Second law of black-hole thermodynamics in Lovelock ...seminar/pdf_2017_kouki/171121Kun… ·...

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Second law of black-hole thermodynamics in Lovelock theories of gravity

Nilay KunduYITP, Kyoto

Reference : 1612.04024 ( JHEP 1706 (2017) 090 )

With : Sayantani Bhattacharyya, Felix Haehl, R. Loganayagam, Mukund Rangamani

2nd law of BH thermodynamics in Lovelock Theories of gravity

Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory

Why Higher derivative theory of gravity ?

Why Higher derivative theory of gravity ?I =

16⇡GN

⇥R+ Lmatter + ↵ LHD

- Low energy limit of any UV-complete theory of quantum gravity is expected to generate higher derivative corrections to the leading two-derivative Einstein-Hilbert action with matter

16⇡GN

- Particular form of the higher derivative correction depends on the particular UV completion, however there are limitations of a fully consistent quantum theory of gravity …

- We need to go beyond classical Einstein’s theory of general relativity - “String theory” - a prominent consistent candidate for a UV-complete theory of gravity

- It also has limitations ===> time dependent processes !!

16⇡GN

Is there any “general principle” to constrain the low energy behavior of the effective theory of gravity ?

16⇡GN

Is there any “general principle” to constrain the low energy behavior of the effective theory of gravity ?

- Dynamical black holes==> They are “theoretical laboratories” for understanding quantum nature of gravity !!

- One such general principle is “2nd law of black hole thermodynamics” : which we can test on solutions of low energy effective theory of gravity.

16⇡GN

A general principle to constrain low energy

effective theory of gravity

16⇡GN

- This is obviously a statement beyond equilibriumWhat is the statement of 2nd law ?

Eq1 ) Eq2, Total Entropy|Eq2 � Total Entropy|Eq1

- This is a non-local statement ==> Depends only on initial and final end points of the time evolution, two equilibrium points.

16⇡GN

- This is obviously a statement beyond equilibriumWhat is the statement of 2nd law ?

- We can ensure this by constructing a local “entropy function”- fn. of the state variables - that is

(a) monotonically increasing under time evolution (b) reduces to familiar notion of equilibrium values at the two end points

- This is a non-local statement ==> Depends only on initial and final end points of the time evolution, two equilibrium points.

16⇡GN

What is the statement of 2nd law in gravity ?

16⇡GN

Equlibrium configuration ==> Metric with a killing horizon

Equilibrium entropy ==> BH entropy on Killing horizon

16⇡GN

metric1 ) metric2, BH Entropy|metric2 � BH Entropy|metric1

16⇡GN

- There is an entropy functional interpolating bw/ two equilibrium metric ==> a local version of the 2nd law

metric1 ) metric2, BH Entropy|metric2 � BH Entropy|metric1

16⇡GN

BH Entropy|metric2 � BH Entropy|metric1

16⇡GN

- BH Entropy ==> Area of the black hole horizon

- 1st law and 2nd law (Hawking’s area theorem) are both known to be satisfied

What happens for Einstein’s general relativity ?

16⇡GN

- BH Entropy ==> Area of the black hole horizon

- 1st law and 2nd law (Hawking’s area theorem) are both known to be satisfied

What happens for Einstein’s general relativity ?

What happens “Beyond” Einstein’s general relativity ?

- Wald Entropy satisfies 1st law for any higher derivative correction to GR

- No general proof of 2nd law

SWald = �2⇡

�L�Rabcd

✏ab✏cd

16⇡GN

No general proof of 2nd law beyond GR for Wald entropyBH Entropy|metric2 � BH Entropy|metric1

16⇡GN

No general proof of 2nd law beyond GR for Wald entropy

BIG GOAL - To prove 2nd law for dynamical BH soln. in general Higher derivative theory of gravity ==> A local version “construct an entropy functional”

16⇡GN

OR- To find a concrete counter example such that we can rule out theories

demanding 2nd law

- To prove 2nd law for dynamical BH soln. in general Higher derivative theory of gravity ==> A local version “construct an entropy functional”BIG GOAL

16⇡GN

What we achieved :

- A small step !! == > We checked things in one “simple” model of higher derivative gravity ==> Lovelock theory

- To prove 2nd law for dynamical BH soln. in general Higher derivative theory of gravity ==> A local version “construct an entropy functional”

- To find a concrete counter example such that we can rule out theories demanding 2nd law

BIG GOALOR

- Constructing a local “entropy function” that is (a) monotonically increasing under time evolution (b) reduces to Wald entropy at the two equilibrium end points of the time evolution.

