welcome to the string theory seminar...
TRANSCRIPT
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Welcome to the String Theory Seminar Series
Tuesdays, 3:00pm, Chamberlin 5280
UW-Madison String Theory Group:
Akikazu Hashimoto
Albrecht Klemm
Fernando Marchesano
Gary Shiu
...
http://uw.physics.wisc.edu/~strings
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Black holes in Godel universes and plane waves
Akikazu Hashimoto
Madison, September 2003
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Godel Universe/Plane−wave
We find the black hole solution in asymptoticallyplane-wave/Godel universe space-time in supergravity
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Solutions of general theory of relativity are classified
according to their asymptotics
• Flat
• Compact
• Freedman-Robertson-Walker
• De-Sitter
• Anti de-Sitter
Black holes in flat space: Schwarzschild solution
Rules of quantum field theory, i.e. scattering, are very
different depending on the asymptotics
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AdS5 × S5 is an important asymptotic geometry
ds2 =R2
cos2 ρ(−dt2 + dρ2 + sin2 ρdΩ2
3)
+R2dΩ25
F5 = N(dΩ5 + ∗dΩ5)
Very different from asymptotically flat space because it
has a boundary
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While still a conjecture, string theory in this background
can be reformulated as gauge theory living on the
boundary
Extremely interesting because it relates the (unknown)
quantum theory of gravity to the (familiar) theory of
glons and gluinos
Black holes in anti de-Sitter spaces corresponding to field
theories at finite temperature
Calculations on the string theory side was very very hard!
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Plane waves are also important
backgrounds in string theory
There are space-times of the
form -6
-4
-2
0
2
-2
0
2
4
6
-2
0
2
4
6
-6
-4
-2
0
2
-2
0
2
4
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ds2 = −dx+dx− + Bij(x+)xixj(dx+)2 + (dxi)2
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Penrose Limit
Plane waves are ubiquitous in geometry
Take any null geodesics and take the near geodesic limit
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This limit can be applied to null geodesics in AdS5 × S5.
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This gives rise to a plane wave geometry with
Background 5-form
F+1234 = F+5678 = const
For this plane wave:
• The string theory in plane wave is much easier
• This limit has interpretation in gauge theory (BMN)
Lots of computations are doable.
• Source of great excitement!
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Godel Universes: ds2 = −(dt + jr2dφ)2 + dr2 + r2dφ2
r0 Cr>
rr 0
t
P
Strange: closed time like curves through every point
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• Gauntlett et.al: Godel Universes are supersymmetric
solutions of supergravity (September 2002)
• Boyda et.al: Godel Universe are related to plane waves
by T-duality/dimensional reduction (December 2002)
−dt2 + dy2 − j2r2(dt + dy)2 + dr2 + r2dφ2
φ = φ− j(t + y)−(dt + jr2dφ)2 + dr2 + r2dφ2 + (dy2 − jr2dφ)2
In string theory, planes wave and Godel universes should
be considered to be in the same class
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Both plane waves and Godel universes are
non-hyperbolic.
That is to say the space do not admit a foliation in terms
of everywhere space-like hypersufaces
• Meaning of initial value problems are obsecure
• Issue of causality and chronology protection is obsecure
• Notion of Hilbert-space of functions obsecure
• Notion of for fields living in these spaces obsecure
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We are concerned with solving the equation of motion oftype IIA/B supergravity
DMPM =124
κ2GMNPGMNP
DPGMNP = PPG∗MNP −
23iκFMNOPQRGPQR
−RMN = PMP ∗N + P ∗
MPN +16κ2FMP1...P4F
P1...P4N
+18κ2(GM
PQG∗NPQ + G∗
MPQGNPQ
−16gMNGPQRG∗
PQR)
Eq. (13.1.27), page 320 of GSW
Why? They have something to do with string theory
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Immediately following BMN many people asked about
black holes in pp-wave backgrounds
• Black hole solution in AdS space taught us alot about
SYM at finite temperature
However, direct analysis did not go very far because the
supergravity ansatz was too complicated.
• It has become a kind of an obsession: a goal with its
own merit
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In this talk, I would like to report on a breakthrough
which lead to the discovery of exact solution of type II
supergravity which descibes a black hole in Godel
universes (black strings in plane waves)
E. G. Gimon and A. Hashimoto, Phys. Rev. Lett. 91 (2003) 021601
hep-th/0304181.
