Supergeometry of Three Dimensional Black Holes. Alan R. Steif.

Transcription

1 UCD-PHY-95- Suergeometry of Three Dimensional Black Holes PostScrit rocessed by the SLAC/DESY Libraries on 4 Ar 995. Alan R. Steif Deartment of Physics University of California Davis, CA 9566 steif@dirac.ucdavis.edu ABSTRACT: The suergeometry of static 2 + dimensional black hole solutions is discussed. The geometry of the black hole solutions corresonds to the grou manifold SL(2;R) with certain discrete identications. After imbedding SL(2;R) in the suergrou OS(j 2; R), the generators of the action of the suergrou which commute with these identications are found. These yield the suersymmetries for the black hole as found in recent studies. It is also shown that in the limit of vanishing cosmological constant, the M = black hole vacuum becomes a null orbifold, a solution reviously discussed in the context of string theory. HEP-TH-9542

2 . Introduction Gravity in 2 + dimensions has no local dynamics. Classical solutions to the theory in the absence of matter are locally at [], or have constant curvature, if a cosmological constant is resent [2]. Nevertheless, non-trivial global eects are ossible and can yield interesting solutions such as black holes [3]. A useful way of describing vacuum solutions to 2 + gravity is in terms of a quotient construction. One begins with a simle symmetric sace S ~ and identies oints under the action of a discrete subgrou, I, of its isometry grou, G, to obtain a sacetime, S. The xed oints of the grou action corresond to singularities of S. The residual symmetry grou of the sacetime is the subgrou H G that commutes with I. In this aer, we study the suergeometry of the static black hole solutions. In Section 2, we review the 2 + dimensional black hole solutions focusing attention on their construction as quotients from the grou manifold SL(2;R). We also discuss how in the limit of vanishing cosmological constant, the M = black hole becomes the null orbifold of string theory. In Section 3, the solutions are imbedded in the suergrou OS(j 2; R). The generators of the action of the isometry suergrou which commute with the black hole identications are found. The even generators yield Killing vectors. The odd generators can be ut into corresondence with two comonent sinors. We obtain the same number of Killing vectors and sinors as found in studies of their suersymmetric roerties [4][5] in the context of 2 + dimensional anti-desitter suergravity [6]. Factors of G are included. To comare with [3][4][5], set G == Dimensional Black Hole Solutions 2 + dimensional black holes [3] are solutions to Einstein's equations with a negative cosmological constant,, G +g =; <: (2:) The metric for the static black hole solutions is given by ds 2 = ( r2 l 2 8GM)dt 2 +( r2 l 2 8GM) dr 2 + r 2 d 2 ; <<2 (2:2) where l ( ) =2 : For various ranges of M, (2.2) describes the following solutions: () M>corresonds to a black hole with event horizon located at r = l 8GM and singularity atr=.

3 (2) M = is the black hole vacuum. (3) 8G <M< are solutions with a naked conical singularity atr=. (4) M = is anti-desitter sace. 8G Every solution to (2.) in 2 + dimensions corresonds to three dimensional antidesitter sace locally. However, since one is still free to make discrete identications, the solution can dier globally. Three dimensional anti-desitter sace is most easily described in terms of the three dimensional hyersurface imbedded in the four dimensional at sace with metric T 2 + X 2 W 2 + Y 2 = l 2 (2:3) ds 2 = dt 2 + dx 2 dw 2 + dy 2 : (2:4) The toology of (2.3) is R 2 S with S corresonding to the timelike circles T 2 + W 2 = const: Anti-deSitter sace is the covering sace obtained by unwinding the circle. The isometry grou of three dimensional anti-desitter sace is the subgrou of the isometry grou of the at sace (2.4) which leaves (2.3) invariant. This is SO(2; 2) with rotations in the T W lane which dier by 2n not identied. The hyersurface (2.3) describing three dimensional anti-desitter sace is the grou manifold of SL(2;R) as can be seen from the imbedding g = T + X Y W l Y + W T X The metric (2.4) is the bi-invariant metric ; det g =(T 2 X 2 +W 2 Y 2 )=l 2 =: (2:5) ds 2 = l2 2 Tr(g dg) 2 : (2:6) In this reresentation, the SO(2; 2) isometries are induced by its two fold cover SL(2;R) L SL(2;R) R : g! AgB; A; B 2 SL(2;R): (2:7) It is a two-fold cover because (A; B) and ( A; B) induce the same element ofso(2; 2). We can choose L 3 = as generators of SL(2;R). ; L + = ; L = (2:8) We now discuss how one can obtain the solutions (2.2) by identifying three dimensional anti-desitter sace under a discrete subgrou of SL(2;R) L SL(2;R) R.* * The black hole solutions as quotients of three dimensional anti-desitter sace by a subgrou of SO(2; 2) were discussed in [3]. 2

