This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 129.123.124.117

This content was downloaded on 17/10/2014 at 01:51

Please note that terms and conditions apply.

Time and dark matter from the conformal symmetries of Euclidean space

View the table of contents for this issue, or go to the journal homepage for more

2014 Class. Quantum Grav. 31 215001

(http://iopscience.iop.org/0264-9381/31/21/215001)

Home Search Collections Journals About Contact us My IOPscience

Time and dark matter from the conformalsymmetries of Euclidean space

Jeffrey S Hazboun and James T Wheeler

Utah State University, Physics Department, Logan, UT 84322, USA

E-mail: jeffrey.hazboun@gmail.com and jim.wheeler@usu.edu

Received 1 May 2014, revised 23 July 2014Accepted for publication 28 July 2014Published 16 October 2014

AbstractStarting with the conformal symmetries of Euclidean space, we construct amanifold where time manifests as a part of the geometry. Though there is nomatter present in the geometry studied here, geometric terms analogous to darkenergy and dark matter appear when we write down the Einstein tensor.Specifically, the quotient of the conformal group of Euclidean four-space byits Weyl subgroup results in a geometry possessing many of the properties ofrelativistic phase space, including both a natural symplectic form and non-degenerate Killing metric. We show that the general solution posessesorthogonal Lagrangian submanifolds, with the induced metric and the spinconnection on the submanifolds necessarily Lorentzian, despite the Euclideanstarting point. Using an orthonormal frame adapted to the phase space prop-erties, we also find two new tensor fields not present in Riemannian geometry.The first is a combination of the Weyl vector with the scale factor on themetric, and determines the timelike directions on the submanifolds. The sec-ond comes from the components of the spin connection, symmetric withrespect to the new metric. Though this field comes from the spin connection, ittransforms homogeneously. Finally, we show that in the absence of Cartancurvature and sources, the configuration space has geometric terms equivalentto a perfect fluid and a cosmological constant.

Keywords: conformal gravity, time, dark matter, biconformal space, generalrelativity, gravitational gauge theoryPACS numbers: 04.20.Cv, 04.50.-h, 04.50.Kd, 02.40.-k, 02.40.Dr, 02.40.Ky

1. Introduction

We show, by basing a gravitational gauge theory on underlying symmetry, how the presenceof a timelike direction can emerge from an initially Euclidean geometry. In addition, we show

Classical and Quantum Gravity

Class. Quantum Grav. 31 (2014) 215001 (34pp) doi:10.1088/0264-9381/31/21/215001

0264-9381/14/215001+34$33.00 © 2014 IOP Publishing Ltd Printed in the UK 1

that it is possible to produce a cosmological constant and cosmological dust as part of aninitial geometry rather than as matter sources. Both of these changes occur as a result ofincreased symmetry. For the first, a new vector field built as the difference of two gaugedependent quantities necessarily gives a timelike direction. The cosmological constant anddust arise in much the same way as the emergence of a cosmological constant in the Mac-Dowell–Mansouri treatment of the de Sitter group [1], with the extra symmetry adding termsto the curvature.

By gauge theory, we typically understand a theory (i.e., the specification of an actionfunctional) which is invariant under a local symmetry group—the gauge symmetry. Thus,there may be many gauge theories having the same gauge group. However, gauge theorieshaving the same gauge group share a common structure: the underlying principal fiber bundlein which the base manifold is spacetime or some other world manifold and the fibers arecopies of the gauge group. Such a principal fiber bundle is most simply constructed as thequotient of a larger group by the symmetry group. Constructed in this way, we haveimmediate access to relevant tensor fields: any group invariant tensors, the curvatures of thebundle, and the vectors of the group representation. Then any functional built invariantly fromthese tensors is a gauge theory. For example, in section 2.3 below, we show how the quotient ofthe Poincaré group by its Lorentz subgroup may be generalized to a principal fiber bundle withLorentz group fibers and a general base manifold having arbitrary Riemannian curvature.Identifying the curvature, solder form, Lorentz metric, and Levi-Civita tensor as tensors withrespect to this local Lorentz symmetry, it is clear that any functional built invariantly from themis a gauge theory. In addition, if we use a linear representation, SO (3, 1) or SL C(2, ), of theLorentz group then the action functional may include vectors or spinors from that representationand their covariant derivatives. For these reasons, we will define a gauging to be the fiberbundle of a specific quotient, along with the identification of its associated tensors. A gaugetheory remains the specification of an action functional invariant on this bundle.

