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b r o u g h t t o y o u b y C O R EV i e w m e t a d a t a , c i t a t i o n a n d s i m i l a r p a p e r s a t c o r e . a c . u k

p r o v i d e d b y C E R N D o c u m e n t S e r v e r

Hamiltonian Formulation of Nonsymmetric Gravitational Theories�

M. A. Claytony

Department of Physics, University of Toronto, Toronto, ON., Canada, M5S 1A7

(September 18, 1995)

The dynamics of a class of nonsymmetric gravitational theories is presented in Hamiltonianform. The derivation begins with the �rst-order action, and treating the generalized connec-

tion coe�cients as the canonical coordinates, and the densitised metric degrees of freedom as

conjugate momenta. All �elds are decomposed into surface tensors by making use of a surfaceadapted coordinate system. The phase space of the symmetric sector is enlarged compared

to the conventional treatments of GR by a canonical pair that represents the metric density

and its conjugate, removable by imposing strongly an associated pair of second class con-straints and introducing Dirac brackets. The lapse and shift functions remain undetermined

Lagrange multipliers that enforce the di�eomorphism constraints in the standard form of theNGT Hamiltonian. Thus the dimension of the physical constraint surface in the symmetric

sector is not enlarged over that of GR. In the antisymmetric sector, all six metric functions

contribute conjugate pairs for the massive theory, and the absence of additional constraintsgives six con�guration space degrees of freedom per spacetime point in the antisymmetric

sector. For the original NGT action (or, equivalently, Einstein's Uni�ed Field Theory), the

U(1) invariance of the action is shown to remove one of these antisymmetric sector conjugatepairs through an additional �rst class constraint, leaving �ve degrees of freedom.

Various issues in the construction that di�er from the conventional GR treatment are

discussed, as is the restriction of the dynamics to GR con�gurations and, in the case of anopen spacetime, the form of the surface terms that arise from the variation of the Hamiltonian.

In the resulting Hamiltonian system for the massive theory, singular behavior is found in

the relations that determine some of the Lagrange multipliers near GR and certain NGTspacetimes.

INTRODUCTION

The Hamiltonian formulation of GR (viewing the metric and the extrinsic curvature of a spacelike hyper-surface evolving in time as a �rst order system) has become popular in GR in recent years primarily dueto various attempts to canonically quantize GR. However the importance of the Hamiltonian picture goesbeyond this formal procedure. In any classical �eld theory, the Hamiltonian formulation clari�es many dy-namical issues of the system. The Bergmann-Dirac canonical analysis begins with a classical action, tellsone whether it is consistent, and identi�es the constraints and allowed choices of gauge. This identi�es thephysical constraint surface in phase space, the dimension of which gives the number of degrees of freedom inthe system under consideration. The resulting �rst-order system (Hamilton's equations) is then ideally suitedfor numerical investigations of the initial value problem. As little is known of a general nature about solutionsof the �eld equations in NGT and phenomenology tends to rely heavily on the properties of the static spher-ically symmetric solutions1 , it is hoped that some results of a more general nature will be accessible throughHamiltonian methods, or at the very least numerical investigations.In the case at hand, one is looking for a description of a class of Nonsymmetric Gravitational Theories

(NGT will refer either to this type of theory in general or to the `old', or massless theory, whereas mNGTwill refer to the massive theory recently introduced in [6{8]) that will tell one which �elds in the metric

�UTPT-95-20yclayton@medb.physics.utoronto.ca1 [1] gives the general form of the spherically symmetric solution within the context of Einstein's Uni�ed Field

Theory, a special case of which was popular in the early works of NGT [2]; another is the Wyman solution [3], recentlyresurrected for both the `old' [4] and massive [5] versions of NGT

2

are involved in the dynamical evolution of the system, and which (if any) are given by constraints on theinitial Cauchy surface or left as freely speci�able Lagrange multipliers. The analysis performed here will treatthe action for NGT in �rst-order form, treating the symmetric and antisymmetric connection coe�cients ascanonical coordinates, the components of the spatial metric showing up as (weakly equivalent to) the conjugatemomenta. Hamilton's equations for the system are then similar to those of GR given in [9], with phase spacesuitably enlarged to accommodate the antisymmetric sector, and the algebraic compatibility conditions leftunimposed. Although it is possible in principle to treat the action as a second-order system (depending onlyon the fundamental tensor and its derivatives) by generalizing the inversion formula of Tonnelat [10,11], thealgebraic complications inherent in the solution of the generalized compatibility conditions make this approachunfeasible.The action for mNGT [6{8] has recently been introduced in order to make the antisymmetric sector well-

behaved when considered as a perturbation of a GR background. The necessity of making this shift away fromthe original formulation of NGT was hinted at in some early works in NGT [12,13], but made a necessity by thework of Damour, Deser and McCarthy [14,15] in showing the bad asymptotic behavior of NGT perturbationsabout asymptotically at GR backgrounds (reviewed in [8]). The action for mNGT is designed so that the skewperturbation equations have the form of massive Kalb-Ramond theory on a GR background, with additionalcurvature coupled potential terms. Thus in the case where the GR background is �xed, one �nds threepropagating degrees of freedom in the skew sector, and three algebraic relations that couple the remainingmodes locally to the source (and background curvature), e�ectively removing what would be negative energymodes had they propagated. The results found here will indicate that in general, all six antisymmetric metriccomponents exist as independent degrees of freedom in the theory.The bulk of this paper (Sections II-IV) will be spent developing the formalism and recasting the dynamics

of NGT in Hamiltonian form. This development is not particularly elegant, as the Hamiltonian is burdenedby a large number of Lagrange multipliers, and the familiar patterns of the GR analysis become obscured.Where possible, parallels to the analogous GR analysis have been made, in order to not lose sight of the (insome ways very simple) overall structure. Indeed, although the algebra is sometimes rather messy, the analysispresented here is a relatively straightforward application of the Bergmann-Dirac constraint algorithm.Upon concluding the constraint analysis, in Section V the number of con�guration space degrees of freedom

per spacetime point in the old version of NGT is shown rigorously to be �ve, clearing up some confusion thathas occurred in the literature [13]. This however is not a new result, as it has long been known in the contextof Einstein's Uni�ed Field Theory [16]. (Their analysis is not altered by the reinterpretation of the skewsector or the introduction of the source coupling into the action that de�nes NGT [2].) The Cauchy analysisperformed in the aforementioned works also determined that the Uni�ed Field Theory (hence also NGT)`su�ers' from multiple light cones in that there are three causal boundaries within which di�erent modes ofthe gravitational system propagate. It is expected that these `causal metrics' will be slightly di�erent formassive NGT, and in fact the considerations of Section VI would seem to indicate that at least one of thesehas no pure GR limit (in the sense that as g[ ] ! 0, the related causal metric also vanishes). The presence ofmore than one physical light cone confuses the issue of what constitutes the proper description of a Cauchysurface, and is discussed in more detail in Section IIA.

I. THE ACTION AND FIELD EQUATIONS

The formalism developed in Section 3 of [8] allows one to easily write the action for theories with non-symmetric metric and connection coe�cients in a surface adapted coordinate system suitable for the Hamil-tonian analysis presented here. All covariant derivatives and curvatures have been de�ned with respect to atorsion-free connection �ABC (which in general is not required to be compatible with any metric tensor in thetheory), whereas what is normally treated as the antisymmetric part of the connection coe�cients is consid-ered as an additional tensor �ABC . Despite the fact that this split introduces more objects into the formalism,the reduction to GR is made slightly more transparent, and allows the antisymmetric part of the torsion-freeconnection to be identi�ed with the structure constants of a general basis in the standard way2. Although

2If one had instead de�ned a covariant derivative that was not torsion-free as in [17], one would have to introduce

additional �elds into the action, and the split between the e�ects of torsion and those due to the choice of coordinates

3

the connection will not initially be assumed torsion-free in the action, a tensor of Lagrange multipliers (LABC )will be introduced to ensure vanishing torsion at the level of the �eld equations. In fact this is not necessary,as one could impose the torsion-free conditions from the outset and remove the appropriate connection coe�-cients from the action. This in fact is what will be done for most of the torsion-free conditions, as a judiciouschoice of arbitrary torsion terms added into the action causes the related connection components to disappearaltogether. However, there are three conditions that have not been imposed in this way, as it was felt thatthe formalism was simpli�ed slightly by not doing so (discussed further in Section III A).The action given in [8] (G = c = 1) extended by the above mentioned Lagrange multiplier term is then3:

SNGT=

ZM

d4xE[�gABRNS

AB � gABre[A[W ]B]

+12�g

(AB)WAWB + lA�A + 14m

2g[AB]g[AB] + LABC TCAB ]: (1.1)

Each contribution to this will be denoted by corresponding Lagrangians (in order of appearance), each tobe separately decomposed in Section III A: LNS

R ;LrW ;LW2 ;Ll;Lm;LT . Densities with respect to the de-

terminant of the metric gAB, will be indicated by boldface (eg. gAB :=p�det(g)gAB). The action (1.1)

encompasses that for mNGT when � = 3=4, `old' NGT (� = 0;m = 0) (equivalent to Einstein's Uni�ed FieldTheory), and recovers GR in the limit that all antisymmetric metric functions are set to zero. For simplicity,and to avoid the ambiguities inherent in de�ning the antisymmetric contributions to the matter stress-energytensor, source-free spacetimes will be dealt with exclusively in this work.The Ricci-like tensor in the action has been split up into two contributions: RNS

AB = RAB +R�AB. The �rst

is identi�ed as the Ricci tensor (i.e. it reduces to the GR Ricci tensor in the limit of vanishing antisymmetricsector):

RAB = eC [�CBA]� eB [�CCA]� 1

2eA[�CBC] +

12eB [�

CAC] + �EBA�

CCE � �ECB�

CEA + �E[AB]�

CEC ; (1.2a)

and the second contains contributions from the antisymmetric tensor �eld � (�A := �BAB):

R�AB = reC [�]

CAB +re[A[�]B] + �CAD�

DBC : (1.2b)

