Post on 20-Dec-2015
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… 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]
Cont… Black holes in KK
• 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]
Cont… Black holes in KK
• 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
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
Cont… Black holes in RS
• 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]
Cont… Black holes in RS
• 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!
Cont… Black holes in KK
• 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