We develop a gauging based on the conformal group of a Euclidean space, and show thatits group properties necessarily lead to a symplectic manifold with Lagrangian submanifoldsof Lorentzian signature. Though we deal almost exclusively with the homogeneous quotientspace, we always have in mind the class of biconformal gauge theories presented insection 4.2. This theory has been studied extensively [2]. In particular, we note from the fieldequations given in [2] that for specific relations of the action coefficients the homogeneousspace is a vacuum solution. We find that these vacuum solutions carry both a cosmologicalconstant and a cosmological perfect fluid as geometric generalizations of the Einstein tensor.In curved models, this geometric background may explain or contribute to dark matter anddark energy. To emphasize the purely geometric character of the construction, we give adescription of our use of the quotient manifold method for building gauge theories. Our use ofthe conformal group, together with our choice of local symmetry lead to several structures notpresent in other related gauge theories. Specifically, we show the generic presence of asymplectic form, that there exists an induced metric from the non-degenerate Killing form,demonstrate (but do not use) Kähler structure, and find natural orthogonal, Lagrangiansubmanifolds. All of these properties arise directly from group theory.

In the remainder of this introduction, we give a brief historical overview of techniquesleading up to, related to, or motivating our own, then describe the layout of our presentation.

As mathematicians began studying the various incarnations of non-Euclidean geometry,Klein started his Erlangen program in 1872 as a way to classify all forms of geometries thatcould be constructed using quotients of groups. These homogeneous spaces allowed forstraightforward classification of the spaces dependent on their symmetry properties. Much ofthe machinery necessary to understand these spaces originated with Cartan, beginning with

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

2

his doctoral dissertation [3]. The classification of these geometries according to symmetryforeshadowed gauge theory, the major tool that would be used by theoretical physicists as thetwentieth century continued. We will go into extensive detail about how these methods areused in a modern context in section 2. Most of the development, in modern language, can befound in [4].

The use of symmetries to construct physical theories can be greatly credited to Weylʼsattempts at constructing a unified theory of gravity and electromagnetism by adding dilata-tional symmetry to general relativity. These attempts failed until Weyl looked at a U (1)symmetry of the action instead, thus constructing the first gauge theory of electromagnetism.These efforts were extended to non-abelian groups by Yang and Mills [5], including all SU(n)and described by the Yang–Mills action. The success of these theories as quantum pre-cursorsinspired relativists to try and construct general relativity as a gauge theory. Utiyama [6]looked at GR based on the the Lorentz group, followed by Kibble [7] who first gauged thePoincaré group to form general relativity.

Standard approaches to gauge theory begin with a matter action globally invariant undersome symmetry group . This action generally fails to be locally symmetric due to thederivatives of the fields, but can be made locally invariant by introducing an -covariantderivative. The connection fields used for this derivative are called gauge fields. The final stepis to make the gauge fields dynamical by constructing their field strengths, which may bethought of as curvatures of the connection, and including them in a modified action.

In the 1970s the success of the standard model and the growth of supersymmetric gravitytheories inspired physicists to extend the symmetry used to construct a gravitational theory.MacDowell and Mansouri [1] obtained general relativity by gauging the de Sitter or anti-deSitter groups, and using a Wigner–Inönu contraction to recover Poincaré symmetry. As a pre-cursor to supersymmetrizing Weyl gravity, two groups [8–11] looked at a gravitational theorybased on the conformal group, using the Weyl curvature-squared action. These approachesare top-down, in the sense that they often start with a physical matter action and generalize toa local symmetry that leads to interactions. However, as this work expanded, physicistsstarted using the techniques of Cartan and Klein to organize and develop the structuressystematically.

In [12, 13] Neʼeman and Regge develop what they refer to as the quotient manifoldtechnique to construct a gauge theory of gravity based on the Poincaré group. Theirs is thefirst construction of a gravitational gauge theory that uses Klein (homogeneous) spaces asgeneralized versions of tangent spaces, applying methods developed by Cartan [14] tocharacterize a more general geometry. In their 1982 papers [15, 16], Ivanov and Niederleexhaustively considered quotients of the Poincaré, de Sitter, anti-de Sitter, and Lorentzianconformal groups (ISO (3, 1), SO (4, 1), SO (3, 2) and SO (4, 2)) by various subgroupscontaining the Lorentz group.

