welcome to the string theory seminar...

1

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

2

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

4

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

5

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

6

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!

7

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

6

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

9

This limit can be applied to null geodesics in AdS5 × S5.

10

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!

11

Godel Universes: ds2 = −(dt + jr2dφ)2 + dr2 + r2dφ2

r0 Cr>

rr 0

t

P

Strange: closed time like curves through every point

12

• 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

13

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

14

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

15

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 =√

32

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 =√

32

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

20

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.

21

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

22

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

)

24

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

25

Twist

Make the periodic identification

(y, φ) = (y + L, φ + αL) .

26

Equivalently, make a change of variables

φ = φ− y

and identify

(y, φ) = (y + L, φ) .

• This adds new terms to the line element)

27

Melvin Universe

Space-time of the form (1963)

ds2 = −dt2 +1

1 + α2r2dy2 +

α4r4

4(1 + α2r2)σ2 + r2dΩ2

7

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.

28

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)

29

This is like taking the Penrose limit of the Melvin

universe

30

“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

31

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

32

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

33

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

34

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

35

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?