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Transcript of Second law of black-hole thermodynamics in Lovelock ...seminar/pdf_2017_kouki/171121Kun… ·...
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
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
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 ?
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 ?I =
1
16⇡GN
Zddx
p�g
⇥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
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 ?I =
1
16⇡GN
Zddx
p�g
⇥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
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 ?I =
1
16⇡GN
Zddx
p�g
⇥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
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 ?I =
1
16⇡GN
Zddx
p�g
⇥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
- 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 …
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
- 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 !!
Why Higher derivative theory of gravity ?I =
1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
Is there any “general principle” to constrain the low energy behavior of the effective theory of gravity ?
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
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.
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
- 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.
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
- 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.
Eq1 ) Eq2, Total Entropy|Eq2 � Total Entropy|Eq1
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
Eq1 ) Eq2, Total Entropy|Eq2 � Total Entropy|Eq1
What is the statement of 2nd law in gravity ?
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
Eq1 ) Eq2, Total Entropy|Eq2 � Total Entropy|Eq1
Equlibrium configuration ==> Metric with a killing horizon
Equilibrium entropy ==> BH entropy on Killing horizon
What is the statement of 2nd law in gravity ?
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
Eq1 ) Eq2, Total Entropy|Eq2 � Total Entropy|Eq1
Equlibrium configuration ==> Metric with a killing horizon
Equilibrium entropy ==> BH entropy on Killing horizon
metric1 ) metric2, BH Entropy|metric2 � BH Entropy|metric1
What is the statement of 2nd law in gravity ?
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
Eq1 ) Eq2, Total Entropy|Eq2 � Total Entropy|Eq1
Equlibrium configuration ==> Metric with a killing horizon
Equilibrium entropy ==> BH entropy on Killing horizon
- 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
What is the statement of 2nd law in gravity ?
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
BH Entropy|metric2 � BH Entropy|metric1
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
- 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 ?
BH Entropy|metric2 � BH Entropy|metric1
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
- 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
BH Entropy|metric2 � BH Entropy|metric1
SWald = �2⇡
Z
H
�L�Rabcd
✏ab✏cd
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
No general proof of 2nd law beyond GR for Wald entropyBH Entropy|metric2 � BH Entropy|metric1
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
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”
BH Entropy|metric2 � BH Entropy|metric1
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
No general proof of 2nd law beyond GR for Wald entropy
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
BH Entropy|metric2 � BH Entropy|metric1
2nd law of BH thermodynamics in Lovelock Theories of gravity
Higher derivative theories of gravity—> Lovelock / Gauss Bonnet Theory
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
A general principle to constrain low energy
effective theory of gravity
No general proof of 2nd law beyond GR for Wald entropy
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
BH Entropy|metric2 � BH Entropy|metric1
- 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⇡
Z
H
�L�Rabcd
✏ab✏cdI =1
16⇡GN
Zddx
p�g
R+ Lmatter + ↵ LHD
�
- 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
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⇡
Z
H
�L�Rabcd
✏ab✏cdI =1
16⇡GN
Zddx
p�g
R+ Lmatter + ↵ LHD
�
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,!,↵]
↵
↵
- 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
Wald Entropy|metric2 � Wald Entropy|metric1
SWald = �2⇡
Z
H
�L�Rabcd
✏ab✏cdI =1
16⇡GN
Zddx
p�g
R+ Lmatter + ↵ LHD
�
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
- 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
Wald Entropy|metric2 � Wald Entropy|metric1
SWald = �2⇡
Z
H
�L�Rabcd
✏ab✏cdI =1
16⇡GN
Zddx
p�g
R+ Lmatter + ↵ LHD
�
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
- 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
Wald Entropy|metric2 � Wald Entropy|metric1
SWald = �2⇡
Z
H
�L�Rabcd
✏ab✏cdI =1
16⇡GN
Zddx
p�g
R+ Lmatter + ↵ LHD
�
Our Aim :
Stotal[a,!,↵]
Our analysis : ↵ ⌧ 1, !ls ⌧ 1, a > 0
- 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
Wald Entropy|metric2 � Wald Entropy|metric1
SWald = �2⇡
Z
H
�L�Rabcd
✏ab✏cdI =1
16⇡GN
Zddx
p�g
R+ Lmatter + ↵ LHD
�
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 ?
- 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
Wald Entropy|metric2 � Wald Entropy|metric1
SWald = �2⇡
Z
H
�L�Rabcd
✏ab✏cdI =1
16⇡GN
Zddx
p�g
R+ Lmatter + ↵ LHD
�
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.