- Modification of Wald entropy away from equilibrium

Our Aim :Wald Entropy|metric2 � Wald Entropy|metric1

SWald = �2⇡

�L�Rabcd

✏ab✏cdI =1

16⇡GN

R+ Lmatter + ↵ LHD

Our Aim :Wald Entropy|metric2 � Wald Entropy|metric1

Let us organize things a little better : - Higher derivative terms in the action comes with a characteristic length scale - We have dimension-less coupling - We start with an initial equilibrium configuration : A stationary black hole - We then perturb it slightly - Perturbations are denoted by two parameters ==> (a) amplitude “a” , (b) frequency “w”- The entropy function should have knowledge about these three

parameters ==> (a) amplitude “a” , (b) frequency “w”, and, (c) coupling “ ”↵

Stotal

SWald = �2⇡

�L�Rabcd

✏ab✏cdI =1

16⇡GN

Our Aim :

What is known so far in literature : - For any coupling “ ”, but with “a=0” : Wald entropy may be constructed as

the desired entropy function - For f(R) theories, in finite range of coupling “ ”, but arbitrary “a” and “w”,

the entropy function can be constructed - For small amplitude expansion (a << 1), considering 4-derivative theories of

gravity the entropy function can be constructed - In the context of holographic EE, particular correction to Wald entropy has

been constructed, but again in small amplitude expansion (a << 1)

Stotal[a,!,↵]

Wald Entropy|metric2 � Wald Entropy|metric1

SWald = �2⇡

�L�Rabcd

✏ab✏cdI =1

16⇡GN

Our Aim :

Stotal[a,!,↵]

- We would work perturbatively in higher derivative interactions, treating the correction to Einstein’s GR in a gradient expansion.

- The small parameter is the dimensionless number with arbitrary amplitude away from the equilibrium.

!ls ⌧ 1

We work in a different expansion : - In our work we aim to construct the entropy function in the frequency

expansion, but for arbitrary amplitude

↵ ⌧ 1, !ls ⌧ 1

SWald = �2⇡

�L�Rabcd

✏ab✏cdI =1

16⇡GN

Our Aim :

We work in a different expansion : - In our work we aim to construct the entropy function in the frequency

expansion, but for arbitrary amplitude

Stotal[a,!,↵]

- In other words, we allow for arbitrary time evolution away from equilibrium, as long as this time evolution is sensibly captured by the low-energy effective action.

- Geometrically, We allow fluctuations of BH horizon with at the horizon is small compared to the curvature scales.

- We assume that the Classical gravity description is valid ==> no loop correction etc. enter in the game.

↵ ⌧ 1, !ls ⌧ 1

SWald = �2⇡

�L�Rabcd

✏ab✏cdI =1

16⇡GN

Our Aim :

Stotal[a,!,↵]

Our analysis : ↵ ⌧ 1, !ls ⌧ 1, a > 0

SWald = �2⇡

�L�Rabcd

✏ab✏cdI =1

16⇡GN

Our Aim :

Stotal[a,!,↵]

Our analysis : ↵ ⌧ 1, !ls ⌧ 1, a > 0

Question ?? If two derivative gravity, i.e. Einstein’s GR, dominates at the horizon, is it possible at all that we need to modify the entropy away from equilibrium ?

