Commun. Math. Phys. Sitter Spaces t Center for Theoretical Physics, Th - PDF document

Commun. Math. Phys. Sitter Spaces t Center for Theoretical Physics, The University Austin, Austin, TX A 0 0 whose group of the action transformation would would These generators should obey the 0(3,2) algebra. (iii) The boundary conditions should include the asymptotically anti-de Sitter solutions the Kerr-anti-de Kerr-anti-de since otherwise, they would be too restrictive. (iv) Lastly, the boundary conditions should preferably be expressed directly in terms of the spacetime metric components, which are the mimic the reconsider asymptotically asymptotically which were expressed in terms of the Ricci tensor components. The canonical on the given in have been been Based on arguments similar to those of the A = 0 Anti-de Sitter considering the supersymmetry transformation transformation These arguments remain formal in the A 0 should obey obey under O (3, 2)] O (3, over the the O(r 2)] and a detailed analysis is necessary. That analysis is provided here for N = 1 canonical generators are shown to be close in Dirac bracket proof of the same gravity with different radii section, in mechanism that coupling to of an an The hamiltonian formalism is shown to be well defined even when one considers different values of A simultaneously and the appropriate surface integrals are given. We find that when gravity coupled the same the geometric properties have been been 16]. II. Asymptotically Anti-de Sitter Spacetimes As we mentioned in the introduction, the boundary conditions at spatial infinity should be devised so as as a metric which it is reasonable to O (3, 2); 2) finite. Henneaux and and ()(). (II.1) Here, R is curvature, related to R = of the well as given in One way O, O) ) + 2) , along the the (A.8a-c)]. [It is assumed throughout that O,O(r")= O(r"-1) when needed.] The above by the vector fields ~ - 2 ~ ~ 0 along the b = are ten (II.3) is the transformations ~ = 0, the anti-de of the where the also angle-dependent translations translations A similar difference between A =0 and A :t=0 :t=0 and possesses ten conformal Killing vectors. These vectors correspond to the ten A = 0, 0, When A = 0, conditions at infinity in supertranslations. These can be the metric metric or asymptotic conditions on the magnetic part of the Weyl tensor [18]. (In the former case, not all supertranslations are eliminated, but because the remaining ones need to the natural are accordingly 2) finite. finite. should be pointed out here that one can in fact relax (II.2d, f,g) and assume that the radial-angular components of the perturbation tend to zero at infinity as r-2. This would not not 6, 4]. In order to 2 , r = I - is 1 - + 1 = 1 - to and the induced 2 + 2 + + 6, 4], it remains to study the behavior of the Riemann tensor close to I. This is most easily carried out in the non-holonomic flame o 2 = = r sin h a just one 1 (e.g., R°a ...) generically terms of only that zero at the same obey the the (if one assumes in addition that I is turn that conditions of of 6] on the surface at infinity. This surface appears here surface, since to I, on the the )We close this section by reformulating the canonical consider here t = obtained from Anti-de Sitter n4O(x) = = Appendix E for a discussion of the preservation of the boundary conditions by the time evolution and for the need to strengthen somewhat (II.2e-j) and (II.9).] HI. Surface Integrals for the Anti-de Sitter Generators The hypersurface deformations defined by by n[~] = N3x~Yf, + ½~2ob. (II1.1) Here, the ~tfu (# = ±, 1, 2, 3) are the usual constraint generators of general relativity relativity ~4~¢3_ : g = - canonical formalism the canonical canonical In other words, its variation should be given only by a volume integral, 3H[ {] = ~ ~ ()()+ Bij(x)c~7"ciJ(x)3 , (III.3) where the variations 3gij(x), 6niJ(x) 6niJ(x) and can be found in [22].) Now, if one computes the variation of the volume integral .