E. G. Gimon, A. Hashimoto, V. E. Hubeny, O. Lunin and
M. Rangamani, JHEP 08 (2003) 035 hep-th/0306131.
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Main Encouragement
Gauntlett et.al. broke the problem down to
1) find the Godel solutuion in 5d minimal supergravity
2) embed the 5d solution into 11d supergravity
Enough to find Godel black hole in 5d theory (gµν, Aµ).
In fact, Gauntlett et.al. found all supersymmetric
solutions
Herdeiro: one of these solutions was the extreme (zero
temperature) black hole in 5d Godel universe.
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5d Godel universe:
ds2 = −(dt + jr2σ)2 + dr2 + r2dΩ23
A =√
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jr2σ
r2σ
2= (x1dx2 − x2dx1 + x3dx4 − x4dx3)
Try an ansatz
ds2 = −f(r) dt2 − g(r) rσ3Ldt− h(r) r2(σ3
L)2
+k(r) dr2 + r2dΩ23,
A =√
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jr2σ3L .
Mimicing the form of rotating black hole in 5d
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In[1]:= eq = << eq.m
Out[1]= 9-H128 g@rD4 k@rD2 + 32 j2 r2 f@rD2 k@rD H-1 + H-1 + 4 h@rDL k@rDL + r2 H-1 + 4 h@rDL k@rD f¢ @rD2 -
8 r2 g@rD k@rD f¢ @rD g¢ @rD + 4 r g@rD2 H-r f¢ @rD k¢ @rD + 2 k@rD Hf¢ @rD + r f¢¢ @rDLL +f@rD H-8 g@rD2 k@rD H-1 + 4 H-1 + 4 j2 r2 + 4 h@rDL k@rDL +
16 r g@rD k@rD g¢ @rD + r Hr H-1 + 4 h@rDL f¢ @rD k¢ @rD +k@rD Hf¢ @rD H6 - 24 h@rD - 4 r h¢ @rDL + 2 r H4 g¢ @rD2 + H1 - 4 h@rDL f¢¢ @rDLLLLL H4 r2 Hf@rD + 4 g@rD2 - 4 f@rD h@rDL k@rD2L, 0, 0,HCos@thD Hr2 H1 - 4 h@rDL k@rD f¢ @rD g¢ @rD + r g@rD k@rD f¢ @rD H-3 + 12 h@rD + 8 r h¢ @rDL +
4 g@rD3 H-4 k@rD + 8 H1 + 4 j2 r2 - 4 h@rDL k@rD2 + r k¢ @rDL +4 r g@rD2 Hr g¢ @rD k¢ @rD - 2 k@rD H4 g¢ @rD + r g¢¢ @rDLL +f@rD Hg@rD H-8 H1 + 4 j2 r2 - 4 h@rDL H-1 + 4 h@rDL k@rD2 +
k@rD H-2 + 32 j2 r2 + 8 h@rD - 4 r h¢ @rDL + r H1 - 4 h@rDL k¢ @rDL + r Hr H1 - 4 h@rDLg¢ @rD k¢ @rD + k@rD Hg¢ @rD H-6 + 24 h@rD - 4 r h¢ @rDL + 2 r H-1 + 4 h@rDL g¢¢ @rDLLLLL H4 r Hf@rD + 4 g@rD2 - 4 f@rD h@rDL k@rD2L, Hr2 H1 - 4 h@rDL k@rD f¢ @rD g¢ @rD +
r g@rD k@rD f¢ @rD H-3 + 12 h@rD + 8 r h¢ @rDL +4 g@rD3 H-4 k@rD + 8 H1 + 4 j2 r2 - 4 h@rDL k@rD2 + r k¢ @rDL +4 r g@rD2 Hr g¢ @rD k¢ @rD - 2 k@rD H4 g¢ @rD + r g¢¢ @rDLL +f@rD Hg@rD H-8 H1 + 4 j2 r2 - 4 h@rDL H-1 + 4 h@rDL k@rD2 +
