Stress tensor for large- D membrane at subleading orders
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Springer
Received: January 2, Revised: May 10, Accepted: June 29, Published: July 17,
2020 2020 2020 2020
Parthajit Biswas School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni 752050, India
E-mail: [email protected] Abstract: In this note, we have extended the result of [1] to calculate the membrane 1 stress tensor up to O D localized on the co-dimension one membrane world volume propagating in asymptotically flat/AdS/dS spacetime. We have shown that the subleading order membrane equation follows from the conservation equation of this stress tensor. Keywords: Black Holes, Classical Theories of Gravity ArXiv ePrint: 1912.00476
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)110
JHEP07(2020)110
Stress tensor for large-D membrane at subleading orders
Contents 1 Introduction 1.1 Final result 1.2 Strategy
1 3 5 6 6 8 10
3 Linearized solution: inside (ψ < 1)
12
4 Stress tensor 4.1 Outside (ψ > 1) stress tensor 4.2 Inside (ψ < 1) stress tensor 4.3 Membrane stress tensor
15 15 16 16
5 Conservation of the membrane stress tensor
18
6 Conclusions
19
A Calculation of integrals (2.4) at linear order
20
B Some details of linearized calculation B.1 Outside (ψ > 1) B.2 Inside (ψ < 1)
23 23 36
C Some details of stress tensor calculation
46
D Important identities
49
E Notations
50
1
Introduction
Recently, it has been shown that in large number of dimensions D, the classical black hole dynamics simplify a lot [2–9].1 In this limit, the effect of the black hole gets confined around its event horizon in a parametrically thin shell whose thickness is proportional to the inverse of the number of spacetime dimensions — which we will refer to as ‘membrane’. The 1
See [10–33] for subsequent developments related to large D expansion.
–1–
JHEP07(2020)110
2 Linearized solution: outside (ψ > 1) 2.1 Large-D metric upto sub-subleading order: linearized 2.2 Change of gauge condition 2.3 Change of subsidiary condition
membrane is characterized by its shape (one variable) and a unit normalized velocity field on the membrane world volume (D −2 variables). There is a one-to-one correspondence between the dynamics of the black hole and the dynamics of the co-dimension one membrane propagating in the asymptotic spacetime of the black hole [6–9]. The Einstein’s equations 1 determine the effective equation of the membrane in an expansion in D . Up to the sublead1 ing order in D expansion, the membrane equations of motion take the following form [9] Pβα
(1.1)
These are D − 1 set of equations for as many variables and (1.1) defines a well-posed initial value problem for membrane dynamics. Here, Greek indices denote coordinates on the (D−1) — dimensional membrane worldvolume, uµ and Kµν denote the unit normalized velocity field and extrinsic curvature on the membrane surface respectively. All the quan(ind) tities in (1.1) are constructed using the induced metric gµν on the membrane — where the membrane is embedded in the b
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