asymptotic flatness in higher dimensional spacetimesasymptotic flatness in higher dimensional...

Asymptotic flatness in higher

dimensional spacetimes

Tetsuya Shiromizu

Department of Physics, Kyoto University

KIPMU ACPseminar 20th June 2012

Based on

K.Tanabe, N. Tanahashi, T.Shiromizu, J.Math.Phys, 51, 062502(2010), ibid, 52, 032501(2011),

K.Tanabe, S.Kinoshita, T.Shiromizu, Phys. Rev. D84, 044055(2011), 1203.0452[PRD85(2012)]

K.Tanabe(Barcelona Univ. ), N.Tanahashi(UC Davis) S.Kinoshita(Osaka City Univ. )

Content

1. Introduction/black holes in 4D

2. Higher dimensional black holes

3. Asymptotics

4. Applications

5. Summay

1. Introduction/4D BH

Why higher dimensions?

String theory

Gauge/gravity correspondence

Black holes in LHC(?)

A famous cartoon? - “4D” pocket, Doraemon -

BHs in 4D

After gravitational collapse, black holes will

be formed.

Then spacetime will be stationary.

The black hole is described by the Kerr

solution.

The Golden age in 4D

The topology of stationary black holes is 2-

sphere. [Hawking 1972]

It is shown that stationary black holes are

axisymmetric(rigidity). [Hawking 1972]

Stationary and axisymmetric black hole is

unique. The Kerr solution! [Carter 1973,…]

The Schwarzschild solution is unique static

BH. [Israel 1967]

The second Golden age?

A lot of remaining issues for

higher dimensional BHs.

It is a chance to have a “gold”.

Focus on higher dimensional GR

Vacuum Einstein equation

Asymptotically flat

Cosmic censorship conjecture

[no naked singularities]

3. Higher dimensional black holes

~ brief review ~

Recent reviews

Eds. K.-I. Maeda, T. Shiromizu, T. Tanaka

Higher dimensional black holes,

Progress of Theoretical Physics Supplement No. 189, 2011

Ed. G. Horowitz,

Black holes in higher dimensions,

Cambridge Univ. Press 2012

Schwarzschild in higher dimensions

・spherical symmetric

・topology: (D-2)-sphere

・any dimensions

Tangherlini 1963

Kerr in higher dimensions

Myers-Perry black holes

・rotating

・topology: (D-2)-sphere

・any dimensions

Myers&Perry 1986

Black rings Discovered by Emparan & Reall 2001

・D=5

・

・rotation is important

21 SS

Black object zoo

Black Saturn Elvang and Figueras, 2007

Di-Ring Iguchi and Mishima, 2007

Bi-Cycling Ring Izumi, 2008, Elvang and Rodriguez, 2008

and so on

Rigidity theorem

Hawking 1972 4D, stationary, rotating, asymptotically flat black holes are

axisymmetric

Hollands, Ishibashi & Wald 2006 any dimensions, stationary, rotating, asymptotically flat black

object spacetimes are axisymmetric.

Topology

Hawking 1972 The topology of 4D, stationary, rotating, asymptotically flat

black hole is 2-sphere

Galloway & Schoen 2005

In higher dimensions,

"0" )2(

S

D RdS

sum connected , , :5 213 SSSD

:Ricci scalar of BH cross section RD )2(

“Uniqueness” ≈ classification

Gibbons, Ida & Shiromizu 2002

Higher dimensional Schwarzschild spacetime is unique static

vacuum BH.

Morisawa & Ida 2004

In D=5, the Myers-Perry solution is unique stationary vacuum

BH if the topology is 3-sphere and 2 rotational symmetries are.

…, Hollands & Yazadjiev 2007

In D=5, stationary BH with 2-rotational symmetries can be

classified by mass, angular momentum and rod

structure(~topology)

Stability

Schwarzschild BH is stable[Ishibashi & Kodama 2003]

Ultraspinning Myers-Perry(MP) solution (D>5) was

conjecture to be unstable [Emparan & Myers 2003]

confirmed by a numerical study[Shibata & Yoshino 2009-2010]

settle down to BH with lower angular momentum.

