Generalized Entropy and

higher derivative Gravity

Joan CampsDAMTP, Cambridge

based on 1310.6659, see also 1310.5713 by Xi Dong

Oxford, 27 May 2014

Entropy in General Relativity

�S � 0

dM = TdS + !dJ

I =1

16⇡G

Z p�gd

DxR

Bekenstein, Hawking

S =A4G

AdS/CFTMaldacena

gµ⌫ AdSD

CFTD�1

N ! 1� ! 1

h |O| iZAdSD = ZCFTD�1

ZAdSD ⇡ e�IE [gµ⌫ ]

Entanglement in AdS/CFTRyu, Takayanagi

| i

A

S =1

4Gmin(A)

Amongst many virtues, this formula makes transparent some properties of entanglement otherwise hidden

• Such corrections are predicted in String Theory (essential to the great success in microscopic accounts of entropy)

• Deformations of AdS/CFT: playground, robustness, universality

• It is an interesting problem per se

Why higher derivatives

⌘/s c 6= a

I =1

16⇡G

Z p�gd

Dx

�R+ ↵

0Riem2 + . . .

�

Entropy for higher derivative theories

�S � 0

dM = TdS + !dJ

I =

Z p�gd

DxL(Rµ⌫⇢�,rµ, gµ⌫)

S = �2⇡

Z

W

p�dD�2� ✏µ⌫✏⇢�

�L�Rµ⌫⇢�

����W

Wald

???

What this talk is about

• Review of an understanding of the origin of the Ryu-Takayangi proposal

• Use this understanding to derive a formula for higher derivative gravity

• Warning: This talk is about a calculation

Overview

• Generalized entropy

• Replica trick

• Gravity dual of the replica trick

• Action of regularised cones

• Comments on the new entropy

Generalized EntropyLewkowycz, Maldacena

Entanglement entropy| i

AA

⇢A = trA| ih |

SA = �tr (⇢A log ⇢A)

Entanglement entropy| i

AA

⇢A = trA| ih |

A A

| i = 1p2|�iA|+iA +

1p2|+iA|�iA

⇢A =

✓1/2 00 1/2

◆SA = log 2

SA = �tr (⇢A log ⇢A)

Setup

A

⇢

AA

| iCFT

CFT

Setup

A

| iCFT

A

Casini, Huerta, Myers

A

⇢CFT

HorizonA

Replica trickS = �tr(⇢ log ⇢)Entanglement Entropy

Trick: Consider Rényi entropies Sn =

log(tr⇢n)

1� n

S = lim

n!1Sn = � @n log tr (⇢

n)|n=1

A

⇢

Path Integral calculation 1(x)

2(x) h 1|⇢| 2i✓

Field theory directions

Euclidean time

Quantum amplitude

Path Integral calculation 1(x)

2(x) h 1|⇢| 2iX

i

h i|⇢| ii = tr⇢✓ ✓

Path Integral calculation 1(x)

2(x) h 1|⇢| 2iX

i

h i|⇢| ii = tr⇢✓ ✓

✓

tr⇢2 =X

i

X

j

h i|⇢| jih j |⇢| ii

Path Integral calculation 1(x)

2(x) h 1|⇢| 2iX

i

h i|⇢| ii = tr⇢

tr⇢2

✓ ✓

✓ ✓

=

Gravity dual

✓

Z’FT’ = tr⇢ ⇡ e�I1

Gravity dual

n = 1

n = 2

n = 3tr (⇢n)

(tr⇢)n⇡ e�In

e�nI1

Generalized entropy

tr (⇢n)

(tr⇢)n⇡ e�In

e�nI1

In

I1

S = lim

n!1Sn = � @n log

tr (⇢n)

(tr⇢)n

����n=1

Entanglement entropy:Replica trick

Gravity saddlepoint

Gravitational entropy S = @n (In � nI1)|n=1

Replica manipulations

�n⇥ = �

= �

In I1n

Z 2⇡

0d✓ =

Z 2⇡n

0d✓

S = @n (In � nI1)|n=1

Replica manipulations

� �( )

S = @n (In � nI1)|n=1

Replica manipulations

�( ) +

O�(n� 1)2

�

S = @n (In � nI1)|n=1

Replica manipulations

�( )S = @n (I[ ])|n=1

+

O�(n� 1)2

�

S = @n (In � nI1)|n=1

Recap

• Replica trick: EE and loops in eucl. time

• Assume ‘holography’

• Entanglement entropy becomes:

S = @n (I[ ])|n=1

Z’FT’ ⇡ e�In

The calculation

S = @n (I[ ])|n=1

First, assume euclidean time independence

Adapted coordinates

�a

W

�ab

x

1

x

2

✓

✓ ⇠ ✓ + 2⇡

ds2 = dr2 + r2d✓2 + �abd�ad�b + . . .

Adapted coordinates

✓ ⇠ ✓ + 2⇡n

(n� 1)

�a

W

�ab

x

1

x

2

ds2 = dr2 + r2d✓2 + �abd�ad�b + . . .

