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Diving into Traversable Wormholes

Journal Club@Osaka University July 26, 2017

Tokiro Numasawa

Osaka University Particle Physics TheoryBased on arXiv:1704.05333 J. Maldacena, D.Stanford and Z. Yang

see also “Traversable Wormholes via a Double Trace Deformation” arXiv:1608.05687, P. Gao , D.Jafferis and A.Wall (Iizuka-san’s JC)

“Conformal symmetry and its breaking in two dimensional Nearly Anti de-Sitter space” arXiv:1606.01857 J. Maldacena, D.Stanford and Z. Yang

2

This paper talks about…

・Traversable wormholes in Nearly AdS2 gravity・Application to information problems

3

This paper talks about…

・Traversable wormholes in Nearly AdS2 gravity・Application to information problems

(1)Introduction/Review of traversable wormholes (2)Nearly AdS2 gravity(3)Traversable wormholes in Nearly AdS2 gravity(4)Application to information problem (5)Conclusion

4

Horizonis described

https://www.sciencenews.org/blog/context/new-einstein-equation-wormholes-quantum-gravity

1.Introduction

by

HorizonMatter

Realistic Black Holes are created by collapsing matters.

1.Introduction

Horizon BH

4d Schwarzschild: ds2 = �(1� 2M

r)dt2 +

1

1� 2Mr

dr2 + r2d⌦22

Realistic Black Holes are created by collapsing matters.

1.Introduction

Horizon

Realistic Black Holes are created by collapsing matters.

BH

But Einstein eq. also permits eternal Black holes solutions:

Horizon

4d Schwarzschild: ds2 = �(1� 2M

r)dt2 +

1

1� 2Mr

dr2 + r2d⌦22

ds2 = �32M3

re�

r2M dUdV + r2d⌦2

2

:just a coordinate transformation

1.Introduction

HorizonThis approximately describes

https://www.sciencenews.org/blog/context/new-einstein-equation-wormholes-quantum-gravity

Horizon

time slice

Einstein-Rosen Bridge(Wormhole)

1.Introduction

Horizon

time slice

https://ja.wikipedia.org/wiki/ブラックホール

Einstein-Rosen Bridge(Wormhole)

1.Introduction

Horizon

time slice

https://ja.wikipedia.org/wiki/ブラックホール

Einstein-Rosen Bridge(Wormhole)

1.Introduction

Horizon

time slice

Einstein-Rosen Bridge(Wormhole)

1.Introduction

Horizon

Einstein-Rosen bridge

time slice

Can we go to the other side?

1.Introduction

Horizon

Einstein-Rosen bridge

time slice

Can we go to the other side?

→No.

1.Introduction

If we through matter with negative energy(violate Averaged Null Energy Condition)

BH

If we through matter with negative energy(violate Averaged Null Energy Condition)

BH

→He can escape from Black Holeshorizon radius decreases

Traversable wormhole via double trace deformation[Gao-Jefferis-Wall, 16]

CFTL CFTR・CFTL and CFTR are decoupled:

H = HL +HR

・Assume　AdS/CFT

Traversable wormhole via double trace deformation[Gao-Jefferis-Wall, 16]

CFTL CFTR・CFTL and CFTR are decoupled

・States are entangled|TFDi = 1p

Z(�)

X

n

e�2 En |EniL ⌦ |EniR

H = HL +HR

・Assume　AdS/CFT

Traversable wormhole via double trace deformation[Gao-Jefferis-Wall, 16]

CFTL CFTR・CFTL and CFTR are decoupled

・States are entangled|TFDi = 1p

Z(�)

X

n

e�2 En |EniL ⌦ |EniR

H = HL +HR

to obtain traversability…・put interaction term Hint = g

Z t+�t

tOL(t)OR(t)

・Assume　AdS/CFT

Traversable wormhole via double trace deformation[Gao-Jefferis-Wall, 16]

CFTL CFTR・CFTL and CFTR are decoupled

・States are entangled|TFDi = 1p

Z(�)

X

n

e�2 En |EniL ⌦ |EniR

H = HL +HR

to obtain traversability…・put interaction term Hint = g

Z t+�t

tOL(t)OR(t)

・effectively compactly radial direction to circle →negative energy by Casimir effect

・Assume　AdS/CFT

(positive/negative depend on the sign of g)

Traversable wormhole via double trace deformation[Gao-Jefferis-Wall, 16]

CFTL CFTR・CFTL and CFTR are decoupled

・States are entangled|TFDi = 1p

Z(�)

X

n

e�2 En |EniL ⌦ |EniR

H = HL +HR

to obtain traversability…・put interaction term Hint = g

Z t+�t

tOL(t)OR(t)

・effectively compactly radial direction to circle →negative energy by Casimir effect

・Assume　AdS/CFT

(positive/negative depend on the sign of g)

Traversable wormhole via double trace deformation[Gao-Jefferis-Wall, 16]

