DAVID C. ROBINSON

GEOMETRY, NULL HYPERSURFACESAND NEWVARIABLES*

Abstract. Hamiltonian formulations of general relativit y employing null hypersurfaces as constant timehypcr surfaces are discussed. New variabl es approaches to the canonical formal ism for general relativityare reviewed .

I. INTRODUCTION

Over the last fifty years the Hamiltonian formulation of general relativity has beenthe subject of many investigations. These have been intimately related to the study ofinitial value problems, cosmological models, asymptotically flat systems, gravita­tional waves and more recently black holes. However, the principal preoccupation ofmuch of the research in this area has been the attempt to formulate a quantum theoryof gravity. Attempts to construct a comprehensive quantum theory which would natu­rally incorporate gravity, such as M-theory (Gibbons 1998) and its precursor super­symmetric string theory, are currently being vigorously and profitably pursued .However, in the last decade, there have also been significant advances in a newapproach to the canonical quantization of general relativity. Research initiated byAshtekar in the 1980's, and his introduction of new canonical variables, focussedattention on connection rather than metric dynamics (Ashtekar 1986, 1987, 1988,1991). Ashtekar's reformulation of Hamiltonian general relativity encouraged theapplication to gravity of techniques which had become well established in gauge the­ory. Development of these ideas has led to the establishment of loop quantum gravityand advances in a rigorous approach to a non-perturbative theory of quantum gravity.For recent reviews see (Rovelli 1997, 1997a). This programme has generated a signif­icant amount of new and interesting activity in the canonical quantization approach toquantum general relativity. Currently it provides the weightiest complement to theambitious M-theory programme.

The new variables canonical formalism , introduced by Ashtekar, can be obtainedby a complex canonical transformation from a geometrodynamical approach toHamiltonian general relativity. Alternatively it can be constructed, via a Legendretransformation, by starting with a complex , chiral , first order Lagrangian (Ashtekar1986, 1987, 1988, 1991; Peldan 1994). The new canonical variables are the compo ­nents of a triple of weight one vector densities and an SO(3)-valued connection on athree manifold . The Hamiltonian is a polynomial of low degree but complex-valued.

349

A. Ashtekar et al. (eds.), Revisiting the Foundations ofRelativistic Physics , 349-360.© 2003 Kluwer Academic Publishers.

350 D AVID C. R OBINSON

It is completely constrained . The seven first class constraint terms appearing in it gen­erate canonical transformations corresponding to space-time diffeomorphisms andinternal rotations. Their Poisson bracket algebra is not a Lie algebra. In order torecover the real phase space, and real general relativity, reality conditions have to beimposed. However both the constraint equations and the reality co nditions are lowdegree polynomial equations. When the reality conditions are satisfied and the realvector densities are linearly independent, the three manifold can be embedded as aspace-like hypersurfac e in a real Lorentzian four manifold. The connecti on can thenbe interpreted as the self-dual part of the Levi-Civita (spin) connection of the metric .The polynomial nature of the formalism, together with the emph asis on connectionrather than metric dynamic s, have suggested new developments in both classical andquantum theory. In recent years, real Lorentzian and Euclidean connection based ver­sions of the formalism have been employed in the quantum gravity programme. How­ever only the (initially) compl ex formali sm, which is geometrically natural in theLorentzian context, will be discussed in this paper.

A central aim of this paper is to review a version , based on null rather than space­like hypersurfaces, of Ashtekar 's approach to classical canonical general relativit y(Goldberg, Robinson, and Soteriou 1991a, 1992; Goldberg and Soteriou 1995; Gold­berg and Robinson 1998). Salient reasons for constructing the null hypersurface newvariables formalism include the successful use of null hypersurfaces in canonical for­mulations of field theories on Minkowski space-time (Sundermeyer 1982; Henneauxand Teitelboim 1992). Of course, in general relativity space-time geometry is dynam ­ical, not merely flat and kinematical as it is Minkowski space-time. Hence it does notplay merely a background role. Thi s gives rise to technical difficulti es as well as con­ceptual problems, such as the identification of observables and the problem of time(Ashtekar and Stachel 1991). Nevertheless the use of null hypersurfaces has alsoproven extremely useful in general relativity. Noteworthy and relevant here are stud­ies of gravitational radiation and asymptotically flat systems. Useful comments onwork using null hypersurfaces can be found in the recent paper by Bartnik (Bartnik1998). A key reason for the successful use of null hypersurfaces in the study of fieldtheories is that constraints arising in the relevant characteristic initial value problemscan be integrated more easily than those arising in corresponding Cauchy problems.The true degree s of freedom can be exposed more explicitl y when null hypersurfaces,rather than space-like hypersurfaces, are used.

