ˇ Black Holes and the Hoop Conjecture Black Holes n Supergravty and M/Superstrng Theory Penn. State Unversty 9 th September 200 Chrs Pope Based on work wth Gary Gbbons and Mrjam Cvetč
When Does a Black Hole Form? In 972, Kp Thorne formulated the Hoop Conjecture, whch proposes that: Horzons form when and only when a mass E s compacted nto a regon whose crcumference C n every drecton s such that C 4πE. The conjecture s clearly reasonable n some sense. For example, the horzon of a Schwarzschld black hole of mass E has radus r + = 2E, and hence crcumference C = 4πE. Thus the conjecture essentally says that f and only f a gven mass s contaned n a regon equal to or smaller than ts Schwarzschld radus, then t must be a black hole. The stpulaton of every drecton s an mportant one, snce clearly a long, thn object can have arbtrarly large mass wthout formng a black hole. However, n ts orgnal form, the conjecture s rather vague. Concentrate for now on the only when, leavng the when for later.
The Brkhoff Invarant Gary Gbbons proposed n four dmensons to use the Brkhoff Invarant to make the hoop conjecture precse. If the domnant energy condton s satsfed, the apparent horzon s an S 2, wth nduced metrc g. Let f be a functon on {S 2, g} R wth one mnmum and one maxmum. Each level set f (c) has a length l(c). For gven f, defne β(f) = max c l(c), and then the Brkhoff Invarant β s β = mn f β(f). Essentally, we choose a folaton, and fnd ts maxmum crcumference (the wast ). Then we mnmse over all choces for the folaton. Gary Gbbons statement of a Hoop Conjecture n four dmensons s then that β 4πE. Brkhoff showed there exsts at least one closed geodesc wth length equal to β. Thus f l(g) s the length of the shortest non-trval closed geodesc, then Gbbons conjecture mples l(g) 4πE. Fndng any non-trval closed geodesc wth l 4πE would support the conjecture.
Example; 4-Charge Rotatng Black Hole Four-charge rotatng black hole n ungauged supergravty (Cvetč & Youm) ds 2 4 = ρ2 2mr W ( (dt + Bdφ) 2 dr 2 + W + dθ2 + sn2 θ ρ 2 2mr dφ2 B = 2ma(rc 234 (r 2m)s 234 ) sn 2 θ ρ 2, 2mr W 2 = r r 2 r 3 r 4 + a 4 cos 4 θ + a 2 [2r 2 + 2mr s 2 + 8m2 c 234 s 234 4m 2 (s 2 23 + s2 24 + s2 34 + s2 234 + 2s2 234 )] cos2 θ, = r 2 2mr + a 2, ρ 2 = r 2 + a 2 cos 2 θ, r = r + 2ms 2, s n = s s n, c n = c c n, s = snh δ, c = cosh δ Now g φφ s largest at θ = 2π, and so the hoop length s ( 2π (r+ 2 + a2 ) r 2 + c + a 2 ) s β = r + j[r+ 2 c2 + a2 s 2. ]/4 ),
The mass s E = 4 m (c 2 + s2 ). Defnng ã = a/r + (where a s the rotaton parameter, and r + s the horzon radus), the hoop bound s satsfed f We have LHS (c 2 + s2 ) 4[ c + ã 2 s ] j[c 2 j + ã2 s 2 0. j ]/4 [ [ c 2 4 ] c j[c 2 j ]/4 + [ [ s 2 4ã2 ] s j[ã 2 s 2 j ]/4 ( ) /2 ] ( = c 2 4 c + s 2 4 whch s ndeed non-negatve, snce the arthmetc mean of nonnegatve quanttes s greater than or equal to the geometrc mean: n n x = ( n = x ) /n. s ) /2 ] So the 4-charge four-dmensonal rotatng black holes obey the hoop conjecture.
Antpodal Map If the horzon {S 2, g} admts an antpodal map (a Z 2 sometry wth no fxed ponts), then we can get an upper bound for the shortest closed geodesc l(g) by takng twce the shortest length from a pont to ts antpode. Pu has shown that n such cases l(g) πa(g), where A(g) s the area of the S 2. The Penrose Area Inequalty states that πa(g) 4πE, and so combnng the two, the Hoop Conjecture holds for all four-dmensonal cases where the apparent horzon admts an antpodal map. The Cosmc Censorshp Bound (generalsaton of Penrose nequalty to asymptotcally AdS 4 backgrounds) states that 4πE πa(g) + ( Λ) 2π 2 [πa(g)]3/2, and thus the Hoop Conjecture s a fortor satsfed for asymptotcally AdS 4 stuatons wth an antpodal map.
