Hamiltonian treatment of asymptotically anti-de Sitter spaces

Volume 142B, number 5,6 PHYSICS LETTERS 2 August 1984

HAMILTONIAN TREATMENT OF ASYMPTOTICALLY ANTI-DE SITTER SPACES

Marc HENNEAUX 1 and Claudio TEITELBOIM Center for Theoretical Physics, The University o f Texas at Austin, Austin, TX 78712, USA

Received 19 March 1984

The hamiltonian formalism for general relativity in an asymptotically anti-de Sitter space is developed. The generators of the asymptotic 0(3, 2) symmetry group are expressed as surface integrals in terms of the canonical variables. A set of O(3, 2) invariant boundary conditions on the (off-shell) canonical variables which makes the surface integrals finite is ex- hibited. As an immediate consequence the surface integrals are shown to obey the 0(3, 2) algebra. The extension to supergravity is also discussed.

The need to reconcile supergravity, which apparent- ly needs a large cosmological constant with the ex- tremely low experimental upper bound on A in our universe makes it necessary to study in depth the dy- namics of the gravitational field when A :~ 0.

It is one virtue of supergravity that it excludes (at least naively) positive values for A since the 0(4, 1 ) group cannot be graded. Therefore we will discuss only the A < 0 case. To begin with we will consider ordina- ry general relativity and will subsequently comment on the extension to supergravity.

The solution of the matter free Einstein equations with A < 0 possessing the maximum number of sym-

metries is the anti-de Sitter spacetime, which takes the role played by Minkowski space when A = 0. There- fore if one imagines path integrating over gravitational fields when A < 0 one must admit in the path integral, or for that matter in the classical action principle, all off-shell configurations which approach the anti- de Sitter geometry at spacelike infinity. It becomes then necessary to specify precisely what is meant by "approaching asymptotically the anti-de Sitter geome-

try". The choice of boundary conditions is not a techni-

cality but it is an issue inextricably related with the crucial feature in the problem, the symmetries at spacelike infinity * 1

I On leave from Facult~ des Sciences, Universit~ Libre de Bruxelles, Brussels, Belgium.

0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Indeed, from a canonical point of view a symmetry exists in a given theory only if one can define its generators. In a gauge theory the generators of the associated "global" symmetries are given by integrals over a two-dimensional surface at infinity. One must then select boundary conditions which make these surface integrals finite. Furthermore the boundary conditions must be invariant under the asymptotic symmetry since otherwise a symmetry transformation would mapp an allowed configuration onto a non-allowed one In this sense the boundary conditions are dictated by

the symmetry itself.

In practice the way in which one arrives at the boundary conditions is to some extent a procedure of trial and error by which one puts together consistently the following mutually related elements: (a) The space of fields must contain those solutions which are ex- pected to give the typical behavior at infinity of a generic configuration. In the asymptotically flat case that prototype is the Kerr-metric. In the present case

,1 The question of boundary conditions in asymptotically anti-de Sitter space has been previously investigated in refs. [1-3]. As a complement we analyze here the problem from the hamiltonian point of view following the methods of refs. [4-6]. The relation between both approaches is briefly discussed in the main text below. We hope to in- clude a detailed study of this issue in a forthcoming paper which will present a fuller account of the results reported in this note.

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it is the Kerr-anti-de Sitter solution. (b) The surface integrals associated with the asymptotic symmetry should be finite. These integrals are determined by the demand that the hamiltonian should have well defined functional derivatives. (c) The boundary conditions should be invariant under the asymptotic symmetry.

By going through this procedure one naturally arrives at the following boundary conditions for the gravitational canonical variables [which are the metric gij of a generic spacelike hypersurface and its conjugate momentum 7rq, related to the extrinsic curvature K i / b y *r i] = - g l / 2 ( K i ] - Kg//)l,

grr = grr + r - 5 f r r ( O, ~) + O ( r - 6 ) ,

gro = r -4 f ro (0, ~) + O ( r - S ) ,

gr¢~ = r-4fre~( 0 , (~) + O( r - s ) ,

goo = g, oo + r - l f o o ( O, ¢P) + O(r-2) ,

goe~ = r - l f o ¢ ( O, dp) + O ( r - 2 ) ,

g¢4) = g,e~¢; + r - l f¢e~(O, (~) + O ( r - 2 ) ,

~rrr = r - l p r r ( o ' (p) + O ( r - 2 ) ,

nro = r - 2 p r O ( o , (~) + O ( r - 3 ) ,

lrr~ = r-2pOe~(O, dp) + O ( r - 3 ) ,

Iroo = r-SpOO (O ' ~) + O ( r - 6 ) ,

noe~ = r -SpOt (O, (p) + O ( r - 6 ) ,

Irea e~ = r - S pea e~ ( O ' (~ ) + O ( r - 6 ) .