[ d3x¢"o~ of (Ill.l), (Ill.l), 6 ~ u = ~ ~ dZ kl + ~lT~Jk)rgjk} , l ~l/2[r~ikr~jl /~il/,jk 2,~ij~,kl~ term can can ~ k i = 2 °Ukl ± -2g° o deal with identity at value on when the constraints surface term, term, by a different approach based on pseudo-tensors and superpotentials, if the latter are reexpressed in terms of canonical variables. The use in (III.7) of the covariant derivative with respect to the the anti-de up to arbitrary constants h u for the first case, one has u = Anti-de Sitter and formula _ sin20)-5/2 ro = charges are given by by 16rcmR -14 '2: --2Li[~-~-~f j c~2" (III.10b) They both vanish when m = �R-- oo), becomes the Henneaux and O (2, (2, H[r/]] = H[[~, r/]]. (IV.l) The bracket bracket q] is given here by the "algebra of surface-deformations" [24], [ ~, rl] = - i - - rl]' = -- 9iJ(tl± ¢ ±, j-- #±~±,j) + ({Jrti,a -- rlJ{ ', y). (IV.2b) (That [H[~], H[t/]] has also well defined functional derivatives once H[{] and H[t/] are improved can be checked by straightforward, but involved calculations.) From (IV.2), one can can q] in in t/] also takes the form (II.3) and that, furthermore, the asymptotic part [~, t/]~ is given in in ~ ,~,lab__ i'~ab }cd.ef (IV.3) ~ tt_lco -- v., cdef~mSlco . This implies that the O (3, 2) 2) - which reduce to the surface term Jab when the constraints hold - indeed obey the 0(3, 2) algebra (in the Poisson bracket). It is sometimes useful in applications to fix the gauge. In the case at hand this may be done by imposing conditions on 9ij and Tc ij whose preservation under surface deformations destroys the possibility of an arbitrary ~u inside. Thus, once the gauge time they they 23] associated with the chosen gauge condition. This is so because neither the given above, J,b close 1 Supergravity conceptual points the theory. N = 1 A 0 a a 27]. The boundary conditions for ho~ which accompany (II.2) turn out to be 3/211 - - - 7(1)]Zo(t, O, ~) + O(r-- = r - 3/211 - 7(1)]Z¢(t, indices are ~ = ~ + ±±~(~.) ~p~, ± complete determination that they U = (1 + (V.4) = Henneaux and f, g) O, ~ = = UT~Pr + F?rtY~ + conditions invariant which are which preserve for the that the to the once the one can only extra term = 1 are the f,~l/2urr~,i ~j-](~ _~ g g 7j]s(a )(b)tpje(a)mbe(b)m + g Anti-de Sitter Sitter (V.12) an expression formally equal to A = 0 0 [~)* is equal to transposition. We close in in QA] = QB(N,b),A, QB(N,b),A, Q~] = ~ to resort S = S g = = It is given by coupling the gravitational field to a completely antisymmetric gauge field Auto according to S:Sd4x(-g) 1/2 (4)R+ 1 F,,a~F,~o~, (VI.2) z.'~: I where the F is = OtuA,~,l" cosmological constant. constant. case A � 0 it is the relationship H = ~ the sum A = 0) f~ld _ - " The function the action for the for the into the cosmological constant, of integration one per Einstein equations), that constant and without and without A is is a AR2= -3. key point - well integral determines is a hold, the want to so as and not ~ = 2 = u , R = (VI.9a) is R = (unspecified) R term at new variations variations h~j=0, rdJ=0, rc °~ = (2 Sitter spaces [the variation H will to be easy to (II.3) - the usual term plus made away 2 o C a when the the factor also dimensionless has the of an when the ~ 2 By taking equal to 1 R 3 (VI. 11), + ~ij?ij) is to minus the configurations possess A is as we algebra to Aij k Indeed, the 1 3 /-" k = - The transformation 4 ± = 34 ± the algebra of of property makes (VI.13) unique] and they coincide with ordinary deformations for the (VI.13) a O (3, 2) must be be = ~(1)ij -I- ~(2)ij. (VI. 16a) 0 3 j . Sitter case only to 1 _ the term term ¢-L = 3R-2~ -L + O(Q-2)]. The need for ~(2)ij, which yields finite surface integrals and only affects Aij k to order 1/0, is more subtle. It results from the fact that the total surface term arising from the variation of the volume piece of the hamiltonian, which is finite once (VI.t6b) is included, is not an "exact differential", i.e., the variation of a surface integral, if ~(2) u = ((2) j vanishes as as g'ijk~°° tl t t.,/ Uabllhmn ) . Henneaux and and is included, the surface term in the variation of S dax(~Ud/gu + ~ijy + ~ 2 2 ijkl ± ± + + needed