Amongst the geometrical features which distinguish null hypersurface based for­mulations from space-l ike hypersurface based approaches to Hamiltonian gravity, arethe following. First, the geometry of null hypersurfaces, unlike that of space-likehyper surfaces, is not metric. The singular metric induced on a null hypersurface fromthe space-time has maximal rank two, and the null geodesic generators of the nullhypersurface are characteri stic vector fields of the singular metric. Connections arealso induced onto null hypersurfaces from connections on space-time, but they arenot Levi-Civita connections of a three metric . In the initial value or Hamiltonian for­malism the relevant structure group becomes the (4 real parameter) null rotation sub­group of the Lorentz group, rather than the three parameter specia l orthogonal group.Partially as a consequence, the formalism can appear less manifestly covariant than it

G EOMETRY, N ULL HYP ERSURFACES ... 35 1

is in the space-like case. The focussing effect of gravity causes the null geodes ic gen­erators of null hypersurfaces to cross over and to form caustics. At such points nullhypersurfaces fail to be (immersed) smooth submanifolds of space-time (Friedrichand Stewart 1983; Hasse, Kriele, and Perlick 1996). Generically, topologically sim­ple null hypersurfaces, such as the null planes and null cones of Minkowski space­time, are not available . Problems related to the differential and metric structure arisealso in the space-like hypersurface formalisms, but they tend to be more overt in thenull hypersurface case.

This paper will deal mainly with local aspects of canonical formalisms. Thus hyper­surfaces will be treated as sub-manifolds with a fixed differential structure. Althoughthe emphasis will be on local theory, it should be noted that it is usually assumed (atleast implicitly) in null hypersurface Hamiltonian general relativity, that the global con­text is one in which the null hypersurface leaves of the space-time foliation are non­compact. It is also implicilly assumed here that the condition of asymptotic flatness (orasymptotically cosmological boundary conditions) can be imposed, and that future nullinfinity can be defined (Chrusciel, MacCallum, and Singleton 1995). Normally here,fields are considered to be defined on outgoing null hypersurfaces.

In the next section a brief review of research related to null hypersurface canoni­cal formulations of general relativity will be presented. Thi s aims only to provide acontext within which the results of later sections can be placed. It does not pretend tobe comprehensive. The third section will contain an outline of Ashtekar's new vari­ables Hamiltonian formulation of general relativity. Thi s will be presented in such away that the similarities to, and differences from, the null hypersurface based newvariables formali sm are highlighted. It will suffice in this paper to consider detail s ofthe two formalisms when there are no matter fields coupled to gravity. The nullhypersurface new variables canonical formalism will be discussed in the final section.In particular a Hamiltonian system which is completely equivalent to Einstein' s vac­uum equations will be presented. The structure of the constraints and reality condi ­tions will also be reviewed . Finally the space-time interpretation will be outlined. Thebody of results on the null hypersurface based new variables canonical formalism isneither as large, nor as developed, as that obtained within the conventional frame ­work where the level sets of the time function are space-like. However the null hyper­surface new variable s Hamiltonian formulati on is more complete than other nullhypersurface based canonical formulations of general relativity.

The notation and conventions of (Goldberg, Robinson, and Soteriou 1992) will befollowed without major excepti on. Lower case Latin indices i , i. k sum and rangeover I, 2, 3 and indicate comp onents with respect to coordinate bases. Upper caseLatin indices A, B sum and run over I, 2, 3 and label components of SO(3) Lie-alge ­bra valued geometrical objects. Lorentzian manifolds will have four-metrics with sig­nature (I , -I , -I , - I).