There are a number of more recent implementations of Cartan geometry in the modernliterature. One good introduction is Wiseʼs use of Cartan methods to look at the MacDowell-Mansouri action [17]. The ‘waywiser’ approach of visualizing these geometries is advocatedstrongly, and gives a clear geometric way of undertsanding Cartan geometry. The use ofCartan techniques in [18] to look at the Chern–Simons action in +2 1 dimensions provides anice example of the versatility of the method. This action can be viewed as having eitherMinkowski, de Sitter or anti-de Sitter symmetry and Cartan methods allow a straightforwardcharacterization of the theory given the various symmetries. The analysis is extended to lookfirst at the conformal representation of these groups on the Euclidean surfaces of the theory(two-dimensional spatial slices). The authors then look specifically at shape dynamics, whichis found equivalent to the case when the Chern–Simons action has de Sitter symmetry.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

3

Tractor calculus is yet another example using a quotient of the conformal group, in which theassociated tensor bundles are based on a linear, +n( 2)-dim representation of the group. Thisis a distinct gauging from the one we study here, but one studied in [19].

Our research focuses primarily on gaugings of the conformal group. Initially motivatedby a desire to understand the physical role of local scale invariance, the growing prospects oftwistor string formulations of gravity [20] elevate the importance of understanding its low-energy limit, which is expected to be a conformal gauge theory of gravity. Interestingly, thereare two distinct ways to formulate gravitational theories based on the conformal group, firstidentified in [15, 16] and developed in [2, 19, 21]. Both of these lead directly to scale-invariant general relativity1. This is surprising since the best known conformal gravity theoryis the fourth-order theory developed by Weyl [22–26] and Bach [27]. When a Palatini stylevariation is applied to Weyl gravity, it becomes second-order, scale-invariant general rela-tivity [19].

The second gauging of the conformal group identified in these works is the biconformalgauging. Leading to scale-invariant general relativity formulated on a n2 -dimensional sym-plectic manifold, the approach took a novel twist for homogeneous spaces in [28]. There it isshown that, because the biconformal gauging leads to a zero-signature manifold of doubleddimension, we can start with the conformal symmetry of a non-Lorentzian space while stillarriving at spacetime gravity. We describe the resulting signature theorem in detail below, andconsiderably strengthen its conclusions. In addition to necessarily developing a direction oftime from a Euclidean-signature starting point, we show that these models give a group-theoretically driven candidate for dark matter.

In the next section, we describe the quotient manifold method in detail, providing anexample by applying it to the Poincaré group to produce Cartan and Riemannian geometries.Then, in section 3, we apply the method to the conformal group in the two distinct waysoutlined above. The first, called the auxiliary gauging, reproduces Weyl gravity. Focusing onthe second, we identify a number of properties posessed by the homogeneous space of thebiconformal gauging. In section 4, we digress to complete both gaugings by modifying thequotient manifold and connections, then writing appropriate action functionals, therebyestablishing physical theories of gravity. We return to study the homogeneous space of thebiconformal gauging in section 5, developing the Maurer–Cartan structure equations in anadapted basis. Then, in the next section, we transform a known solution to the structureequations into the adapted basis and identify the properties of the resulting space. This revealstwo previously unknown objects, one a tensor of rank three, and the other a vector. Insection 6 we find the form of the connection and basis forms when restricted to the config-uration and momentum submanifolds. This reveals the possibility of Riemannian curvature ofthe submanifolds, even though the Cartan curvature of the full space vanishes. Corre-spondingly, the Einstein tensor of the submanifolds does not vanish, but includes twoadditional geometric terms. The vanishing trace of the Cartan curvature therefore leads to thevanishing of the generalized Einstein tensor, i.e., the Einstein tensor plus these geometricterms. Imposing the form of the solution, we find the vanishing generalized Einstein tensordescribes a spacetime with both a cosmological constant and cosmological dust. Finally, wesummarize our results.

1 Scale-invariant general relativity refers to a theory where the field equation is the dilatationally covariant Einsteinfield equation. This is equivalent to a trivial Weyl geometry, i.e. one in which the dilatational curvature is zero, withthe Einstein field equation dictatating the physics of the manifold.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

4

2. Quotient manifold method

We are interested in geometries—ultimately spacetime geometries—which have continuouslocal symmetries. The structure of such systems is that of a principal fiber bundle with Liegroup fibers. The quotient method starts with a Lie group, , with the desired local symmetryas a proper Lie subgroup. To develop the local properties any representation will giveequivalent results, so without loss of generality we assume a linear representation, i.e. a vectorspace +n 2 on which acts. Typically this will be either a signature p q( , ) (pseudo-)Eucli-dean space or the corresponding spinor space. This vector space is useful for describing thelarger symmetry group, but is only a starting point and will not appear in the theory.