k@rD H-2 + 32 j2 r2 + 8 h@rD - 4 r h¢ @rDL + r H1 - 4 h@rDL k¢ @rDL + r Hr H1 - 4 h@rDLg¢ @rD k¢ @rD + k@rD Hg¢ @rD H-6 + 24 h@rD - 4 r h¢ @rDL + 2 r H-1 + 4 h@rDL g¢¢ @rDLLLL H4 r Hf@rD + 4 g@rD2 - 4 f@rD h@rDL k@rD2L,
8 j2 H-4 g@rD2 k@rD + f@rD H2 + H-1 + 4 h@rDL k@rDLL
f@rD + 4 g@rD2 - 4 f@rD h@rD -Hr2 H1 - 4 h@rDL2 k@rD f¢ @rD2 + 16 r2 g@rD H1 - 4 h@rDL k@rD f¢ @rD g¢ @rD +32 g@rD4 Hk@rD + 3 r k¢ @rDL + 4 r2 g@rD2HH1 - 4 h@rDL f¢ @rD k¢ @rD + k@rD H8 g¢ @rD2 + 8 f¢ @rD h¢ @rD + 2 H-1 + 4 h@rDL f¢¢ @rDLL -32 r g@rD3 H-r g¢ @rD k¢ @rD + 2 k@rD Hg¢ @rD + r g¢¢ @rDLL +2 r f@rD2 HH-1 + 4 h@rDL H-3 + 12 h@rD + 2 r h¢ @rDL k¢ @rD +
4 k@rD HH2 - 8 h@rDL h¢ @rD + 2 r h¢ @rD2 + r H1 - 4 h@rDL h¢¢ @rDLL +
f@rD Hr2 H-1 + 4 h@rDL HH-1 + 4 h@rDL f¢ @rD k¢ @rD + 2 k@rD H4 g¢ @rD2 + H1 - 4 h@rDL f¢¢ @rDLL -8 r g@rD Hr H-1 + 4 h@rDL g¢ @rD k¢ @rD +
2 k@rD Hg¢ @rD H1 - 4 h@rD + 4 r h¢ @rDL + r H1 - 4 h@rDL g¢¢ @rDLL -
8 g@rD2 H2 r H-3 + 12 h@rD + r h¢ @rDL k¢ @rD + k@rD H-1 + 4 h@rD - 8 r h¢ @rD - 4 r2 h¢¢ @rDLLLL I4 r2 Hf@rD + 4 g@rD2 - 4 f@rD h@rDL2k@rDM, 0, 0, 0,
-Hr H-1 + 4 h@rDL k@rD f¢ @rD - 8 r g@rD k@rD g¢ @rD -4 g@rD2 H4 k@rD + 4 H-1 + 8 j2 r2 - 4 h@rDL k@rD2 - r k¢ @rDL +f@rD H4 H-1 + 8 j2 r2 - 4 h@rDL H-1 + 4 h@rDL k@rD2 + 4 k@rD H-1 + 4 j2 r2 + 4 h@rD + r h¢ @rDL +
r H1 - 4 h@rDL k¢ @rDLL H8 Hf@rD + 4 g@rD2 - 4 f@rD h@rDL k@rD2L, 0,
0, -H-8 r g@rD k@rD g¢ @rD HSin@thD2 + 2 r Cos@thD2 h¢ @rDL - r H-1 + 4 h@rDL k@rDH-4 r Cos@thD2 g¢ @rD2 + f¢ @rD H-1 + 4 Cos@thD2 h@rD + 2 r Cos@thD2 h¢ @rDLL -2 g@rD2 H8 H-1 + 2 j2 r2 - 6 j2 r2 Cos@2 thD + 2 H1 + 8 j2 r2 Cos@thD2 + 3 Cos@2 thDL h@rD -
16 Cos@thD2 h@rD2L k@rD2 + 2 r H-1 + 4 Cos@thD2 h@rD + 2 r Cos@thD2 h¢ @rDL k¢ @rD +k@rD H9 + 96 j2 r2 Cos@thD2 + Cos@2 thD - 40 Cos@thD2 h@rD - 40 r Cos@thD2 h¢ @rD -
4 r2 h¢¢ @rD - 4 r2 Cos@2 thD h¢¢ @rDLL + f@rD H4 H-1 + 4 h@rDL H-1 + 2 j2 r2 -6 j2 r2 Cos@2 thD + H2 + 8 j2 r2 + H6 + 8 j2 r2L Cos@2 thDL h@rD - 16 Cos@thD2 h@rD2L
k@rD2 + r H-1 + 4 h@rDL H-1 + 4 Cos@thD2 h@rD + 2 r Cos@thD2 h¢ @rDL k¢ @rD +2 k@rD H-2 - 4 j2 r2 - 12 j2 r2 Cos@2 thD - 32 Cos@thD2 h@rD2 + r H5 + 3 Cos@2 thDL h¢ @rD +
4 r2 Cos@thD2 h¢ @rD2 + r2 h¢¢ @rD + r2 Cos@2 thD h¢¢ @rD + 4 h@rD H3 + 8 j2 r2 +Cos@2 thD + 8 j2 r2 Cos@2 thD - 8 r Cos@thD2 h¢ @rD - 2 r2 Cos@thD2 h¢¢ @rDLLLL H8 Hf@rD + 4 g@rD2 - 4 f@rD h@rDL k@rD2L, HCos@thD H-16 j2 r2 k@rDH4 g@rD2 H-3 + k@rD - 4 h@rD k@rDL + f@rD H-1 + 4 h@rDL H2 + H-1 + 4 h@rDL k@rDLL +