352 D AVID C. R OBINSON

2. NULL HYPERSURFACE CANONICAL FORMALISMS

Dirac pointed out in (Dirac 1949), that it is physically natural to use a time parameterwith level sets which are null hypersurfaces. A particularly relevant example of theuse of this method of describing time evolution in Minkowski space-time is providedby one approac h to the canonical formalis m for source free Maxwell and Yang-Mill stheory. This uses a first order complex Lagrangian, with the potential (connection)and self-dual part of the field (self-dual curvature) as field variables. A Hamiltonian isconstructed via a 3+ I decomposition of Minkowski space- time using outgoing nullcones or null hyperplanes. It incorporates many of the features which are encounteredin the general relativistic null hypersurface new variables formalism, some of whichwill be discussed in the four th section. It is also sufficiently simple that it can be car­ried out completely which, to date, is not the case for general relativity. Consequentlyit is instructive to outlin e the results in the case of the Yang-Mills field, with internalsymmetry group a real n-parametcr semi-simple Lie group (Soteriou 1992). A 3+ Idecomposition of the first order Lagrangian, using a null hyperplane foliation, leadsby inspection to a compl ex Hamiltonian defined on a null hyperplane. There are 2nfirst class constraints and 4n seco nd class constraints. The Yang-Mills field equationsare equivalent to the constraint and evolution (Hamilton's) equations. The canonicaltransformations which correspond to the infinitesimal internal gauge transformationsare generated by the first class constraints. The first class constraint algebra repre­sents the internal symmetry Lie algebra. Second class constraints arise because nullhypersurfaces are being used. T he seco nd class constraint equations relate physicallyredund ant variables. Solution of these leads to the elimination of an equal number ofcanonically conjugate pairs of phase space variables. On the resulting partiallyreduced phase space there remain only the first class constrai nts, reflecting the gaugeinvariance of the theory, An alternative and equivalent way of handl ing seco nd classconstraints is to introduce Dirac brackets, either directly or via thc starred variablesprocedure of Bergmann and Komar, see e.g. (Sundermeyer 1982). This can be sim­pler than solving the second class constraint equations directly and is the procedurefollowed in (Soteriou 1992). There the completely reduced phase space is realized asa Poisson spacc by gauge fixing so that all constraints become seco nd class. ThePoisson structure on the completely reduced 2n dimensional phase space is definedby the Dirac bracket and the constraints can be sct equal to zero in the Hamiltonian.The dynamic s arc determined by the Dirac brackets of the 2n indepe ndent degrees offreedom with the reduced Hamiltonian. The imposition of reality conditions enablesthe rcal theory to be rccovered and a reduced phase space quantization to be effected.For details in the similar but simpler case of the Maxwell field see (Goldberg, Robin ­son, and Soteriou 1991; Goldberg 1991, 1992) .

Early investigations of null hypersurface based canonical formulations of generalrelativity were carried out mainly within geometrodynamical frameworks and dealtwith the vacuum equations; for example see (Aragone and Chcla-Flores 1975; Gam­bini and Restuccia 1978; Goldberg 1984, 1985, 1986; Torre 1986; Aragone and Kou­deir 1990). Since a null hypersurface time parametrization was used there were secondclass constraints as well as the first class constraints associated with gauge freedom.

G EOMETRY, N ULL H YPERSURFACES ... 353

They were non-polynomial in nature . These non-polynomial canonical formali sms,their constraint structures and their dynamics, were not as straightforward to analyzeas had been hoped. The complete formulation of the dynamics on the completelyreduced phase space was not satisfactorily achieved. Because this work used as a coor­dinate t , a time function with level sets which were assumed to be null hypersurfaces,consequently the inverse metric components gil were assumed to be zero. Hence theEinstein equations, Gil = 0 could not be naturally derived, either in the 3+1 decom­posed Lagrangian or in the Legendre transformation related Hamiltonian formali sm.This research did however provide some useful technical and conceptual insights. Itsuffices to mention here two significant differences which were revealed between theconstraint structure of spacelike hypersurface based and null hypersurface basedcanonical formali sms. Torre, using a 2+2 formali sm in an investigation of a null hyper­surface-based geometrodynamical formulation, gave a particularly clear and compre­hensive discussion of these (Torre 1986). In the space-like case the Hamiltonian is alinear combination of first class constraints. In the null-hypersurface based case it wasfound that, while the Hamiltoni an is again a linear combination of constraints, not allof them are first class. Furthermore in the space-like case the scalar constraint whichgenerates time translations out of the hypersurface is first class. However there are nocompact deformations which map a null hypersurface to a neighboring null hypersur­face. The choice of a null time parameter is effectively a choice of gauge. As a conse­quence it was found that the scalar constraint was second class.