The Ricci tensor is de�ned from the two independent contractions of the geometric curvature tensor derivedin the standard way from the connection that de�nes parallel transport of the general basis vectors [8] by:reA[e]B = �CABeC . The basis vectors are related in the usual way to a coordinate basis via a vierbeineA = EA

�@�, although the vierbein in this case is not necessarily mapping the coordinate basis into a Lorentzframe. As mentioned above, the torsion tensor:

TABC = �A [reB [e]C �reC [e]B � [eB ; eC]] = �ABC � �ACB � CBCA; (1.3)

will be constrained to vanish by variation of the Lagrange multipliers L, allowing one to determine theantisymmetric part of the connection from the vierbeins.Variation of both the connection coe�cients (�) and the antisymmetric tensor (�) will result in the com-

patibility conditions:

CABC := reC [g]

AB � gDB�ACD � gAD�BDC + 23��

[AC �

B]D g(DE)WE = 0; (1.4)

where equations (4.14) and (4.28) of [8] have been written in terms of the densitised inverse metric components.The remaining �eld equations are4:

�A = 0; (1.5a)

lA = 13�g

(AB)WB ; (1.5b)

reB [g][AB] = �g(AB)WB; (1.5c)

RAB = RNS

AB + re[A[W ]B]� 1

2�WAWB � 14m

2MAB = 0; (1.5d)

would be obscured.3For the translation of the conventions used here for a coordinate basis with that of [2,6], see Section 1 of [8].4Note that one of (1.5a) and (1.5c) is redundant, as l has been replaced in the compatibility conditions deriveddirectly from the action using (1.5b), to give the form (1.4).

4

where the mass tensor appearing in the last of these is:

MAB = g[AB] � gCAgBDg[CD] + 12gBAg

[CD]g[CD]: (1.6)

Neither (1.4) nor any of (1.5) will be used in order to pass to the second-order form of the action; they willinstead be used in surface decomposed form to check that Hamilton's equations (as derived from the NGTHamiltonian developed in the following section) are equivalent to the Euler-Lagrange equations quoted here.To date there has been no complete physical interpretation of the antisymmetric structure, nor will anything

of the sort be attempted here. The action (1.1) will be considered as the fundamental starting point of theHamiltonian analysis, and the goal of the present line of research is be to make some headway towardsunderstanding the dynamics of the system. Thus in general one is left with many ambiguities as to how tomake measurements on the skew sector, and the meaning of the various additional antisymmetric tensors aswell as the additional contributions to the symmetric sector (GR) quantities is unclear. However, once oneunderstands what data is con�gurable (Cauchy data), the �eld equations �x the evolution uniquely (up todi�eomorphisms), allowing one to determine the physical implications of the presence of the antisymmetricsector in cases where the interpretation of the symmetric sector is essentially that of GR. This will be thecase in the asymptotic region of spacetimes that are asymptotically dominated by the symmetric sector, aswell as in perturbative scenarios (discussed in Section VI) where initial data is close to GR con�gurations.The next step in the analysis is to introduce the foliation of the spacetime manifoldM, and decompose all

objects into components perpendicular and parallel to the surfaces.

II. METRIC AND COMPATIBILITY IN A SURFACE COMPATIBLE BASIS

Normally one would assume at this point that spacetime is globally hyperbolic, so that a Cauchy surface�0 exists in M (a closed achronal set without edge, intersected by any smooth causal curve that is futureand past inextensible). Spacetime may then be viewed as the evolution of 3-geometries, that is, (M; g)emerges from the set of �eld con�gurations on spacelike hypersurfaces that make up a foliation of spacetime:f(�t; gt)jt 2 I � R1g. Then one may show that di�eomorphically equivalent initial data on �0 generatespacetimes that are di�eomorphically equivalent, and a physical spacetime is then identi�ed as an equivalenceclass of such solutions (equivalent up to di�eomorphisms).For NGT however, making a sensible de�nition of hyperbolicity is somewhat more complicated. In Chapter

IV of [18], it is demonstrated that the �eld equations of Einstein's Uni�ed Field Theory propagate informationalong three separate light cones (the same may be seen in covariant wave equations for particles of spin greaterthan or equal to one [19]). These cones are de�ned by symmetric metric tensors, the set of which will bedenoted: fgcg, and referred to as the causal metrics of NGT5. Thus the statement that a vector is timelike(spacelike) with respect to fgcg wil indicate that it is timelike (spacelike) with respect to all of the causalmetrics in fgcg. Global hyperbolicity would then require that �0 be achronal with respect to fgcg, and thecausal curves would of course also be causal with respect to fgcg. This would ensure that none of fgcg wouldbecome degenerate anywhere in spacetime, and hence that �0 is in fact a Cauchy surface. At this stage it isnot known whether one can require such a condition on the fundamental tensor6. However even if this is notpossible, the Hamiltonian system still exists in a less global context. One may instead specify initial data ona large enough S0 � �0 in order that the con�guration on St � �t (St is a closed achronal set) is determineduniquely. This requires that fgcg is known in order to determine D+(S0) or D

�(St) (the future (past) Cauchy

5In the Uni�ed Field Theory, fgcg = fl; h; g where l represents g( ) and its inverse, h represents g( ) and its inverse,

and is de�ned by �� = 2hgh���l�� where h = det[h�� ] and g = det[g�� ]. These metrics were found to be compatible

in the sense that there is a largest and smallest speed of light. The set of causal metrics is not known explicitly formNGT at this time, and until it is, strictly speaking one cannot discuss the causal structure of any given spacetime.6What is commonly referred to as the `metric' of NGT is the tensor determined directly by the �eld equations

(appearing in (1.1)), the determinant of which de�nes the spacetime volume element in the action, and from which onederives the causal metrics fgcg. The presence of antisymmetric components indicates that it is in fact not a metric (in

the usual sense) at all, and will therefore be referred to as the `fundamental tensor' (in accordance with Lichnerowicz

[16]). Note however that one may consider is as a Hermitian metric in a complex or hyperbolic complex space [20{22],or in more general spaces [23].

5

development or domain of dependance of S0 (St)), to ensure that S0 � D+(S0) or S0 � D�(St). For thepurposes of this work, it will su�ce to discuss the global problem, as one may restrict the results to in astraightforward manner once the causal metrics of mNGT are known.

A. The Surface Adapted Basis

We begin by assuming that there is a time function t that is used to foliate M into hypersurfaces ofconstant time �t. This requires that each (3-)surface of constant time, �t, be spacelike with respect to fgcg,and rtt = 1 de�nes a vector t that is timelike with respect to fgcg. This ensures that the degrees of freedomthat travel along each of the light cones in fgcg will evolve forward in what has been chosen as time.In GR, one may then introduce a coordinate basis on the surface and a unit normal, such that the metric is

of the form: g = �? �? � ab�a �b. This e�ectively reduces the metric to a Riemannian metric on �, andthe projection of dt perpendicular and parallel to the surface are the lapse and shift functions respectively[24]. It is the components of this 3-d metric on � that are taken to be the Canonical coordinates in theHamiltonian approach to GR, and all other spacetime tensors are decomposed onto the surface by consideringthe components perpendicular and parallel to the surface as separate tensors on �.On attempting to generalize this to NGT, one �nds that the presence of more than one physical metric on

M implies that there is no physically well-motivated or natural choice of metric on � that would play thesame role. One may introduce a coordinate basis on the surface in the usual way, but in general each of themetrics in fgcg will provide a di�erent de�nition of the unit normal vector e?, and the resulting decompositionof tensors onto � will be inequivalent. However as this is merely a choice of parameterization, the physicalcontent of the system will be independent of the choice of metric used to de�ne the surface decomposition.In this work, the components of g( ) will be used to de�ne the unit normal e?, therefore making a particularchoice of decomposition:

g�1( ) = e? e? � (ab)ea eb; (2.1a)

g�1= e? e? +Bae? ea �Baea e? � abea eb: (2.1b)

This choice is made in order to simplify as much as possible the form of the action (1.1), as the fundamentaltensor appears in the form gAB.The time vector is decomposed as t = tAeA = Ne? + Naea, where (N;Na) are the lapse (component

of t along e?) and shift (projection of t on the surface �t). This basis can therefore be written in terms

of the coordinate basis (@t; @a) by: e? = 1N@t � Na

N@a, ea = @a, and the covector bases by: �? = Ndt,

�a = dxa +Nadt. This de�nes the vierbein through �A = EA�dx

�, the determinant of which is:

E := det[EA�] = N: (2.2)

Note that ea is a coordinate basis on � and can be therefore be thought of as acting on tensor �elds as an

ordinary derivative @a. It is also useful to denote surface vectors (not just the components) by ~V := V aea andcovectors by ~! := !a�

a. It is not di�cult to check that in the coordinate basis (@t; @a), g�1( ) and its inverse

take on the usual ADM form [9].The metric is found from the inverse of (2.1b):

g = F�? �? � (�a � �a)�? �a � (�a + �a)�a �? �Gab�

a �b; (2.3a)

where:

F := 1 +G(ab)BaBb; (2.3b)

�a:= G[ab]Bb; (2.3c)

�a:= G(ab)Bb; (2.3d)

from which one �nds �aBa = 0. The spatial part of the fundamental tensor (G) is given as the inverse of

� B B:

( ac �BaBc)Gcb = Gbc( ca � BcBa) = �ab : (2.3e)

6

Note that invertibility of the fundamental tensor (2.3) is required in order that the volume element inthe action be nondegenerate, and this requires that G exist. Given that the fundamental tensor, g, is anondegenerate solution of the �eld equations in no way guarantees that fgcg are all nondegenerate, allowing forthe possibility that a regular solution of the �eld equations has regions in spacetime where NGT perturbations`see' a singular spacetime.Some useful metric identities may be derived from (2.3e):

(ac)G[cb] + [ac]G(cb) + Ba�b= 0; (2.4)

(ac)G(cb) + [ac]G[cb] �Ba�b= �ab ; (2.5)

(ac)�c + [ac]�c= 0; (2.6)

(ac)�c + [ac]�c= FBa: (2.7)

