arXiv:0806.4602] [INSPIRE].
S. Bhattacharyya, R. Loganayagam, I. Mandal, S. Minwalla and A. Sharma, Conformal Nonlinear Fluid Dynamics from Gravity in Arbitrary Dimensions, JHEP 12 (2008) 116 [arXiv:0809.4272] [INSPIRE].
P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].
P. Romatschke, New Developments in Relativistic Viscous Hydrodynamics, Int. J. Mod. Phys. E 19 (2010) 1 [arXiv:0902.3663] [INSPIRE].
M.M. Caldarelli, R.G. Leigh, A.C. Petkou, P.M. Petropoulos, V. Pozzoli and K. Siampos, Vorticity in holographic fluids, PoS(CORFU2011)076 [arXiv:1206.4351] [INSPIRE].
A. Mukhopadhyay, A.C. Petkou, P.M. Petropoulos, V. Pozzoli and K. Siampos, Holographic perfect fluidity, Cotton energy-momentum duality and transport properties, JHEP 04 (2014) 136 [arXiv:1309.2310] [INSPIRE].
P.M. Petropoulos, Gravitational duality, topologically massive gravity and holographic fluids, Lect. Notes Phys. 892 (2015) 331 [arXiv:1406.2328] [INSPIRE].
J. Gath, A. Mukhopadhyay, A.C. Petkou, P.M. Petropoulos and K. Siampos, Petrov Classification and holographic reconstruction of spacetime, JHEP 09 (2015) 005 [arXiv:1506.04813] [INSPIRE].
P.M. Petropoulos and K. Siampos, Integrability, Einstein spaces and holographic fluids, in proceedings of the Workshop in honour of the 65th birthday of Professor Philippe Spindel, Mons, Belgium, 4-5 June 2015, N. Boulanger and S. Detournay eds., Université de Mons, Mons Belgium (2017) [arXiv:1510.06456] [INSPIRE].
A.C. Petkou, P.M. Petropoulos and K. Siampos, Geroch group for Einstein spaces and holographic integrability, PoS(PLANCK 2015)104 [arXiv:1512.04970] [INSPIRE].
I. Antoniadis, J.-P. Derendinger, P.M. Petropoulos and K. Siampos, Isometries, gaugings and \( \mathcal{N} \) = 2 supergravity decoupling, JHEP 11(2016) 169[arXiv:1611.00964] [INSPIRE].
S. Alexandrov, S. Banerjee and P. Longhi, Rigid limit for hypermultiplets and five-dimensional gauge theories, JHEP 01 (2018) 156 [arXiv:1710.10665] [INSPIRE].
T. Damour, Black Hole Eddy Currents, Phys. Rev. D 18 (1978) 3598 [INSPIRE].
T. Damour, Quelques propriétés mécaniques, électromagnétiques, thermodynamiques et quantiques des trous noirs, Thèse de Doctorat d’Etat, Université Pierre et Marie Curie, Paris VI, Paris France (1979) and online pdf version at http://www.ihes.fr/∼damour/Articles/these1.pdf.
S. de Haro, K. Skenderis and S.N. Solodukhin, Gravity in warped compactifications and the holographic stress tensor, Class. Quant. Grav. 18 (2001) 3171 [hep-th/0011230] [INSPIRE].
I. Bredberg, C. Keeler, V. Lysov and A. Strominger, From Navier-Stokes To Einstein, JHEP 07 (2012) 146 [arXiv:1101.2451] [INSPIRE].
G. Compère, P. McFadden, K. Skenderis and M. Taylor, The Holographic fluid dual to vacuum Einstein gravity, JHEP 07 (2011) 050 [arXiv:1103.3022] [INSPIRE].
G. Compère, P. McFadden, K. Skenderis and M. Taylor, The relativistic fluid dual to vacuum Einstein gravity, JHEP 03 (2012) 076 [arXiv:1201.2678] [INSPIRE].
M.M. Caldarelli, J. Camps, B. Goutéraux and K. Skenderis, AdS/Ricci-flat correspondence, JHEP 04 (2014) 071 [arXiv:1312.7874] [INSPIRE].
N. Pinzani-Fokeeva and M. Taylor, Towards a general fluid/gravity correspondence, Phys. Rev. D 91 (2015) 044001 [arXiv:1401.5975] [INSPIRE].
C. Eling, A. Meyer and Y. Oz, The Relativistic Rindler Hydrodynamics, JHEP 05 (2012) 116 [arXiv:1201.2705] [INSPIRE].
G. Arcioni and C. Dappiaggi, Exploring the holographic principle in asymptotically flat space-times via the BMS group, Nucl. Phys. B 674 (2003) 553 [hep-th/0306142] [INSPIRE].
G. Arcioni and C. Dappiaggi, Holography in asymptotically flat space-times and the BMS group, Class. Quant. Grav. 21 (2004) 5655 [hep-th/0312186] [INSPIRE].