The comparative tractability of constraints on null hypersurfaces, the low degreepolynomial nature of the new variables formalism first introduced by Ashtekar, andthe focus on connection rather than metric dynamics, suggested that a new variablesformalism based on null hypersurfaces might be more successful than previous nullhypersurface Hamiltonian constructions. Unlike the Hamiltonian systems mentionedearlier, all the Einstein equations arise naturally in the new formul ation (Goldberg,Robinson, and Soteriou 1991, 1992; Goldberg and Soteriou 1995; Goldberg and Rob­inson 1998). Thi s is achieved by adding Lagrange multiplier terms to a Lagran giandensity derived from the self-dual first order Lagrangian mentioned above (Ashtekar1986, 1987, 1988; Ashtekar and Tate 1991; Peldan 1994). The corresponding EulerLagrange equations then include all the Einstein (and matter) field equations plusequations which identify level sets of constant time as null hypersurfaces. T he Hamil­tonian, obtained by a Legendre transformation, consequently also contains all thisinformation. It is compl etely constrained but not all the constraint s are first class.There are a large number of second class constraints as there always are in nullhypersurface based formali sms. Analysis of the constraint structure reveals it to berather different from that of the space-like hypersurface based Hamiltoni an formal­ism. The first class constraints form a Lie algebra and the scalar constraint is secondclass. This is a distincti ve feature of this formali sm; the Lie algebra can be interpretedas a representation of the semi-direct product of the null rotation sub-group of theLorentz group and the diffeomorphi sms within the null hypersurface. Both con­straints and reality conditions are polynomial of low degree. In addition to the workon the vacuum case , coupled Einstein-matter systems have been analyzed by Soteriou(1992). He extended the vacuum formalism to gravita tional systems which included

354 D AVID C. R OBINSON

scalar fields, Yang-Mills fields and Grassmann valued Dirac fields. In the latter a two­component spinor formalism was used. Such extensions are esse ntially unproblem­atic, as had previously been shown in the space- like case (Ashtekar and Tate 1991;Ashtekar, Romano, and Tate 1989) . Boundary conditions have been examined (Gold­berg and Soteri ou 1995), and some progress has been made towards the eliminationof the second class constraints (Go ldberg and Robinson 1998). The construct ion ofthe completely reduced phase space (ibid.), has yet to be completed as success fully asit has been in, for example, the Yang-Mills canonical formalism discussed above.Consequently, the linearization of the theory about a Minkowski space-time wasinvestigated , using both Minkowski null hyperplanes and outgoing Minkowski nullcones (Soteriou 1992). For a geometrody namical formulation see (Evans, Kunstatter,and Torre 1987). In this case , as might be expected, the programme could be com­pleted along lines simil ar to those outlined above for Yang-Mill s fields in Minkowskispace-time. Despite the fact that not all problems have been resolved in practice, thiscanonical formalism has been developed more fully than any other null hypersurfacebased Hamiltonian formul ation of general relativity; in part this is because of itspolynomi al nature . In the first approaches to the null hypersurfac e, new variable scanonical formalism (Goldberg, Robinson, and Soteriou 1991a, 1992; Soteriou1992), the full Dirac-Bergmann algorithm (Sundermeye r 1982; Henneaux and Teitel­boim 1992), was applied. A large phase space resulted . The framework used wasbroad enough to ensure that the complete structure of the new formulation could beproperly analyzed. Once this was clearly understood a streamlined vers ion, whichemploys only connection and vector density triad components as canonica lly conju ­gate variables, could be used (Goldberg and Soteriou 1995; Goldberg and Robin son1998). It is that formalism which will be discussed in more detail in section 4.

In recent years there have been a number of other approaches to new variablesformulations of general relativity on a null hypersurface. These include the applica­tion of multi sympl ectic techniques, as used in jet bundle formul ations of classicalfield theories, to the new variables null hypersurface formalism. In this work thepropert ies of multi -momentum maps and their relation to the constraint analysis arediscussed (Esposito and Storn aiolo 1997; Stornaiolo and Esposito 1997). The Hamil­tonian formulation of general relativity on a null hypersurface has also been consid­ered within the conte xt of teleparallel geometry (Maluf and da Rocha-Neto 1999).Finally it shou ld be noted that a 2+2 (Lagrangian) formulation using new variabl eshas been considered by d 'Inverno and Vickers (1995) . Thi s work focuses on theimportant case of a double null foliation of space-time and their discussion mirrors,to an extent, the null hypersurface new variables formali sm. It includes a discussionof the relationsh ip between different formalisms.