The quotient method, laid out below, is identical in many respects to the approaches of[17, 18]. The nice geometric interpretation of using a Klein space in place of a tangent spaceto both characterize a curved manifold and take advantage of its metric structure are alsoamong the motivations for using the quotient method. In what follows not all the manifoldswe look at will be interpreted as spacetime, so we choose not to use the interpretation of aKlein space moving around on spacetime in a larger ambient space. Rather we directlygeneralize the homogeneous space to add curvatures. The homogeneous space becomes alocal model for a more general curved space, similar to the way that n provides a local modelfor an n-dim Riemannian manifold.

We include a concise introduction here, but the reader can find a more detailed expositionin [4]. Our intention is to make it clear that our ultimate conclusions have rigorous roots ingroup theory, rather than to present a comprehensive mathematical description.

2.1. Construction of a principal -bundle B ; π;;M0ð Þ with connection

Consider a Lie group, , and a non-normal Lie subgroup, , on which acts effectively andtransitively. The quotient of these is a homogeneous manifold, M0. The points of M0 are theleft cosets,

= ′∣ ′ = ∈g g g gh h{ for some }

so there is a natural −1 1 mapping ↔g . The cosets are disjoint from one another andtogether cover . There is a projection, π → M: 0, defined by π = ∈g g M( ) 0. There isalso a right action of , g , given for all elements of by right multiplication.

Therefore, is a principal -bundle, B π( )M, , , 0 , where the fibers are the leftcosets. This is the mathematical object required to carry a gauge theory of the symmetrygroup . Let the dimension of be m, the dimension of be k. Then the dimension of themanifold is = −n m k and we write M n

0( ). Choosing a gauge amounts to picking a cross-

section of this bundle, i.e., one point from each of these copies of . Local symmetryamounts to dynamical laws which are independent of the choice of cross-section.

Lie groups have a natural Cartan connection given by the one-forms, ξ A, dual to thegroup generators, GA. Rewriting the Lie algebra in terms of these dual forms leads imme-diately to the Maurer–Cartan structure equations

ξ ξ ξ= − ∧cd1

2, (1)A

BCA B C

where c BCA are the group structure constants, and ∧ is the wedge product. The integrability

condition for this equation follows from the Poincaré lemma, =d 02 , and turns out to beprecisely the Jacobi identity. Therefore, the Maurer–Cartan equations together with theirintegrability conditions are completely equivalent to the Lie algebra of .

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

5

Let ξa (where a = 1,…,k) be the subset of one-forms dual to the generators of thesubgroup, . Let the remaining independent forms be labeled χ α. Then the ξa give aconnection on the fibers while the χ α span the co-tangent spaces to M n

0( ). We denote the

manifold with connection by ξ= ( )M ,n n A0( )

0( ) .

2.2. Cartan generalization

For a gravity theory, we require in general a curved geometry, n( ). To achieve this, thequotient method allows us to generalize both the connection and the manifold. Since theprincipal fiber bundle from the quotient is a local direct product, this is not changed if weallow a generalization of the manifold, →M Mn n

0( ) ( ). We will not consider such topological

issues here. Generalizing the connection is more subtle. If we change ξ ξ χ= α( ),A a to a newconnection ξ ω ξ ω χ ω→ → →α α, ,A A a a arbitrarily, the Maurer–Cartan equation is alteredto

ω ω ω Ω= − ∧ +cd1

2,A

BCA B C A

where ΩA is a two-form determined by the choice of the new connection. We needrestrictions on ΩA so that it represents curvature of the geometry ω= ( )M ,n n A( ) ( ) and not

of the full bundle B. We restrict ΩA by requiring it to be independent of lifting, i.e.,horizontality of the curvature.

To define horizontality, recall that the integral of the connection around a closed curve inthe bundle is given by the integral of ΩA over any surface bounded by the curve. We requirethis integral to be independent of lifting, i.e., horizontal. It is easy to show that this means thatthe two-form basis for the curvatures ΩA cannot include any of the one-forms, ωa, that spanthe fiber group, . With the horizontality condition, the curvatures take the simpler form

Ω ω ωΩ= ∧αβα β1

2.A A

More general curvatures than this will destroy the homogeneity of the fibers, so we would nolonger have a principal -bundle.