(la)

( lb )

( lc )

(ld)

( l e )

( l f )

(2a)

(2b)

(2c)

(2d)

(2e)

(2f)

Here gi] denotes the spatial components of the anti- de Sitter metric in standard coordinates, whose only non-vanishing components are,

grr = [1 + ( r / R ) 2] -1 , (3a)

require examination of the behavior of the "pure spin two" content of the field (1), (2) on the boundary of the hemisphere of the Einstein universe which corre- sponds to the anti-de Sitter space upon conformal re- scaling. Alternatively one might fall back on the analysis of ref. [3] and examine the asymptotic behavior of the curvature tensor derived from ( t ) , (2). We plan to discuss these issues in a forthcoming paper. Here we just note that the fields (1), (2) obey the symmetry criteria numbered (ii), (iv) in ref. [3].

Once conditions (1), (2) are adopted one can con- struct unambiguously the gravitational field hamilton- Jan. It takes the form,

(4)

where the ~gu (/J = ±' 1 ,2, 3) are the usual constraint generators of general relativity, the ~u are the normal and tangential components of the deformation that connects two neighboring three-surfaces and the Jab = --Jba (a, b, = 1 .... ,5 ) are surface integrals over a two. dimensional surface at infinity.

In order to preserve the boundary conditions (1), (2) the deformation ~u(x), while being arbitrary "inside" (gauge invariance) should become asymptotically a linear combination of the anti-de Sitter Killing vectors UZab,

~u !~abrru ' 0 (5) - - 2 g "~ab

r.--~ oo

and one may also verify that this is the most general possibility .2 . The precise fall off of (5) also follows from (1), (2). It turns out that its I and r components must vanish as r -4 and r -2 respectively whereas the 0 and ¢ ones should go as r -5 .

As stated above, from a canonical viewpoint what determines the surface integrals is the requirement that the hamiltonian (4) should have well defined functional derivatives when the canonical variables obey (1), (2) and the deformation vector satisfies (5).

g00 = r2 , g ~ = r2sin20 • (3b, c)

The radius of curvature R is related to the cosmological constant A by R = ( 3 / - A ) 1/2.

The question immediately arises here as to the relation between (1), (2) and the boundary conditions of refs. [ 1 - 3 ] . A direct answer to that question would

4-2 There is therefore no analog for A < 0 of the additional "angle dependent translations" that appear in the asymptotically fiat case (see Appendix of ref. [4] ). Thus it would appear that the "BMS group" coincides with O(3, 2) in thi.~ case. This conclusion is supported by the observation that here null infinity is a timelike surface and its group of conformal motions is just 0(3, 2) (see ref. [7] ).

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This procedure yields the following expressions, which coincide with those obtained previously by a different method in ref. [8],

Jab = f d2~ "r~i/klrH± " ± ~'i L~" tV ab6kl/] -- Uab/j(gkl - gkl)]

+ 2Ukabnki}. (6)

In (6) the bar denotes covariant differentiation in the spatial anti-de Sitter background (3) and ~i / k l = ~ k l / 2 ( ~ i k ~ f l + ~i l~jk _ 2~i/~kl) .

Once the surface integral (6) is included in the hamilton±an according to (4) the hamilton±an becomes capable of generating an arbitrary surface deformation which becomes asymptotically an 0(3, 2) transformation. This is so because the functional derivatives of H[~] with respect to gi! and n0" become well defined. As a consequence the Poisson bracket of H[~] and H[r/] for ~ and r /of the form (5) becomes also well defined and obeys,

[H[~] ,h [r~]] =H[[~, 7711 , (7)

where [~, r/] is given by the "algebra of surface-deformations" [9],

[~, 1)] ± = - - (~i~l , i -- ~i~±,i) , (8a)

[ ~, ~] i = _ g ij (~,L~.L,j _ ~±n.L,/.) + (~ jrli,j _ r~j~i,j). (8b)

From (8) one can express the asymptotic part of [g, r/] in terms of the asymptotic parts of ~ and r/. One then finds using (5) and (1) that [~, r~] also takes the form (5) and that,, furthermore, the asymptotic part [~, 7)] ~ is given in terms of ~ab and r/ab according to the Lie-algebra of 0(3, 2). In this way one establishes that the hamilton±an formulation based on (4) is not only gauge invariant (i.e. allows for ~ arbitrary inside) but is also invariant under the asymptotic (non-gauge) 0(3, 2) symmetry.

It is sometimes useful in applications, especially in quantum mechanics, to fix the gauge. In the case at hand this may be done by imposing conditions on gii and IriJ whose preservation under surface deformations destroys the possibility of an arbitrary ~u inside. Thus, once the gauge is fixed, giving ~ determines ~u everywhere.