G EOMETRY, N ULL HYPERSURFACES ...

3. NEW VARIABLES AND SPACELIKE HYPERSURFACES

355

Let M be a real three dimensional manifold with local coo rdinates xi . Let L~ a i bethree weight one vector densities tangent to M, and let A/ d x i be a so(3)-valuedconnectio n I-form on M with curvature 2-form R

A. The new variables Hamiltonian

for vacuum general relativity can be written as

(I )

Here N, N iand SA are Lagrange multipliers, with N a weight minus one density.

The canonicall y conju gate dynamical variables are taken to be the pair ( AiA, L~ ) and

the only non-zero Poisson brackets between these variables are

{ AtCr), Lj (y )} = of%( x, y) . (2)

The Hamiltonian density, Lagrange multipliers and dynamical variables can beregarded as being complex valued so that the phase space is complex. The scalar, vec­tor and Gauss constraint equati ons are

(3)

Here

A Ai j A AB C i .R = -I I2Rij dx s d x = (aiA j + f BC Ai A j )dx s dx' ; (4)

Di is the covariant derivative of the connection A/ and f ABC = f[ABC] withf 123 = I. In the basis used the SO(3)- invariant metric is given by

11AB = diag(-I , -I , -I ). (5)

(6)

All these constraints are first class although the constraint algebra is not a Lie alge­bra. Appropriately combin ed they generate canonical transformations corre spondin gto space-time diffeomorphi sms and (small internal ) gauge rotations. They are trans­parently low degree polynomials in the canonical variables. Dynamical evolution isdetermined by Hamilton 's equations

a,L Ai

= {L Ai,

H}

a, A i A = {A /, H} .

When the vector densities L Ai are linearly independent they determine a 3-metric on

M, with inverse metric

where

ij -2 AB~ i~ jg = -v 11 ~A ~B ' (7)

356 DAVID C. R OBINSON

(10)

2 . i j kV = -/ EijkIl I 2 I 3 . (8)

In this case, on the four manifold M x ~ with local coordinates (t, x\ the tetrad of4-vector fields

- I - I · . - I ieo = v N (a , -N'a i) and eA = - IV I Aa; (9)

form s an orthonormal basis for a 4-metric with inverse eo@ eo+ 11ABeA@ eB, so

2 2 -2 -2 -2 - 2 i )a l as = V N a, @a , -v N N(a ,@ai +ai@u,

+ I l2(v - 2N- 2NiN j - v - 211 ABI AiI / )( u;@ uj + Uj@ ai) .

• 2 1 2 3The volume 4-form IS v Ndt A dx A dx A dx . The I-form,

A A i AI' = Ai dx + B dt , (II)

corresponds to the self-dual part of the Levi-Civita (spin) connecti on of this metric .The constraint and evolution equations given above then correspond to Einstein'svacuum equations for the metric . The pull back of the curvature of this connection, toa leaf of the foliation by level sets of t, can then be identified with the 2-form R

A.

Real Lorentzian general relativity can be recovered when polynomial reality condi­tions, given by requirin g that I A'I /ll

ABand {IAiI /ll

AB, H} be real, are imposed

on the phase space variables. Agreement with the standard ADM geometrodynamicalversion of Hamiltoni an relativity follows when the constraint, evolution and realityequations are satisfied. The standard geometrodynamical field variables can then beidentified with the three metric gij (with Levi-Civita so(3) -valued connection

Ad ' ) d h .. . K - 1 '" k( A A A)Yi X , an t e symmetnc extn nsic curvature tensor ij = -v g ik~A j + Yj ,

on a space-like hypersurface given by a level set of t (Ashtekar 1986, 1987, 1988;Ashtekar and Tate 199I).

4. NEW VARIABLES CANONICAL FORMALISM FORGENERAL RELATIVITY ON A NULL HYPERSUR FACE

In the new variables, null hypersurface Hamiltonian formalism, presented in (Gold­berg and Soteriou 1995), the canonically conjugate phase space variables are againdefined by the components of a triple of weight one vector densities I A and the com­ponents of an so (3) valued connection one form AA on a three manifold M . ThePoisson bracket relations are given as in eq. (2). The complex Hamiltonian is, how­ever, now given by

3 . A . 2H = fd x( N 'If +N' 'If; - B GA-f!i@""-p a ) , (12)

The functions N (a scalar density of weight minus one), Ni, B

A, v', f!; and pare

treated as Lagrange multipli ers and

G EOMETRY, N ULL HYPERS URFACES .. . 357

(13)

i , j Z j i'If: v (Rij L3 + Rij L] ) : V <Pi '

A· .'lfi : -Rij Li , GA: D/L A' ) ,

rr:-i i i\l:< : L Z + av.