Varyingp�g with respect to the densitised components of the inverse of the fundamental tensor results in:

�p�g = � 2

2 � F �a�Ba +

1

2� F Gba� ab: (2.8)

This will be used order to compute the variation of the Hamiltonian of NGT with respect to the same densitisedcomponents, as well as applied as an ordinary derivative relation in the surface-decomposed compatibilityconditions in Appendix A 1. Note that the condition F = 2 corresponds to the case where

p�g is independentof (Ba; ab), and F 6= 2 will be assumed throughout this work.The non-vanishing structure constants for this basis are found to be:

[e?; ea]= C?a?e? + C?a

beb; (2.9)

C?a?= ea[ln(N )]; C?a

b =1

Nea[N

b]; (2.10)

and if one were to require that the connection � be torsion-free using (1.3), the antisymmetric components ofthe connection would be:

�?[?a] =1

2ea[ln(N )]; �b[?a] =

1

2Nea[N

b]; �?[ab] = 0; �a[bc] = 0: (2.11)

A �nal useful property of this basis is:

eb[Ca?b] + ea[C?b

b] + Ca??Cb?

b + Ca?bC?b

? = 0; (2.12)

which can either be derived directly from a contraction of the Jacobi identity, or proved by a brute forceinsertion of the structure constants (2.9). This will be useful when identifying the di�eomorphism constraints,linking them to the algebraic �eld equations in Appendix A 2.In beginning with the components of the inverse of the fundamental tensor in (2.1), the con�guration of the

system is being described by the degrees of freedom N;Na;Ba; ab (the latter two are included as densitiessince they will turn up later on as conjugate momenta in this form). As one can see from the form of (2.3),di�erent parameterizations of the metric will have a large e�ect on the details of the analysis presented here.If one had begun by requiring that the symmetric part of the fundamental tensor took on the ADM form,then the inverse (that appears in the action) would have had nontrivial g(0a) components, and the analysiswould have been more complicated. In NGT, one must be slightly more careful to be consistent in whatone is calling the independent degrees of freedom, as the antisymmetric degrees of freedom will mix with thesymmetric sector as one moves from spatial metric to inverse spatial metric.

B. Surface Decomposition

Four-dimensional covariant derivatives are decomposed as usual [24] into surface covariant derivatives (writ-

ten as r(3)a ) whose action on basis vectors in T� is given by r(3)

a [e]b = �cabec, and contributions from deriva-tives o� of � which will be written in terms of the surface tensors:

7

� : = �???; (2.13)

ca := �?a?; aa : = �??a; �a := �a??; (2.14)

wab := �a?b; uab : = �ab?; kab := �?ab: (2.15)

(Both k and �cab will initially be considered as non-symmetric, allowing for the possibility of non-zero torsion.The torsion-free conditions will be imposed for these components very early in the development, and fromthen on they will be assumed to be symmetric.) In GR most of these are related by algebraic compatibilityconditions (�a = abab; ca = 0;� = 0; uab = ackbc), and the absence of torsion would allow one to furtherrelate a and (u;w) to the structure constants. The skew tensor �:

ba := �?a?; jab := �?ab; vab := �a?b; �abc := �abc; (2.16)

the torsion Lagrange multipliers:

La := La?? ; Lab := Lab? ; Lab := La?b ; Lbca := Lbca ; (2.17)

and the vector �elds: WA = (W;Wa) and lA = (l; la), are all decomposed in a similar fashion. Traces are

denoted as v := vaa, and the traceless part: vaTb := vab � 1=3�abv. The components of �A are given by�? = v;�a = ba+�

bab, and jab is an antisymmetric surface tensor by de�nition. It also will be useful to de�ne

a few combined quantities that will appear:

Kab:= kab + jab; (2.18a)

kab:=12 (

acKbc + caKcb) = (ac)kbc + [ac]jbc; (2.18b)

jab:=12 (

acKbc � caKcb) = [ac]kbc + (ac)jbc; (2.18c)

uab:= 12( (ac)ubc + (bc)uac + [ac]vbc + [bc]vac); (2.18d)

vab:= 12( [ac]ubc � [bc]uac + (ac)vbc � (bc)vac): (2.18e)

Aside fromK, which will be useful as a notational tool to combine the results of both sectors into one relation,these combinations correspond to symmetric (k; u) and antisymmetric (j; v) sector contributions once an indexhas been `raised' by the spatial metric ab.

C. Lie Derivatives

A Lie derivative on � (to be denoted by J ) is required that properly represents the specialization of the

spacetime Lie derivative to the 3-surface (in the case that ~X; ~Y 2 T�), as well as generalizing properly toderivatives o� the surface. It is easy to see that:

J ~X[~Y ]= [ ~X; ~Y ] = (Xaea[Y

b]� Y aea[Xb])eb; (2.19a)

J ~X[Y ?]= Xaea[Y

?]; (2.19b)

since in the �rst case one has the standard de�nition of the Lie derivative, and the second is the Lie derivativeof a surface scalar (the ? index on the perpendicular components of the vectors and covectors has beenretained in this section). The derivative of a vector and a scalar perpendicularly o� of � that properly takesinto account the vector nature of the former is [24]:

JX?e? [~Y ]= X?(e?[Yb] + Y aC?a

b)eb; (2.19c)

JX?e? [Y ?]= X?e?[Y?]: (2.19d)

The structure constant in the �rst of these takes into account the possibility that the surface basis may`move' on � as one moves perpendicularly o� the surface. Note that these are not just projected onto thesurface; Y ? is considered to be a scalar with respect to the surface derivatives, not a vector pointing normalto the surface. The Lie derivative of covectors is treated in a similar way:

8

J ~X[~!]= (Xaea[!c] + !aec[X

a])�c; (2.20a)

J ~X[!?]= Xaea[!?]; (2.20b)

JX?e?[~!]= X?(e?[!b] + !aCa

b? )�b; (2.20c)

JX?e?[!?]= X?e?[!?]; (2.20d)

and the de�nition is extended to arbitrary tensors in the standard manner.From the spacetime Lie derivatives of tensor densities:

$X [T]=p�g$X [T ] + T$X [

p�g ]; (2.21a)

$X [p�g ]= 1

2gCB$X [gBC ] = �1

2gCB$X [g

BC ]; (2.21b)

and using the de�ned surface Lie derivatives, one �nds:

@t[p�g] = Jt[

p�g] = NJe?[p�g] + J ~N

[p�g] = N (e?[

p�g]�p�gC?aa) +r(3)a [N]a: (2.21c)

Time derivatives of arbitrary tensors or tensor densities are given as in GR by: @t[Z] = Jt[Z] = NJe?[Z] +J ~N

[Z], where Z is any tensor density.One would now like to identify the extrinsic curvature of the surface � under consideration. Although

there are a variety of equivalent ways of describing it in the context of Riemannian geometry, none of theseis equivalent for the type of nonsymmetric theory considered here. One way is to compute the perpendicular

component of the parallel transport of a surface vector along the surface: �?hr ~M

[ ~N ]i= �?abM

aN b =

kabMaN b, in conjunction with the fact that the surface projection is just the surface transport result in GR

�ahr ~M

[ ~N ]i= �a

hr(3)

~M[ ~N ]

i[25]. One could equivalently describe it by the change in the unit normal as it is

parallel transported along the surface ra[e?] [26]. In GR, the magnitude of this vector does not change whentransported along the surface (since �?a? = ca = 0), and the result is a vector tangent to the surface describingthe amount that the normal vector must `rotate' in order to remain normal ra[e?] = �ba?eb = ubaeb. Metriccompatibility relates these two de�nitions since kab = bcu

ca. In the case of NGT, this de�nition is clouded by

the fact that in general ca 6= 0, and the length of the unit normal will not be preserved under (this de�nitionof) parallel transport. One could also identify the extrinsic curvature of � with the change of the 3-metricperpendicularly o� the surface re?[g][ea eb] = �Je? [ ]ab + cbu

ca + acu

cb = 0, giving Je? [ ]ab = 2kab (or

the equivalent derived from the inverse metric).Although equivalent in GR, none of these de�nitions coincide in nonsymmetric theories in general. Indeed,

since the compatibility conditions cannot entirely be written in terms of a metric compatible connection, thereis no truly natural choice of parallel transport, and any choice of `extrinsic curvature' based upon a covariantderivative is conventional at best. The approach followed here will be the �rst of those described above,essentially due to convenience. It turns out that with the choice of decomposition made here, �?ab = kab isa canonical coordinate, and as such plays a more central role than u (which will turn out to be a derivedquantity, as it acts as a determined Lagrange multiplier), or the relation derived from the evolution of theinverse spatial metric.The stage is now set, as one now has the tools necessary to decompose spacetime tensors and covariant

derivatives into spatially covariant objects, and identify time derivatives in the action. The con�gurationspace variables have also been chosen to be: (N;Na;Ba; ab). In Appendix A 1 the compatibility conditions(1.4) are decomposed, and the �eld equations (1.5) in Appendix A 2. In both cases, algebraic conditions thatwill determine Lagrange multipliers on � have been identi�ed, as well as those responsible for time evolution.The reader is again reminded that all of these will be derived from the Hamiltonian, and are only given here inorder to introduce some de�ned quantities (C;G;Z) that will help to simplify the presentation of the algebra,and ensure that one is �nding results that are in accord with the Lagrangian variational principle.

III. DETERMINING THE HAMILTONIAN

This section is concerned with obtaining the form of the Hamiltonian for NGT. The various terms inthe Lagrangian (1.1) will be decomposed into surface compatible form, and the �elds that appear as timederivatives identi�ed. After this, writing down the form of the Hamiltonian turns out to be a rather simpletask, however the work is far from done. The Hamiltonian is riddled with Lagrange multipliers, and theconstraint analysis is where one begins to see the full structure of NGT emerge.

