Gravitation in flat spacetime from entanglement
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Springer
Received: September 20, 2019 Accepted: November 22, 2019 Published: December 6, 2019
Victor Godeta and Charles Marteaub a
Institute for Theoretical Physics, University of Amsterdam, 1090 GL Amsterdam, Netherlands b CPHT, Ecole Polytechnique, CNRS UMR 7644, Universit´e Paris-Saclay 91128, Palaiseau, France
E-mail: [email protected], [email protected] Abstract: We explore holographic entanglement entropy for Minkowski spacetime in three and four dimensions. Under some general assumptions on the putative holographic dual, the entanglement entropy associated to a special class of subregions can be computed using an analog of the Ryu-Takayanagi formula. We refine the existing prescription in three dimensions and propose a generalization to four dimensions. Under reasonable assumptions on the holographic stress tensor, we show that the first law of entanglement is equivalent to the gravitational equations of motion in the bulk, linearized around Minkowski spacetime. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence, Classical Theories of Gravity, Models of Quantum Gravity ArXiv ePrint: 1908.02044
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2019)057
JHEP12(2019)057
Gravitation in flat spacetime from entanglement
Contents 1 Introduction
1
2 Working assumptions on flat holography
2 4 4 9 12 15
4 Flat 3d gravity from entanglement 4.1 General strategy 4.2 Linearized gravitational equations
17 17 18
5 Holographic stress tensor in flat spacetime 5.1 AdS3 in Bondi gauge 5.2 Flat limit and Carrollian geometry
22 22 23
6 Generalization to 4d 6.1 Ryu-Takayanagi prescription in 4d Minkowski 6.2 General 4d prescription 6.3 Linearized gravitational equations
27 27 31 36
7 Conclusion
41
A Bulk Rindler transformation
42
B Precisions on the general strategy
45
C Alternative proof in 3d
46
1
Introduction
The AdS/CFT correspondence has been a fruitful avenue to understand quantum gravity in asymptotically AdS spacetimes. A question of interest is whether the holographic principle makes sense in more general spacetimes, such as our own universe. Some proposals have been made for de Sitter [1], Kerr [2] or warped AdS [3, 4]. The asymptotically flat case is particularly interesting because it can be obtained as a flat limit of AdS [5, 6]. Other approaches to flat space holography exist, such as applying AdS/CFT on hyperbolic foliations of Minkowski spacetime [7] or using the recently discovered equivalence between BMS Ward identities and Weinberg’s soft theorems [8]. The flat space limit of AdS is an ultra-relativistic limit, or Carrollian limit, of the dual field theory. Already at the level of the symmetries, one can show that the conformal Carroll
–1–
JHEP12(2019)057
3 Ryu-Takayanagi prescription in 3d Minkowski 3.1 Review of the 3d prescription 3.2 General 3d prescription 3.3 First law of entanglement 3.4 Positivity constraints
group is the BMS group [9], which is the symmetry group of asymptotically flat gravity [10]. More precise
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