The evolution equations are given by Hamilton 's equations, as in eq. (6).Th e Hamiltonian density is a linear sum of the produ ct of Lagran ge multipliers

with term s which all vanish when constra int equati ons and multiplier equations aresatisfied. Th e constraints, ct = 0 , a Z

= 0 , and the multipli er equation ~l/ = 0 ,arise from Lagrange multipli er terms originating in the Legendre transformationrelated Lagrangian formali sm. Added to a chiral Lagrangian for Einstein's vacuumequations they ensure that this is a correctly set null hypersurface based formali sm,and that the canonical formalism contains all the Einstein gravitational (and, whenpresent matter) field equations. Here a is not treated as an independent dynamicalvariable; to do so would merely result in further trivial second class constraints whichwould lead to its ultimate elimination from a reduced phase space. It is set equal tozero after Poisson brackets have been computed. Its vanishing mirrors the fact thatthis is a null hypersurface based formalism and implies, together with the cons traints,(§'i = 0 , the vector density degeneracy condition, L/ = O. Th e latter conditionmirrors the degeneracy of a basis of self-dual 2-forms pulled back to a null hypersur­face in space-time. Th e seco nd multipli er equation above, ~i v i = 0 , is required inorder to obtain all the Einstein equations from the formalism. It arises naturall y fromthe .Lagrangian formali sm. The vanishing of <Pi ' the coefficie nts of the multipliersN Vi, results in three constraint equations. The scalar, vecto r and Gauss constraintequations are given by 'If =: 0, 'lfi =0 and GA = 0, as in section 3. It can be seenfro m the above that 'If = Vi<Pi and L / <Pi = L3' 'lfi. Hence there are eleven indepen­dent constraints above. Propagation of the constra ints leads to secondary conditionson Lagran ge multipliers. An important example of the latter, exemplify ing differ­ences from the results of section 3, follows from the propagation of (§" = 0 , that isevolution of the vector density degeneracy condition. Thi s leads to three secondaryconditions on the multipli ers / and 8

3, given by

Xi:2b/Dj{NvliLA}](b B'b/ +b/b, A)}_2A/ NliL / J_83L / = O. (14)

Th e eleven constraints comprise six second class con straints and five first class con­straints. The independent first clas s constraints can be identified with

(15)

There are 18 phase space variables, 6 seco nd class constraints and 5 first cla ss con­straints. Co nsequently, at each point of M, there are 2 independent degrees of phasespace freedom. The canonical transformations generated by the constraints functionsG , and Gz correspond to self dual null rotations, reflectin g the fact that the formal­ism is chiral (self-dual), complex and null hypersurface based. Th e constraints 'If'igenerate diffeomorphisms in M. Th e scalar constraint is second class. A null hyper-

358 D AVID C. R OBINSON

surface time gauge is effecti vely set within the formali sm and the problem of timedoes not appear in the way it does in the space-like formalism. The algebra of firstclass constraints is a genuin e Lie algebra, in contrast to the space like based formal­ism in section 3. Th e formalism is agai n polynomial.

As was the case in the previous sect ion, when a non-degeneracy condition is satis­fied, the constraint , evolution and mult iplier equations are equivalent to the Einsteinvacuum equations. More explicitly, when the weight one vector density fields tangentto M, via;, L / a;, L/ a;, are linearly independent so that

(16)

is non-zero, a (non-degenerate) four metric can be constructed on M x R. Its inverseis given by

and

2 2 2 - 2 ( - 2 ; ; ) ( )v a l as = -2aN a, ®a, + 2aN N -NL 1 a,®a; +a;®a,-2 ; j ; j - I ; j

+ (aN NN - VL 3 -N L, N )(a;® a;+ aj ® a;).