9

A. The Surface Decomposed Action

The simplest Lagrangian density terms in (1.1) are decomposed as:

Lm= �Hm = 12m

2NBa�a +14m

2N [ab]G[ab]; (3.1a)

Ll= N lvaa +N la(ba + �bab); (3.1b)

LT= 2N [LabK[ab] + Labc �c[ab] + Lab(uba �wba +C?a

b) + La(ca � aa +C?a?)]; (3.1c)

LW2= 12�N [

p�g(W )2 � (ab)WaWb]; (3.1d)

LrW= NBa[r(3)a [W ]� Je? [W ]a +W (aa � ca)] + N

[ab]r(3)

[a [W ]b]: (3.1e)

The �rst of these de�nes Hm, which will be useful in Section IVC. The remaining contribution LRNS , will befurther split into contributions from each of the symmetric and antisymmetric sectors as LR and LR� , andfurther split to give contributions from each component of the tensor (for example, LR(ab)

corresponds to the

contribution from N (ab)R(ab)). Note that the fact that g(?a) = 0 in this decomposition implies that thereis no contribution from LR(?a)

, which is essentially why the choice of surface adapted basis (2.1) was made.

Making a choice of parameterization in which g(?a) 6= 0 would have explicitly introduced a term involving _Ba

into the action (see (A5b)). The remaining contributions are:

LR�??

= �Np�gvabvba;LR�

(ab)= N

(ab)[babb � jacvcb � jbcvca + �dac�cbd];

LR�[?a]

= �NBa[Je?[�]bab �Je? [b]a + 2r(3)b [v]ba �r(3)

a [v]

+2bbuba � 2bau+ 2(ab � cb)vba � (aa � ca)v � 2ubc�

cba + 2jab�

b];

LR�[ab]

= N [ab][Je?[J ]ab +r(3)

c [�]cab +r(3)

[a [b]b] +r(3)

[a [�]cb]c

�jacucb � jcbuca + (� + u)jab � kcavcb + kcbvca + aabb � abba + ac�

cab];

LR??= �Np�g[�Je? [u] +r(3)a [�]a + �u+ �a(aa � 2ca)� uabu

ba];

LR(ab)= N

(ab)[R(3)

(ab) + Je? [k](ab) �r(3)

(a [a]b) � aaab + (� + u)kab � kcbuca � kacu

cb]; (3.1f)

where R(3)

(ab) is the surface Ricci tensor given by (1.2a) determined from surface covariant derivatives, and the

identity R[AB] = 0 (see Section IV in [8]) has been used.The terms that result in time derivatives in the action are then:

�NBaJe? [W ]a �NBaJe? [�]bab + NBaJe?[b]a + Np�gJe?[u] + N

abJe?[K]ab; (3.2)

and so de�ning the vector �eld:

W a = �Wa � �bab + ba; (3.3)

will simplify the remaining analysis. Rewriting the above in terms of the time derivative gives:

BaJt[W ]a +p�gJt[u] +

abJt[K]ab

�BaJ ~N[W ]a �

p�gJ ~N[u]�

abJ ~N[K]ab;

where:

J ~N[u]= Nar(3)

a [u]; (3.4a)

J ~N[W ]a= N br(3)

b [W ]a +W br(3)a [N ]b; (3.4b)

J ~N[K]ab= N cr(3)

c [K]ab +Kcbr(3)a [N ]c +Kacr(3)

b [N ]c: (3.4c)

Although these have been written in surface covariant form, all dependence on the spatial connection coe�-cients vanish in these. As is it is clear that W will be a canonical coordinate, (3.3) will henceforth be used toreplace W everywhere in the remainder of this work.

10

Gathering together terms in the action that are multiplied by the same metric function and identifying thetime derivatives as above, the Lagrangian density for NGT can now be written:

LNGT = �Hm +BaJt[W ]a +p�gJt[u] +

abJt[K]ab �BaJ ~N[W ]a �

p�gJ ~N[u]�

abJ ~N[K]ab

+Np�g[lv + la(ba + �bab) + 2La(ca � aa +C?a

?)

+12�(W )2 � vabv

ba �r(3)

a [�]a � �u� �a(aa � 2ca) + uabuba]

+NBa[r(3)a [W ] +W (aa � ca)� 2r(3)

b [v]ba +r(3)a [v] (3.5)

�2bbuba + 2bau� 2(ab � cb)vba + (aa � ca)v + 2ubc�cba � 2jab�

b]

+N [ab][�r(3)

[a [W ]b] +r(3)c [�]cab + 2r(3)

[a [b]b]

�jacucb � jcbuca + (� + u)jab � kcavcb + kcbvca + aabb � abba + ac�

cab]

+N (ab)[R

(3)

(ab) �r(3)

(a [a]b) � aaab + (� + u)kab � kcbuca � kacucb

�12�(W a + �cac � ba)(W b + �dbd � bb) + babb � jacv

cb � jbcv

ca + �dac�

cbd]: (3.6)

Up to this point, the torsion-free conditions have been applied arbitrarily in order to simplify the Lagrangiandensity as much as possible. At this stage, the �elds k[ ] and �[ ] have been dropped along with their associated

torsion Lagrange multipliers Lab and Labc , as they would appear only in these torsion Lagrange multiplierterms. From this point onwards, both kab and �cab will refer to symmetric objects. The Lagrange multiplierterm imposing the torsion-free condition:

wab = uab +C?ba; (3.7)

has also been dropped, as once all terms (in the action as well as compatibility conditions and �eld equations)are written in terms of Je?[ ], w will not appear either, and may be determined by (3.7) once uab is known.One would also like to impose the torsion conditions to solve for aa explicitly, however this would introducea number of terms involving the structure constants C?a

? into the action, compatibility conditions and �eldequations, and it was felt that avoiding this simpli�ed the algebra. Although there is nothing preventing onefrom doing this, the presentation is slightly simpler if the torsion condition is not imposed in this case.

B. Legendre Transform

In the form (3.5), the �elds with conjugate momenta are trivial to identify (these are densities by de�nition,and will not be written in boldface)7:

p:=�LNGT

� _u=p�g; (3.8a)

pa:=�LNGT

�_W a

= Ba; (3.8b)

�ab:=�LNGT

� _Kab

= ab: (3.8c)

One could in principle treat all �elds in the action as canonical coordinates, generating many cyclic momentaand hence many second class constraints that are trivially removed by the introduction of Dirac brackets[27]. Instead the more economical approach of treating any �eld in the action whose time derivative does notappear anywhere as a Lagrange multiplier �eld will be followed8.

7Note that the Canonical coordinates have been taken to be connection components (and Wa) resulting in the

densitised metric components appearing as conjugate momenta. By discarding a total time derivative in the action,

one could easily arrange for the metric degrees of freedom to play their more traditional role as Canonical coordinates.8It is common to include the lapse and shift functions and their conjugate momenta in phase space, particularly if

one is attempting to realize all of the generators of the spacetime di�eomorphism group on phase space [28,29]. This

will not be done in this work simply to avoid the unnecessary algebra, as it is a trivial matter to extend the formalismlater if necessary.

11

The extended Hamiltonian density is written as:

H� = H0 +N�ab(�ab �

ab) + 2N�a(pa �Ba) +N�(p�p�g); (3.9a)

where the form of the Lagrange multiplier terms that enforce the algebraic conditions (3.8) have been writtenas the reduction of the covariant form: E�AB(�

AB � gAB). The basic Hamiltonian density is derived fromthe Lagrangian density in the standard manner: H0 = p _q � L, and is found to be (equivalent to droppingtime derivatives in (3.5) and taking its negative):

H0 = Hm +p�gJ ~N

[u] +BaJ ~N[W ]a +

abJ ~N[K]ab

�Np�g[lv + la(ba + �bab) + 2La(ca � aa + C?a?)

+12�(W )2 � vabvba �r(3)

a [�]a � �u� �a(aa � 2ca) + uabuba]

�NBa[r(3)a [W ] +W (aa � ca)� 2r(3)

b [v]ba +r(3)a [v]

�2bbuba + 2bau� 2(ab � cb)vba + (aa � ca)v � 2jab�

b + 2ubc�cba]

�N [ab][�r(3)

[a [W ]b] +r(3)c [�]cab + 2r(3)

[a [b]b]

�jacucb � jcbuca + (� + u)jab � kcav

cb + kcbv

ca + aabb � abba + ac�

cab]

�N (ab)[R

(3)(ab) �r

(3)(a [a]b) � aaab + (� + u)kab � kcbu

ca � kacucb

�12�(W a + �cac � ba)(W b + �dbd � bb) + babb � jacvcb � jbcvca + �dac�

cbd]: (3.9b)

Hamiltonians and Hamiltonian densities will be related everywhere by (for example): H� =R�d3xH�, where

H will refer to the Hamiltonian, and H to the related Hamiltonian density.The canonical phase space variables that describe the con�guration of the system at any time are:�

(P I ; QI)=�(p; u); (pa;Wa); (�

ab;Kab); (3.10)

and the usual canonical Poisson brackets have been assumed:

fQI(x); QJ(y)g = fP I(x); P J(y)g = 0; (3.11a)

fQI [fI]; P J [gJ ]g =

Z�

d3z f I(z)gI (z): (3.11b)

The notation QI [fI ] is used to denote the smearing of the object by an appropriately weighted test tensor

(eg. pa[fa] =R�d3zpa(z)fa(z)) leaving a scalar on �, and the sub/superscript `I' indexes the sets in (3.10).

The Poisson brackets are standard:

fF [Q;P ];G[Q;P ]g=Z�

d3zXI

��F

�QI(z)

�G

�P I(z)� �F

�P I(z)

�G

�QI(z)

�; (3.12)

where (F;G) are scalar functions of the canonical coordinates.The �elds (p; u), although symmetric sector �elds, do not show up in the standard Hamiltonian formulations

of GR as canonical variables. There one has u = abkab, and this is used in the action to either remove thetime derivative from u (second-order action) or write: _u = ab _kab + kab _

ab [30]. The point being that innonsymmetric theories, one cannot replace all terms in the action by various combinations of the spatialmetric and extrinsic curvature in a straightforward manner, and it seems one must be content to carry aroundthe 103 Lagrange multipliers:

fLg:= f(La); (l; la)g; (3.13)

f�g:= fW; ba; vab; �abcg; (3.14)

f�g:= f�; ca; aa; �a; wab; uaTb;�abcg; (3.15)

f�g:= f�; �a; �(ab); �[ab]g; (3.16)

fNg:= fN;Nag; (3.17)

f g:= fBa; abg; (3.18)

where the Lagrange multipliers have been split up into smaller subsets for later reference.