In addition to horizontality, we require integrability. Again using the Poincaré lemma,ω ≡d 0A2 , we always find a term ω ω ω∧ ∧c cB C

ADE

B C D E1

2 [ ] which vanishes by the Jacobi

identity, ≡c c 0B CA

DEB

[ ] , while the remaining terms give the general form of the Bianchiidentities,

ω ΩΩ + ∧ =d c 0.ABC

A B C

2.3. Example: pseudo-Riemannian manifolds

To see how this works in a familiar example, consider the construction of the pseudo-Riemannian spacetimes used in general relativity, for which we take the quotient of the ten-dim Poincaré group by its six-dim Lorentz subgroup. The result is a principal Lorentz bundleover 4. Writing the one-forms dual to the Lorentz ( )M b

a and translation ( )Pa generators asξ b

a and ωa, respectively, the ten Maurer–Cartan equations are

ξ ξ ξ

ω ω ξ

= ∧

= ∧

d

d

,

.

ba

bc

ca

a bb

a

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

6

Notice that the first describes a pure gauge spin connection, ξ Λ Λ= − d¯b

ab

cc

a where Λ ca is a

local Lorentz transformation. Therefore, there exists a local Lorentz gauge such that ξ = 0ba .

The second equation then shows the existence of an exact orthonormal frame, which tells usthat the space is Minkowski.

Now generalize the geometry, ξ ω→( ) ( )M M, ,A A04 4 where =M0

4 4 and we denote

the new connection forms by ω ω= ( )e,Ab

a b . In the structure equations, this leads to thepresence of ten curvature two-forms,

ω ω ωω

= ∧ += ∧ +

d R

de e T

,

.

ba

bc

ca

ba

a bb

a a

Since the ω ba span the Lorentz subgroup, horizontality is accomplished by restricting the

curvatures to

= ∧

= ∧

R

T

R e e

T e e

1

21

2

ba

bcda c d

abca b c

that is, there are no terms such as, for example, ω ∧R eb dea c

cd e1

2or ω ω∧T b d

a c ec

be

d1

2.

Finally, integrability is guaranteed by the pair of Bianchi identities,

ω ωω

+ ∧ − ∧ =+ ∧ + ∧ =

dR R R

dT T e R

0,

0.

ba

bc

ca

ca

bc

a bb

a bb

a

By looking at the transformation of R ba and Ta under local Lorentz transformations, we find

that despite originating as components of a single Poincaré-valued curvature, they areindependent Lorentz tensors. The translations of the Poincaré symmetry were broken whenwe curved the base manifold (see [7, 12, 13], but note that Kibble effectively uses a 14-dimensional bundle, whereas ours and related approaches require only ten-dim.) Werecognize R b

a and Ta as the Riemann curvature and the torsion two-forms, respectively.Since the torsion is an independent tensor under the fiber group, it is consistent to consider thesubclass of Riemannian geometries, =T 0a . Alternatively (see section 4 below), vanishingtorsion follows from the tetradic Palatini action.

With vanishing torsion, the quotient method has resulted in the usual solder form, ea, andrelated metric-compatible spin connection, ω b

a ,

ω− ∧ =de e 0,a bb

a

the expression for the Riemannian curvature in terms of these,

ω ω ω= − ∧R d ,ba

ba

bc

ca

and the first and second Bianchi identities,

∧ ==

e R

DR

0

0.

bb

a

ba

This is a complete description of the class of Riemannian geometries.Many further examples were explored by Ivanov and Niederle [15, 16].

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

7

3. Quotients of the conformal group

3.1. General properties of the conformal group

Physically, we are interested in measurements of relative magnitudes, so the relevant group isthe conformal group, , of compactified n. The one-point or light-cone compactification atinfinity allows a global definition of inversion, with translations of the point at infinitydefining the special conformal transformations. Then has a real linear representation in

+n 2 dimensions, +n 2 (alternatively we could choose the complex representation +2 n[( 2) 2]

for + +Spin p q( 1, 1)). The isotropy subgroup of n is the rotations, SO p q( , ), togetherwith dilatations. We call this subgroup the homogeneous Weyl group, and require ourfibers to contain it. There are then only three allowed subgroups: itself; the inhomogeneousWeyl group, , found by appending the translations; and together with special con-formal transformations, isomorphic to . The quotient of the conformal group by eitherinhomogeneous Weyl group, called the auxiliary gauging, leads most naturally to Weylgravity (for a review, see [19]). We concern ourselves with the only other meaningful con-formal quotient, the biconformal gauging: the principal -bundle formed by the quotient ofthe conformal group by its Weyl subgroup. To help clarify the method and our model, it isuseful to consider both these gaugings.

