Traversable Wormholes Supported by GUP Corrected Casimir ...
Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with...
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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
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2
This paper talks about…
・Traversable wormholes in Nearly AdS2 gravity・Application to information problems
![Page 3: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/3.jpg)
3
This paper talks about…
・Traversable wormholes in Nearly AdS2 gravity・Application to information problems
![Page 4: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/4.jpg)
(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
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Horizonis described
https://www.sciencenews.org/blog/context/new-einstein-equation-wormholes-quantum-gravity
1.Introduction
by
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HorizonMatter
Realistic Black Holes are created by collapsing matters.
1.Introduction
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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
![Page 8: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/8.jpg)
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
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HorizonThis approximately describes
https://www.sciencenews.org/blog/context/new-einstein-equation-wormholes-quantum-gravity
![Page 10: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/10.jpg)
Horizon
time slice
Einstein-Rosen Bridge(Wormhole)
1.Introduction
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Horizon
time slice
https://ja.wikipedia.org/wiki/ブラックホール
Einstein-Rosen Bridge(Wormhole)
1.Introduction
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Horizon
time slice
https://ja.wikipedia.org/wiki/ブラックホール
Einstein-Rosen Bridge(Wormhole)
1.Introduction
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Horizon
time slice
Einstein-Rosen Bridge(Wormhole)
1.Introduction
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Horizon
Einstein-Rosen bridge
time slice
Can we go to the other side?
1.Introduction
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Horizon
Einstein-Rosen bridge
time slice
Can we go to the other side?
→No.
1.Introduction
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If we through matter with negative energy(violate Averaged Null Energy Condition)
BH
![Page 17: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/17.jpg)
If we through matter with negative energy(violate Averaged Null Energy Condition)
BH
→He can escape from Black Holeshorizon radius decreases
![Page 18: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/18.jpg)
Traversable wormhole via double trace deformation[Gao-Jefferis-Wall, 16]
CFTL CFTR・CFTL and CFTR are decoupled:
H = HL +HR
・Assume AdS/CFT
![Page 19: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/19.jpg)
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
![Page 20: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/20.jpg)
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
![Page 21: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/21.jpg)
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)
![Page 22: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/22.jpg)
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)
![Page 23: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/23.jpg)
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)
![Page 24: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/24.jpg)
(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
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2.Nearly AdS2 gravity dynamics
・exact AdS2 does not permit finite energy excitation
For example, → Tµ⌫ = 0Z
pgR+ Smatter
![Page 26: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/26.jpg)
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
![Page 27: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/27.jpg)
・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
![Page 28: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/28.jpg)
Horizon
AdS2
・Consider a model of following part.
![Page 29: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/29.jpg)
Horizon
AdS2
AdS2 boundary
・Consider a model of following part.
→AdS2 are cut at some trajectory in AdS2.
![Page 30: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/30.jpg)
→ 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.
![Page 31: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/31.jpg)
→ 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.
![Page 32: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/32.jpg)
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
![Page 33: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/33.jpg)
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
![Page 34: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/34.jpg)
(1)Introduction/Review of traversable wormholes (2)Nearly AdS2 gravity(3)Traversable wormholes in Nearly AdS2 gravity(4)Application to information problem (5)Conclusion
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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
![Page 36: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/36.jpg)
not traversable
BH solution
![Page 37: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/37.jpg)
eig�L(tL)�R(tR)
eigh�L(tL)�R(tR)i
⇠ e�igVpot
Vpot
⇠ e�m⇢
⇢ :AdS distance
put two side interaction
![Page 38: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/38.jpg)
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
![Page 39: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/39.jpg)
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
![Page 40: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/40.jpg)
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
![Page 41: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/41.jpg)
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
![Page 42: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/42.jpg)
Back reaction
![Page 43: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/43.jpg)
Back reaction
kicked outward
![Page 44: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/44.jpg)
Back reaction
Vpot
⇠ e�m⇢
⇢ :AdS distance→becomes long by back reaction
![Page 45: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/45.jpg)
(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
![Page 46: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/46.jpg)
4.Application to information problem
Problem: Can we extract the information behind the horizon from Hawking radiations?
[Hayden-Preskill,07]
![Page 47: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/47.jpg)
4.Application to information problem
BH
Consider one side BH
Bob
Horizon
![Page 48: Diving into Traversable Wormholes - Particle …journal/jc2017_slides/...If we through matter with negative energy (violate Averaged Null Energy Condition) BH →He can escape from](https://reader031.fdocuments.us/reader031/viewer/2022020319/5c8b9da209d3f27d138b993b/html5/thumbnails/48.jpg)
4.Application to information problem
BH
Horizon
Consider one side BH
Bob
Bob collects Hawking radiations (entangled with BH)…
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4.Application to information problem
BH
Consider one side BH
Bob
After half evaporation…
Horizon
Bob collects Hawking radiations (entangled with BH)…
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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
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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
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Then, we obtain
Bob
BHoriginal BH
Bob’s 2nd BH
connected by a wormhole
BH2
4.Application to information problem
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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
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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
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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
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(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
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・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
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Appendix
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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
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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
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| 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]
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CFT1
Conformal symmetry → Tµµ = 0
In 1d , and there are no finite energy excitation
T 00 = H = 0
In 1d, QFT becomes quantum mechanics