CFTL CFTR・CFTL and CFTR are decoupled

・States are entangled|TFDi = 1p

Z(�)

X

n

e�2 En |EniL ⌦ |EniR

H = HL +HR

to obtain traversability…・put interaction term Hint = g

Z t+�t

tOL(t)OR(t)

・effectively compactly radial direction to circle →negative energy by Casimir effect

・Assume　AdS/CFT

(positive/negative depend on the sign of g)

(1)Introduction/Review of traversable wormholes (2)Nearly AdS2 gravity(3)Traversable wormholes in Nearly AdS2 gravity(4)Application to information problem (5)Conclusion

24

2.Nearly AdS2 gravity dynamics

・exact AdS2 does not permit finite energy excitation

For example, → Tµ⌫ = 0Z

pgR+ Smatter

Horizon

AdS2 ⇥ Y UV geometry

・exact AdS2 does not permit finite energy excitation

・But AdS2 × Y type geometry appears from near horizon limit of near extremal BHs

For example, → Tµ⌫ = 0Z

pgR+ Smatter

2.Nearly AdS2 gravity dynamics

・exact AdS2 does not permit finite energy excitation

・But AdS2 × Y type geometry appears from near horizon limit of near extremal BHs

For example, → Tµ⌫ = 0Z

pgR+ Smatter

Horizon

AdS2 ⇥ Y UV geometry

・middle geometry can be changed by finite energy excitation

2.Nearly AdS2 gravity dynamics

Horizon

AdS2

・Consider a model of following part.

Horizon

AdS2

AdS2 boundary

・Consider a model of following part.

→AdS2 are cut at some trajectory in AdS2.

→ include the term that perturb from AdS21

16⇡G

Z�pg(R+ 2) +

1

8⇡G

Z

bdy�bK

Horizon

AdS2

AdS2 boundary

・We want to include the effect of finite energy excitation

[Almheiri-Polchinski 14][Jackiw 85][Teitelboim 83]

・Consider a model of following part.

→AdS2 are cut at some trajectory in AdS2.

→ include the term that perturb from AdS21

16⇡G

Z�pg(R+ 2) +

1

8⇡G

Z

bdy�bK

Horizon

AdS2

→AdS2 are cut at some trajectory in AdS2.

AdS2 boundary

・We want to include the effect of finite energy excitation

・We call this nearly AdS2 gravity

[Almheiri-Polchinski 14][Jackiw 85][Teitelboim 83]

・Consider a model of following part.

Nearly AdS2 gravity

S = �C

Zdu{f(u), u}

{f(u), u} =f 000

f 0 � 3

2

⇣f 00

f 0

⌘2

Schwarzian action:

describes the dynamics of boundary cutoff curveDynamics are encoded the motion of boundary curve !

1

16⇡G

Z�pg(R+ 2) +

1

8⇡G

Z

bdy�bK

SYK model (1d Nearly CFT)

Nearly AdS2 gravity

S = �C

Zdu{f(u), u}

{f(u), u} =f 000

f 0 � 3

2

⇣f 00

f 0

⌘2

Schwarzian action:

describes the dynamics of boundary cutoff curveDynamics are encoded the motion of boundary curve !

low energy

1

16⇡G

Z�pg(R+ 2) +

1

8⇡G

Z

bdy�bK

(1)Introduction/Review of traversable wormholes (2)Nearly AdS2 gravity(3)Traversable wormholes in Nearly AdS2 gravity(4)Application to information problem (5)Conclusion

34

BH solution

cutoff curve in global AdS2 (described by Schwarzian action)

Horizon

(BH/Rindler patch)

3.Traversable wormhole in Nearly AdS2

What we need is to see the dynamics ofboundary curves

not traversable

BH solution

eig�L(tL)�R(tR)

eigh�L(tL)�R(tR)i

⇠ e�igVpot

Vpot

⇠ e�m⇢

⇢ :AdS distance

put two side interaction

eig�L(tL)�R(tR)

eigh�L(tL)�R(tR)i

⇠ e�igVpot

Vpot

⇠ e�m⇢

⇢ :AdS distance

pushed

g > 0: attractive force (wormhole becomes traversable)

:Approximate

eig�L(tL)�R(tR)

eigh�L(tL)�R(tR)i

⇠ e�igVpot

:Approximate

g > 0: attractive force (wormhole becomes traversable)

Vpot

⇠ e�m⇢

⇢ :AdS distance

pushed

eig�L(tL)�R(tR)

eigh�L(tL)�R(tR)i

⇠ e�igVpot

:Approximate

g > 0: attractive force (wormhole becomes traversable)g < 0: repulsive force (not traversable)

Vpot

⇠ e�m⇢

⇢ :AdS distance

eig�L(tL)�R(tR)

eigh�L(tL)�R(tR)i

⇠ e�igVpot

:Approximate

g > 0: attractive force (wormhole becomes traversable)g < 0: repulsive force (not traversable)

Vpot

⇠ e�m⇢

⇢ :AdS distance

Back reaction

Back reaction

kicked outward

Back reaction

Vpot

⇠ e�m⇢

⇢ :AdS distance→becomes long by back reaction

(1)Introduction/Review of traversable wormholes (2)Nearly AdS2 gravity(3)Traversable wormholes in Nearly AdS2 gravity(4)Application to information problem (5)Conclusion

45

4.Application to information problem

Problem: Can we extract the information behind the horizon from Hawking radiations?

