Abstract
We propose that the intrinsic geometry of holographic screens should be described by the Newton-Cartan geometry. As a test of this proposal, we show that the evolution equations of the screen can be written in a covariant form in terms of a stress tensor, an energy current, and a momentum one-form. We derive the expressions for the stress tensor, energy density, and momentum one-form using Brown-York action formalism.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. ’t Hooft, Dimensional reduction in quantum gravity, in the proceedings of the Conference on Highlights of Particle and Condensed Matter Physics (Salamfest 1993), March 8-12, Trieste, Italy (1993), gr-qc/9310026 [INSPIRE].
L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
R. Bousso, A covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].
E.E. Flanagan, D. Marolf and R.M. Wald, Proof of classical versions of the Bousso entropy bound and of the generalized second law, Phys. Rev. D 62 (2000) 084035 [hep-th/9908070] [INSPIRE].
R. Bousso, The holographic principle, Rev. Mod. Phys. 74 (2002) 825 [hep-th/0203101] [INSPIRE].
R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Proof of a quantum Bousso bound, Phys. Rev. D 90 (2014) 044002 [arXiv:1404.5635] [INSPIRE].
R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Entropy on a null surface for interacting quantum field theories and the Bousso bound, Phys. Rev. D 91 (2015) 084030 [arXiv:1406.4545] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
R. Bousso, Holography in general space-times, JHEP 06 (1999) 028 [hep-th/9906022] [INSPIRE].
R.M. Wald, General relativity, Chicago University Press, Chicago, U.S.A. (1984).
R. Bousso and N. Engelhardt, New area law in general relativity, Phys. Rev. Lett. 115 (2015) 081301 [arXiv:1504.07627] [INSPIRE].
R. Bousso and N. Engelhardt, Proof of a new area law in general relativity, Phys. Rev. D 92 (2015) 044031 [arXiv:1504.07660] [INSPIRE].
F. Sanches and S.J. Weinberg, Holographic entanglement entropy conjecture for general spacetimes, Phys. Rev. D 94 (2016) 084034 [arXiv:1603.05250] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
Y. Nomura, N. Salzetta, F. Sanches and S.J. Weinberg, Spacetime equals entanglement, Phys. Lett. B 763 (2016) 370 [arXiv:1607.02508] [INSPIRE].
Y. Nomura, N. Salzetta, F. Sanches and S.J. Weinberg, Toward a holographic theory for general spacetimes, Phys. Rev. D 95 (2017) 086002 [arXiv:1611.02702] [INSPIRE].
R. Bousso and M. Moosa, Dynamics and observer-dependence of holographic screens, Phys. Rev. D 95 (2017) 046005 [arXiv:1611.04607] [INSPIRE].
D. Bigatti and L. Susskind, Review of matrix theory, in the proceedings of Strings, branes and dualities, May 26-June 14, Cargese, France (1997), hep-th/9712072 [INSPIRE].
I. Booth and S. Fairhurst, Horizon energy and angular momentum from a Hamiltonian perspective, Class. Quant. Grav. 22 (2005) 4515 [gr-qc/0505049] [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].
E. Poisson, A relativist’s toolkit: the mathematics of black-hole mechanics, Cambridge University Press, Camrbidge U.K. (2004).
E. Gourgoulhon, A generalized Damour-Navier-Stokes equation applied to trapping horizons, Phys. Rev. D 72 (2005) 104007 [gr-qc/0508003] [INSPIRE].
E. Gourgoulhon and J.L. Jaramillo, Area evolution, bulk viscosity and entropy principles for dynamical horizons, Phys. Rev. D 74 (2006) 087502 [gr-qc/0607050] [INSPIRE].
E. Gourgoulhon and J.L. Jaramillo, New theoretical approaches to black holes, New Astron. Rev. 51 (2008) 791 [arXiv:0803.2944] [INSPIRE].
S.A. Hayward, General laws of black hole dynamics, Phys. Rev. D 49 (1994) 6467 [INSPIRE].
A. Ashtekar and B. Krishnan, Dynamical horizons: energy, angular momentum, fluxes and balance laws, Phys. Rev. Lett. 89 (2002) 261101 [gr-qc/0207080] [INSPIRE].
A. Ashtekar and B. Krishnan, Dynamical horizons and their properties, Phys. Rev. D 68 (2003) 104030 [gr-qc/0308033] [INSPIRE].
A. Ashtekar and B. Krishnan, Isolated and dynamical horizons and their applications, Living Rev. Rel. 7 (2004) 10 [gr-qc/0407042] [INSPIRE].
E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Ann. Sci. É cole Norm. Supér. 40 (1923) 325.
E. Cartan, Sur les variétés à connexion affine, et la théorie de la relativité généralisée (deuxième partie), Ann. Sci. É cole Norm. Supér. 42 (1925) 17.
D.T. Son, Newton-Cartan geometry and the quantum Hall effect, arXiv:1306.0638 [INSPIRE].
M. Geracie, D.T. Son, C. Wu and S.-F. Wu, Spacetime symmetries of the quantum Hall effect, Phys. Rev. D 91 (2015) 045030 [arXiv:1407.1252] [INSPIRE].
R. Banerjee, A. Mitra and P. Mukherjee, Localization of the Galilean symmetry and dynamical realization of Newton-Cartan geometry, Class. Quant. Grav. 32 (2015) 045010 [arXiv:1407.3617] [INSPIRE].
K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime, arXiv:1408.6855 [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Schrödinger invariance from Lifshitz isometries in holography and field theory, Phys. Rev. D 92 (2015) 066003 [arXiv:1409.1522] [INSPIRE].
M. Geracie, K. Prabhu and M.M. Roberts, Fields and fluids on curved non-relativistic spacetimes, JHEP 08 (2015) 042 [arXiv:1503.02680] [INSPIRE].
R. Banerjee and P. Mukherjee, Torsional Newton-Cartan geometry from Galilean gauge theory, Class. Quant. Grav. 33 (2016) 225013 [arXiv:1604.06893] [INSPIRE].
K. Jensen, Aspects of hot Galilean field theory, JHEP 04 (2015) 123 [arXiv:1411.7024] [INSPIRE].
A. Mitra, Weyl rescaled Newton-Cartan geometry and non-relativistic conformal hydrodynamics, arXiv:1508.03207 [INSPIRE].
J. Hartong and N.A. Obers, Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP 07 (2015) 155 [arXiv:1504.07461] [INSPIRE].
P. Hořava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].
M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Torsional Newton-Cartan geometry and Lifshitz holography, Phys. Rev. D 89 (2014) 061901 [arXiv:1311.4794] [INSPIRE].
M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography, JHEP 01 (2014) 057 [arXiv:1311.6471] [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Lifshitz space-times for Schrödinger holography, Phys. Lett. B 746 (2015) 318 [arXiv:1409.1519] [INSPIRE].
G. Hayward, Gravitational action for spacetimes with nonsmooth boundaries, Phys. Rev. D 47 (1993) 3275.
C.R. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nucl. Phys. B 546 (1999) 52 [hep-th/9901021] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1702.06960
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Moosa, M. Non-relativistic geometry of holographic screens. J. High Energ. Phys. 2017, 38 (2017). https://doi.org/10.1007/JHEP06(2017)038
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2017)038