12

IV. CONSTRAINT ANALYSIS

Now begins the analysis of the constraints and conditions that follow from variations of the Hamiltonianwith respect to the Lagrange multipliers. It will be found that all of the compatibility conditions and �eldequations (1.3,1.4,1.5) are reproduced from the Hamiltonian, leaving fNg as undetermined Lagrange multi-pliers enforcing the di�eomorphism constraints in the standard form of the Hamiltonian. Two second classconstraints will also be found which could be used to strongly remove the conjugate pair (p; u) from the phasespace of the theory, leaving the symmetric phase space with the same dimension as in GR. As the results areunsurprising and the algebra not unduly di�cult but rather lengthy, the results will be given as sparsely aspossible.

A. Constraints Enforced by f�g, fLg, f�g and f�g

The Lagrange multipliers f�g are designed to enforce the conjugate momenta conditions (3.8):

�H�

��ab� N (�ab �

ab) � 0; (4.1a)

�H�

��a� 2N (pa �Ba) � 0; (4.1b)

�H�

��� N (p�p�g) := N 1 � 0: (4.1c)

These conditions clearly determine f g, with one constraint left over: 1, which has been identi�ed in (4.1c).From fLg, the Lagrange multiplier enforcing the remaining torsion constraint from (2.11) gives:

�H�

�La� �2N (ca � aa +C?a

?) � 0; (4.2)

which will be used to determine aa. The constraints resulting from the variation of lA are:

�H�

�l� �Nv � 0; (4.3a)

�H�

�la� �N (ba + �bab) � 0: (4.3b)

These are the components of (1.5a), and will be used to determine v and �bab respectively.The conditions arising from the variation of f�g are now calculated, �rst varying H� with respect to W

and then vba:

�H�

�W� NC [?a]

a � 0; (4.4a)

�H�

�vba� �2N (C

[?a]b � 1

3C [?a]a )� N�ab (l� 1

3r(3)c [B]c) � 0: (4.4b)

Using the result that v � 0 from (4.3a), the trace of the last of these gives l � 13r

(3)c [B]c � �

3W, so that

C[?a]b � 0. Next varying ba gives:

�H�

�ba� 2NC

[ab]b �N [la � 1

3� (ab)Wb] � 0; (4.4c)

and using the above conditions one �nds that:

�H�

��cab� �NC [ab]

c � 0; (4.4d)

which, combined with the previous result, yields la � �3 (ab)Wb. At this point, all of the skew sector algebraic

compatibility conditions ((A2c) and (A2d)) as well as the �eld equations (1.5b) have been reproduced.

13

Varying f�g, one �nds:

�H�

��� NC(?a)

a =: N 2 � 0; (4.5a)

�H�

��a� �NC(??)

a � 0; (4.5b)

�H�

�ca� �2N [La + �a +Bbvab � 1

2BaW ] � 0; (4.5c)

where the �rst of these yields a constraint (which has been identi�ed as 2 in (4.5a)) discussed further inSection IVB, and the last is used to determine the Lagrange multiplier La as:

La � ��a �Bbvab +1

2BaW: (4.5d)

Using these, one computes:

�H�

�aa� N [C

(ab)

b �C(?a)

? ] � 0; (4.5e)

and the trace-free part of the variation with respect to uba gives the variation with respect to ubTa:

�H�

�ubTa� �2N [C

(?a)b � 1

3�abC

(?c)c ] � 0; (4.5f)

which yields C(?a)b � 0 when the constraint (4.5a) is taken into account. Finally, the variation with respect

to the surface connection coe�cients yields:

�H�

��cab� �NC(ab)

c � 0; (4.5g)

and all but C� of the symmetric sector algebraic compatibility conditions ((A1c), (A1e) and (A1f)) have beenfound, and the Lagrange multiplier La determined by (4.5d).

B. Time Evolution of the Canonical Fields and the constraints f g

Time evolution of the canonical variables is determined as usual (from (3.11) and (3.12)) by Hamilton'sequations:

_QI= fQI ;H�g ��H�

�P I; (4.6a)

_P I= fP I;H�g � ��H��QI

: (4.6b)

These evolution equations should reproduce the same dynamics as given by (1.3, 1.4, 1.5). It is convenientat this point to show that the canonical momenta evolve in accordance with the dynamical compatibilityconditions, and that the evolution of the canonical coordinates also allows one to identify conditions thatmust be found in order to reproduce the NGT �eld equations (1.5d). This is why this calculation is performedat this stage, even though strictly speaking it is not part of the constraint analysis.The functional derivatives of H� with respect to the canonical coordinates will be necessary:

�H�

�u� �r(3)

a [N]a +N (�� u� 2Baba); (4.7a)

�H�

�W a

� �Bar(3)b [N ]b +Bbr(3)

b [N ]a � N br(3)b [B]a

+N (r(3)b [ ][ab] +

[ab]C?b? + � (ab)(W b � 2bb)); (4.7b)

14

�H�

�kab� � (ab)r(3)

c [N ]c + (ac)r(3)

c [N ]b + (cb)r(3)

c [N ]a �N cr(3)c [ ](ab)

�N ( (ab)(� + u)� 2uab); (4.7c)

�H�

�jab� � [ab]r(3)

c [N ]c + [ac]r(3)

c [N ]b + [cb]r(3)

c [N ]a � N cr(3)c [ ][ab]

+N (Ba�b �Bb�a � [ab](� + u) + 2vab): (4.7d)

Time evolution of the conjugate momenta can be shown to be weakly equivalent to the dynamical compati-bility conditions (A1a, A1b, A2a, A2b):

_p= ��H��u

� Jt[p�g]� NC??? ; (4.8a)

_pa= � �H�

�W a

� Jt[B]a �NC [?a]

? ; (4.8b)

_�ab= � �H�

�Kab

� Jt[ ]ab + NCab

? ; (4.8c)

since in all cases (heuristically) _P � Jt[ ], and one is left with the result that C? � 0 in evolution. Using(A5) and the evolution of the coordinates:

_u=�H�

�p� N�; (4.9a)

_W a=�H�

�pa� 2N�a; (4.9b)

_Kab=�H�

��ab� N�ab; (4.9c)

allows one to de�ne the objects fRg:R:= N�� J ~N

[u]�NZ?? + 14m2NM??; (4.10a)

Ra:= 2N�a � J ~N[W ]a + 2NZ[a?] � 1

2m2NM[a?]; (4.10b)

Rab:= N�ab � J ~N[K]ab + NZab � 1

4m2NMab: (4.10c)

In order for Hamilton's equations (4.6) to reproduce the dynamical �eld equations (1.5d), one must �nd thatfRg � 0, correctly determining f�g.There are two constraints f g that have appeared from the variations so far ((4.1c) and (4.5a) respectively):

1� p �p�g[pa; �ab] � 0; (4.11a)

2� pu� �abKab � 0; (4.11b)

where the density has been written as functionally dependent on the other momenta in order to stress thatit must be considered as a functional of those �elds alone (as in (2.8)) when calculating Poisson brackets.Although all of the Lagrange multiplier variations have not been dealt with yet, the conditions that resultfrom requiring that these constraints are preserved in time will prove useful at this stage. Requiring that 1be preserved in time results in:

_ 1 = f 1;H�g � ��H�

�u� 2

2� F�a�H�

�W a

+1

2� FGba

�H�

�Kab

� � 4

2� FNC� � 0; (4.12)

where (2.8) has been used. This directly gives the remaining algebraic compatibility condition (A4a). Thepreservation of 2 gives:

_ 2 = f 2;H�g � �p� �ab�ab � u�H�

�u+Kab

�H�

�Kab

� � 4

F � 2

p�gR+ 2NG??; (4.13)

where R is de�ned in (4.10) and G?? by (A10a).

15

C. Variations of f g and fNg

In order to compute the variations with respect to f g, it is convenient to rewrite the extended Hamiltoniandensity (3.9a) in the form (fZg is de�ned in (A8)):

H�� Hm +p�g(NZ?? + J ~N

[u]� N�) +Ba(�2NZ[a?] + J ~N[W ]a � 2N�a)

+ ab(�NZab + J ~N[K]ab � N�ab); (4.14)

where � refers to the fact that some conditions and constraints have been imposed in H�, provided that theresults of the variations are not weakly a�ected. (Note that this form holds for variations with respect to f gonly.) The variation of Hm as de�ned in (3.1a) can be written (once again making use of (2.8)):

�Hm= �14Nm

2MAB�gAB

= �1

2Nm2(M[?a] �

1

2� F �aM??)�Ba +

1

4Nm2(Mab � 1

2� FGbaM??)�

ab; (4.15)

and one then computes the conditions enforced by f g to be (using (4.10)):

�H�

�Ba� �Ra + 2

2� F�aR � 0; (4.16a)

�H�

� ab� �Rab � 1

2� F GbaR � 0: (4.16b)

As it stands, the Hamiltonian H�, is not in the standard form:R�(NM + NaHa). It may be put in this

form by the removal of the surface term:

E� :=

Z@�

dSa[NbW bB

a + 2N bkab � 2NLa]; (4.17)

so that

H� =

Z�

d3x(NH+ NaHa) + E�: (4.18)

(Further discussion of surface terms will occur in the next section.) One then �nds directly the Hamiltonianconstraint H

H :=�H�

�N� �2G?? � 0; (4.19a)

which has been weakly identi�ed with the algebraic �eld equation (A10a) by making use of previous results.This constraint, combined with (4.13) and (4.16), establishes that fRg � 0. The supermomentum constraintsare:

Ha:=�H�

�Na= 2Bbr(3)

[a [W ]b] �W ar(3)b [B]b +

bcr(3)a [K]bc +

p�gr(3)a [u]� 2r(3)

b [k]ba

� pbea[W b]� eb[pbW a] + �bcea[Kbc] + pea[u]� 2eb[kba]

� �2G?a � 0; (4.19b)

which has been identi�ed with the related algebraic �eld equations (A10b). The identity (A7) has been used inorder to make the identi�cation ofH andHa with G

?? andG?a respectively. The form of the super-Hamiltonian

constraint is not given explicitly in (4.19a) as it can essentially be read o� from (3.9a) or (A10a) directly.Unlike the case of the supermomentum constraint, there seems to be no relatively simple formulation of thesuper-Hamiltonian constraint in terms of canonical variables alone. This leads to computational problems(as will be discussed in Section IVE), as Poisson brackets with H cannot be calculated in a straightforwardmanner.With these last constraints, the Euler-Lagrange equations of NGT have been fully reproduced by Hamilton's

equations combined with all of the algebraic compatibility conditions and �eld equations resulting from thevariation of the Lagrange multipliers.