to bring the to the form by the preservation ~ is It contains R z = R 3 x ,j the charge of an an is quite analogous to a surface deformation (II.3) with a non-zero ~. On the other hand, if for a given ~-~ the flux (VI.20) is zero (as it happens when ~:i leaves the potential A~k invariant everywhere is a can have k ~ ~, ~, )by an the latter permissible in the complete y-abj ± ~ are exactly the when the cosmological By the above, they can be shown to the right-hand O (3, 2) to have for the appropriate combination deformations and "physical space" space" and has been studied by various authors (see, e.g., [17, 30, 31] and references therein). The above analysis can be extended to N = 1 transformation. The has thus to be when the cosmological one can varied enables one to These (supersymmetric) same system the same one can the energy according to there was A = 0 A + also have zero also arrived at Killing Vectors, Killing 2 : - 2 + [1 b = 1 ( ~ = - 0 + ) / 2 1'2 ) / s = + ~ 1 + 02) 0 - to the t = Sitter Spaces 1 is R-~U2~, R-~U3~, U31, U41). O (3, (3, Uj = C~I.b~eU~s (A.6a) with ef & ef fe ef fe +, +, are two ] " 4 ° = 4 r = O A = ~ U A play an an (&Pu = Vue-(2R)-lyue [27] vanishes at infinity). IV, is here the spinor covariant derivative without torsion. The Dirac y-matrices are taken to be real, and so are the spinor fields ("Majorana spinors"), yo is M. Henneaux general solution solution (()(()()o .o .¢ • (cos~ +sm~y(,y(2))(cos} +sln~,(2)7(3)) "c sin-~ y(o) ~3, • (cos~- 2 ) spinor whose whose (II.8)]. Different choices of that constant spinor define the different UA. For definiteness we take here (~A) B = 6aBRI/2. The solution (A.10) is double-valued (e.g., it changes sign under ¢-~¢ + 2~). the spinor 1 + + (1) is the radial leg of the tetrad] acting on some o-independent spinor and hence, they belong to the eigenspace of 7m with - I the one with where the = - M = - a � the Killing Killing vector U A = U = - a � F a = b , , 34] and references therein.) The set of Killing vectors-Killing spinors of anti-de Sitter space is thus closed under the operations (A.6), (A.11), as defining Anti-de Sitter these objects Appendix B. The Kerr-Anti-de Sitter Sitter The metric is explicitly given by + a2 2 = - + a 2 2 0 + R z 2 ~ ~ dr2 +(?2 +a2 cos O)](f2+a2)(~/R)-~f22rnf+a2[_ ~- 1 _ • 2 2 +(~2+a2)( 1 ~_)ld~ .a2 2 (B.1) +sin UL~2 + a2 c0s2 0 The parameter a, which will be related to the angular momentum per unit mass, obeys [a] R. When a + coordinate transformation to bring coordinate transformation implicitly defined 0 = f cos 0, Sitter form. to be m 4: the anti-de a 2 a 2 c~-s20, + a 2 2 03 03 ()] [(f2+a2)(~/R)2+f:_2mP+a2 ] , 4 0 the new the leading = ~ 2 0) - = 2am ho4 - = ~ 2 a 2 ~ = t = o + Nij~ + z 0 K~O - = r 2 = 0 r 2 position to variables are Asymptotically Anti-de U l contain an 2 o the 2~d 2~d = 0@ -4) and ~0r= 0], SO that J41 vanishes. Similarly, J s 1 = R 0 - (1 = - (1_~2) :. k~r = 5 - £~0~ = (C.2) is by terms 0 1 r - 2 2 the upper left index (4) emphasizes that one is dealing with spacetime N = 1 a is obtained by ~ = 0 when the the [We would like to point out here that when A =0 the asymptotic super- algebra is infinite dimensional due just the Poincar6 algebra Anti-de Sitter We have have We prove prove imply ours. We begin the analysis by assuming, as in [4], that one can conformally rescale the "physical" metric ds 2, dgZ = ~"~2 ds 2 (D. I) located where the the new new as will be apparent in the sequel, one only needs these as in in - to prove the equivalence]. Lastly, in order to simplify the mathematical discussion as much as possible, we also take the "unphysical" metric to be the problem + 2 (see e.g. e.g. )Here and everywhere in this appendix, the semi-column denotes the covariant derivative in the "unphysical" metric 0~6, whereas A2f2 and All2 stand for respectively. Besides, we refer the components of tensors to frames which are smooth in the vicinity of I [unlike the orthonormal frame (II.8) of Sect. II]. The metric ~a and the function f2 are by hypothesis regular close to and