Here the natural basis of vector fields is a null tetrad given by

(17)

(18)

- I N-1( ; ) -I - I ( - I - I ; - I ;eo = v at -Na;, e, = - av N at + o.v N N -v L, )a;,(19)

and the volume 4-form is - iNv2dt 1\ dx 1

1\ di 1\ dx3

• When a = 0, the level setsof t are null hypersurfaces and the null tetrad (and its null co-frame) is adapted to thenull hypersurface foliati on with e , tangent to the null geodesic generators of the nullhypersurfaces. The connection I-form, given as in eq. ( 11), is the self-dual part of theLevi-Civita (spin) connection of the metric. When the remaining multiplier equa­tions, the constraint and the evolution equations are satisfied, the metr ic satisfies Ein­stein's vacuum equations. Real relativity can be extracted from the compl ex phasespace description by impo sing appropriate polynom ial reality conditi ons (Goldberg,Robinson , and Soteriou 1992; Soteri ou 1992). It suffices to record here that / canthen be chosen to be the compl ex conj ugate of L3 ; ; L I ;' v and N can be chosen to bepure imaginary; finally Nt can be chosen to be real. The null tetrad then satisfies thestandard reality conditions: eo and e l real, e2 and e3 compl ex conj ugates.

Startin g from the Hamiltoni an formali sm the properties of the constructed space­times emerge from those of the phase-space geometry. The form of the Hamiltoniandetermines whether M should be embedded in a space-time as a space-like or nullhypersurface. Space-time properties such as asy mptotic flatness, and hypersurfacerelated properti es such as crossovers and caustics, arise from solutions of the differ­ential equations of the canonical formalism subject to boundary conditions. The latterhave not been discussed in this paper but extension of this work to includ e fields on

G EOMETRY , N ULL HYPERSURFACES ... 359

future null infinity and hence a complete global formul ation is, in principle, straight­forward , (c.f. calculations by Goldberg in a geometrodynamical framework in (Gold­berg 1986) and discussions of boundary conditions in (Goldbe rg and Soteriou 1995)).The formalisms discussed in sections 3 and 4 encompass theories which may be realor complex and which allow the possibility of degenerate inverse metr ic densities.The latter arise when y

Z = 0 and can be identified with the expressions fory ZaZ

/ ai in eq. (10) and eq. (18). This possibil ity has been discussed in the space­like case, e.g. in (Bengtsson and Jacobson 1997), and merits investigation in the nullhypersurface formalism.

In order to obtain a complete and satisfactory canonical formalism the constraintsmust be solved and Dirac brackets calculated in one of the ways indicated in section2. The progress that has been made in this area (Goldberg and Soteriou 1995; Gold ­berg and Robinson 1998) indicates that the canonically conjugate reduced compl exphase space variable s are AZ

Band LD

z, with B = 3 and D = 3. In the linearizedtheory the imposition of the reality conditions, and the compl ete fixing of coordinateand gauge freedom , leads to the breaking of the chiral nature of the formalism. Thecompletely reduced phase space and final Hamiltoni an are real and the Poisson struc­ture is provided by the Dirac bracket. These calculations, by Soteriou (1992), supportthe conclusion that, after removal of the gauge and coo rdinate freedom , the two realdegrees of freedom on the completely reduced phase space can be identified with thereal and imaginary parts of the connection component A/ The latter has a space­time interpretat ion as the shear of the null hypersurface geodes ic ray generators.Work in progress, on the geometry of null hypersurfaces and connections with valuesin the Lie algebra of the group of null rotations, may help clarify these matters.

The approach to the new variables, null hypersurface cano nical formalis m pre­sented above incorporates the components of a self-dual connectio n as dynamic alvariables. The Lagrange multiplier techniques used in (Goldberg, Robinson, andSoteriou 1992; Goldberg and Soteriou 1995) could also be employed to develop acomplete, real, null hypersurface based, geometrodynamical canonical formalism. Itwould expected that this would be related to the above by a complex canonical trans­formation.

King 's College London

ACKNOWLEDGEMENTS

I would like to thank Lee McCulloch, Chrys Soteriou and Josh Goldberg for usefulcomments.

NOTE

* Joh n Stac hel has contributed significantly to the study of many technical. conceptual and historicalproblems in general relativity. His hospita lity in Boston enabled me to start investing this approach toHami ltonian general relativity. It is a pleasure to dedicate this paper to him.