16

D. Surface Terms

Up to this point all surface terms have been consistently ignored. This is �ne if � is closed (@� = ;) andthere is no surface to have terms on, but in general one must add a surface contribution to the action (1.1)designed to cancel o� any variations of H� that do not vanish on the @�. This ensures that the �eld equationsthat have been generated follow correctly from the Hamiltonian [31].To begin with, one may add to H� the surface contribution:

ELM =

Z@�

dSa[N�(bc)�abc �N�(ab)�cbc � Np�a � N�(ab)ab

+N�[bc]�abc + 2N�[ab]bb � 2Npbvab +Npav +NpaW ]; (4.20)

which ensures that the variation with respect to any of the determined Lagrange multipliers never result inany surface terms. In addition, there are surface terms involving variations of f g and fNg to consider,resulting from variations of ELM as well as the variations computed earlier in this section. These will be splitup into contributions involving the lapse and shift functions respectively, and the full Hamiltonian for NGTincluding surface contributions will be written:

HNGT :=

Z�

d3x(NH +NaHa) +E� + ELM +EN + E ~N; (4.21)

where E� is given by (4.17). The contribution from the shift function must satisfy:

�E ~N� �

Z@�

dSa[Nap �u+Na�bc�Kbc + kab�N

b +Napb�W b + paW b�Nb]; (4.22a)

and is given solely in terms of Cauchy data and the undetermined Lagrange multipliers fNg. The conditionthat the lapse contribution must satisfy is more complicated:

�EN � �Z@�

dSa�N (�abc��

(bc) � �cbc��(ab) � �a�p� ab��

(ab)

+�abc��[bc] + �[ab]�W b + 2bb��

[ab] � 2vab�pb +W�pa)

+(�(bc)�abc � �(ab)�cbc � p�a � �(ab)ab � 2La

+�[bc]�abc + 2�[ab]bb � 2pbvab + paW )�N�: (4.22b)

Note that this is a weak condition, and all determined Lagrange multipliers are properly expressed solely interms of Cauchy data and fNg.Just as is the case for GR, one cannot determine the form of these surface terms in general, but must

be content to determine EN and E ~Nfrom (4.22) for each speci�c situation. As a simple example, one may

assume an asymptotically at spacetime in which all of the antisymmetric functions fall o� fast enough asr ! 1 so that they will not contribute any surface terms. Assuming the fall-o� of the symmetric sectorvariables as given in [31], one �nds that there are no additional contributions from (4.22). Evaluating thenon-vanishing surface terms:

R@� dSaN

p�g( (bc)�abc� (ab)�cbc), assuming the asymptotically Schwarzschildform given in [32], yields the Schwarzschild mass parameter as expected.

E. Completing the Constraint analysis

It remains to be shown that the super-Hamiltonian and supermomentum constraints are preserved undertime evolution (they lead to no further constraints), and that they can both be taken to be �rst class, leavingf g (4.11) second class. It would then be possible to de�ne Dirac brackets and impose f g strongly9. In

9This would be accomplished quite simply by replacing u strongly everywhere using (4.11b), and considering both p

andp�g as functions of Ba and ab, as determined by (4.11a) and (2.8). Thus (p; u) would no longer be included in

phase space, nor would they appear anywhere in the Hamiltonian.

17

order to examine the supermomentum constraints, it is useful to note (and can be explicitly checked) thatthey do indeed generate spatial di�eomorphisms on �:

fQI;Ha[Na]g� J ~N

[Q]I; (4.23)�P I ;Ha[N

a]� J ~N

[P ]I; (4.24)

and, as this may obviously be extended to any tensor �eld on � built strictly out of the canonical variablesas: fT;Hc[N

c]g � J ~N[T ], one �nds (by explicit calculation, or using the above results for the last three):

f 1[f ]; 2[g]g� �Z�

d3z fgp4

2 � F; (4.25a)

f 1[f ];Ha[Na]g� �

Z�

d3z 1J ~N[f ] � 0; (4.25b)

f 2[f ];Ha[Na]g� �

Z�

d3z 2J ~N[f ] � 0; (4.25c)

�Ha[Ma];Hb[N

b]� �

Z�

d3zHaJ ~N[M ]a � 0: (4.25d)

It has also been checked by explicit calculation that the time evolution of the supermomentum constraint(4.19b) is weakly vanishing _Ha � 0.Thus the only condition required to show that the constraint algebra closes, is that fH[M ];H[N ]g is some

linear combination of the super-Hamiltonian and supermomentum constraints, and therefore vanishes weakly.This is extremely di�cult to check explicitly since (as noted before) H is not given in a simple form in termsof canonical variables alone, but instead depends on Lagrange multipliers which are di�cult to solve for ingeneral10. This calculation however, need not be performed explicitly.Considering the dynamics of NGT within the larger arena of hyperspace (the manifold of all spacelike

embeddings [33]), one may consider the surfaces �t as embedded in the Riemannian manifold (M; g�1( ) ). A

particular foliation de�nes a path in hyperspace, the tangent to which is the vector t, as de�ned in Section IIA.The hypersurfaces �t can then be viewed as deformations of an initial hypersurface �0 along the vector �eldt, and the Hamiltonian must then generate the evolution of the �elds under these hypersurface deformations.This implies that one must be able to write the Hamiltonian in the standard form:

R�d3x (NH + NaHa)

[24]. It can then be proved from the principle of path independence of dynamical evolution [34], that theHamiltonian closing relations are constrained to take on the form:

fH[M ];H[N ]g=Z�

d3zHa ab(Meb[N ]�Neb[M ]); (4.26)

�H[M ];Hb[Nb]= �

Z�

d3zHJ ~N[M ]; (4.27)

�Ha[Ma];Hb[N

b]= �

Z�

d3zHaJ ~N[M ]a: (4.28)

This principle essentially requires that the data on �0 evolve uniquely to �t, regardless of how �0 is deformedinto �t (i.e., independently of how one foliates the spacetime in-between), or equivalently, that evolutionalong di�erent paths in hyperspace (between identical initial and �nal points) yield identical results. Thisalso ensures that the collection of all possible �t's is can be interpreted as describing di�erent slicings of(M; g�1( )

), and is therefore a re ection of the di�eomorphism invariance of spacetime11.

10Most of these Lagrange multipliers can in fact be ignored in Poisson brackets if H is kept in a form very similar to

that appearing in H�, as their variations will vanish weakly using the algebraic compatibility conditions. The variationof La (or ca if aa has been strongly removed from the system) will only vanish if the smearing function is exactly

the lapse function N . This is not the case in general, and one would then require the variation of ba with respect to

canonical variables in order to compute Poisson brackets containing H; a variation that is not easy to compute.11In fact, a close inspection of the arguments in [35] adapted to this case, shows that one only knows (4.26) up to a

18

The Hamiltonian for NGT is thus found to be of the form (4.21), consistently generating the �eld equationsand compatibility conditions necessary to make Hamilton's equations (4.6) reproduce the �eld equations (1.3,1.4, 1.5) derived from the action (1.1) via the Euler-Lagrange equations. The number of degrees of freedomfor mNGT may now be easily computed using the standard algorithm as given by equation (1.60) of [36]. Inthis case, one �nds that the number of con�guration space degrees of freedom per spacetime point is: the # ofcanonical coordinates [fQIg in (3.10)] (13) � the # �rst class constraints [H and Ha in (4.19)] (4) � 1

2� the

# of second class constraints [f g in (4.11)] (12 � 2) = 8. Since all of the constraints exist in the symmetricsector, two of these are the propagating modes of GR, and the remaining six occur in the NGT sector. Inthe next section, the dynamics of pure GR con�gurations is reduced to that of GR (with a subtlety), and thelimit to old NGT (Einstein's Uni�ed Field Theory) is discussed.

V. THE REDUCTION TO GR AND `OLD' NGT

It is worthwhile at this point to make further contact with GR by considering how one reduces NGT to GRcon�gurations, as well as considering the dynamics of old NGT.

