Black Holes in Extra Dimensions

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

2. Outline of talk Can we test extra dimensions using strong gravity? Eg. with astrophysical observations of black holes • 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! ?

3. 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. 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

5. Cont…Gregory-Laflamme instability [GL ’94] • Uniform strings are unstable in infrared • s-wave metric perturbation; dgmn = e W t e i k z fmn(r) k < kc k = kc k > kc Static End state of decay unknown [Horowitz, Maeda ‘01; Choptuik et al;’03] New non-uniform strings! • Myers-Perry solution may have GL-like instability • [Emparan, Myers ‘03]

6. 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 Homogenous component 4d graviton Vector field Dilaton/radion (massive KK modes)

7. 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]

8. 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

9. 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 Identify Unstable Stable • Question: Is this the only solution at large masses?

10. Cont…Black holes in KK • And have compact non-uniform strings Stable Turn on GL static mode… Critical uniform string • First numerically constructed perturbatively [Gubser ’01] • Then non-pert [TW ’02] Unstable

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

12. Cont…Non-uniform strings

13. Cont…Non-uniform strings

14. Cont…Non-uniform strings

15. Cont…Non-uniform strings

16. Cont…Non-uniform strings

17. Cont…Non-uniform strings

18. Cont…Non-uniform strings

19. Cont…Non-uniform strings

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23. Cont…Non-uniform strings

24. 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]

25. 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]

26. Cont…Black holes in KK • Kol conjectures; Cone geometry Non-uniform strings Black holes

27. Cont…Black holes in KK Preliminary!

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31. Cont…Black holes in KK Preliminary!

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37. Cont…Black holes in KK Preliminary!

38. 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 Mass Cone transition (Stable) Very interesting mass range ~ M Uniform strings Black holes (Stable?) Non-uniform Strings (Unstable) [TW] (Unstable) [Kol ’02]

39. Randall-Sundrum [RS ’99] • Now `non-compact’ extra dimension • Put brane into AdS L = - 1/L^2 gives scale • Warping produces effective compactification ds2 = 1/z2 ( hmn dxm dxn + dz2 ) Warp factor Similar to KK but z = [1,) Brane at z = 1 • 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

40. Cont…Compare KK and RS • For static matter source R >> L L L KK RS demand asymptotic AdS

41. Black holes in RS • Also exists a black string solution; z =  Singularity far from brane • Unphysical, and also suffers GL instability … • [Chamblin, Hawking, Reall; Gregory] … so expect to find localised solution Asymptotically AdS

42. 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

43. Cont…Black holes in RS • AdS-CFT says that for a black hole Rhorizon >> L ; Hawking radiation 4d black hole with CFT matter Large number of fields Including 1-loop effects Dual 4d geometry induced on brane • No tests of conjecture in this regime many unclear issues • BUT implication; No static(large)black hole on brane! • [Tanaka; Emparan,Fabbri,Kaloper]

44. 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 3 2 Lifetime Mass black hole 1mm ~ 100 1 year 1 Solar mass L • Typical galactic black holes evaporate too slowly, but theory predicts no smaller mass objects!

45. 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!

46. Cont…Black holes in KK • Define l = ½ ( Rmax/Rmin – 1) Appear to be a maximum mass Stable Non-uniform Uniform strings Also unstable uniform strings cannot decay to non-uniform ones Unstable TW ‘02

47. Cont…Gregory-Laflamme instability [GL ’94] • Perturb black string • Shooting problem; regular horizon, asymptotic flatness dgmn(t,r,z) = e W t e i k z fmn(r) GL instability Stable oscillating modes Critical wavenumber kc