Fluid-gravity and membrane-gravity dualities. Comparison at subleading orders
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Springer
Received: February 24, 2019 Accepted: April 25, 2019 Published: May 9, 2019
Sayantani Bhattacharyya, Parthajit Biswas, Anirban Dinda and Milan Patra National Institute of Science Education and Research, HBNI, Bhubaneshwar 752050, Odisha, India
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: In this note, we have compared two different perturbation techniques that could be used to generate solutions of Einstein’s equations in the presence of negative cosmological constant. One of these two methods is derivative expansion and the other is an expansion in inverse powers of dimension. Both the techniques generate space-time with a singularity shielded by a dynamical event horizon. We have shown that in the appropriate regime of parameter space and with an appropriate choice of coordinates, the metrics and corresponding horizon dynamics, generated by these two different techniques, are exactly equal to the order the solutions are known both sides. This work is essentially an extension of [1] where the authors have shown the equivalence of the two techniques up to the first non-trivial order. Keywords: Black Holes, Classical Theories of Gravity ArXiv ePrint: 1902.00854
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)054
JHEP05(2019)054
Fluid-gravity and membrane-gravity dualities. Comparison at subleading orders
Contents 1 1 2 4 4
2 Hydrodynamic metric and its large D limit 2.1 Hydrodynamic metric up to 2nd order in derivative expansion
6 6
3 Large-D metric and membrane equation
9
4 Implementing part-1: the split of the hydrodynamic metric ¯ A ∂A 4.1 The null geodesic O 4.2 The mapping functions and the ‘split’ of the hydrodynamic metric
11 12 13
5 Implementing part-2: large-D metric in terms of fluid data 5.1 Determining ψ ¯A 5.2 Fixing the normalization of O 5.2.1 Large-D metric in terms of fluid data rest and W rest 5.3 Comparison between Gµν µν 5.3.1 (1/D) expansion of the functions appearing in hydrodynamic metric
16 16 17 18 22 22
6 Implementing part-3: equivalence of the constraint equations
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7 Discussion and future directions
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A Comparison up to O ∂ 2 ,
1 D
following [1] exactly
B Large-D limit of the functions appearing in hydrodynamic metric B.1 Within the membrane region B.1.1 F (y) B.1.2 H1 (y) B.1.3 K1 (y) B.1.4 K2 (y) B.1.5 L(y) B.1.6 H2 (y) B.2 Outside membrane region B.2.1 F (y) B.2.2 K1 (y) B.2.3 H1 (y) B.2.4 H2 (y)
–i–
28 32 33 33 33 34 34 35 35 39 39 40 40 41
JHEP05(2019)054
1 Introduction 1.1 Strategy 1.1.1 Part-1 1.1.2 Part-2 1.1.3 Part-3
C Relation between horizon ρH (y) in Y A (≡ {ρ, y µ }) coordinates and H(x) in X A (≡ {r, xµ }) coordinates 41 D Identities
43
E Notations
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Introduction
Classical evolution of the space-time is governed by Einstein’s equations, which are a set of nonlinear partial differential equations. Till date, it has been impossible to solve these equations in full generality, particularly when the geometry has
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