Remaining issues

Stability

Although there are many efforts…

Complete classification

requirement of rod structure(≈topology,…) is too strong…

Construction of exact solution in D>5

need a new way to find (exact) solution…

blackfold approach…? Emparan et al 2009

Asymptotic structure

3. Asymptotics

Asymptoics

Naïve concept

)||/1( 3 D

ijij xOg

2

2

||

1D

xt

Og null infinity

t

x

spatial infinity

curries the information of dynamics

How to analyze

…, Ashtekar & Hansen 1978, Hollands & Ishibashi 2005

conformal completion method

finite0

g~2

0 infinity

Minkowski spacetime

spatial infinity

null infinity

0 2g~

embemded into the Einstein static universe

rturt, vuv ,)1()1(4Ω 1212

uvuv 1111 tantan ,tantan

)sin(~ 222222 dddddxdxg

)0 , ,(

Einstein static universe

Conformal treatment in general

rg /1~ ,g~ 2

Near null infinity , where hg )/1( 12/ DrOh

12/~ Dh

g~in coordinate a be toused is

-The half-integer for odd dimensions.

-It is supposed that the conformally transformed spacetime is regular.

-Without solving the Einstein equation, one discusses the asymptotics in the physical

spacetime.

-As a result, we encounter the bad behavior because of the half-integer for odd

dimensions.

Conformal treatment in general

The conformal treatment

does not work for odd

dimensions…

Introduce the Bondi coordinate and then solve

the Einstein equation near the null infinity.

Bondi coordinate

sphere-)2(unit of volume:

,)det(

2

2

D

h

D

DIJ

))((2 222 duCdxduCdxhrdudreduAeds JJII

IJ

BB

)(

)(),(),(1

)12/(

2/)2()12/(

D

IJIJ

DIDD

rOh

rOCrOBrOA

Vacuum Einstein equation

constr constu

infinity null :r

Definition of asymptotic flatness

2)det( DIJh


IJ

BB

)(

)(),(),(1

)12/(

2/)2()12/(

D

IJIJ

DIDD

rOh

rOCrOBrOA

The spacetime is said to be asymptotically flat if the

metric in the Bondi coordinate behaves like

Bondi coordinate

Bondi mass ))((2 222 duCdxduCdxhrdudreduAeds JJII

IJ

BB

22/

0

2/5

312/

)1(

)/1(),(

1Dk

k

D

D

I

kD

kB

uu rOr

xum

r

AAeg

)1()1( k

IJ

JIk hDDA

2

16

2)(

DSBondi md

DuM

of because integral surface the tocontributenot does )1( kA

Bondi mass loss law ))((2 222 duCdxduCdxhrdudreduAeds JJII

IJ

BB

22/

0

2/5

312/

)1(

)/1(),(

1Dk

k

D

D

I

kD

kB

uu rOr

xum

r

AAeg

2

16

2)(

DSBondi md

DuM

derivative total)2(2

1 )1()1(

IJ

uIJuu hhD

m

032

1)(

2

)1()1(

dhhuM

du

dDS

IJ

uIJuBondi

The Einstein equation implies

Flux of gravitational waves

Bondi mass loss law

032

1)(

2

)1()1(

dhhuM

du

dDS

IJ

uIJuBondi

Flux of gravitational waves

constu

infinity null :r

spatial infinity

u

initialfinalBondi

ADMinitialBondi

MMM

MMM

)(

0)(

Gravitational wave curries the

positive energy

Asymptotic symmetry


IJ

BB

g

0 ,0 ,0 IJ

IJ

rIrr gggg

)(

)(),(),(1

)12/(

2/)2()12/(

D

IJIJ

DIDD

rOh

rOCrOBrOA

Bondi coordinate condition

)(2

,),(

),,(

2

I

II

Ir

J

IJB

III

Iu

fDfDCD

r

fDhr

edrxuf

xuf

2)det( DIJh

Asymptotic symmetry is defined to be the transformation group which

preserves the asymptotic structure at null infinity

IJID w.r.t.derivativecovariant :

