Membrane dynamics and Gravity
Based on arXiv:1712.09400
with S. Kundu, S. Mazumdar, S. Minwalla, A. Mishra, A. Saha

Yogesh Dandekar
Tata Institute of Fundamental Research, Mumbai

Gong Show, Strings 2018

Yogesh Dandekar

Membrane dynamics and Gravity

1/6Large D membrane paradigm[2015 - ongoing]

Main point: Dynamics of black hole → Dynamics of membrane
codimension-1 timelike non-gravitational hypersurface
embedded in unperturbed (e.g. flat) background spacetime.
Membrane variables: shape and velocity field uµ (D − 1)
Membrane equations obtained in a perturbation series in 1/D.
Membrane localized Stress tensor and Entropy current
16πTµν = K Pµν − 2σµν + (Kµν − Kgµν ) + (. . .)uµˆ µ J µ = 1 σαβ σ αβ + (. . .)JSµ =+ (. . .)=⇒ ∇S
42K(D) Spacetime Stress tensor conservation:ˆ µ Tµν = 0 =⇒ membrane equations at Large D(D − 1): ∇µν(1): K Tµν = 0 identically order by order=⇒ Formulation consistent strictly at large D.
Local form of 2nd law at large D
Yogesh Dandekar

Membrane dynamics and Gravity

2/6‘Improved’ arbitrary D membrane[arXiv:1712.09400]‘Improved’ membrane Stress tensor and Entropy currentuµ41ˆ µJ µ =∇σαβ σ αβ
S2WJSµ =16πTµν = W Pµν − 2σµν + (Kµν − Kgµν )W ≡K 2 − K µν Kµν + 2K µν σµν
K + u.K.u

in any background that solves Einstein equations.
K µν Tµν = 0 identically =⇒ Formulation consistent for arbitrary D
Exactly obeys local form of 2nd law
Reduces to previous Tµν , JSµ at leading order in large D=⇒ Reproduces black hole physics “at least” at large D.
=⇒ So, a consistent finite D completion
Note: ρ = 0,Yogesh DandekarW
p = 16π,

WT = 4π,s = 14 ,1
η = 16π,

Membrane dynamics and Gravity

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η1
s = 4πStationary solutionskµˆ = 0.
for a Killing vector k µ . Also, σαβ = 0, ∇.uuµ = √−k.k2µνˆ µ Tµν =⇒ K −Kµν K∇
K+u.K.u0= √4πT: Shape equation−k.kConsider action:
ZZ√√
24πT01√Λ−G +−g K −S=
16π D − 2−k.kVM
δS = 0 under change in shape =⇒ Shape equation
Stationary stress tensor Tµν = − √2−g δgδSµνAction onshell reduces toS=E− Sent = − ln Z
T0Note ∂β S = E onshell
Yogesh Dandekar

Membrane dynamics and Gravity

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Surprises at finite D
Energy, Entropy and Temperature of static spherical membrane in
AdS/Flat/dS matches at finite D with Schwarzschild black hole.
Consider slowly varying planar membrane in AdS. Calculate the linearized
gravitational field sourced by Tµν . Find the boundary stress tensor. After
field redefinitions
d(σσ + ωω + ωσ + v.Dσ)Tµν = p (ηµν + d vµ vν ) − 2ησµν + 2η
4πT
dd−14πT
14πT1,η =, (d = D − 1)
where p =16πd16πdMatches with Fluid-Gravity upto first order in derivative expansion,even at finite d!
Agrees with Fluid-Gravity at second order only in large d limit,with subleading deviations.
Yogesh Dandekar

Membrane dynamics and Gravity

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SummarySpecial kind membrane dynamics, well defined at arbitrary D.
Simple shape equation and Action for stationary membranes. Action onshell
reduces to thermal partition function.
Reproduces black hole physics better than expected for finite D!
Can simulations capture key features of black hole dynamics (e.g. mergers,Gregory-Laflamme instabilities)?
Can we construct a membrane formulation such that it can reproduce all thestationary black hole solutions exactly?
Can we understand dynamical Second law of black holes thermodynamics forhigher derivative theories of gravity?

Thank you!
I invite you to see my poster
Yogesh DandekarMembrane dynamics and Gravity

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