4 2.. M>: Black Hole Solutions Consider the coordinates (~ t; ~r; ~ ) [7] dened by the imbedding T =~rcosh ~ ; X =~rsinh ~ ; W = ~r 2 l 2 sinh ~ t=l; Y = ~r 2 l 2 cosh ~ t=l; ~t; ~ 2 ( ; ) valid for l<~r<and T =~rcosh ; ~ X =~rsinh ; ~ W = l 2 ~r 2 cosh t=l; ~ Y = l 2 ~r 2 sinh t=l; ~ ~t; ~ 2 ( ; ) (2:9) (2:) valid for < ~r <l. Note that ~ and ~ t have innite range because they arameterize boosts in the T X sace and W Y sace. The coordinates break down at ~r = l. In terms of these coordinates (2.9) and (2.), the metric for three dimensional anti-desitter sace (2.4) becomes ds 2 = ( ~r2 l 2 )d~ t 2 +( ~r2 l 2 ) d~r 2 +~r 2 d ~ 2 ; ~ t; ~ 2 ( ; ): (2:) The black hole is now obtained by identifying ~ with eriod 2 8GM where M is the mass of the black hole. Rescaling the coordinates ~r = r= 8GM ; ~ t = 8GM t; ~ = 8GM ; (2:2) one obtains the black hole metric (2.2) where has the canonical eriod 2. Translations in ~ by are seen from (2.9) to corresond to boosts in the T and from (2.5) can be exressed in grou theoretic form (2.7) as g! ex (=2) ex ( =2) g ex (=2) ex ( =2) X lane : (2:3) Thus, the black hole is obtained by identifying oints in three dimensional anti-desitter sace under the action of the discrete subgrou, I by I = f(a n ;A n );nintegerg SL(2;R) L SL(2;R) R A= ex ( 8GM L 3 )= ex 8GM ex 8GM ; M> (2:4) g A n ga n ; n integer (2:5) where L 3 is given in (2.8). The set of xed oints of g under the identication (2.5) is ft = X =g, and from (2.) and (2.2) is seen to corresond to the singularity r =. 3

5 2.2. M =Black Hole The M = black hole solution is given by ds 2 = r2 l 2 dt2 + l2 r 2 dr2 + r 2 d 2 ; <<2: (2:6) We rst obtain its! (l! ) limit which should describe a locally at metric. Dene the new coordinate v =2t+2l 2 =r; (2:7) which arameterizes outgoing null curves. The metric (2.6) then becomes ds 2 = r2 4l 2 dv2 dvdr + r 2 d 2 ; <<2: (2:8) Now, as l!, (2.8) has the smooth limit ds 2 = dvdr + r 2 d 2 ; <<2: (2:9) (2.9) is the metric for a null orbifold and has been considered reviously in the context of string theory [8]. It has zero curvature and, as we now recall, can be obtained by identifying three-dimensional Minkowski sace under the action of a discrete subgrou of the Lorentz grou. The metric for three-dimensional Minkowski sace in terms of null coordinates U = T X; V = T + X is ds 2 = dudv + dy 2 : (2:2) Consider the Lorentz transformation, N E, dened by U! U = U N E : V! V = V +2EY + E 2 U Y! Y = Y + EU: (2:2) N E is called a null boost or null rotation and can be obtained by conjugating a Euclidean rotation of angle by a boost of velocity v in the simultaneous limit that v! c and! with E = = v 2 =c 2 held xed. N E mas each U = const: null lane into itself and leaves the null line L = fu = Y =g (2:22) 4