SWald = �2⇡

�L�Rabcd

✏ab✏cdI =1

16⇡GN

Our Aim :

Stotal[a,!,↵]

Our analysis : ↵ ⌧ 1, !ls ⌧ 1, a > 0

Question ?? If two derivative gravity, i.e. Einstein’s GR, dominates at the horizon, is it possible at all that we need to modify the entropy away from equilibrium ?

Naive answerThough the leading area contribution is large, it’s variation may be anomalously small and contribution from higher derivative terms may overcome it.

SWald = �2⇡

�L�Rabcd

✏ab✏cdI =1

16⇡GN

↵m `2m�2s Lm + Lmatter

Lm = �µ1⌫1···µm⌫m⇢1�1···⇢m�m

R⇢1µ1

�1⌫1 · · · R⇢m

µm�m

⌫m .

The action for the Lovelock theory

↵m = dimensionless numbers, coupling

ls = Some scale at which the higher derivative terms become important

We restrict to the Gauss-Bonnet theory, “m=2”

- We are neglecting the matter part, as it will not play any role in our analysis. Or we need to impose NEC.

✓R + ↵2l

L2 =LGB = R2 �Rµ⌫Rµ⌫ + Rµ⌫⇢�R

µ⌫⇢�

↵m `2m�2s Lm + Lmatter

Lm = �µ1⌫1···µm⌫m⇢1�1···⇢m�m

R⇢1µ1

�1⌫1 · · · R⇢m

µm�m

⌫m .

The action for the Lovelock theory

↵m = dimensionless numbers, coupling

ls = Some scale at which the higher derivative terms become important

- We start with one equilibrium/stationary metric of a BH with a regular horizon, by fixing a coordinate chart

We restrict to the Gauss-Bonnet theory, “m=2”

ds2 = 2 dv dr � f(r, v,x) dv2 + 2 kA(r, v,x) dv dxA + hAB(r, v,x) dxAdxB

f(r, v,x)��H+ = kA(r, v,x)

��H+ = @rf(r, v,x)

��H+ = 0

The null hypersurface of the horizon H+ is the locus r = 0

Spatial section (const. v slices) of H+ = ⌃v

a�nely parametrized null generator of H+ = @v

- We are neglecting the matter part, as it will not play any role in our analysis. Or we need to impose NEC.

✓R + ↵2l

µ⌫⇢�

“r = 0” surface � Horizon

∂r - generator

∂v - generator

∂A - generator

Schematics of Horizon coordinates

f(r, v,x)��H+ = kA(r, v,x)

��H+ = @rf(r, v,x)

��H+ = 0

The geometry has a horizon, a null hypersurface at r=0 H

The coordinates on the horizon = {v, xA}

The coordinates on the constant v-slices of horizon ⌃v ) {xA}

Away from the horizon the coordinate “r” ==> Affinely parametrized along null

geodesics piercing the horizon at an angle (@v, @r)

��H+ = 1, (@r, @A)

��H+ = 0

“r = 0” surface � Horizon

∂r - generator

∂v - generator

∂A - generator

Schematics of Horizon coordinates

f(r, v,x)��H+ = kA(r, v,x)

��H+ = @rf(r, v,x)

��H+ = 0

The geometry has a horizon, a null hypersurface at r=0 H

The coordinates on the horizon = {v, xA}

The coordinates on the constant v-slices of horizon ⌃v ) {xA}

Away from the horizon the coordinate “r” ==> Affinely parametrized along null

geodesics piercing the horizon at an angle (@v, @r)

��H+ = 1, (@r, @A)

��H+ = 0

We define these quantities KAB =

2@vhAB |r=0, KAB =

2@rhAB |r=0

) KAB vanishes at equilibrium

The extrinsic curvature on the horizon slice

We are working withGauss-Bonnet theory

We want to construct an entropy functional

Condition 1 : @vStotal � 0

Condition 2 : Stotal reduces to Wald entropy

SWald in equilibrium

✓R + ↵2l

µ⌫⇢�

The Wald entropy needs to be modified away from equilibrium

Stotal = SWald + Scor

Scor|equilibrium = 0

✓R + ↵2l

µ⌫⇢�

How is 2nd law proved then ?