16 r2 g@rD k@rD g¢ @rD h¢ @rD + r H-1 + 4 h@rDL k@rDH-4 r g¢ @rD2 + f¢ @rD H-1 + 4 h@rD + 2 r h¢ @rDLL - 4 g@rD2 H4 H1 - 4 h@rDL2 k@rD2 +r H1 - 4 h@rD - 2 r h¢ @rDL k¢ @rD + k@rD H-5 + 20 h@rD + 20 r h¢ @rD + 4 r2 h¢¢ @rDLL +
godeldemo.nb 1
f@rD H4 H-1 + 4 h@rDL3 k@rD2 - r H-1 + 4 h@rDL H-1 + 4 h@rD + 2 r h¢@rDL k¢@rD + 4 k@rD H1 +16 h@rD2 - 4 r h¢@rD - 2 r2 h¢@rD2 - r2 h¢¢@rD + 4 h@rD H-2 + 4 r h¢@rD + r2 h¢¢@rDLLLLL
H8 Hf@rD + 4 g@rD2 - 4 f@rD h@rDL k@rD2L, -H-16 r2 g@rD k@rD g¢@rD h¢@rD -r H-1 + 4 h@rDL k@rD H-4 r g¢@rD2 + f¢@rD H-1 + 4 h@rD + 2 r h¢@rDLL -4 g@rD2 H4 H1 + 4 j2 r2 - 4 h@rDL H-1 + 4 h@rDL k@rD2 + r H-1 + 4 h@rD + 2 r h¢@rDL k¢@rD +
k@rD H5 + 48 j2 r2 - 20 h@rD - 20 r h¢@rD - 4 r2 h¢¢@rDLL +f@rD H4 H1 - 4 h@rDL2 H1 + 4 j2 r2 - 4 h@rDL k@rD2 + r H-1 + 4 h@rDL
H-1 + 4 h@rD + 2 r h¢@rDL k¢@rD + 4 k@rD H-1 - 8 j2 r2 - 16 h@rD2 + 4 r h¢@rD +2 r2 h¢@rD2 + r2 h¢¢@rD + 4 h@rD H2 + 8 j2 r2 - 4 r h¢@rD - r2 h¢¢@rDLLLL
H8 Hf@rD + 4 g@rD2 - 4 f@rD h@rDL k@rD2L, 2 !!!!3 ji
k
jjjjjjj8 j
"###########################################################################################Hf@rD + 4 g@rD2 - 4 f@rD h@rDL k@rD+
Hg@rD Hr H1 - 4 h@rDL k@rD f¢@rD + 4 g@rD2 H-6 k@rD + r k¢@rDLL + f@rD H2 r H-1 + 4 h@rDLk@rD g¢@rD + g@rD Hk@rD H-6 + 24 h@rD - 4 r h¢@rDL + r H1 - 4 h@rDL k¢@rDLLL
Ir Hf@rD + 4 g@rD2 - 4 f@rD h@rDL2 k@rD2My
zzzzzzz, 0, 0, 0,
-I2 !!!!3 j H8 g@rD2 k@rD H8 g@rD2 k@rD - r f¢@rDL +f@rD2 H4 H1 - 4 h@rDL2 k@rD2 + 4 k@rD H-1 + 4 h@rD - r h¢@rDL + r H1 - 4 h@rDL k¢@rDL +f@rD Hr H-1 + 4 h@rDL k@rD f¢@rD + 8 r g@rD k@rD g¢@rD -
4 g@rD2 H4 k@rD + 8 H-1 + 4 h@rDL k@rD2 - r k¢@rDLLLMIr2 Hf@rD + 4 g@rD2 - 4 f@rD h@rDL2 k@rD2M=
In[2]:= sol = DSolve@8f@rD 1 - 2* m r^2, g@rD j*r + m *lr^3, h@rD j^2*Hr^2 + 2* mL - m *l^22r^4,k@rD 1H1 - 2* m r^2 + 16*j^2* m^2r^2 + 8*j* m *lr^2 + 2* m *l^2r^4L<, 8f,g, h, k<, rD;
In[3]:= Simplify@eq . solD
Out[3]= 880, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0<<
godeldemo.nb 2
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It is remarkable that one finds
f(r) = 1− 2m
r2
g(r) = 2jr
h(r) = j2 (r2 + 2m)
k(r) =(
1− 2m
r2 +16j2m2
r2
)−1
is a solution
5d Godel Black Hole (Black string in 6d pp wave)
For how we found this solution, please ask privately.