Asymptotic symmetry


IJ

BB

)/1( ),/1( ),/1( ),/1( 32/222/12/ D

IJ

D

ur

D

rI

D

uu rOgrOgrOgrOg

)(),(),(),(1 )12/(2/)2()12/( D

IJIJ

DIDD rOhrOCrOBrOA

The asymptotic behavior

IJ

K

KJIu

I

IIJ

K

KIJJI

I

uD

fDDfDDfDfD

D

fDfDfDf

2 ,)2(,

2

2 ,0

)/1(2

22

2 e.g., 32/2

D

IJ

K

KJIIJ

K

KIJJIIJ rO

D

fDDfDDr

D

fDfDfDrg

Asymptotic symmetry is Poincare

The asymptotic symmetry is generated by

IJ

K

KJIu

I

IIJ

K

KIJJI

I

uD

fDDfDDfDfD

D

fDfDfDf

2 ,)2(,

2

2 ,0

Iff and

group Lorentz

ntranslatio

If

f

D=4 is special

IJ

K

KJIu

I

IIJ

K

KIJJI

I

uD

fDDfDDfDfD

D

fDfDfDf

2 ,)2(,

2

2 ,0

)/1(2

22

2 32/2

D

IJ

K

KJIIJ

K

KIJJIIJ rO

D

fDDfDDr

D

fDfDfDrg

group Lorentz

ntranslatio

If

f

)/1(, 32/ D

IJ rOg )(rOgIJ

)(rO

+supertranslation

This gives us troublesome when one defines angular momentum at null infinity

4. Application

Stability of BHs

The symmetry which exact BH solutions have are

not enough to analyse the stability.

Together with the general properties of dynamical

BHs, we may be able to have some indications.

Stability of BHs

area)(A

fixed J

Figueras, Murata & Reall 2011, Hollands & Wald 2012

mass)(M

solution a :)(AMM BH

Area theorem

Bondi mass loss

Perturbed initial data

Stable

unstable

Classification

RC nsor Riemann te ofpart free trace

)/1( 5

4

)(

3

),(

2

)()(

rOr

C

r

C

r

C

r

CC

IDIIIIIN

Weyl tensor：

contains information of spacetime(gravity)

Peeling property in 4D

Algebraic classification of Weyl tensor (Petrov type) ⇔ asymptotic behavior

In Type D the equations will be significantly simplified and then

solved completely. The Kerr solution, C-metric ,….

Using our result, a peeling properties has been discussed in higher dimensions

[Godazgar & Reall 2012]. ] odd)[/1(or ]even )[/1( 2/32/22/

12/

)(

2/

)(

12/

)(

DrODrOr

C

r

C

r

CC DD

D

G

D

II

D

N

5 )( Di

5 )( Dii)/1( 4

2/7

)(

3

)(

2/5

)(

2/3

)(

rOr

C

r

C

r

C

r

CC

GNIIN

Classification

Similar to four dimensions, the classification of

the Weyl tensor and the peeling property may

give us a hint to classify and/or construct exact

solutions in higher dimensions.

5. Summary and future issues

Summary

For odd dimensions, we had to solve the Einstein equation

to formulate the asymptotics at null infinity.

We could show the Bondi mass loss law and the presence

of the Poincare symmetry at null infinity.

We also defined the momentum and angular momentum.

There are a few efforts to discuss the stability and

classification of higher dimensional BHs/spacetimes.

Remaining issues

Still…

Stability…

Classification…

Construction…

New issues?

Asymptotically anti-deSitter spacetimes

⇔ always unstable? Final fate? [Dias,Horowitz,Santos,2011]

Asymptotics in spacetimes with compact space

⇔ final fate of black string?

[Latest numerical simulation:Lehner & Pretorius 2010]

YITP Molecule-type workshop on

Non-linear massive gravity

23rd July to 8th August

Core participants :

C. de Rham, S. Mukohyama, …, T.Shiromizu(Chair), A. Tolley, …

Appendix

Constraint and evolution eqs.

In Bondi coordinate,

))((222 duCdxduCdxdudreduAeds JJII

IJ

BB

time u

equations constraint 0 RRr

equations evolution 0,0 IJuu RR

Constraint equations


IJ

BB

JLIK

KLIJrr hhhhD

rBR

)2(4 0

Rr

eBh

r

eBBh

r

e

CCher

Cr

D C

r

ArDR

hB

J

IJ

I

B

JI

IJB

JI

IJ

BI

I

I

ID

D

)(

2

)()(

2

)()(

2

2)()(

2

3

)(2

2

)2(2)()2( 0

IJ

Jh

II

J

IJ

BD

DrI hBr

DBCher

rR

)()()(

2

2)(

1 0

determined are , , surface,const - initial on thegiven are Once I

IJ CBAuh

Constraint equations


IJ

BB

)dimensions oddfor 2 even,for ( 0

)12/()1( ZkZkrhhk

kDk

IJIJIJ

　

s tracelesis )12/()det( )1(

2

Dkhh k

IJDIJ

)1()1()1(2/3)2()1(

16

1 ),/1( KLIJ

JLIKDD hhBrOrBB

IJk

J

IkDDk

kD

II

kD

IkI h

kDkD

kDCrO

r

xuJ

r

CC )1()()1(2/1

12/

012/

)1(

)22)(2(

)22(2 ),/1(

),(

IJk

JI

kDDk

kD

I

kD

k

hkDkDkD

kDArO

r

xum

r

AA )1()()()1(2/5

22/

0312/

)1(

)42)(22)(2(

)42(4 ),/1(

),(1

Evolution equations


IJ

BB

)dimensions oddfor 2 even,for ( 0

)12/()1( ZkZkrhhk

kDk

IJIJIJ

　

