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Black Holes in Extra Dimensions:

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:

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:

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:

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]:

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! dgmn = e W t e i k z fmn(r)

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

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 = gmn(x) dxm dxn + Am(x) dxm dz + f(x) dz2 + harmonics in z Dilaton/radion Vector field 4d graviton Homogenous component (massive KK modes)

Cont… KK theory:

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:

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:

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:

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 strings:

Cont… Non-uniform strings Fix asymptotic compactification length, L S3

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Non-uniform strings:

Cont… Non-uniform strings

Cont… Cone at `waist’:

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:

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]