360 D AVID C. R OBINSON

REFERENCES

Aragone, C. and 1. Chela-Aores. 1975. Nuovo Cimento 25B:225.Aragone, C. and A Koud eir. 1990. Class. Quantum Grav. 7:1291.Ashtekar , A 1986. Phys. Rev. Lell. 57: 2244 .--. 1987. Phys. Rev. 036: 1587.- - -. 1988. New Perspectives in Canonical Gravity. Naples: Bibliopolis.Ashte kar, A, J. D. Romano, and R. S. Tate. 1989. Phys. Rev. D40:2572 .As htekar , A and J. Stachel, eds. 1991. Conceptual problems of quant um gravity. Boston: Birkhauser,Ashteka r, A. and R. Tate. 1991. Lectures on Non-Perturbative Canonical Gravity. Singapore: World Scien -

tific.Bartnik. R. 1997. Class . Quantum Grav. 14:2185.Bengt sson, 1.and T. Jacobson. 1997. Class. Quantum Grav. 14:3 109.Chrusciel, P.T.. M. A H. MacCallum, and D. B, Singleton , 1995. Phil. Trans. Roy. Soc. Lond. A 350:113.Dirac, P. A M. 1949. Rev. Mod. Phys. 2 1:392.Esposi to, G. and C. Stornaiolo. 1997. NU(M ) Cimento B 112:1395.Stornaiolo, C. and G. Esposito. 1997. Nucl . Phys. Proc. Suppl. 57:241.Evans. D.• G. Kunstatter, and C. Torre. 1987. Class. Quantum Grav. 4 :1503.Friedrich, H. and 1. M. Stewart. 1983. Proc. R. Soc. A 385:345.Gambini, R. andA. Restuccia. 1978. Phys . Rev. D I7:3150.Gib bons. G. W 1998. Quantum GravitylS tringlM- theory as we approa ch the 3rd Millennium, gr-qc/

9803065. [Procee dings of Genera l Relativity and Gravitation, GR 15. Puna . India 1997.]Goldberg, 1. N. 1984. Found . Phys . 14:121 1.---.1985. Found. Pliys . 15:439.---. 1986. In Gravita tional Collapse and Relativity (Proceedings of the XIV Yamada Confe rence) . eds.

H. Sato and T. Nakamura. Singapore: World Scientific.--- .1991. Gen. ReI. and Grav. 23:1403.--- . 1992. J. Geom.and Phys. 8:163.Goldberg, J. N., D. C. Robinson, and C. Soteriou. 199 1. In 9th Italian Conference on General Relati vity

and Gravitati onal Physics, Capri (Napoli), September 25 - 28, 1990 (Peter G. Bergmann Festschrift ),ed. R. Cianci et al. Singapore : World Scientific.

--- . 1991a, in Gravitation and Modern Cosmology: vol. in honor Peter Gabriel Bergmann's 75 . birth-day, eds. A Zichichi, V. de Sabbata, and N. Sanc hez. New York: Plenum.

--- .1 992. Class. Quantum Grav. 9:1309.Goldberg, 1. N. and C. Soteriou . 1995. Class . Quantum Grav. 12:2779.Goldberg . 1. N. and D. C. Robinson . 1998. Acta Phys. Polonica B. 29:849.Hasse, W , M. Kriele , and V. Perlick . 1996. Class. and Quantum Grav. 13:116 1.Henneaux, M. and C. Teitelboim. 1992. Quantisation of Gauge Systems. Princeto n, N.J .: Princeton U.

Press.d'In verno, R.A and J. A Vickers. 1995. Class. Quantum Grav. 12:753.Maluf', J. W and 1. F. da Rocha-Neto. 1999. Gen. Rei. and Grav. 31:/73.Peldan, P. 1994. Class. Quantum Grav. I I :1087.Rovelli, C. 1997. Strings, loops and others: a crit ical survey of the present approaches to quantum gravity,

gr-qc/9803024. [Proceedings of General Relativity and Gravita tion. GR I5, Puna, India 1997.]--- . 1997a. "Loop Quantum Grav ity," gr-qc/97 10008,

http ://www.livingreviews.org/ArticieslVolum el / 1998-1rovelli/.Soterio u. C. 1992. Canonical Formalisms, New Variables and Null Hypersurfa ces. Ph.D. Thesis . Univer­

sity of London.Sundermeyer, K. 1982. "Constrained Dynamics: with Applications to Yang-Mills Theory, General Relativity,

Classical Spin, Dual String Model." Lecture Notes in Physics 169. Berlin/New York: Springer-Verlag.Torre, C. 1986. Class. Quantum Grav. 3:773.