The generators of the resulting overall "global" transformation are then just the surface integrals Jab by themselves. However this time they must be under-

stood as acting in terms of Dirac brackets [10] associated with the chosen gauge condition. This is so because neither the volume integral nor the surface integral in (4) have separately well defined Poisson brackets but after the gauge is fixed the ~gu become "strongly" (i.e. identically) zero and the Jab acquire well defined (Dirac) brackets ,a

As a consequence of the arguments immediately following (8) one sees [without explicit computation of the change of the surface integral under a surface deformation obeying (5)] that the Dirac brackets of the Jab close according to the algebra of 0(3, 2). That discussion also shows that the algebra of the Jab is independent of the particular gauge condition chosen.

Lastly we touch upon the extension of the preced- ing analysis to supergravity. The framework and general conceptual points made above remain applicable but it becomes necessary to give also boundary conditions for the other fields appearing in the theory. Further- more 0(3, 2) is replaced by its appropriate graded extension as the asymptotic symmetry group. Additional surface integrals then appear which act as generators of the new symmetries. In those surface integrals Killing spinors [8] play for the supersymmetries the role that Killing vectors play in (5), (6).

In the simplest case o f N = 1 supergravity a canoni- cally self-conjugate vector spinor ~<.is present as the fermionic partner of the pair (gi/, 7r~l) • The boundary conditions for ~i which accompany (1), (2) turn out to be

~ r = r - 7 / 2 [ l + 7 ( r ) ] X r ( O , ¢ ) + O ( r - 9 / 2 ) , (9a)

tb 0 =r-3/2[1 - 7 ( r ) ] X o ( O , ¢ ) + O ( r - 5 / 2 ) , (9b)

4¢ =r -3/2 [1 - 7(r)] X¢(0, ¢) + O(r-5/2) • (9c)

Here the spinor indices are referred to the local ortho- normal spherical frame and 7(r) is the "radial" 7-ma- trix (whose square is unity).

On account of the supersymmetry an extra term of the form

f d3x ~ASA + ~QA (10)

• 3 A surface integral can act as a generator of a transformation defined all over space because the Dirac bracket is non-local in space and hence it is necessary to know the values of the fields everywhere to evaluate the bracket of the surface integral with the dynamical variables.

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is added to the hamiltonian (4), where the S A are the constraint-generators of local supersymmetry transfor- mations and QA (A = 1 . . . . . 4) is the supercharge surface integral.

In order to preserve the boundary conditions (1), (2), (9) the supersymmetry parameter ~A (x) must become asymptotically a linear combination of the four linearly independent anti,de Sitter Killing spinors UAB

~jA _ ~ B u A ' 0 . (11) r - - ~ oo

(The lower index in U A labels the different spinors of

the basis.) It turns out that (11) must fall off as r -5 /2[1 + 7(r)] •(0, ¢).

The surface integrals QA are determined by de- manding that (10) should have well defined functional derivatives. The resulting expression, which coincides with that given in ref. [8] , is

QA = f d 2Si(g)l/2 u T [7i, 7 / ] if] , (12)

where T denotes transposition. The generators (6), (12) close in Dirac brackets according to the algebra of

OSp(1,4) . Again here it is not necessary to resort to explicit computat ion to establish this algebra since it follows from the asymptotic form of the local supersymmetry algebra ,4

This work was supported in part by the US National Science Foundat ion under Grant PHY- 8 216715 to The University of Texas at Austin. M.H. would like to express his gratitude to Professor J.A. Wheeler for his kind hospital i ty at Austin.

References

[1] S.J. Avis, C.J. Isham and D. Storey, Phys. Rev. D18 (1978) 3565.

[2] P. Breitenlohner and D.Z. Freedman, Ann. Phys. (NY) 144 (1982) 249.

[3] S.W. Hawking, Phys. Lett. 126B (1983) 1751 [4] T. Regge and C. Teitelboim, Ann. Phys. (NY) 88 (1974)

286. [5] R. Benguria, P. Cordero and C. Teitelboim, Nucl. Phys.

B122 (1977) 61. [6] C. Teitelboim, Phys. Lett. 69B (1977) 240. [ 7] R~ Penr0se, in: Relativity-gr6ups aiid tb-pol0gy (Go-rdbn

and Breach, New York, 1964). [8] L. Abbott and S. Deser, Nucl. Phys. B195 (1982) 76. [9] C. Teitelboim, Ann. Phys. (NY) 79 (1973) 542; in:

General relativity and gravitation one hundred years after the birth of Albert Einstein, Vol. 1 (Plenum, New York, 1980).

[10] P.A.M. Dirac, Lectures on quantum mechanics (Academic Press, New York, 1964); A. Hanson, T. Regge and C. Teitelboim, Constrained hamiltonian systems (Accademia Nazionale dei Lincei, Rome, 1976).

,4 If one chooses an asymptotic field configuration ("background") with less symmetries then there will be less surface integrals added to the hamiltonian. The reason is that for each "missing symmetry" the corresponding parameter [~aO in (4), ~A in (10)] will have to vanish at infinity in order to preserve the boundary conditions. In that case the gauge invariance is still present but the "global" symmetry is absent.

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Hamiltonian treatment of asymptotically anti-de Sitter spaces