- 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
Wald Entropy|metric2 � Wald Entropy|metric1
SWald = �2⇡
Z
H
�L�Rabcd
✏ab✏cdI =1
16⇡GN
Zddx
p�g
R+ Lmatter + ↵ LHD
�
I =1
4⇡
Zddx
p�g
R+
1X
m=2
↵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.
I =1
4⇡
Zddx
p�g
✓R + ↵2l
2s L2
◆
L2 =LGB = R2 �Rµ⌫Rµ⌫ + Rµ⌫⇢�R
µ⌫⇢�
I =1
4⇡
Zddx
p�g
R+
1X
m=2
↵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.
I =1
4⇡
Zddx
p�g
✓R + ↵2l
2s L2
◆
L2 =LGB = R2 �Rµ⌫Rµ⌫ + Rµ⌫⇢�R
µ⌫⇢�
“r = 0” surface � Horizon
∂r - generator
∂v - generator
∂A - generator
Schematics of Horizon coordinates
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 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
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 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 =
1
2@vhAB |r=0, KAB =
1
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
I =1
4⇡
Zddx
p�g
✓R + ↵2l
2s L2
◆
L2 =LGB = R2 �Rµ⌫Rµ⌫ + Rµ⌫⇢�R
µ⌫⇢�
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
The Wald entropy needs to be modified away from equilibrium
Stotal = SWald + Scor
Scor|equilibrium = 0
I =1
4⇡
Zddx
p�g
✓R + ↵2l
2s L2
◆
L2 =LGB = R2 �Rµ⌫Rµ⌫ + Rµ⌫⇢�R
µ⌫⇢�
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
Stotal = SWald + Scor
Scor|equilibrium = 0
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
I =1
4⇡
Zddx
p�g
✓R + ↵2l
2s L2
◆
L2 =LGB = R2 �Rµ⌫Rµ⌫ + Rµ⌫⇢�R
µ⌫⇢�
Condition 1 : @vStotal � 0
Condition 2 : Stotal reduces to Wald entropy
SWald in equilibrium
Stotal = SWald + Scor
Scor|equilibrium = 0
Strategy :
I =1
4⇡
Zddx
p�g
✓R + ↵2l2
s L2
◆
L2 =LGB = R2 � 4Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�
(1) Define entropy density : Stotal =
Z
⌃v
dd�2xph ⇢total
(2) Define ⇥ ) @vStotal =
Z
⌃v
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
Condition 1 : @vStotal � 0
Condition 2 : Stotal reduces to Wald entropy
SWald in equilibrium
Stotal = SWald + Scor
Scor|equilibrium = 0
Strategy :
I =1
4⇡
Zddx
p�g
✓R + ↵2l2
s L2
◆
L2 =LGB = R2 � 4Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�
(1) Define entropy density : Stotal =
Z
⌃v
dd�2xph ⇢total
(2) Define ⇥ ) @vStotal =
Z
⌃v
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
Condition 1 : @vStotal � 0
Condition 2 : Stotal reduces to Wald entropy
SWald in equilibrium
Stotal = SWald + Scor
Scor|equilibrium = 0
Strategy :
(1) Define entropy density : Stotal =
Z
⌃v
dd�2xph ⇢total
(2) Define ⇥ ) @vStotal =
Z
⌃v
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
I =1
4⇡
Zddx
p�g
✓R + ↵2l2
s L2
◆
L2 =LGB = R2 � 4Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�
Strategy : (1) Define entropy density : Stotal =
Z
⌃v
dd�2xph ⇢total
(2) Define ⇥ ) @vStotal =
Z
⌃v
dd�2xph ⇥
(3) To obtain @vStotal � 0, show that @v⇥ 0
How does it work for Einstein GR (Hawking’s area increase theorem)
I =1
4⇡
Zddx
p�g R(1) SWald =
Z
⌃v
dd�2xph
(2) ⇥Einstein =1
2hAB@vhAB = KA
A
(3) @v⇥Einstein = �KABKAB �Rvv = �KABKAB � Tvv 0
(We used EOM Rvv = Tvv, and NEC Tvv � 0)
Strategy : (1) Define entropy density : Stotal =
Z
⌃v
dd�2xph ⇢total
(2) Define ⇥ ) @vStotal =
Z
⌃v
dd�2xph ⇥
(3) To obtain @vStotal � 0, show that @v⇥ 0
How does it work for Einstein GR (Hawking’s area increase theorem)
I =1
4⇡
Zddx
p�g R(1) SWald =
Z
⌃v
dd�2xph
(2) ⇥Einstein =1
2hAB@vhAB = KA
A
(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?