Brkhoff Invarant n Fve Dmensons? For black holes wth S 3 horzons, we can consder a functon f : S 3 R, and level sets f = c of fxed topologcal type and fxed sngular sets. Possbltes are S 2 or S S (Clfford tor) level sets. For example (round S 3 for smplcty): dω 2 3 = dθ2 + sn 2 θ dω 2 2, for whch the S 2 hoop has maxmum area at θ = 2π, or dω 2 3 = dθ2 + sn 2 θ dφ 2 + cos 2 θ dψ 2, for whch the S S hoop has maxmum area at θ = 4 π. In general, defne hoop 2-volume β = mn f max c vol(f (c)). S S hoops are well adapted to 5-dmensonal black holes wth (dstorted S 3 ) horzons, snce the metrc s always a cohomogenety- folaton by (homogeneous) Clfford tor. For S S hoop we conjecture (wth equalty for Schwarzschld) β 6π 3 E. Bound s satsfed by the known fve-dmensonal black holes.
What s a Hoop n Hgher Dmensons? The natural generalsaton of the four-dmensonal Brkhoff nvarant requres folatng the n-dmensonal sphere wth (n )-dmensonal surfaces. However, the known black holes n D > 5 dmensons (Kerr- AdS) have S D 2 horzon topologes. The S D 2, whch s dstorted, s folated by Clfford tor T [(D )/2], and the horzon metrc has cohomogenety [(D 2)/2]. Thus, beyond D = 5, ths doesn t work. In some cases there s a natural folaton of S D 2 by S p S q, wth D 2 = p + q +. For example, for the round S D 2, we may wrte dω 2 D 2 = dθ2 + sn 2 θ dω 2 p + cos 2 θ dω 2 q. The (p + q)-volume of the hyperhoop s maxmsed at sn θ = p p + q, cos θ = q p + q. We can get dfferent values for the Brkhoff nvarant ((D 3)- volume) dependng on the choce of sweep out.
S S D 4 Sweep Outs n D = 2N + Kerr-AdS If all the rotatons are equal, the metrc has a smple form ds 2 = ( + g2 r 2 )dt 2 + Udr2 Ξ V 2m + r2 + a 2 Ξ [(dψ + A)2 + dσ 2 N ] + 2m UΞ 2 [dt a(dψ + A)]2, U = (r 2 + a 2 ) N, V = ( + g2 r 2 )(r 2 + a 2 ) N r 2, Ξ = g 2 a 2 wth A a potental for the Kähler form of the Fubn-Study metrc dσ 2 N on CP N, whch can then be folated by squashed S 2N 3, va the nestng dσ 2 N = dξ2 + sn 2 cos 2 ξ (dτ + B) 2 + sn 2 ξ dσ 2 N 2 The maxmum wastlne volume s V max 2N 3 = (N ) 2 (N ) N 2 N Ω 2N 3 wth Ω 2N 3 the volume of a unt round S 2N 3. The volume of the S S 2N 3 hyperhoop on the horzon s β(s S 2N 3 ) = 2π ( r 2 + + a 2 ) N 2 (N ) 2 (N ) N 2 N Ω 2N 3 r + Ξ
The mass s E = mω 2N 4πΞ N ( N Ξ 2 and the S S 2N 3 hoop nequalty (saturated for Schwarzschld) s 32πE 2N (N ) 2 (N+) N 2 N β(s S 2N 3 ) 0 ) For ths to hold for Kerr-AdS we must then have ( + a2 ) /2 ( + g 2 r+ 2 ) ( ) 2N Ξ 0 Ξ 2N r 2 + whch s clearly satsfed.