6 invariant. The orbits of N E ;E 2 R form arabolas in the U = const: 6= lanes. The arabolas become steeer as U!, and in the U = lane, the orbits become lines arallel to the V -axis V (E) =V +2EY ; Y (E)=Y ; Y 6=: (2:23) The metric (2.9) describes three-dimensional Minkowski sace identied under the action of N E : To see this, dene coordinates consistent with the action of N E following sense. in the For a given U 6=, associate to each oint (U; V; Y ), the coordinates (u; v; ) U V Y This denes the coordinate transformation A = u v U = u A (2:24) V = v + u 2 (2:25) Y = u between (u; v; ) and (U; V; Y ). arameterizes orbits of the grou action. Now, identifying oints under the action of I = fn E ;E=2n; n integerg corresonds to making erodic in 2. Substituting (2.25) into (2.2), one obtains ds 2 = dudv + u 2 d 2 ; <2 which is indeed (2.9) with r relaced by u. The invariant null line L (2.22) corresonds to the singularity u =. The null orbifold shares some features of Misner sace [9]. The action of the grou of identications, I, is not roerly discontinuous and the resulting manifold is not Hausdorf. By excising either the fu = ; Y > g or fu = ; Y < g halflane, one obtains the maximal Hausdorf extension, but at the cost of geodesic incomleteness. Also like Misner sace, as a result of the identications, there exist closed null curves on the U = lane. It should be noted that it is also ossible to construct a distinct solution from the same grou of identications I by identifying only those oints which lie on the U =;Y> half-lane. This yields a solution describing a article moving at the seed of light []. Let us return to the case of the M = black hole (2.8). Like the null orbifold, the M = black hole can also be obtained by identifying oints under the action of a null 5

7 boost, but now in three dimensional anti-desitter sace rather than at sace. Consider coordinates in three dimensional anti-desitter sace dened by the following imbedding U T X = r V T + X = v W = vr 2l Y = r: l rv 2 4l 2 + r2 (2:26) Translations (! + ) corresond to null boosts (2.2) in (U; V; Y ) with W xed. r in (2.26) labels the U = const: null surfaces which N E leaves invariant. Identifying oints under the action of corresonds to making eriodic in 2. I = fn 2n ;nintegerg (2:27) Substituting (2.26) into (2.4), we obtain the M = black hole solution (2.8). Translations in v also reserve the metric (2.8) and corresond to null boosts in the (U; V; W ) sace with Y xed. The set of xed oints of I are L = fu = Y =;W = lg (2:28) and from (2.26) is seen to corresond to the singularity r =. (2.28) is a null geodesic in three dimensional anti-desitter sace. Hence, we nd that the M = black hole vacuum has a null singularity. From (2.5), the null boost (2.2) takes the form (2.7) g! E g : (2:29) E The M = black hole is thus obtained by identifying under the action of the discrete subgrou I = f(a n ;B n );nintegerg SL(2;R) L SL(2;R) R ; M = 2 (2:3) A= ex 2L + = ; B = ex 2L = 2 by g A n gb n ; n integer: (2:3) 6