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Instead, let me describe a general solution generating
technique which reproduces this answer (and generalize
alot) (GHHLR)
We were led to seek such a technique because we wanted
to find analogous solutions in higher dimensions.
In that case, the supergravity equations did not truncate
to a simple (gµν, Aµ) system.
and the equations were a mess
There had to be a better way
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The better way: starting with Minkowski space, and
1. Compactify
2. T-duality
3. Twist: xµ → Λµνx
ν
4. T-duality
5. Decompactify
is a solution generating technique for Melvin universe
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T-duality
Consider compactification on TN
Consider components of the metric Gij and the
Kalb-Ramond field Bij polarized along TN .
Assemble them into a 2N × 2N matrix
M =
(G−BG−1B BG−1
−G−1B G−1
)
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T-duality:
If M is a solution to the equation of motion, so is
M ′ = ΛMΛT , Λ ∈ O(N,N,Z)
plus appropriate transformation for other fields (again,
look up the details in books)
Nice because this manipulation is algebraic. It is much
easier than solving PDE’s
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Twist
Make the periodic identification
(y, φ) = (y + L, φ + αL) .
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Equivalently, make a change of variables
φ = φ− y
and identify
(y, φ) = (y + L, φ) .
• This adds new terms to the line element)
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Melvin Universe
Space-time of the form (1963)
ds2 = −dt2 +1
1 + α2r2dy2 +
α4r4
4(1 + α2r2)σ2 + r2dΩ2
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B =αr2
2(1 + α2r2)σ ∧ dy
σ is an one form
r2σ = xTΛdx
This is a geometry is supported by the flux which is
localied near r = 0.
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Back to the better way: starting with Minkowski space,
and
1. Compactify
2. T-duality
3. Twist: xµ → Λµνx
ν
4. T-duality
5. Decompactify
is a solution generating technique for Melvin universe
6. Infinite boost
gives rise to plane wave (if the boost and twist is scaled)
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This is like taking the Penrose limit of the Melvin
universe
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“T-dualizing and twisting along a null direction”
(Alishahiha, Ganor)
Do the same thing with a black string
0. un-Boost
1. Compactify
2. T-duality
3. Twist: xµ → Λµνx
ν
4. T-duality
5. Decompactify
6. Boost
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This turns out to give rise to a black string in plane
waves!
Rank of Λ determines the dimension of the plane wave.
ds2str = −
f(r)(1 + β2 r2
)k(r)
dt2 − 2 β2 r2 f(r)k(r)
dt dy
+(
1− β2 r2
k(r)
)dy2 +
dr2
f(r)+ r2 dΩ2
7 −β2 r4 (1− f(r))
4 k(r)σ2
f(r) = 1− M
r6, k(r) = 1 +
β2M
r4
β → 0: Schwarzschild black string
M → 0: vacuum plane wave
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We call this solution generating technique the Null
Melvin Twist
Important disclaimer:
• Null Melvin Twist generates spacetimes supported by
the 3-form background field strength. Not the same as
BMN background supported by the 5-form background
field strength.
• Works for black strings but not for black holes (because
we need at least one isometry direction to do the twist)
Nontheless, very interesting solution generating technique
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What might one say about the physical properties of the
black hole?
• Area of the horizon
This is easy, and comes out be independent of β
• Surface gravity
This is subtle, because this depends on the choice of
normalization of the time like Killing vector.
If we take the time-like Killing vector to be ∂∂t, then the
answer is again β independent
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String/Black hole correspondence principle:
Entropy of a black hole
SBH =1G9
(G9M)7/6
Entropy of a string
Ss = lsM
match regardless of gs
SBH = Ss =R9
g2sls
when the horizon radius is comperable to the string scale
R6H ∼ G9M ∼ l6s
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In plane waves, the string entropy is changed
Ss =ls
1 + 8πlsβ
(Pando-Zayas, Vaman; Greene, Schalm, Shiu; Brower, Lowe, Tan)
Black string entropy is unchanged, so the matching point
shifts to
RH =ls
1 + 8πβls
Correspondence principle broken?