Antpodal Maps and Geodescs n Hgher Dmensons Kerr-AdS metrc n dmenson D (Gbbons, Lü, Page & Pope): ds 2 = W ( + g 2 r 2 ) dt 2 + 2m U + N = ( W dt N = r 2 + a 2 µ 2 Ξ dφ2 + U dr2 N+ɛ V 2m + g 2 W ( + g 2 r 2 ) (N+ɛ = r 2 + a 2 Ξ µ dµ = ) 2, a µ 2 dφ ) 2 Ξ r 2 + a 2 Ξ where D = 2N + + ɛ, ɛ =0 or, and N+ɛ µ 2 =, W N+ɛ = µ 2 Ξ, V r ɛ 2 ( + g 2 r 2 ) U r ɛ N = N+ɛ = µ 2 r 2 + a 2 N j= (r 2 + a 2 j ), dµ 2 (r 2 + a 2 ), Ξ g 2 a 2. It satsfes R µν = (D ) g 2 g µν. The horzon s located at r = r +, where r + s the largest root of V (r) = 2m.
D = 2N + 2 Dmensons N azmuthal angles φ and N + lattude coordnates µ. Takng µ = ν sn θ, ( N), µ N+ = cos θ, wth N ν 2 =, we can consder a path on the horzon from the pont θ = 0 to the antpodal pont θ = π. Ths can be bounded above, takng ν and φ fxed, by ds 2 [ r 2 + sn2 θ + We can, w.o.l.o.g., assume N = r+ 2 + a2 ] ν 2 cos 2 θ dθ 2. Ξ a 2 a2 2 a2 3 a2 N, and hence by takng ν =, ν = 0 for 2, we have ds 2 r2 + + a2 Ξ dθ 2, so the path from the pont to ts antpode and back has length L 2π (r2 + + a2 )/2 Ξ /2. Gves an upper bound for the shortest non-trval closed geodesc.
The mass of the Kerr-AdS black hole n D = 2N + 2 dmensons s (Gbbons, Perry, Pope) E = mω D 2 4π Ξ N j= Ξ j. The geodesc hoop conjecture (saturated for Schwarzschld) s L 2π ( 8π E N Ω D 2 ) 2N, where L s the length of the shortest non-trval closed geodesc. Thus for Kerr-AdS n D = 2N + 2 dmensons, the bound s establshed f ( + g 2 r+ ( 2 ) + a2 ) /2 N= (r + 2 + a2 ) r+ 2 (r+ 2 + a2 )N, whch s clearly true. ( N j= Ξ Ξ j ) Ξ ( N N k= Ξ k )
D = 2N + Dmensons The stuaton s very dfferent n odd dmensons. Now have N azmuthal angles φ and N lattudes µ. Cannot move from a pont to ts antpode at fxed φ. One choce of path now s φ φ + π (all ) at fxed µ. On horzon, ds 2 = Z j dφ dφ j wth Z j = r2 + + a2 Ξ δ j + 2ma a j U Ξ Ξ j µ 2 µ2 j, and the length L to antpode and back on the torus T N s L = 2π ( Z + 2 <j Z j ) /2. The geodesc hoop bound, saturated for Schwarzschld, s L 2π ( 6π E ) 2N 2, (2N )Ω D 2 whch s satsfed by Kerr-AdS n D = 2N + dmensons f ( + g 2 r 2 + ) (r 2 + + a2 ) Ξ 2 (2N ) True f D = 5, but not for D 7. ( 2 j ) (r2 + + a2 )2N 2 Ξ j r+ 2N 4 Ξ 2N 2.
Conclusons Thorne s orgnal Hoop Conjecture n four dmensons can be gven a precse meanng n terms of the Brkhoff nvarant β = mn f max c vol(f (c)). The bound β 4πE holds for all know four-dmensonal black holes. In hgher dmensons, there are varous ways to generalse a hoop. One possblty s a hgher-dmensonal Brkhoff type nvarant; a closed mnmal (D 3) surface that fts over the wast of the (D 2)-dmensonal sphercal horzon. Snce the horzons n Kerr-AdS black holes are folated by Clfford T [(D )/2] tor ths works well n D = 4 and 5, but not hgher. Another possblty s to consder the shortest non-trval closed geodesc on the horzon as a hoop. Ths can be bounded above by the shortest path to and from an antpode, f there s an antpodal Z 2 symmetry. Even and odd dmensons are very dfferent. The geodesc hoop bound s satsfed for Kerr-AdS n all even dmensons. In odd dmensons greater than 5, the natural choce of antpodal path volates the bound. It s not clear f there exsts a more optmal choce of path for whch t works. We have checked some other geometres, such as black holes n magnetc backgrounds (à la Melvn). Black rngs?