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 ds 2 (5d) = ds 2 (4d Schwz) + dz 2 z Not asymptotically flat

Cont… Gregory-Laflamme instability [GL ’94] Uniform strings are unstable in infrared –s-wave metric perturbation; k < k c k = k c k > k c 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 5 th coordinate, period L –New dynamic gravity scale; M = L/G 4 Ie. The mass of a black hole with radius L ds 2 = g MN dx M dx N = g  (x) dx  dx + A  (x) dx  dz +  (x) dz 2 + 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 / r 2 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/G 4 ? If L as large as 0.1 mm M ~ kg –Ie. Matter confined to branes Without matter confinement M ~ 10 8 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 UnstableStable 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 strings Fix asymptotic compactification length, L S3S3

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]

Cont… Black holes in KK But should also be non-wrapping black hole! –At least should exist for R horizon << 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 Kol conjectures; Cone geometry Non-uniform strings Black holes

Cont… Black holes in KK Preliminary!

Cont… Black holes in KK Preliminary!

Cont… Black holes in KK Preliminary!

Cont… Black holes in KK Preliminary!

Cont… Black holes in KK Preliminary!

Cont… Black holes in KK Preliminary!

Cont… Black holes in KK Preliminary!

Cont… Black holes in KK Preliminary!

Cont… Black holes in KK Preliminary!

Cont… Black holes in KK Preliminary!

Cont… Black holes in KK Preliminary!

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?) Uniform strings Non-uniform Strings (Unstable) [TW] (Unstable) (Stable) 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 ds 2 = 1/z 2 (   dx  dx + dz 2 ) Similar to KK but z = [1,  ) Warp factor Z non-compact so continuum of bulk modes Propagator on brane ~ 1/r ( 1 + L 2 /r 2 + … ) 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 L L KKRS 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; R horiz < L –But numerics fail for moderate sizes ; R horiz ~ L

Cont… Black holes in RS AdS-CFT says that for a black hole R horizon >> L ; 4d geometry induced on brane 4d black hole with CFT matter Large number of fields Including 1-loop effects Dual Hawking radiation 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 LifetimeMass black hole 1 Solar mass 1mm L Typical galactic black holes evaporate too slowly, but theory predicts no smaller mass objects! 1 year ~

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 kg solutions very different to 4-d, with complicated evapouration RS astrophysical black holes may classically radiate away. –For L = 0.1 mm masses < kg may evapourate fast!

Cont… Black holes in KK Define = ½ ( R max /R min – 1) Unstable Stable Non-uniform Appear to be a maximum mass Uniform strings 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 k c