All parts of this construction work for any p q( , ) with = +n p q. The conformal group isthen + +SO p q( 1, 1) (or + +Spin p q( 1, 1) for the twistor representation.) The Maur-er–Cartan structure equations are immediate. In addition to the −n n( 1)

2generators β

αM ofSO p q( , ) and n translational generators αP , there are n generators of translations of a point atinfinity (‘special conformal transformations’) αK , and a single dilatational generator D. Dualto these, we have the connections ξ χ πβ

α αα, , ,δ, respectively. Substituting the structure

constants into the Maurer–Cartan dual form of the Lie algebra, equation (1) gives

ξ ξ ξ π χΔ= ∧ + ∧βα

βμ

μα

νβαμ

μνd 2 , (2)

χ χ ξ δ χ= ∧ + ∧α ββα αd , (3)

π ξ π δ π= ∧ − ∧α αβ

β αd , (4)

δ χ π= ∧ααd , (5)

where Δ δ δ δ δ≡ −νβαμ

να

βμ αμ

νβ( )1

2antisymmetrizes with respect to the original p q( , ) metric,

δ = … − … −μν diag (1, , 1, 1, , 1). These equations, which are the same regardless of thegauging chosen, describe the Cartan connection on the conformal group manifold. Beforeproceeding to the quotients, we note that the conformal group has a non-degenerate Killingform,

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Δδ

δ≡ = =( )K tr G G c c

0

01

.AB A B ADC

BCD

dbac

ba

ba

This provides a metric on the conformal Lie algebra. As we show below, when restricted to0, it may or may not remain non-degenerate, depending on the quotient.

Finally, we note that the conformal group is invariant under inversion. Within the Liealgebra, this manifests itself as the interchange between the translations and special conformaltransformations δ↔α αβ

βP K along with the interchange of conformal weights, → −D D. The

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

8

corresponding transformation of the connection forms is easily seen to leave equations (2)–(5)invariant. In the biconformal gauging, below, we show that this symmetry leads to a Kählerstructure.

3.2. Curved generalizations

In this sub-section and section 4 we will complete the development of the curved auxiliaryand biconformal geometries and show how one can easily construct actions with the cur-vatures. In this sub-section, we construct the two possible fiber bundles, , where ⊆ .For each, we carry out the generalization of the manifold and connection. The results in thissub-section depend only on whether the local symmetry is = or = . In sections 5and 6 we will return to the un-curved case to present a number of new calculations char-acterizing the homogenous space formed from the biconformal gauging.

The first sub-section below describes the auxiliary gauging, given by the quotient of theconformal group by the inhomogeneous Weyl group, .

Since is a parabolic subgroup of the conformal group, the resulting quotient can beconsidered a tractor space, for which there are numerous results [29]. Tractor calculus is aversion of the auxiliary gauging where the original conformal group is tensored with + +p q( 1, 1). This allows for a linear representation of the conformal group with

+n( 2)-dimensional tensorial (physical) entities called tractors. This linear representation,first introduced by Dirac [30], makes a number of calculations much easier and also allows forstraightforward building of tensors of any rank. The main physical differences stem from theuse of Diracʼs action, usually encoded as the scale tractor squared in the +n 2-dimensionallinear representation, instead of the Weyl action we introduce in section 4.

In section 3.2.2 below, we quotient by the homogeneous Weyl group, giving thebiconformal gauging. This is not a parabolic quotient and therefore represents a less con-ventional option which turns out to have a number of rich structures not present in theauxiliary gauging. The biconformal gauging will occupy our attention for the bulk of oursubsequent discussion.

3.2.1. The auxiliary gauging: ¼ . Given the quotient , the one-forms ξ π δβα

μ( ), ,

span the -fibers, with βα spanning the co-tangent space of the remaining n independentdirections. This means that n

0( ) has the same dimension, n, as the original space.

Generalizing the connection, we replace ξ χ π δ ω ω ω→βα α

α βα α

α( ) ( )e, , , , , , and the Cartanequations now give the Cartan curvatures in terms of the new connection forms,

ω ω ω ω ΩΔ= ∧ + ∧ +βα

βμ

μα

νβαμ

μν

βαd e2 , (6)

ω ω= ∧ + ∧ +α ββα α αde e e T , (7)

ω ω ω ω ω= ∧ − ∧ +α αβ

β α αd S , (8)

ω ω Ω= ∧ +ααd e . (9)

Up to local gauge transformations, the curvatures depend only on the n non-verticalforms, αe , so the curvatures are similar to what we find in an n-dim Riemannian geometry. Forexample, the SO p q( , ) piece of the curvature takes the form Ω Ω= ∧β