A. GR

There is, of course, an important surface in phase space on which all of the antisymmetric metric functionsare identically zero. If one sets up the system identically on this surface, it should remain on it since thedynamics there are identically those of GR (there is a complication, which will be discussed below). One couldeasily just eliminate all skew objects from the action, and derive the ADM results directly. In the contextof NGT-type theories one would prefer to view the GR dynamics as occurring in the larger phase space ofNGT, and view the reduction of dynamics as the imposition (by hand) of a constraint on the initial Cauchysurface that puts all skew Cauchy data weakly to zero. Consistent dynamics on this surface will require thatthis constraint be preserved in time.Considering �rst the case when � 6= 3=4, one sets up �elds on the initial surface such that all skew sector

Cauchy data (pa; �[ab]; jab;Wa) vanish identically; the algebraic compatibility conditions require that vab = 0and �abc = ba = 0 ((A2c) and (A2d) respectively). The time evolution conditions ((A2a) and (A2b)) (or

the equivalent in (4.8)) guarantee that the skew metric functions themselves (Ba and [ab] , respectively) willremain zero at later times. The skew sector conjugate momenta vanish by the antisymmetric part of (4.1a)and (4.1b), and remain zero in evolution once the previous results are inserted into (4.9) through the equationsde�ning � (4.10). The skew sector is thus consistently eliminated from any future dynamics once all skewCauchy data is set to zero on the initial hypersurface. The symmetric compatibility conditions (A1) reduceto those of GR, as do the symmetric time evolution equations in (4.8) for (what is then unambiguously) theextrinsic curvature of the surface k. The super-Hamiltonian and supermomentum constraints (4.19a, 4.19b)

then take on the familiar forms [26] (R(3) = abR(3)ab ):

H� kabkba � kaak

bb �R(3) � �2G?? � 0; (5.1a)

Ha� �2rb[kba � �bak] � �2G?a � 0: (5.1b)

Thus given � 6= 3=4, one �nds that if at any time, all of the antisymmetric sector canonical variables vanish,then the ensuing dynamics of the system is identically that of GR. The case where � = 3=4 is slightly di�erent,in that when one imposes (A2d) (and the remaining conditions), ba is left undetermined. Then, although onestill �nds that _�[ab] � 0 from (4.8), the evolution of pa is determined from _pa � 3

2N�(ab)bb, and is therefore

c-number (i.e. independent of the canonical variables). This c-number has been assumed to be zero in this case, as

it would not lead to further constraints, but to an inconsistent system of equations. It is generally believed that the

action for NGT generates a consistent set of �eld equations, and conservation laws (A11) have been derived based ondi�eomorphism invariance that imply that this c-number vanishes.

19

undetermined. Similar behavior occurs with the canonical coordinates jab;Wa, as there are nonvanishingcontributions from ba to (4.9). Thus the skew sector only remains trivial provided that b is chosen to be zeroon every surface, and the possibility exists that the system may pass through a GR con�guration withoutremaining there, in contrast to the case above12.

B. old NGT

The reduction to old NGT (or Einstein's Uni�ed Field Theory) is accomplished by setting � = 0 (settingm = 0 will not a�ect results, and can easily be left nonzero). Note that one should take � = 0 in thecompatibility conditions and �eld equations before solving for any Lagrange multipliers, as these are theequations that are properly derived from the action. The algebraic compatibility conditions in this case maybe solved for the determined Lagrange multipliers using similar methods to those in [10,11,37], with only mildconditions on the components of the fundamental tensor. However (A3c) is now a constraint related to theU (1) invariance that the action will now possess [2]. It is straightforward to show that this constraint is �rstclass (its Poisson bracket with all other constraints is weakly vanishing) and generates U (1) transformationson �: fW a; eb[p

b][�]g � ea[�]. There is then left �ve con�guration space degrees of freedom per spacetimepoint, consistent with the results for the Uni�ed Field Theory [16] as well as NGT [38,39] and [13]. In thelast of these, ad hoc constraints are applied in Section 4 so as to remove Wa from the dynamics, guaranteeinga weakly positive-de�nite Hamiltonian for the linearized theory. These constraints cannot in fact be imposedeven in the linearized theory [8], since lack of gauge invariance in the full action [40,41] manifests itself asa non-conserved source for the skew tensor �eld, exciting all �ve modes even at the linearized level. Thisis essentially equivalent to the results of Damour, Deser and McCarthy [15] who avoid discussing the sourceterms directly by demonstrating that the same e�ect is generated through the coupling of the skew �eld to aGR background.

VI. A CLOSER LOOK AT THE COMPATIBILITY CONDITIONS

The symmetric sector algebraic constraints are easily solved for in terms of the skew sector lagrange mul-tipliers and Cauchy data. � is determined by (A4a), ca is determined by (A4b), aa is determined by (4.2),�a is determined by (A1c), uab is determined by (A1e), the trace of which is the constraint 2. The relation

(A1f) can be used to uniquely solve for �abc, provided the inverse of (ab) exists. In the skew sector, vab isdetermined by (A2c), where the trace gives (A3c), determining the Lagrange multiplier W . The remainingequations for �abc and ba: (A2d), are complicated since one must �rst replace the symmetric sector Lagrangemultipliers, and then solve for the Lagrange multipliers in terms of Cauchy data. The behavior relevant tothis sector occurs in (A3d) alone, leaving the trace of (A2d) to determine �aTbc. (A3d) can be written as analgebraic relation for ba solely in terms of Cauchy data (using (A4b) and (A1e)):

p[(43�� 1) (ab)+BaBb � 1

3� [ac]G[cd]

(db)]bb

� eb[�[ab]] + kbbp

a � kabpb + 2

3��(ab)W b + �[ab]jbcB

c � 16��

[ab]�bW � 16��

[ab]G[bc] (cd)W d; (6.1)

which will be written as (with obvious de�nitions):

pOabbb � �a: (6.2)

The issue at hand is whether or not the operator O is invertible in general. Clearly if the skew sector is weakenough and � 6= 3=4, then (ab) will dominate the operator, and O�1 will exist. More generally there mayexist only a subspace on which it is invertible, in which case the component(s) of ba in this space can be solvedfor, and the other(s) are left undetermined. This will also imply that there are constraints corresponding toany undetermined components of ba of the form: �i � 0. One would then need to check that this constraint

12Note that requiring _pa � _�[ab] � 0 on the initial surface is actually a stronger condition than jab;Wa � 0, as itimplies that ba � 0 initially.

20

is preserved in time _�i � 0, in order to determine whether it is possible for the system to remain on thisconstraint surface. As discussed below in the case where the Cauchy data is dominated by the symmetricsector, it appears more likely that these are momentary con�gurations that the system may pass through.For the case of mNGT (� = 3=4), one can easily see that the operator O is not dominated by the sym-

metric sector, and that in the absence of any skew metric components, it disappears altogether. Considering(Ba; [ab]) as a perturbation on (ab), all metric quantities may be expanded to lowest order in powers of(Ba; [ab]):

G(ab)= (ab); (6.3)

G[ab]= � [ab] = � (ac) (bd) [cd] ; (6.4)p�g= p ; (6.5)

�a= Ba = (ab)Bb; (6.6)

�a= � [ab]Bb: (6.7)

If one assumes that ba is of order (Ba; [ab]), then (6.2) becomes three constraint equations �a � 0, further

generating _�a � 0 for consistency, and one has obtained the dynamics of massive a Kalb-Ramond �eld on aGR background consistent with the linearized treatment of the �eld equations given in [6{8]. This is not thegeneral case, as one does not have the freedom to assume the behavior of the Lagrange multipliers, it mustinstead be derived from the compatibility conditions. In particular, (6.2) determines ba in terms of Cauchydata, and keeping the dominant terms, one must �nd an inverse for the operator Oab:

Oab := BaBb + 14 [ac] [cd]

(db) = BaBb � (ab) � + a b; (6.8)

where in the second form the following have been introduced:

[ab]= 2p �abc c; a :=

1

4p �abc

[bc]; (6.9)

[ab]= 21p �abc

c; a :=

p

4�abc [bc]: (6.10)

The Levi-Civita tensor density of weight +1: �abc and �1: �abc = (ad) (be) (cf)�def have been used (�123 =�123 = +1), and the notation � = a a has been introduced. (Note that these are not quite duals; theextra numerical factor is introduced for convenience.) The inverse of O is found to be:

O�1ab = � 1

�

� (ab) �

1

�B (Ba b + aBb) +B �B � � ( �B)2 a b

�: (6.11)

(Although the nonperturbative inversion of the operator O in (6.2) may be accomplished using similar tech-niques, the general result is not particularly enlightening and will not be given here.) If � = 0 ( = 0assuming that ( ) is nondegenerate), then assuming that B 6= 0, one �nds two constraints coming fromprojecting perpendicular to B, and one relation for b from that parallel to B. If 6= 0 and either B = 0 or � B = 0 then there is one constraint from projecting parallel to , and two conditions determining b fromthe parallel component. (The remaining operator when B = 0 projects out directions transverse to ). Ithas not been possible so far to �nd any of these cases which can be maintained in evolution, and so they areexpected to be momentary con�gurations and not surfaces on which the system may evolve consistently.The solution (6.11) for ba implies that ba � O�1ab �

b is not O( [ ]), but in fact O([ [ ]]�1). (This does nothappen in Einstein Uni�ed Field Theory (or old NGT) since � = 0 leaves a term that allows one to solve forba at lowest order.) This in turn shows up in the evolution equations of the antisymmetric sector canonicalvariables ((4.8) and (4.9)), causing them to evolve arbitrarily quickly as the skew initial data is made smaller.It is possible to have � � O(( [ ])3), in which case b will again reduce to O( [ ]), and this behavior is avoided.However, this is a condition on the canonical variables alone, and since one could easily set up initial datathat does not satisfy this condition, it would have to be realized dynamically. At this point it is not knownwhether this is a reasonable expectation, though it seems unlikely that it would occur for any initial data

21

con�guration in which the skew sector is small13.This sort of behavior in the skew sector must be better understood, as the arbitrarily large time derivatives

cause one to worry about whether GR spacetimes would be unstable in mNGT. It is also not hard to see thatthe same situation may occur near NGT spacetimes as well. As a speci�c example, consider the solution of(6.1) specialized to spherically symmetric systems. In the general case where both of (B1; [23]) are non-zero,the solution is:

b1 � 1

2

11W 1

(B1)2+ 2

k22B1

: (6.12)

Thus one �nds that in regions of spacetime where B1 is vansihingly small (for example, perturbative situations,or in the asymptotic region of asymptotically at spacetimes), b1 becomes arbitrarily large, which in turn

may cause ( _p1; _W1; _j23) to become very large. Although the Wyman sector solution [5,8] (which is becomingthe basis for most of the phenomenology in mNGT [6,42{44]) assumes that both B1 and W 1 vanish globally,if one considers perturbations of these �elds on �0, the above behavior reappears.This is essentially the same e�ect as was found in [45], where the e�ect of gravitational dynamics on the

constraints of various derivative coupled vector �elds was studied. It was found that constraints on the vector�eld may be lost when GR is considered as evolving concurrently with the vector �eld (as opposed to thevector �eld evolving on a GR background). This manifested itself as an increase in the number of degrees offreedom in the vector �eld, and singular behavior in the evolution equations when approaching asymptotically at spacetimes. There one �nds no evidence of this when considering the vector �eld dynamics on a �xed GRbackground.This is intimately linked to degenerate solutions of the Euler-Lagrange Equations. If one considers linearized

perturbations to such a solution, the �eld equations require that the perturbations vanish, whereas the analysisof the full dynamics near the degenerate solution may result in quite di�erent behavior. This degeneracyalso results in the loss of hyperregularity (Ch. 7.4 of [46]) for the Lagrangian system, and equivalence ofthe Hamiltonian system is not guaranteed. However in this case Hamilton's equations correctly reproducethe Euler-Lagrange equations provided the system avoids con�gurations corresponding to these degeneratesolutions.