8 2.3. M<Solutions with Naked Singularities For the M < solution, it is convenient to use static coordinates dened by the imbedding T = ~r 2 + l 2 cos t=l; ~ W = ~r2 + l 2 sin ~ t=l; X =~rcos ; ~ Y =~rsin ; ~ (2:32) in terms of which the metric (2.4) for three dimensional anti-desitter sace takes the form ds 2 = ( ~r2 l 2 +)d ~ t 2 +( ~r2 l 2 +) d~r 2 +~r 2 d ~ 2 : (2:33) ~t and ~ now arameterize rotations in the T W and X Y lanes. The solution is now obtained by identifying ~ eriodically with eriod 2 8GjMj: Rescaling the coordinates ~r = r= 8GjMj; ~ t = 8GjMj t; ~ = 8GjMj ; (2:34) one obtains (2.2) where has canonical eriod 2. From (2.5), a rotation of angle in the X Y lane takes the form (2.7) g! cos =2 sin =2 sin =2 cos =2 g cos =2 sin =2 sin =2 cos =2 : (2:35) Hence, the M < solution is obtained by identifying oints in three dimensional antidesitter sace under the action of the discrete subgrou I = f(a n ;A n ); nintegerg SL(2;R) L SL(2;R) R ; M< A= ex 8GjMj(L + L )= cos 8GjMj sin 8GjMj sin 8GjMj cos 8GjMj : (2:36) by g A n ga n : (2:37) The xed oints of the grou action are fx = Y =g, and from (2.32) is seen to corresond to the singularity r =. These solutions are the anti-desitter analog of the conical solution [] and were rst constructed in [2]. 7

9 3. Suergeometry In this section, we study the suergeometry of the black hole solutions. After imbedding the black hole sacetime in the suergrou OS(j 2; R), one nds the generators of the isometry grou of the suergrou which commute with the black hole identications. The even generators yield the usual Killing vectors. However, in addition, there are odd generators of the isometry grou of OS(j 2; R) which are consistent with the black hole identications. These can be ut into corresondence with two-comonent sinors. We nd the same number of these Killing sinors as were found in studies of their suersymmetric roerties [4][5]. Let us now review the construction of the suergrou OS(j 2; R). 3.. OS(j 2; R) Consider a Grassmann algebra, A, generated by one Grassmann element, A = fz = a + b; a; b 2 R; 2 =g: (3:) a and b are the even and odd arts of z: OS(j 2; R) is the set of linear transformations of ( ; 2 ;x) of the form OS(j 2; R) = a b c d A ; a;::: even; ;::: odd (3:2) which reserve dl 2 = ab a b + x 2 (3:3) and where ; 2 are Grassmannian satisfying = 2 2 =; f ; 2 g=; f; a g =: (3:4) The condition that M reserves the line element (3.3) imlies the relations ad bc = c a = (3:5) d b = : 8

10 Since these are three relations for 8 arameters, OS(j 2; R) is ve dimensional. OS(j 2; R) contains SL(2;R) as a subgrou SL(2;R)'S(2;R) ' g a b c d A ; ad bc = OS(j 2; R): (3:6) Consider the following basis for the Lie algebra os(j2; R). The even generators are those in the sl(2;r) subalgebra and are given by (2.8) L 3 A ; L+ and the odd generators are They satisfy the algebra Q + A ; Q = A ; L = [L 3 ;L + ]=L + ; [L 3 ;L ]= L ; [L + ;L ]=L A A : (3:8) [L 3 ;Q + ]=Q + ; [L + ;Q + ]=; [L ;Q + ]=Q [L 3 ;Q ]= Q ; [L + ;Q ]=Q + ; [L ;Q ]= (3:9) fq + ;Q + g= 2L + ; fq ;Q g=2l ; fq + ;Q g=l 3 : As we now show, the adjoint action of the SL(2;R) subgrou induces an SO(2; ) transformation on the sl(2;r) subalgebra and an SL(2;R) transformation on the odd generators Q. Consider the adjoint action by an element h a b c d A 2 SL(2;R) (3:) on the Lie algebra os(; 2jR). On the sl(2; R) subalgebra, the adjoint action ad h : L! h Lh (3:) induces the transformation on the basis (2.8) L 3! (ad + bc)l 3 +2bdL + L +! cdl 3 + d 2 L + c 2 L L! abl 3 b 2 L + + a 2 L : 9 2acL (3:2)