Adapted coordinates

✓ ⇠ ✓ + 2⇡n

(n� 1)

�a

W

�ab

x

1

x

2

ds2 = dr2 + r2d✓2 + �abd�ad�b + . . .

⇤

Regulated cones

A

1

r

⇤

0

✓ ⇠ ✓ + 2⇡n

⇤vol = ⇤

2

r2d✓2/n2

r2d✓2

ds2 = dr2 + r2✓1� n� 1

nA(r2/⇤2)

◆2

d✓2 + �abd�ad�b + . . .

Entropy

�Rµ⌫⇢� =1

⇤2

n� 1

n

1

y

d2(yA(y2))

dy2✏µ⌫✏⇢�

⇤ ! 0

S = �2⇡

Z

W

p�dD�2� ✏µ⌫✏⇢�

�L�Rµ⌫⇢�

����W

S = @n (I[ ])|n=1

r = ⇤y

�I =�I

�Riem�Riem

Comments

• Wald’s entropy

• Regulation independent

• Assumed Euclidean time stationarity: No extrinsic curvature

More generally...

W

�ab

�ax

1

x

2

✓

ds2 =

⇥�ab � 2

�Kab

1 r cos ✓ +Kab2 r sin ✓

�⇤d�ad�b

+dr2 + r2d✓2 + . . .

More generally...

W

�ab

�ax

1

x

2

✓

ds2 =

�ab � 2

�Kab

1 r cos ✓ +Kab2 r sin ✓

� ⇣ r

⇤

⌘(n�1)B(r2/⇤2)�d�ad�b

+dr2 + r2✓1� n� 1

nA(r2/⇤2

)

◆2

d✓2 + . . .

Why the new termsKab

1 r cos ✓⇣ r

⇤

⌘(n�1)B(r2/⇤2)⇡ Kab

1 r cos ✓⇣ r

⇤

⌘(n�1)

✓ = n✓ ✓ ⇠ ✓ + 2⇡

⇤ x

1= r cos

˜

✓

⇤x

1x

2

Why the new termsKab

1 r cos ✓⇣ r

⇤

⌘(n�1)B(r2/⇤2)⇡ Kab

1 r cos ✓⇣ r

⇤

⌘(n�1)

✓ = n✓ ✓ ⇠ ✓ + 2⇡

An analogous analysis applies to terms beyond extrinsic curvature (higher powers of r)

⇤

n = 2

x

1= r cos

˜

✓

Kab1 r

ncos(n

˜

✓)

⇤

n�1�! Kab

1 (x

1)

2 � (x

2)

2

⇤

⇤x

1x

2

Entropy contributions

⇤ ! 0

�Riem ⇠ (n� 1)K

⇤

�S ⇠Z

p�dD�2�

@2L@Riem2K

2

W

�ab

�ax

1

x

2

✓

The subtlety

limn!1

@n

Z 1

0dy (n� 1)2y2n�3e�y2

limn!1

@n

✓(n� 1)2

�(n� 1)

2

◆=

1

2

�S ⇠Z

p�dD�2�

@2L@Riem2K

2

S = @n (I[ ])|n=1

A(y2)

Comments on the new formula

• It reduces to Wald’s for stationary cases (trivially)

• For Lovelock Gravity, it gives the Jacobson-Myers entropy functional

• Disagrees with Wald-Iyer’s

S ⇠Z

W

p�dD�2�

✓@L

@Riem+

@2L@Riem2K

2

◆

de Boer et alMyers et al

Fursaev et al

Final remarks

• First principles derivation of Ryu-Takayanagi

• Euclidean space essential: Lorentzian?

• Multiple regions?

• Holographic emergence of space?

• Non-stationary entropy? Second law?

The final formulaS =

Z

W

p� dD�2�

✓�(1)Rµ⌫⇢�

�L�Rµ⌫⇢�

+ �(2)Rµ⌫⇢�⌧⇡⇠⇣@2L

@Rµ⌫⇢� @R⌧⇡⇠⇣

◆����W

?⌫�⇡⇣=?⌫⇡?�⇣ + ?⌫⇣?⇡� � ?⌫�?⇡⇣

?⌫�⇡⇣ =?⌫⇡ ✏�⇣+ ?�⇣ ✏⌫⇡

W�ab

�a

?µ⌫

✏µ⌫

Wald New�(1)Rµ⌫

⇢� = �2⇡✏µ⌫✏⇢�

�(2)Rµ⌫⇢�

⌧⇡⇠⇣ =4⇡

⇣K[µ

[⇢|i| ?⌫]�]

[⇡[⇣K⌧ ]

⇠]j ?ij +K[µ[⇢|k|?⌫]

�][⇡

[⇣K⌧ ]⇠]l✏kl

⌘

A further refinementZ

p�dD�2�K2 @2L

@Riem2 �!Z

p�dD�2�K2

X

↵

1

1 + q↵

@2L@Riem2

����↵

q↵ =

1

2

(# of Kabi s) +1

2

(# of Raijk s) + (# of Rab(ij) s)

Dong

Orthogonal Parallel

limn!1

@n

Z 1

0dy (n� 1)2y2n�3e�y2 �

yn�1K�t

=(K)t

1 + t/2