SfinalWald � Sinitial

Wald =

Z final

initial@vStotal dv � 0

The Wald entropy needs to be modified away from equilibrium

✓R + ↵2l

µ⌫⇢�

Strategy :

✓R + ↵2l2

L2 =LGB = R2 � 4Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�

(1) Define entropy density : Stotal =

dd�2xph ⇢total

(2) Define ⇥ ) @vStotal =

dd�2xph ⇥

(3) Considering ⇥ ! 0, as v ! 1) calculate @v⇥, and show that @v⇥ 0,

(4) We get ⇥ > 0, for all v

(5) In tern we get @vStotal � 0

Strategy :

✓R + ↵2l2

L2 =LGB = R2 � 4Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�

dd�2xph ⇢total

dd�2xph ⇥

Strategy :

dd�2xph ⇢total

dd�2xph ⇥

✓R + ↵2l2

L2 =LGB = R2 � 4Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�

Strategy : (1) Define entropy density : Stotal =

dd�2xph ⇢total

dd�2xph ⇥

(3) To obtain @vStotal � 0, show that @v⇥ 0

How does it work for Einstein GR (Hawking’s area increase theorem)

p�g R(1) SWald =

dd�2xph

(2) ⇥Einstein =1

2hAB@vhAB = KA

(3) @v⇥Einstein = �KABKAB �Rvv = �KABKAB � Tvv 0

(We used EOM Rvv = Tvv, and NEC Tvv � 0)

dd�2xph ⇢total

dd�2xph ⇥

How does it work for Einstein GR (Hawking’s area increase theorem)

p�g R(1) SWald =

dd�2xph

(2) ⇥Einstein =1

2hAB@vhAB = KA

(3) @v⇥Einstein = �KABKAB �Rvv = �KABKAB � Tvv 0

(We used EOM Rvv = Tvv, and NEC Tvv � 0)

What is the problem with higher derivative gravity?

16⇡GN

@v⇥ = �KABKAB �Rvv = �KABKAB � Tvv,

Rvv = Tvv

This equation will be changed due to the higher derivative term

dd�2xph ⇢total

dd�2xph ⇥

16⇡GN

@v⇥ = � KABKAB � Tvv,

Rvv = Tvv

dd�2xph ⇢total

dd�2xph ⇥

16⇡GN

@v⇥ = �hAA0hBB0

KA0B0⇥KAB + ↵l2s@r@vKAB

⇤� Tvv

For example we can have a situation where

@v⇥ = � KABKAB � Tvv,

Rvv = Tvv

KAB ⇠ ↵l2s@r@vKABAnd

@r@vKAB May remain unsuppressed and violate the proof

In a perturbative amplitude expansion things do work out up to linearized order

SWald =1

dd�2xph⇥1 + ↵ ⇢HD

p�g (R+ ↵LHD)

@vSWald =1

dd�2xph⇥@v(log

ph)(1 + ↵ ⇢HD) + ↵ @v⇢HD

SWald =1

p�g (R+ ↵LHD)

@vSWald =1

dd�2xp

@v(logp

h)(1 + ↵ ⇢HD) + ↵ @v⇢HD| {z }=⇥

@v⇥ = �Tvv + ↵⇥rvrv⇢HD � ⇢HDRvv + EOMvv

SWald =1

p�g (R+ ↵LHD)

@vSWald =1

dd�2xp

@v(logp

h)(1 + ↵ ⇢HD) + ↵ @v⇢HD| {z }=⇥

@v⇥ = �Tvv + ↵⇥rvrv⇢HD � ⇢HDRvv + EOMvv

⇤| {z }

==>2nd law is valid up to linear order in amplitude@v⇥ = �Tvv < 0

Let us examine the Gauss-Bonnet case

(1) Obtain ⇥eq ) @vSWald =

dd�2xph ⇥eq

(2) Compute @v⇥eq and convince that @v⇥eq 0 not satisfied

STEP 1 :

Conclusion : We need to modify Wald entropy

SWald =1

dd�2xph [1 + 2↵2l

2s Rind]