11 From (2.6) we nd that the inner roduct for the generators is For the basis (2.8), the non-zero inner roducts are <A;B>= l2 Tr(AB): (3:3) 2 <L 3 ;L 3 >=l 2 ; <L + ;L >= l2 2 : (3:4) (3.) ((3.2)) is an SO(2; ) transformation reserving inner roduct (3.3) with h and h inducing the same element of SO(2; ). Under the adjoint action (3.), the odd generators (3.8) transform as Q +! ad h Q + = h Q + h = dq + cq ; Q! ad h Q = h Q h = bq + + aq which can be written as Q+ Q! (h ) t Q+ Q (3:5) : (3:6) If we associate odd generators with real two-comonent sinors Q +! Q! ; (3:7) then under the adjoint action, the sinors transform in the fundamental or sinor reresentation (3.6) of SL(2;R). Avector on the SL(2;R) submanifold of OS(j 2; R) at the oint g (3.6) can be decomosed into a vector w tangent tosl(2;r) and a transverse odd vector eld v = w + ; A (3:8) with ;::: odd and satisfying (3.5). We associate with each odd vector eld (3.8), the sinor eld = ab ; =a; = b: (3:9) A right invariant basis of vector elds on SL(2;R) can be obtained by left multilication of (3.6) by the generators (3.7) and (3.8). The three vector elds obtained from (3.7) are a right invariant basis of vector elds tangent to SL(2;R) while the two odd vectors obtained from (3.8) yields the right invariant basis of odd vector elds given by with corresonding sinors using (3.8) and (3.9). + = Q + g A = Q g A (3:2) c d a b + = = (3:2)

12 3.2. Killing Sinors in Anti-deSitter Sace The isometry grou of OS(j 2; R) isos(j 2; R) L OS(j 2; R) R, while the isometry subgrou which reserves the SL(2;R) submanifold is SL(2;R) L SL(2;R) R. Killing vectors on SL(2;R) generate the orbits of SL(2;R) L SL(2;R) R while the Killing sinors are those odd vector elds which generate the orbits of the odd elements of OS(j 2; R) L OS(j 2; R) R. Anti-deSitter sace therefore has six Killing vectors and four Killing sinor elds. The Killing sinors are generated by the two odd generators, Q, of the left and right OS(j 2; R) factors. The Killing sinors associated with OS(j 2; R) L are L + = Q + g = + ; L = Q g = (3:22) using (3.2) and those associated with OS(j 2; R) R are R + = gq + a ca = a + + c ; R = gq b da = b + + d : (3:23) In terms of two-comonent sinors (3.2), the four Killing sinors (3.22) and (3.23) are thus given by L + = a c R + = ; L = ; R = b d : (3:24) Note that as exected (3.24) reduces to Q at the identity fa = d =;b=c=g. Using the corresondence (2.5) with three dimensional anti-desitter sace the Killing sinors (3.24) take the form L = + T R + = a = T + X; b = Y W c = Y + W; d = T X; ; L = X Y + W ; R = Y W T + X : (3:25) (3:26) One can exress these in any coordinate system fq i g by substituting the corresonding imbedding T = T (q i );W=W(q i ); ::: in (3.26). Recall that these are the sinor comonents with resect to the right invariant basis (3.2). To exress them in a dierent basis such as the one used in [4][5], one erforms the aroriate local Lorentz transformation.