αβμνα α βe e1

2. The

coefficients have the same number of degrees of freedom as the Riemannian curvature of ann-dim Weyl geometry.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

9

Finally, each of the curvatures has a corresponding Bianchi identity, to guaranteeintegrability of the modified structure equations,

Ω Ω ω ω ΩΔ= + ∧ − ∧βα

νβαμ

μν

μν( )D0 2 , (10)

Ω Ω= − ∧ + ∧α ββ

α αDT e e0 , (11)

Ω ω ω Ω= + ∧ − ∧α αβ

β αDS0 , (12)

Ω ω= + ∧ − ∧αα

ααD T e S0 , (13)

where D is the Weyl covariant derivative,

Ω Ω Ω ω Ω ω

ω ω

ω ωΩ Ω

= + ∧ − ∧

= + ∧ − ∧

= − ∧ + ∧=

βα

βα

βμ

μα

μα

βμ

α α ββ

α α

α α αβ

β α

D d

DT dT T T

DS dS S S

D d

,

,

,

.

Equations (6)–(9) give the curvature two-forms in terms of the connection forms. We havetherefore constructed an n-dim geometry based on the conformal group with local symmetry.

We note no additional special properties of these geometries from the group structure. Inparticular, the restriction (in square brackets, [ ], below) of the Killing metric, KAB, to n( )

vanishes identically,

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Δδ

δ= ×( )[0]

01

0 ,

dbac

ba

ba n n

n( )

so there is no induced metric on the spacetime manifold. We may add the usual metric byhand, of course, but our goal here is to find those properties which are intrinsic to theunderlying group structures.

3.2.2. The biconformal gauging: ¼ . We next consider the biconformal gauging, firstconsidered by Ivanov and Niederle [16], given by the quotient of the conformal group by itsWeyl subgroup. The resulting geometry has been shown to contain the structures of generalrelativity [2, 21].

Given the quotient , the one-forms ξ δβα( ), span the -fibers, with χ πα

α( ), spanning

the remaining n2 independent directions. This means that n0(2 ) has twice the dimension of the

original compactified n( ). Generalizing, we replace ξ χ π δ ω ω ω ω→βα α

α βα α

α( ) ( ), , , , , ,and the modified structure equations appear identical to equations (6)–(9). However, thecurvatures now depend on the n2 non-vertical forms, ω ωα

α( ), , so there are far more componentsthan for an n-dim Riemannian geometry. For example,

Ω ω ω ω ω ω ωΩ Ω Ω= ∧ + ∧ + ∧βα

βμνα μ ν

β να μ

μν

βα μν

μ ν1

2

1

2.

The coefficients of the pure terms, Ω βμνα and Ω β

α μν each have the same number of degrees offreedom as the Riemannian curvature of an n-dim Weyl geometry, while the cross-termcoefficients Ω β ν

α μ have more, being asymmetric on the final two indices.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

10

For our purpose, it is important to notice that the spin connection, ξ βα , is antisymmetric

with respect to the original p q( , ) metric, δαβ, in the sense that

ξ ξδ δ= −βα αμ

βν μν .

It is crucial to note that ω βα retains this property, ω ωδ δ= −β

α αμβν μ

ν . This expresses metriccompatibility with the SO p q( , )-covariant derivative, since it implies

ω ωδ δ δ δ≡ − − =αβ αβ μβ αμ

αμ βμD d 0.

Therefore, the curved generalization has a connection which is compatible with a locallyp q( , )-metric. This relationship is general. If καβ is any constant-component metric, itscompatible spin connection will satisfy ω ωκ κ= −β

α αμβν μ

ν . Since we also have local scalesymmetry, the full covariant derivative we use will also include a Weyl vector term.

The Bianchi identities, written as two-forms, also appear the same as equations (10)–(13),but expand into more components.

In the conformal group, translations and special conformal transformations are related byinversion. Indeed, a special conformal tranformation is a translation centered at the point atinfinity instead of the origin. Because the biconformal gauging maintains the symmetrybetween translations and special conformal transformations, it is useful to name thecorresponding connection forms and curvatures to reflect this. Therefore, the biconformalbasis will be described as the solder form and the co-solder form, and the correspondingcurvatures as the torsion and co-torsion. Thus, when we speak of ‘torsion-free biconformalspace’ we do not imply that the co-torsion (Cartan curvature of the co-solder form) vanishes.In phase space interpretations, the solder form is taken to span the cotangent spaces of thespacetime manifold, while the co-solder form is taken to span the cotangent spaces of themomentum space. The opposite convention is equally valid.