C. Dappiaggi, V. Moretti and N. Pinamonti, Rigorous steps towards holography in asymptotically flat spacetimes, Rev. Math. Phys. 18 (2006) 349 [gr-qc/0506069] [INSPIRE].
J. de Boer and S.N. Solodukhin, A Holographic reduction of Minkowski space-time, Nucl. Phys. B 665 (2003) 545 [hep-th/0303006] [INSPIRE].
E.T. Newman and R. Penrose, An Approach to gravitational radiation by a method of spin coefficients, J. Math. Phys. 3 (1962) 566 [INSPIRE].
T.M. Adamo, E.T. Newman and C.N. Kozameh, Null Geodesic Congruences, Asymptotically Flat Space-Times and Their Physical Interpretation, Living Rev. Rel. 12 (2009) 6 [Living Rev. Rel. 15 (2012) 1] [arXiv:0906.2155] [INSPIRE].
T. Mädler and J. Winicour, Bondi-Sachs Formalism, Scholarpedia 11 (2016) 33528 [arXiv:1609.01731] [INSPIRE].
A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].
A. Bagchi, Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories, Phys. Rev. Lett. 105 (2010) 171601 [arXiv:1006.3354] [INSPIRE].
A. Bagchi and R. Fareghbal, BMS/GCA Redux: Towards Flatspace Holography from Non-Relativistic Symmetries, JHEP 10 (2012) 092 [arXiv:1203.5795] [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
G. Barnich, A. Gomberoff and H.A. Gonzalez, The Flat limit of three dimensional asymptotically anti-de Sitter spacetimes, Phys. Rev. D 86 (2012) 024020 [arXiv:1204.3288] [INSPIRE].
A. Bagchi, R. Basu, D. Grumiller and M. Riegler, Entanglement entropy in Galilean conformal field theories and flat holography, Phys. Rev. Lett. 114 (2015) 111602 [arXiv:1410.4089] [INSPIRE].
J. Hartong, Holographic Reconstruction of 3D Flat Space-Time, JHEP 10 (2016) 104 [arXiv:1511.01387] [INSPIRE].
K. Jensen and A. Karch, Revisiting non-relativistic limits, JHEP 04 (2015) 155 [arXiv:1412.2738] [INSPIRE].
O. Baghchesaraei, R. Fareghbal and Y. Izadi, Flat-Space Holography and Stress Tensor of Kerr Black Hole, Phys. Lett. B 760 (2016) 713 [arXiv:1603.04137] [INSPIRE].
T. He, P. Mitra and A. Strominger, 2D Kac-Moody Symmetry of 4D Yang-Mills Theory, JHEP 10 (2016) 137 [arXiv:1503.02663] [INSPIRE].
D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, 2D Stress Tensor for 4D Gravity, Phys. Rev. Lett. 119 (2017) 121601 [arXiv:1609.00282] [INSPIRE].
S. Pasterski, S.-H. Shao and A. Strominger, Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere, Phys. Rev. D 96 (2017) 065026 [arXiv:1701.00049] [INSPIRE].
D. Kapec and P. Mitra, A d-Dimensional Stress Tensor for Mink d+2 Gravity, JHEP 05 (2018) 186 [arXiv:1711.04371] [INSPIRE].
R. Fareghbal and I. Mohammadi, Flat-space Holography and Correlators of Robinson-Trautman Stress tensor, arXiv:1802.05445 [INSPIRE].
J.-M. Lévy-Leblond, Une nouvelle limite non-relativiste du groupe de Poincaré, Ann. Inst. Henri Poincaré III (1965) 1 and online at https://eudml.org/doc/75509.
C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time, Class. Quant. Grav. 31 (2014) 085016 [arXiv:1402.0657] [INSPIRE].
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav. 31 (2014) 092001 [arXiv:1402.5894] [INSPIRE].
C. Duval and P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys. A 42 (2009) 465206 [arXiv:0904.0531] [INSPIRE].
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups, J. Phys. A 47 (2014) 335204 [arXiv:1403.4213] [INSPIRE].
L. Ciambelli, C. Marteau, A.C. Petkou, P.M. Petropoulos and K. Siampos, Covariant Galilean versus Carrollian hydrodynamics from relativistic fluids, accepted for publication in Class. Quant. Grav. (2018), arXiv:1802.05286 [https://doi.org/10.1088/1361-6382/aacf1a] [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].
R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [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].
A. Ashtekar, Geometry and Physics of Null Infinity, in One hundred years of general relativity, L. Bieri and S.T. Yau eds., International Press, Boston U.S.A. (2015), p. 99 [arXiv:1409.1800] [INSPIRE].
A. Bagchi, S. Chakrabortty and P. Parekh, Tensionless Strings from Worldsheet Symmetries, JHEP 01 (2016) 158 [arXiv:1507.04361] [INSPIRE].
B. Cardona, J. Gomis and J.M. Pons, Dynamics of Carroll Strings, JHEP 07 (2016) 050 [arXiv:1605.05483] [INSPIRE].
R.F. Penna, BMS invariance and the membrane paradigm, JHEP 03 (2016) 023 [arXiv:1508.06577] [INSPIRE].