IJ

k

K

Kk

JI

k

KJ

K

I

k

IJ

k

IJIJ

kk

IJIJ

CCkDhh

hkkDDAkDhkR

)1()()1(

)(

)()1(

)

)(

(

)()1(2)(

)1(22)1()2(

)42(2

1)2(

2

1

)]4(46[8

1)42(

2

1)1( 0

s tracelesis )12/()det( )1(

2

Dkhh k

IJDIJ

)22/(2)()22/()()1()1(

2

1

2

5

)2(2

1 0

DD

I

IIJ

IJuu AD

CD

Dhh

DmR

Asymptotic symmetry

Asymptotic symmetry


IJ

BB

g

u

I

II

I

ur

IJ

IJ

J

IJ

uJ

IJ

u

I

B

rI

uB

rr

Cgg

Ceg

eg

)()(

)(

22)(log)(log

02

)(

)(),(),(1

)12/(

2/)2()12/(

D

IJIJ

DIDD

rOh

rOCrOBrOA

Bondi coordinate condition

)(2

,),(

),,(

2

I

II

Ir

J

IJB

III

Iu

fDfDCD

r

fDhr

edrxuf

xuf

2)det( DIJh

Asymptotic symmetry is defined to be the transformation group which

preserves the asymptotic structure at null infinity

IJID w.r.t.derivativecovariant :

Asymptotic symmetry


IJ

BB

)/1(2

22

2

)/1(])2([2

1])2([

2

)/1()2)(2(

22])2([

2

1

)/1(2

))2((2

2

2

2

32/2)(

)()()(

)(

)(2

22/

32/

)1(2)()(2

2

2/

)()(22/

0

)1()(

12/

22/

)1(2)()(

D

IJJIIJ

K

KJIIJ

D

D

J

uIJIu

I

IIIuuI

D

kD

JIDk

k

k

IJu

I

Iur

D

D

I

uIu

I

Iuuu

rOD

ffr

D

ffrg

rOr

fhfDf

DfDf

D

rfrg

rOr

fh

kDD

kDfDf

Dg

rOr

fCfn

Df

D

rg

)(

)(),(),(1

)12/(

2/)2()12/(

D

IJIJ

DIDD

rOh

rOCrOBrOA

2)det( DIJh

IJ

K

KJIu

I

IIJ

K

KIJJI

I

uD

fffDf

D

ffff

2 ,)2(,

2

2 ,0

)()()()()(

)()()(

Asymptotic symmetry

IJ

K

KJIu

I

IIJ

K

KIJJI

I

uD

fffDf

D

ffff

2 ,)2(,

2

2 ,0

)()()()()(

)()()(

)(2

)( )2(: ,0 )( I

I

u

I

I

I

u xuD

xFffDfFf

IJ

K

KJIIJ

K

KJI

D

FF

D

ff

2

2

)()()()(

)()()()(

0))2(( 2

2 2)()(

)()()()(

FD

D

fff IJ

K

KIJJI

JI

IJ

K

KJIIJ

K

KJI

DD

ff

2

2

)()()()(

)()()()(

2on harmonics

scalar theof modes 1

DS

l

2on harmonics

scalar theof modes 1,0

DS

l

0 ,0 .,., ofpart e transver: )()()()()()()( I

trans

JJ

trans

I

Itrans

I

Itrans fffeiff Lorentz

group

time/space translations

Petrov type

Principal null direction

mnmnCmnnlC

mnmlCmlnlCmlmlC

43

*

210

,

,,,

Five complex functions

0,0,1,1 , ***

nnllmmnlmmmmlnnlg v

null tetrad

Transformation of null tetrad and principal null direction

naamaamllanmmnn ***,,

4

4

3

3

2

2

100 464 aaaa

afor equation thesatisfyingby zero made becan 00464 4

4

3

3

2

2

10 aaaa

I (four distinct roots), II(two coincident ), D(two distinct double),

III(three coincident), N(four coincident)

Petrov classification

I (four distinct roots), II(two coincident ), D(two distinct double),

III(three coincident), N(four coincident)

0 I 10

0 II 410

0 D 4310

0 III 4210

0 N 3210

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