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
@v⇥ = �KABKAB �Rvv = �KABKAB � Tvv,
Rvv = Tvv
This equation will be changed due to the higher derivative term
??
Strategy : (1) Define entropy density : Stotal =
Z
⌃v
dd�2xph ⇢total
(2) Define ⇥ ) @vStotal =
Z
⌃v
dd�2xph ⇥
(3) To obtain @vStotal � 0, show that @v⇥ 0
What is the problem with higher derivative gravity?
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
This equation will be changed due to the higher derivative term
@v⇥ = � KABKAB � Tvv,
Rvv = Tvv
??
Strategy : (1) Define entropy density : Stotal =
Z
⌃v
dd�2xph ⇢total
(2) Define ⇥ ) @vStotal =
Z
⌃v
dd�2xph ⇥
(3) To obtain @vStotal � 0, show that @v⇥ 0
What is the problem with higher derivative gravity?
I =1
16⇡GN
Zddx
p�g
⇥R+ Lmatter + ↵ LHD
⇤
This equation will be changed due to the higher derivative term
@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
4⇡
Z
⌃v
dd�2xph⇥1 + ↵ ⇢HD
⇤
I =1
4⇡
Zddx
p�g (R+ ↵LHD)
@vSWald =1
4⇡
Z
⌃v
dd�2xph⇥@v(log
ph)(1 + ↵ ⇢HD) + ↵ @v⇢HD
⇤
In a perturbative amplitude expansion things do work out up to linearized order
SWald =1
4⇡
Z
⌃v
dd�2xph⇥1 + ↵ ⇢HD
⇤
I =1
4⇡
Zddx
p�g (R+ ↵LHD)
@vSWald =1
4⇡
Z
⌃v
dd�2xp
h⇥
@v(logp
h)(1 + ↵ ⇢HD) + ↵ @v⇢HD| {z }=⇥
⇤
@v⇥ = �Tvv + ↵⇥rvrv⇢HD � ⇢HDRvv + EOMvv
⇤
In a perturbative amplitude expansion things do work out up to linearized order
SWald =1
4⇡
Z
⌃v
dd�2xph⇥1 + ↵ ⇢HD
⇤
I =1
4⇡
Zddx
p�g (R+ ↵LHD)
@vSWald =1
4⇡
Z
⌃v
dd�2xp
h⇥
@v(logp
h)(1 + ↵ ⇢HD) + ↵ @v⇢HD| {z }=⇥
⇤
@v⇥ = �Tvv + ↵⇥rvrv⇢HD � ⇢HDRvv + EOMvv
⇤| {z }
O(a2)
==>2nd law is valid up to linear order in amplitude@v⇥ = �Tvv < 0
Let us examine the Gauss-Bonnet case
(1) Obtain ⇥eq ) @vSWald =
Z
⌃v
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
4⇡
Z
⌃v
dd�2xph [1 + 2↵2l
2s Rind]
I =1
4⇡
Zddx
p�g
R + ↵2l2
s(R2 � 4Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�)
�
Let us examine the Gauss-Bonnet case
(1) Obtain ⇥eq ) @vSWald =
Z
⌃v
dd�2xph ⇥eq
(2) Compute @v⇥eq and convince that @v⇥eq 0 not satisfied
STEP 1 :
Conclusion : We need to modify Wald entropy
STEP 2 :
(1) Stotal = SWald + Scor, Scor|equilibrium = 0
(2) Obtain ⇥total = ⇥eq +⇥cor ) @vStotal =
Z
⌃v
dd�2xph⇥⇥eq +⇥cor
⇤
(3) Make sure that @v⇥total 0
Conclusion : 2nd law is satisfied for Gauss-Bonnet theory
SWald =1
4⇡
Z
⌃v
dd�2xph [1 + 2↵2l
2s Rind]
I =1
4⇡
Zddx
p�g
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
4⇡
Z
⌃v
dd�2xph ⇢cor
such that, ⇢cor =1X
n=0
n lns @nv
�↵2l
2s hA
B
�lns @
nv
�↵2l
2s hB
A
�
where, hAB =�AA1A2
BB1B2KB1
A1KB2
A2
SWald =1
4⇡
Z
⌃v
dd�2xph [1 + 2↵2l
2s Rind]
I =1
4⇡
Zddx
p�g
R + ↵2l2
s (R2 � Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�)
�
The final result for the Gauss-Bonnet case
SWald =1
4⇡
Z
⌃v
dd�2xph [1 + 2↵2l
2s Rind]
The Equilibrium quantities :
The non-Equilibrium correction :
Stotal =SWald + Scor
Scor =1
4⇡
Z
⌃v
dd�2xph ⇢cor
such that, ⇢cor =1X
n=0
n lns @nv
�↵2l
2s hA
B
�lns @
nv
�↵2l
2s hB
A
�
where, hAB =�AA1A2
BB1B2KB1
A1KB2
A2
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.
