Black Holes in Extra Dimensions COSMO 2003, Ambleside Toby Wiseman Cambridge (UK) / Harvard Work with B. Kol (Hebrew University) H. Kudoh (Kyoto)

Black Holes in Extra Dimensions

COSMO 2003, Ambleside

Toby Wiseman

Cambridge (UK) / Harvard

Work with B. Kol (Hebrew University)

H. Kudoh (Kyoto)

Outline of talk

• More dimensions more complex behaviour– Uniqueness theorems, black strings, Gregory-

Laflamme instability

• Understanding Kaluza-Klein vacuum solutions– Black holes, black strings …?

• Do Randall-Sundrum black holes exist?– AdS-CFT astrophysical black holes are not static!?

Can we test extra dimensions using strong gravity?Eg. with astrophysical observations of black holes

4-d 5-d ; asymptotically flat case

• In 4-d Schwarzschild and Kerr are unique asymptotically flat, regular vacuum solutions

• Recently Emparan-Reall showed 5-d rotating vacuum black ring solution in addition to Myers-Perry solution

Implies lack of uniqueness in higher dimensions

• Now Schwarzschild is proven unique if static and stable

[Gibbons,Ida,Shiromizu ‘02;Maeda,Ishibashi ‘03]

• But obviously no uniqueness in general and stability not yet solved for stationary cases

4-d 5-d ; non-asymptotically flat case

• Unlike 4-d, in 5-d or above one has black string solutions• Uniform vacuum string is just a product geometry

– Can have any radius horizon

ds2(5d) = ds2

(4d Schwz) + dz2

z

Not asymptotically flat

Cont… Gregory-Laflamme instability [GL ’94]

• Uniform strings are unstable in infrared– s-wave metric perturbation;

k < kc k = kc k > kc

Static

End state of decay unknown

[Horowitz, Maeda ‘01; Choptuik et al;’03]

• Myers-Perry solution may have GL-like instability [Emparan, Myers ‘03]

New non-uniform strings!

g = e t e i k z f(r)

Kaluza-Klein theory [Kaluza ’21, Klein ’26]

• Pure gravity in 5 dimensions– Compactify the 5th coordinate, period L

– New dynamic gravity scale; M = L/G4

• Ie. The mass of a black hole with radius L

ds2 = gMN dxM dxN

= g(x) dx dx + A(x) dx dz + (x) dz2

+ harmonics in z

Dilaton/radionVector field4d graviton

Homogenous component

(massive KK modes)

Cont… KK theory

• At large distances r >> L; 4d Einstein-Dilaton-Maxwell

– Static propagator goes as; 1 / r ( 1 + e- r/L + …. )

Yukawa corrections from massive KK modes

• At small distances r << L; 5d gravity

– Propagator goes as; 1 / r2

• Expt bound on L: for SM matter fields ~ (TeV)-1

for gravity ~ 0.1 mm

• So if introduce branes for matter, L ~ 0.1 mm!(ADD senario) [Arkani-Hamed, Dimopoulous, Dvali

‘98]

Cont… KK theory

• So what is M = L/G4 ?

• If L as large as 0.1 mm M ~ 1024 kg

– Ie. Matter confined to branes

• Without matter confinement M ~ 108 kg

– So L can only be as large as L ~ (TeV)-1

Black holes in KK• Can simply compactify uniform strings• Choose to fix asymptotic radius of compactification, L,

away from symmetry axis• `Thick’ strings now stable as infrared GL instability

projected out by periodic boundary conditions

Unstable Stable

• Question: Is this the only solution at large masses?

Identify

Cont… Black holes in KK

• And have compact non-uniform strings

Critical uniform string

Turn on GL static mode…

Stable

Unstable

• First numerically constructed perturbatively [Gubser ’01]

• Then non-pert [TW ’02]

Cont… Non-uniform stringsFix asymptotic compactification length, L

S3

Cont… Non-uniform strings












Cont… Cone at `waist’

Kol predicts a cone geometry at the string `waist’

There is a unique cone with the correct isometries

[Kol ’02]

Numerical comparison agrees very well

[Kol, TW ‘03]


• But should also be non-wrapping black hole!