on the surface at infinity. Accordingly, the Ricci tensor /~a and = - = - J one on f2--0. This the identity one can prove M. Henneaux C. Teitelboim which vanishes the new the new = - z + 2 + 2 . . [The ten 0(3, 2) conformal Killing vectors of (D.8) are simply obtained from the vectors by subtracting the prime the new = 0 as follows. y" (a u = 0, ~ u = 2 = 2 + Oab(U, b , - ( U , = . = - for the for the identity for G~, the It actually the second mentioned above; consider the (D.14), (D.15), of the show that = - = - = - 1 u - - 2 = 2 ds~a s + p . metric written dictated by of the ours, it u 2 which can arbitrarily at u 4 (D.22), (D.23) are the (D.12) is R x S 2 by the note that that -~ ~ ~apabd2z~b, the "cylinder" x S one can (D.24). [-The equal to with those those It is important to realize that the conservation of the charges charges does not completely follow from the asymptotic behavior (II.2) of the spacetime metric. Indeed, as we we will be conserved only if the "constraint "constraint associated with asymptotic 0(3, 2) Killing vectors preserve the boundary conditions on the canonical variables, and hence, are well defined as canonical generators, only if additional requirements are met met the sense of (II.2e-j), (II.9)] on some spacelike slice, but which do not remain asymptotically anti-de Sitter in the course of evolution. It should of extra example, in in b] and required to vanish at the ~b(a)= ~b(b)= 0 Asymptotic Conditions have shown metric obeying functions such means of N l = go0) - 1/2, N i equal to we restrict a t = = (B. 1) with a = N k Sitter space, N ±, N k N k N k for the N ± = = N~as + N o = = N~.as + Straightforward examination 0 0 demanding that that H[~]] and [9r4, H[~]], respectively) be of of O(r-4)]. The other O~b equations give no constraint on N "t, N k. N ± ± N ~ [r~ "b, "b, = O(n "b) are not two two and not six because two of them, the ~°=O(r-2) and zV°=O(r -2) equations, are automatically satisfied.] Moreover, N r that they requirements at are met will N ± exist. for one to new Asymptotically Anti-de Sitter Spaces can be to be canonical variables the surface shown without to the the mapped allowed configurations on allowed configurations - so the analysis and conclusions in the paper remain valid. Lastly we remark that the considerations of this section may be carried X L L incidentally, guarantees that the volume integral ~ (Nl~£~ + Nk:Kk)d3x 2 + 1 1 In that simpler theory one finds the concise us (C.T.) would like to thank informative discussions the subject this article which place in Princeton in also grateful Weinberg for useful discussions correspondence. This No. PHY-8216715 research funds Texas Center Theoretical Physics, de Estudios Cientificos After completion this work, we were informed asymptotically anti-de Sitter been studied in [32]. Regge, T., Teitelboim, C.: Role surface integrals in relativity. Ann. Phys. (N.Y.) 88, 286 Hawking, S.W., Ellis, G.F.R.: large scale structure spacetime. Cambridge, Cambridge University Press Hamilton-Jacobi and Schr6dinger separable solutions Einstein's equations. Commun. Math. Demianski, M.: Acta Astron. Hawking, S.W.: conditions for gauged supergravity. Phys. Lett. L.F., Deser, gravity with a cosmological constant. Nucl. Phys. B Breitenlohner, P., Freedman, D.Z.: Stability in gauged extended supergravity. Ann. Phys. G.W., Hull, C.M., Warner, N.P.: gauged supergravity. Nucl. Phys. Teitelboim, C.: Surface integrals symmetry generators in supergravity theory. Phys. Lett. Teitelboim, C.: Supergravity positive energy. Phys. Rev. Lett. Witten, E.: A proof of the positive energy theorem. Strominger, A.: Rev. D 27, 2793 (1983) Rev. D 27, 2805 (1983) Hull, C.M.: Rev. D 29, 8 supergraxdty. B 176, Phys. Lett. Phys. Lett. B, 415 (1984) B 122, Storey, D.: Misner, C.W.: How commutators the spacetime 542 (1973). See Teitelhoim, C.: In: General Vol. I. A. ed. University, New York, supergravity. Phys. 38, 1433 supergravity. Phys. 2802 (1977) Nucl. Phys. B 79, 276 (1974) Henneaux, M.: Henneaux, M.: 78B, 80 Henneaux, M.: 25, 1984