I =1
4⇡
Zddx
p�g
R + ↵2l2
s (R2 � Rµ⌫Rµ⌫ + Rµ⌫⇢�Rµ⌫⇢�)
�
(1) Obtain ⇥eq ) @vSWald =
Z
⌃v
dd�2xph ⇥eq
(2) Compute @v⇥eq and convince that @v⇥eq 0 not satisfied
STEP 1 :
Conclusion : We need to modify Wald entropy
STEP 2 :
(1) Stotal = SWald + Scor, Scor|equilibrium = 0
(2) Obtain ⇥total = ⇥eq +⇥cor ) @vStotal =
Z
⌃v
dd�2xph⇥⇥eq +⇥cor
⇤
(3) Make sure that @v⇥total 0
Conclusion : 2nd law is satisfied for Gauss-Bonnet theory
SWald =1
4⇡
Z
⌃v
dd�2xph [1 + 2↵2l
2s Rind]
I =1
4⇡
Zddx
p�g
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
BKA0
B0MBB0
AA0| {z }T3
+ ↵2`2s KA
B @v��BA1A2AB1B2
KB1A1
KB2
A2
�| {z }
T4
+rAYA
| {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
0
A0 + ↵2`2sMBB0
AA0⇤
T3 < T2
(3) T2 + T4 =KAB
⇥KB
A + ↵2`2s @v
��BA1A2AB1B2
KB1A1
KB2
A2
�⇤
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
@v⇥eq = � Tvv|{z}T1
�KABKAB| {z }T2
+↵2`2s KA
BKA0
B0MBB0
AA0| {z }T3
+ ↵2`2s KA
B @v��BA1A2AB1B2
KB1A1
KB2
A2
�| {z }
T4
+rAYA
| {z }T5
,
—> does not contain any v-derivative ..! MBB0
AA0
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
0
A0 + ↵2`2sMBB0
AA0⇤
T3 < T2
(3) T2 + T4 =KAB
⇥KB
A + ↵2`2s @v
��BA1A2AB1B2
KB1A1
KB2
A2
�⇤
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
@v⇥eq = � Tvv|{z}T1
�KABKAB| {z }T2
+↵2`2s KA
BKA0
B0MBB0
AA0| {z }T3
+ ↵2`2s KA
B @v��BA1A2AB1B2
KB1A1
KB2
A2
�| {z }
T4
+rAYA
| {z }T5
,
—> does not contain any v-derivative ..! MBB0
AA0
Let us examine the Gauss-Bonnet case with some explicit expressions
STEP 2 :(1) Stotal = SWald + Scor, Scor|equilibrium = 0
(2) Obtain ⇥total = ⇥eq +⇥cor ) @vStotal =
Z
⌃v
dd�2xph⇥⇥eq +⇥cor
⇤
(3) Make sure that @v⇥total 0
Question : How to decide the correction to wald entropy @v⇥total = @v⇥eq + @v⇥cor 0
@v⇥eq = � Tvv|{z}T1
+rAYA
| {z }T5
�KABKAB
| {z }T2
+↵2l2s KA
B @v
�BA1A2
AB1B2KB1
A1KB2
A2
�
| {z }T4
STEP 2 :(1) Stotal = SWald + Scor, Scor|equilibrium = 0
(2) Obtain ⇥total = ⇥eq +⇥cor ) @vStotal =
Z
⌃v
dd�2xph⇥⇥eq +⇥cor
⇤
(3) Make sure that @v⇥total 0
Question : How to decide the correction to wald entropy @v⇥total = @v⇥eq + @v⇥cor 0
@v⇥eq + @v⇥cor| {z }=@v⇥total
= � Tvv|{z}T1
+rAYA
| {z }T5
� KABKAB
| {z }T2
+↵2l2s KA
B @v
�BA1A2AB1B2
KB1A1
KB2
A2
�
| {z }T4
+ � (↵2l2s)
2 @v
�AA1A2BB1B2
KB1A1
KB2