– At least should exist for Rhorizon << L

– Then geometry ~ higher dim Schwarzschild– Analytic construction only in 4d [Myers ‘87]

Use numerical methods

Work in progress!! [Kudoh, TW]


• Kol conjectures;

Cone geometry

Non-uniform strings

Black holes


Preliminary!


Preliminary!


Preliminary!


Preliminary!


Preliminary!


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Preliminary!


Preliminary!


Preliminary!


Preliminary!


Preliminary!


• If transition picture is correct, very likely to be maximum mass for black hole solutions– For `large’ masses, uniform string unique solution

• No corrections to 4-d

Black holes(Stable?)

Uni

form

str

ings

Non-uniformStrings(Unstable) [TW]

(Uns

tabl

e)(S

tabl

e)

Cone transitionMass

Very interesting mass range ~ M

[Kol ’02]

Randall-Sundrum [RS ’99]

• Now `non-compact’ extra dimension– Put brane into AdS = - 1/L^2 gives scale– Warping produces effective compactification

ds2 = 1/z2 ( dx dx + dz2 )

Similar to KK but z = [1,)Warp factor

• Z non-compact so continuum of bulk modes

Propagator on brane ~ 1/r ( 1 + L2/r2 + … )

• So again L<0.1 mm

NOT Yukawa suppressed!4d

Brane at z = 1

Cont… Compare KK and RS• For static matter source R >> L

LLKK RS

demand asymptotic AdS

Black holes in RS• Also exists a black string solution;

Singularity far from brane

z =

• Unphysical, and also suffers GL instability … [Chamblin, Hawking, Reall; Gregory]

Asymptotically AdS

… so expect to find localised solution

Cont… Black holes in RS

• Constructed analytically in 4-d [Emparan, Horowitz, Myers ‘02]

• Various analytic attempts eg. [Kanti, Olasagasti, Tamvakis ’02 ‘03;

Neves, Vaz ’03…], and general progress on axisymmetric solutions [Harmark, Obers ‘02; Charmousis, Gregory ’03…]

• Recently same numerical methods as for the non-uniform strings applied to find black hole

[Kudoh,Tanaka,Nakamura ‘03]

– Indeed find small black hole solutions; Rhoriz < L

– But numerics fail for moderate sizes ; Rhoriz ~ L


• AdS-CFT says that for a black hole Rhorizon >> L ;

4d geometry inducedon brane

4d black hole with CFT matterLarge number of fieldsIncluding 1-loop effects

Dual

Hawkingradiation

• No tests of conjecture in this regime many unclear issues• BUT implication; No static (large) black hole on brane!

[Tanaka; Emparan,Fabbri,Kaloper]


• So black holes confined to RS brane classically radiate away!

• Can predict lifetime (as understand Hawking radiation in dual theory)– NOT the same Planck constant as usual in 4d

Lifetime Mass black hole

1 Solar mass

1mm

L

• Typical galactic black holes evaporate too slowly, but theory predicts no smaller mass objects!

1 year~ 100

3 2

Conclusions• Black hole phenomenology much more subtle

with extra dimensions. – Haven’t mentioned bulk matter, stabilization…

• There are several KK solutions, that may be elegantly related– At galactic black hole masses probably only one type

(uniform string), no corrections to usual 4-d behaviour– But if L = 0.1 mm, for masses below 1024 kg solutions

very different to 4-d, with complicated evapouration

• RS astrophysical black holes may classically radiate away. – For L = 0.1 mm masses < 1032 kg may evapourate fast!


• Define = ½ ( Rmax/Rmin – 1)U

nsta

ble

Sta

ble

Non-uniform

Appear to be a maximum mass

Uni

form

str

ings

Also unstable uniform strings cannot decay to non-uniform ones

TW ‘02

Cont… Gregory-Laflamme instability [GL ’94]

• Perturb black string– Shooting problem; regular horizon, asymptotic flatness

g(t,r,z) = e t e i k z f(r)

Stable oscillating modes

GL instability

Critical wavenumber kc

Documents

Black Holes in Extra Dimensions COSMO 2003, Ambleside Toby Wiseman Cambridge (UK) / Harvard Work with B. Kol (Hebrew University) H. Kudoh (Kyoto)