[Hayden-Preskill,07]

4.Application to information problem

BH

Consider one side BH

Bob

Horizon

4.Application to information problem

BH

Horizon

Consider one side BH

Bob

Bob collects Hawking radiations (entangled with BH)…

4.Application to information problem

BH

Consider one side BH

Bob

After half evaporation…

Horizon

Bob collects Hawking radiations (entangled with BH)…

BH

Consider one side BH

Bob

After half evaporation…

Bob create 2nd BH by radiations

BH

Horizon

Bob collects Hawking radiations (entangled with BH)…

4.Application to information problem

Consider one side BH

Bob

After half evaporation…

Assume EPR=ER (entanglement = wormhole) [Maldacena-Susskind,13]

Horizon

Bob create 2nd BH by radiations

BHBH

Bob collects Hawking radiations (entangled with BH)…

4.Application to information problem

Then, we obtain

Bob

BHoriginal BH

Bob’s 2nd BH

connected by a wormhole

BH2

4.Application to information problem

Then, we obtain

Bob

BH

Now, Alice throws her message in the original BH

Alice

BH2 Bob’s 2nd BH

original BH

4.Application to information problem

connected by a wormhole

Then, we obtain

Bob

BH

Now, Alice throws her message in the original BH

Alice

BH2 Bob’s 2nd BH

original BH

4.Application to information problem

connected by a wormhole

Then, we obtain

Bob

BH

Now, Alice throws her message in the original BH

Alice

BH2

By traversable wormhole protocol, Bob can extract Alice’s message from radiations(2nd BH)

Bob’s 2nd BH

original BH

4.Application to information problem

connected by a wormhole

(1)Introduction/Review of traversable wormholes (2)Nearly AdS2 gravity(3)Traversable wormholes in Nearly AdS2 gravity(4)Application to information problem (5)Conclusion

56

・They studied traversability of wormholes in nearly AdS2 gravity

・Because of the simple dynamics in nearly AdS2 gravity,we can also see the back reaction of message.

・Assuming ER=EPR, traversable wormholes are used to extract information behind the horizon from Hawking radiations

5.Conclusion

Appendix

Averaged Null Energy Condition(ANEC) and Traversability

ds2 = �r2 � r2hl2

dt2 +l2

r2 � r2hdr2 + r2d⌦2

d�2BH metric:

After perturbation , satisfiesTµ⌫ ⇠ O(✏) hµ⌫ = �gµ⌫ ⇠ O(✏)

d� 2

4[(d� 3)r�2

h +(d� 1)l�2(hUU +@U (UhUU ))� 2r�2h @2

Uh��] = 8⇡GNTUU

at V = 0 in Kruskal Coordinate

V (U) = �(2gUV (0))

Z U

�1dU hUU

8⇡GN

ZdUTUU =

d� 2

4((d� 3)r�2

h + (d� 1)l�2)

ZdUhUU

→ANEC = traversability

AdS2 form Higher dim

・Magnetic brane

ds

24 = gµ⌫dx

µdx

⌫ + �2(x)(dy21 + dy

22)

L =1

16⇡GN

p�g�2R+ �(r�)2 � U(�)

・CGHS modelU(�) = �A�2� = 4

UV: near extremal diatonic BH at 4 or 5 dim

UV:

� = 2 U(�) =B

�2�A�2

| iA ⌦ |EPRiBC

Initial state:Quantum Teleportation

=1

2

4X

k=1

| kiAB ⌦ | ki

| i = ↵ |0i+ � |1i

| 1i =1p2(|0i |1i � |1i |0i) | 2i =

1p2(|0i |1i+ |1i |0i)

| 3i =1p2(|0i |0i � |1i |1i) | 4i =

1p2(|0i |0i+ |1i |1i)

| 1i = ↵ |1i � � |0i | 2i = ↵ |1i+ � |0i

| 3i = ↵ |0i � � |1i | 4i = ↵ |0i+ � |1i

where

Pk = | ki h k|Projection: ,

Alice Bob Charlie

EPR state

| i

Projection Measurement

Classical Comm.

| ki

Uk

Unitary Transf.

| i

X

k

Pk = 1

Uk : k dependent Unitary (indep. from α and β )61

[cf: Bennett-Brassard-Crepeau-Jozsa-Peres-Wootters 93]

CFT1

Conformal symmetry → Tµµ = 0

In 1d , and there are no finite energy excitation

T 00 = H = 0

In 1d, QFT becomes quantum mechanics