A2
�@v
�BA1A2AB1B2
KB1A1
KB2
A2
�
| {z }=@v⇥cor
STEP 2 :(1) Stotal = SWald + Scor, Scor|equilibrium = 0
(2) Obtain ⇥total = ⇥eq +⇥cor ) @vStotal =
Z
⌃v
dd�2xph⇥⇥eq +⇥cor
⇤
(3) Make sure that @v⇥total 0
Question : How to decide the correction to wald entropy @v⇥total = @v⇥eq + @v⇥cor 0
@v⇥eq + @v⇥cor| {z }=@v⇥total
= � Tvv|{z}T1
+rAYA
| {z }T5
� KABKAB
| {z }T2
+↵2l2s KA
B @v
�BA1A2AB1B2
KB1A1
KB2
A2
�
| {z }T4
+ � (↵2l2s)
2 @v
�AA1A2BB1B2
KB1A1
KB2
A2
�@v
�BA1A2AB1B2
KB1A1
KB2
A2
�
| {z }=@v⇥cor
Scor = �
Z
⌃v
dd�2xph (↵2l
2s)
2
�AA1A2BB1B2
KB1A1
KB2
A2
� �BA1A2AB1B2
KB1A1
KB2
A2
�
@v⇥cor = � (↵2l2s)
2 @v
�AA1A2BB1B2
KB1A1
KB2
A2
�@v
�BA1A2AB1B2
KB1A1
KB2
A2
�will produce the desired
STEP 2 :(1) Stotal = SWald + Scor, Scor|equilibrium = 0
(2) Obtain ⇥total = ⇥eq +⇥cor ) @vStotal =
Z
⌃v
dd�2xph⇥⇥eq +⇥cor
⇤
(3) Make sure that @v⇥total 0
Scor = �
Z
⌃v
dd�2xph (↵2l
2s)
2
�AA1A2BB1B2
KB1A1
KB2
A2
� �BA1A2AB1B2
KB1A1
KB2
A2
�
@v⇥cor = � (↵2l2s)
2 @v
�AA1A2BB1B2
KB1A1
KB2
A2
�@v
�BA1A2AB1B2
KB1A1
KB2
A2
�will produce the desired
STEP 2 :(1) Stotal = SWald + Scor, Scor|equilibrium = 0
(2) Obtain ⇥total = ⇥eq +⇥cor ) @vStotal =
Z
⌃v
dd�2xph⇥⇥eq +⇥cor
⇤
(3) Make sure that @v⇥total 0
Scor = �
Z
⌃v
dd�2xph (↵2l
2s)
2
�AA1A2BB1B2
KB1A1
KB2
A2
� �BA1A2AB1B2
KB1A1
KB2
A2
�
@v⇥cor = � (↵2l2s)
2 @v
�AA1A2BB1B2
KB1A1
KB2
A2
�@v
�BA1A2AB1B2
KB1A1
KB2
A2
�will produce the desired
@v⇥eq + @v⇥cor| {z }=@v⇥total
= � Tvv|{z}T1
+rAYA
| {z }T5
�✓
KAB � ↵2l2s
2@v
�AA1A2
BB1B2KB1
A1KB2
A2
�◆2
+ (↵2l2s)
2 (� � 1/4) @v
�AA1A2
BB1B2KB1
A1KB2
A2
�@v
�BA1A2
AB1B2KB1
A1KB2
A2
�
) @v⇥total 0 � 1/4==> if we fix the free parameter ==>
STEP 2 :(1) Stotal = SWald + Scor, Scor|equilibrium = 0
(2) Obtain ⇥total = ⇥eq +⇥cor ) @vStotal =
Z
⌃v
dd�2xph⇥⇥eq +⇥cor
⇤
(3) Make sure that @v⇥total 0
Scor = �
Z
⌃v
dd�2xph (↵2l
2s)
2
�AA1A2BB1B2
KB1A1
KB2
A2
� �BA1A2AB1B2
KB1A1
KB2
A2
�
@v⇥cor = � (↵2l2s)
2 @v
�AA1A2BB1B2
KB1A1
KB2
A2
�@v
�BA1A2AB1B2
KB1A1
KB2
A2
�will produce the desired
@v⇥eq + @v⇥cor| {z }=@v⇥total
= � Tvv|{z}T1
+rAYA
| {z }T5
�✓
KAB � ↵2l2s
2@v
�AA1A2
BB1B2KB1
A1KB2
A2
�◆2
+ (↵2l2s)
2 (� � 1/4) @v
�AA1A2
BB1B2KB1
A1KB2
A2
�@v
�BA1A2
AB1B2KB1
A1KB2
A2
�
) @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 : —