13 3.3. Killing Sinors and Vectors in the Black Hole Backgrounds We now obtain the Killing sinors in the background of the black hole solution. The isometry grou, H, of the identied sace corresonding to the black hole is the subgrou of the full isometry grou that commutes with the identication grou I H OS(j 2; R) L OS(j 2; R) R (3:27) [H; I] =: (3:28) The even elements of the Lie algebra of H yield the Killing vectors while the odd elements yield the Killing sinors. Let us now consider each of the three ranges of M discussed in Section 2. Using (3.9), we nd that for the M> black hole, there are two generators commuting with I (2.4) imlying there are L 3 2 os(j 2; R) L ; L 3 2 os(j 2; R) R (M>) (3:29) 2 Killing vectors and Killing sinors (M >): (3:3) There are no Killing sinors because no linear combination of Q commutes with L 3. For the M = black hole vacuum, using (3.9) we nd that there are four generators commuting with I (2.3) L + 2 os(j 2; R) L ; L 2 os(j 2; R) R (M =) Q + 2os(j 2; R) L ; Q 2 os(j 2; R) R (3:3) imlying 2 Killing vectors and 2 Killing sinors (M =): (3:32) From (3.24), the two Killing sinors in two comonent form are L + = ; R = b d : (3:33) From (3.25) and in the coordinates (2.9) with (2.2), the Killing sinors (3.33) become L = + ; R l r2 =l = 2 8GM ex ((8GM) =2 t=l) (8GM) =2 r ex ((8GM) =2 : (3:34) ) 2

14 Note again, these are the Killing sinors in the right invariant basis (3.2). To exress them in a dierent basis such as the one used in [4][5], one erforms the aroriate local Lorentz transformation. From (3.9), we nd that for the M<solutions, there are two generators commuting with I (2.36) L + L 2 os(j 2; R) L ; L + L 2 os(j 2; R) R ( =8G <M <) (3:35) imlying 2 Killing vectors and Killing sinors (M <): (3:36) There are no Killing sinors because no linear combination of Q commutes with L + L. For all the black hole solutions, the two Killing vectors corresond to linear combinations : By a similar analysis, it should be ossible to obtain the Killing the rotating black hole solutions []. We can also recover the Killing vectors and sinors for the self-dual backgrounds considered in [2]. The grou of identications I for a causally well-behaved self-dual solution is a subgrou of one of the SL(2;R) factors, say SL(2;R) L, generated by a sacelike generator. Since the left and right factors commute, there are two Killing sinors and three Killing vectors coming from OS(; 2jR) R. From SL(2;R) L, there are zero Killing sinors and one Killing vector. Hence, for the self-dual solution there are in total four Killing vectors and two Killing sinors. Acknowledgements Iwould like to thank Gary Gibbons and Paul Townsend for helful discussions and Steve Carli and Yoav Peleg for useful comments on the aer. Iwould also like toacknowledge the nancial suort of NSF grant NSF-PHY at Davis and the SERC at Cambridge. 3

15 References [] S. Deser, R. Jackiw, and G. `t Hooft, Ann. Phys. 52 (984) 22. [2] S. Deser and R. Jackiw, Ann. Phys. 53 (984) 45. [3] M. Banados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett. 69 (992) 849; M. Banados, M. Henneaux, C. Teitelboim, and J. Zanelli, Phys. Rev. D 48 (993) 56. [4] O. Coussaert and M. Henneaux, Phys. Rev. Lett. 72 (994) 83. [5] J.M. Izquierdo and P.K. Townsend, DAMTP rerint R/94/44, gr-qc/958. [6] A. Achucarro and P. Townsend, Phys. Lett. B 8 (986) 89. [7] G.T. Horowitz and D.L. Welch, Phys. Rev. Lett. 7 (993) 328. [8] G. T. Horowitz and A. Steif, Phys. Lett. B, 258 (99) 9. [9] S. W. Hawking and G.F.R. Ellis, The Large Scale Structure ofsace-time, (Cambridge University Press, Cambridge, 973). [] S. Deser and A. Steif, Class. Quantum Grav. L53 (992) 9. [] A. Steif, in rearation [2] O. Coussaert and M. Henneaux, Universite Libre de Bruxelles rerint ULB-TH 4/94, he-th/