CONCLUSIONS

The Hamiltonian formulation of both old (massless) NGT, as well as the massive theory has been given,identifying the former as having �ve antisymmetric sector con�guration space degrees of freedom per spacetimepoint, and the latter six. The symmetric sector of phase space has been enlarged over that of GR by a singlecanonical pair representing the metric density

p�g. The existence of two related second class constraintsallows one to remove these additional coordinates and recover the same number of degrees of freedom as GRin the symmetric sector. The reduction of the system to pure GR �eld con�gurations has been discussed, andthe dynamics of GR recovered.All �elds have been decomposed by making use of a surface basis adapted to g( ), where the surface has

been assumed to be spacelike (and the time vector timelike) with respect to the causal metrics of NGT (fgcg).Con�guration space has been taken to consist of the components of the inverse metric for convenience, brokenup into the lapse and shift functions and the surface degrees of freedom: (N;Na; (ab);Ba; [ab]). The lapseand shift functions remain undetermined Lagrange multipliers, and phase space consists of canonical pairsthat include all of the remaining inverse metric components, as well as an additional redundant pair mentionedabove. The Hamiltonian has been put into standard form, and the lapse and shift functions enforce the NGTsuper-Hamiltonian (H) and supermomentum (Ha) constraints.This examination of nonsymmetric gravitational theories in Hamiltonian form has allowed one to identify

potential problems with the massive action, as singular behavior of the determined Lagrange multipliers ba has

13Note also that although one might expect that H might impose this constraint on the system due to the presence ofterms of the form (ab)baba inH, it can be shown that these terms identically cancel once � = 3=4 and �abc = �aTbc+�

a[bbc]

is employed.

22

been uncovered for con�gurations that are very close to those of GR, as well as certain NGT con�gurations,the Wyman solution in particular. This behavior is not uncommon in derivative coupled theories, where therank of the kinetic matrix changes in certain �eld con�gurations.

ACKNOWLEDGEMENTS

The author would like to thank the Natural Sciences and Engineering Research Council of Canada andthe University of Toronto for funding during part of this work. Thanks also to J. W. Mo�at, P. Savaria, L.Demopoulos, and N. Cornish for discussions related to the work, and especially to J. L�egar�e for a criticalreading of the manuscript.

APPENDIX A: SURFACE DECOMPOSED COMPATIBILITY CONDITIONS AND FIELD

EQUATIONS

The surface decomposition of (1.4) and (1.5d) is straightforward, however as these will be required weakly,the algebraic conditions that determine the Lagrange multipliers, as well as the remaining torsion-free condi-tion, have been applied to write them in the given form. The de�nition of W a given by (3.3) has also beenused throughout.

1. Compatibility Conditions

The symmetric components of (1.4) are found to be:

C(??)? = Je?[

p�g] + �� u� 2Baba; (A1a)

C(ab)? = �Je?[ ](ab) +

(ab)(� + u)� 2uab; (A1b)

C(?a)? = �a �

(ab)ab +Bbvab + [ab]bb; (A1c)

C(??)a = r(3)

a [p�g] + ca + 2jabB

b; (A1d)

C(?a)b = uab �Babb �Bc�acb � kab; (A1e)

C(ab)c = �r(3)

c [ ](ab) + (ab)cc +Bavbc +Bbvac +

[ad]�bdc + [db]�acd; (A1f)

and the skew components:

C[?a]? = Je?[B]a �Bau+Bbuab +

[ab]ab � (1 + 23�)

(ab)bb +13�

(ab)W b; (A2a)

C[ab]? = �Je? [ ][ab] �Ba�b +Bb�a +

[ab](� + u)� 2vab; (A2b)

C[?a]b = r(3)

b [B]a � vab + jab � 13��

abW; (A2c)

C [ab]c = �r(3)

c [ ][ab] �Baubc +Bbuac + [ab]cc +

(ad)�bdc + (db)�acd +

23��

[ac �

b]d

(de)(W e � 2be): (A2d)

The �rst two of each of these are dynamical compatibility conditions, and the rest are algebraic conditionsthat determine Lagrange multipliers. It is useful to have the contractions of a few of these:

C(?a)a = u� kaa; (A3a)

C(ab)b = �r(3)

b [ ](ab) + (ab)cb +Bbvab �

[ab]bb + [bc]�acb; (A3b)

C [?a]a = r(3)

a [B]a � �W; (A3c)

C[ab]b = �r(3)

b [ ][ab] �Bau+Bbuab + [ab]cb + (4

3�� 1) (ab)bb � 2

3� (ab)W b: (A3d)

Note that there is one further algebraic compatibility condition buried in the dynamical compatibilityconditions. By taking appropriate combinations in order to make use of (2.8) (replacing the variation with aderivative o� the surface), one can �nd:

23

C� := ��Baba +16�

(ab)�a(W b � 2bb); (A4a)

giving the remaining algebraic condition that determines �. Repeating this last calculation and consideringa derivative in the surface yields a condition that determines ca:

Cc := ca + jabBb � 1

6��aW � 16�G[ab]

(bc)(W c � 2bc); (A4b)

although this is not independent of the algebraic conditions already found.

2. Algebraic Field Equations and Conservation Laws

The decomposition of the �eld equations (1.5d) yields:

R??= �Je? [u] + Z?? � 14m2M??; (A5a)

R(a?)= �Je? [jabBb] + Z(a?) � 14m2M(a?); (A5b)

R[a?]=12Je?[W ]a + Z[a?] � 1

4m2M[a?]; (A5c)

Rab= Je? [K]ab + Zab � 14m

2Mab; (A5d)

the second of which has made use of the Jacobi identity (2.12). For completeness, the components of themass tensor are:

M??= F (F � 1) � 2 (ab)�a�b +12F

[ab]G[ab]; (A6a)

M(a?)= �12 [bc]G[bc]�a + �c(G[ab]

(bc) �G(ab) [bc]); (A6b)

M(ab)= �(F � 1)G(ab) � 12G(ab)

[cd]G[cd] + 2�a�b � 2�a�b +12

[cd](GcaGbd � GdbGac); (A6c)

M[a?]= ��a + 12

[bc]G[bc]�a � �c(G(ab) (bc) � G[ab]

[bc]); (A6d)

M[ab]= (F � 2)G[ab] +12G[ab]

[cd]G[cd] + 2�a�b � 2�b�a � 12 [cd](GcbGad �GcaGbd); (A6e)

for which a few identities exist:

M?? + abMab= �2(F � 1)� [cd]G[cd]; (A7a)

M(a?) +M[ab]Bb= 0; (A7b)

(ab)M(?b) + [ab]M[?b]= 0: (A7c)

The remaining objects in (A5) fZg are:

Z??= r(3)a [�]a + �u� uabuba + �a(aa � 2ca) + vabv

ba � 1

2�(W )2; (A8a)

Z(a?)= �r(3)a [u] +r(3)

b [u]ba �r(3)a [Bbbb]

+uca � kab�b � (aa � ca)B

bbb + uba(ab � cb) + vbabb + vbc�cab +

12�W (W a � 2ba); (A8b)

Z(ab)= R(3)

(ab) �r(3)

(a [a]b) + (� + u)kab � aaab � kcbuca � kacucb

+babb � jacvcb � jbcvca + �cad�dbc � 1

2�(W a � 2ba)(W b � 2bb); (A8c)

Z[a?]=1

2r(3)a [W ] +

1

2W (aa � ca)�r(3)

b [v]ba � jab�b + (cb � ab)vba + uba � ubabb � ubc�cab; (A8d)

Z[ab]= �r(3)

[a[W ]b] + 2r(3)

[a[b]b] +r(3)

c [�]cab

+(� + u)jab + ac�cab + aabb � abba � jcbu

ca � jacucb � kcav

cb + kcbv

ca: (A8e)

One may construct the tensor density (these are, of course, just particular combinations of the �eld equa-tions):

GAB := 12(gACGBC + gCAGCB) = 1

2(gACRBC + gCARCB � �ABgCDRCD); (A9)

24

where GAB = RAB � 12gBAR is the equivalent of the Einstein tensor for NGT. It is a fairly straightforward

calculation to show that the components G?? and G?a are both algebraic �eld equations:

G??= G?? = 12

�p�gR?? + abRab

�= 1

2

�p�gZ?? + abZab � 1

4m2(p�gM?? +

abMab)� � �u+ uBaba + jabB

a�b + kabuba � jabv

ba; (A10a)

G?a= G?a =p�gR(a?) +BbR[ab]

=p�gZ(a?) + Z[ab]B

b + jab

h��Bb + ubcB

c + 2BbBcbc + [bc]ac � (1 + 4

3�) (bc)bc +

23�

(bc)W c

i; (A10b)

analogous to the Gauss and Codacci relations respectively in GR. The conservation laws for NGT derived in[47] written in a general basis are:

BA := reB [G]BA + 12RBCreA[g]

BC = 0; (A11)

guaranteeing the existence of four Bianchi-type identities related to the above algebraic constraints. Notethat these identities cannot be used here in this form, as the compatibility conditions have been imposed intheir derivation. Although this will not be necessary here, one could in principle derive the form of (A11)that would hold without imposing the compatibility conditions, and could therefore be used here to derivethe closing relations (4.26) directly from the action (1.1); using the assumption of di�eomorphism invariancein a slightly more operational manner.

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