R + ↵2l2

s(R2 � 4Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�)

Let us examine the Gauss-Bonnet case

dd�2xph ⇥eq

STEP 1 :

STEP 2 :

(1) Stotal = SWald + Scor, Scor|equilibrium = 0

(2) Obtain ⇥total = ⇥eq +⇥cor ) @vStotal =

dd�2xph⇥⇥eq +⇥cor

(3) Make sure that @v⇥total 0

Conclusion : 2nd law is satisfied for Gauss-Bonnet theory

SWald =1

dd�2xph [1 + 2↵2l

2s Rind]

R + ↵2l2

s(R2 � 4Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�)

The final result for the Gauss-Bonnet case

The Equilibrium quantities :

The non-Equilibrium correction :

Stotal =SWald + Scor

Scor =1

dd�2xph ⇢cor

such that, ⇢cor =1X

n lns @nv

�↵2l

�lns @

�↵2l

where, hAB =�AA1A2

BB1B2KB1

SWald =1

dd�2xph [1 + 2↵2l

2s Rind]

R + ↵2l2

s (R2 � Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�)

The final result for the Gauss-Bonnet case

SWald =1

dd�2xph [1 + 2↵2l

2s Rind]

The Equilibrium quantities :

The non-Equilibrium correction :

Stotal =SWald + Scor

Scor =1

dd�2xph ⇢cor

such that, ⇢cor =1X

n lns @nv

�↵2l

�lns @

�↵2l

where, hAB =�AA1A2

BB1B2KB1

The conditions for ==>

@v⇥total = @v⇥eq + @v⇥cor 0

An = 2n �2n�1

An�2, for n = �2,�1, 0, 1, · · ·

(for �2 = �1/2, 0 = �1, �1 = �2),

the constraint reads : An 0 for n � �2.

R + ↵2l2

s (R2 � Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�)

dd�2xph ⇥eq

STEP 1 :

STEP 2 :

(1) Stotal = SWald + Scor, Scor|equilibrium = 0

Conclusion : 2nd law is satisfied for Gauss-Bonnet theory

SWald =1

dd�2xph [1 + 2↵2l

2s Rind]

R + ↵2l2

s(R2 � 4Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�)

�Let us examine the Gauss-Bonnet case with some explicit expressions

@v⇥eq = � Tvv|{z}T1

�KABKAB| {z }T2

+↵2`2s KA

B0MBB0

AA0| {z }T3

+ ↵2`2s KA

B @v��BA1A2AB1B2

�| {z }

| {z }T5

,STEP 1 :

! MBB0

AA0 —> does not contain any v-derivative ..

Let us examine the Gauss-Bonnet case with some explicit expressions

STEP 1 :

(1) T1 + T2 0 ) with NEC

(2) T2 + T3 =KABKA0

B0⇥�BA�B

A0 + ↵2`2sMBB0

AA0⇤

T3 < T2

(3) T2 + T4 =KAB

A + ↵2`2s @v

��BA1A2AB1B2

�⇤

This can spoil the proof

- For Einstein’s gravity, things work out nicely .. - We neglect Term-3 compared to Term-2- Term-4 is potentially dangerous and if ``T4 > T2’’ ==> .@v⇥eq � 0

�KABKAB| {z }T2

+↵2`2s KA

B0MBB0

AA0| {z }T3

+ ↵2`2s KA

�| {z }

| {z }T5

—> does not contain any v-derivative ..! MBB0

STEP 1 :

(1) T1 + T2 0 ) with NEC

(2) T2 + T3 =KABKA0

B0⇥�BA�B

A0 + ↵2`2sMBB0

AA0⇤

T3 < T2

(3) T2 + T4 =KAB

A + ↵2`2s @v

��BA1A2AB1B2

�⇤

This can spoil the proof

- For Einstein’s gravity, things work out nicely .. - We neglect Term-3 compared to Term-2- Term-4 is potentially dangerous and if ``T4 > T2’’ ==> .- We need to modify the equilibrium Wald entropy …