R.F. Penna, Near-horizon BMS symmetries as fluid symmetries, JHEP 10 (2017) 049 [arXiv:1703.07382] [INSPIRE].
M.T. Anderson, Geometric aspects of the AdS/CFT correspondence, IRMA Lect. Math. Theor. Phys. 8 (2005) 1 [hep-th/0403087] [INSPIRE].
I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].
S. Fischetti, W. Kelly and D. Marolf, Conserved charges in asymptotically (locally) AdS spacetimes, in Springer Handbook of spacetime, A. Ashtekar and V. Petkov eds., Springer (2014), p. 381 [arXiv:1211.6347] [INSPIRE].
A. Ashtekar and S. Das, Asymptotically Anti-de Sitter space-times: Conserved quantities, Class. Quant. Grav. 17 (2000) L17 [hep-th/9911230] [INSPIRE].
L.D. Landau and E.M. Lifchitz, Physique Théorique. Volume 6: Mécanique des fluides, MIR, Moscow Russia (1969).
L. Ciambelli, A.C. Petkou, P.M. Petropoulos and K. Siampos, The Robinson-Trautman spacetime and its holographic fluid, PoS(CORFU2016)076 [arXiv:1707.02995] [INSPIRE].
M. Humbert, Holographic reconstruction in higher dimension, internship report, École normale supérieure, École polytechnique, Paris France (2017).
B. Coll, J. Llosa and D. Soler, Three-dimensional metrics as deformations of a constant curvature metric, Gen. Rel. Grav. 34 (2002) 269 [gr-qc/0104070] [INSPIRE].
D.S. Mansi, A.C. Petkou and G. Tagliabue, Gravity in the 3 + 1-Split Formalism I: Holography as an Initial Value Problem, Class. Quant. Grav. 26 (2009) 045008 [arXiv:0808.1212] [INSPIRE].
D.S. Mansi, A.C. Petkou and G. Tagliabue, Gravity in the 3 + 1-Split Formalism II: Self-Duality and the Emergence of the Gravitational Chern-Simons in the Boundary, Class. Quant. Grav. 26 (2009) 045009 [arXiv:0808.1213] [INSPIRE].
S. de Haro, Dual Gravitons in AdS 4 /CFT 3 and the Holographic Cotton Tensor, JHEP 01 (2009) 042 [arXiv:0808.2054] [INSPIRE].
I. Bakas, Energy-momentum/Cotton tensor duality for AdS 4 black holes, JHEP 01 (2009) 003 [arXiv:0809.4852] [INSPIRE].
O. Miskovic and R. Olea, Topological regularization and self-duality in four-dimensional anti-de Sitter gravity, Phys. Rev. D 79 (2009) 124020 [arXiv:0902.2082] [INSPIRE].
J.F. Plebanski and M. Demianski, Rotating, charged, and uniformly accelerating mass in general relativity, Annals Phys. 98 (1976) 98 [INSPIRE].
G. Bernardi de Freitas and H.S. Reall, Algebraically special solutions in AdS/CFT, JHEP 06 (2014) 148 [arXiv:1403.3537] [INSPIRE].
I. Bakas and K. Skenderis, Non-equilibrium dynamics and AdS 4 Robinson-Trautman, JHEP 08 (2014) 056 [arXiv:1404.4824] [INSPIRE].
I. Bakas, K. Skenderis and B. Withers, Self-similar equilibration of strongly interacting systems from holography, Phys. Rev. D 93 (2016) 101902 [arXiv:1512.09151] [INSPIRE].
K. Skenderis and B. Withers, Robinson-Trautman spacetimes and gauge/gravity duality, PoS(CORFU2016)097 [arXiv:1703.10865] [INSPIRE].
R. Fareghbal, A. Naseh and S. Rouhani, Aspects of Ultra-Relativistic Field Theories via Flat-space Holography, Phys. Lett. B 771 (2017) 189 [arXiv:1511.01774] [INSPIRE].
A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Flat Holography: Aspects of the dual field theory, JHEP 12 (2016) 147 [arXiv:1609.06203] [INSPIRE].
E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Annales Sci. Ecole Norm. Sup. 41 (1924) 1 [INSPIRE] and online pdf version at http://archive.numdam.org/article/ASENS_1924_3_41_1_0.pdf.
X. Bekaert and K. Morand, Connections and dynamical trajectories in generalised Newton-Cartan gravity II. An ambient perspective, arXiv:1505.03739 [INSPIRE].
J.B. Griffiths and J. Podolský, Exact space-times in Einstein’s general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2009) [ISBN: 9781139481168] [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].
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Ciambelli, L., Marteau, C., Petkou, A.C. et al. Flat holography and Carrollian fluids. J. High Energ. Phys. 2018, 165 (2018). https://doi.org/10.1007/JHEP07(2018)165
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DOI: https://doi.org/10.1007/JHEP07(2018)165