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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