@v⇥eq � 0

�KABKAB| {z }T2

+↵2`2s KA

B0MBB0

AA0| {z }T3

+ ↵2`2s KA

�| {z }

| {z }T5

—> does not contain any v-derivative ..! MBB0

STEP 2 :(1) Stotal = SWald + Scor, Scor|equilibrium = 0

Question : How to decide the correction to wald entropy @v⇥total = @v⇥eq + @v⇥cor 0

| {z }T5

�KABKAB

| {z }T2

+↵2l2s KA

�BA1A2

AB1B2KB1

| {z }T4

@v⇥eq + @v⇥cor| {z }=@v⇥total

= � Tvv|{z}T1

| {z }T5

� KABKAB

| {z }T2

+↵2l2s KA

�BA1A2AB1B2

| {z }T4

+ � (↵2l2s)

�AA1A2BB1B2

�BA1A2AB1B2

| {z }=@v⇥cor

= � Tvv|{z}T1

| {z }T5

� KABKAB

| {z }T2

+↵2l2s KA

�BA1A2AB1B2

| {z }T4

+ � (↵2l2s)

�AA1A2BB1B2

�BA1A2AB1B2

| {z }=@v⇥cor

Scor = �

dd�2xph (↵2l

�AA1A2BB1B2

� �BA1A2AB1B2

@v⇥cor = � (↵2l2s)

�AA1A2BB1B2

�BA1A2AB1B2

�will produce the desired

Scor = �

dd�2xph (↵2l

�AA1A2BB1B2

� �BA1A2AB1B2

@v⇥cor = � (↵2l2s)

�AA1A2BB1B2

�BA1A2AB1B2

Scor = �

dd�2xph (↵2l

�AA1A2BB1B2

� �BA1A2AB1B2

@v⇥cor = � (↵2l2s)

�AA1A2BB1B2

�BA1A2AB1B2

= � Tvv|{z}T1

| {z }T5

�✓

KAB � ↵2l2s

�AA1A2

BB1B2KB1

�◆2

+ (↵2l2s)

2 (� � 1/4) @v

�AA1A2

BB1B2KB1

�BA1A2

AB1B2KB1

) @v⇥total 0 � 1/4==> if we fix the free parameter ==>

Scor = �

dd�2xph (↵2l

�AA1A2BB1B2

� �BA1A2AB1B2

@v⇥cor = � (↵2l2s)

�AA1A2BB1B2

�BA1A2AB1B2

= � Tvv|{z}T1

| {z }T5

�✓

KAB � ↵2l2s

�AA1A2

BB1B2KB1

�◆2

+ (↵2l2s)

2 (� � 1/4) @v

�AA1A2

BB1B2KB1

�BA1A2

AB1B2KB1

) @v⇥total 0 � 1/4==> if we fix the free parameter ==>

- Also the obstruction term (T5) vanishes —rAYA = 4↵2l

2s rArB

KKAB � KA

CKBC � hAB

2(K2 � KCDKCD)

�= 0

- for spherically symmetric cases it indeed vanishes ..

Final comments … 1. This can be generalized to arbitrary orders in alpha

expansion and also for Lovelock families .. 2. Our construction surely works for spherically symmetric

configurations .. 3. The obstruction term should have some geometric

meaning .. need to be explored ..4. This construction is also not unique .. 5. Subtle issues regarding field re-definitions and foliation

dependence .. 6. This method is also indirect .. 7. Possible connections with Holographic entanglement

entropy ..

Final comments … 1. This can be generalized to arbitrary orders in alpha

expansion and also for Lovelock families .. 2. Our construction surely works for spherically symmetric

configurations .. 3. The obstruction term should have some geometric

meaning .. need to be explored ..4. This construction is also not unique .. 5. Subtle issues regarding field re-definitions and foliation

dependence .. 6. This method is also indirect .. 7. Possible connections with Holographic entanglement

entropy ..

— : THANK YOU VERY MUCH FOR YOUR KIND ATTENTION : —

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