Unlike other quotient manifolds arising in conformal gaugings, the biconformal quotientmanifold posesses natural invariant structures. The first is the restriction of the Killing metric,which is now non-degenerate,

⎛

⎝

⎜⎜⎜⎜⎡⎣⎢

⎤⎦⎥

⎞

⎠

⎟⎟⎟⎟⎛⎝⎜

⎞⎠⎟

Δδ

δδ

δ=

×

0

0

1

0

0,

dbac

ba

ba

ba

ba

n n2 2

n(2 )

and this gives an inner product for the basis,

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

ω ω ω ω

ω ω ω ω

δ

δ≡

α β αβ

αβ

α β

βα

αβ

, ,

, ,

0

0. (14)

This metric remains unchanged by the generalization to curved base manifolds.The second natural invariant property is the generic presence of a symplectic form. The

original fiber bundle always has this, because the structure equation (5), shows that χ π∧αα

is exact hence closed, ω =d 02 , while it is clear that the two-form product is non-degeneratebecause χ πα

α( ), together span n0(2 ). Moreover, the symplectic form is canonical

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

11

⎡⎣⎢⎢

⎤⎦⎥⎥Ω

δδ

=−

αβ

βα[ ]

0

0,AB

so that χ α and πα are canonically conjugate, in the sense that they form a canonical basis2 forthe symplectic form Ω. The symplectic form persists for the two-form, ω ω Ω∧ +α

α , as longas it is non-degenerate, so curved biconformal spaces are generically symplectic.

Next, we consider the effect of inversion symmetry. As a ( )11

tensor, the basis

interchange takes the form

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟χ χ

ππχ

δδ

δδ

= =αν

βμ

μ

ν

ανν

βμμI

00 .B

A B

In order to interchange conformal weights, I BA must anticommute with the conformal weight

operator, which is given by

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟χ χ

πχπ

δδ

=−

= +−

μα

βν

μ

ν

α

βW

0

0.B

A B

This is the case: we easily check that = + =I W I W W I{ , } 0BA

CA

BC

CA

BC . The commutator

gives a new object,

⎛⎝⎜

⎞⎠⎟

δδ≡ = − αβ

αβJ I W[ , ] 0

0.B

ABA

Squaring, δ= −J JCA

BC

BA , we see that J B

A provides an almost complex structure. That thealmost complex structure is integrable follows immediately in this (global) basis by theobvious vanishing of the Nijenhuis tensor,

= ∂ − ∂ − ∂ − ∂ =( )N J J J J J J J 0.BCA

CD

D BA

CD

D BA

DA

C BD

B CD

Next, using the symplectic form to define the compatible metric

Ω≡g u v u Jv( , ) ( , )

we find that in this basis⎛⎝⎜

⎞⎠⎟

δ

δ= αβ

αβg

0

0, and we check the remaining compatibility

conditions of the triple Ωg J( , , ),

ω

ϕ ϕ

=

= ω−( ) ( )

u v g Ju v

J u u

( , ) ( , )

( ) ( ) ,g

1

where ϕω and ϕg are defined by

ϕ ωϕ

==

ω u u

u g u

( ) ( , ·)

( ) ( , ·).g

These are easily checked to be satisified, showing that that n0(2 ) is a Kähler manifold.

Notice, however, that the metric of the Kähler manifold is not the restricted Killing metricwhich we use in the following considerations. While Kähler manifolds play a role in stringtheory and geometric quantization, this relationship lies beyond the scope of this work.

2 An examination of the Hamiltonian form of the biconformal action functional 4.2 would take us too far afield.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

12

Finally, a surprising result emerges if we require n0(2 ) to match our usual expectations

for a relativistic phase space. To make the connection to phase space clear, the preciserequirements were studied in [28]. There it was required that the flat biconformal gauging ofSO p q( , ) in any dimension = +n p q have Lagrangian submanifolds that are orthogonalwith respect to the 2n-dim biconformal (Killing) metric and have non-degenerate n-dimrestrictions of the metric. This is found to be possible only if the original space is Euclidean orsignature zero ∈( ){ }p n0, ,n

2. The signature of the Lagrangian submanifolds is severely

limited → ±p p( 1), leading in the two Euclidean cases to Lorentzian configuration space,and hence the origin of time. For the case of flat, eight-dim biconformal space we paraphrasethe following theorem from [28].

Flat eight-dim biconformal space is a metric phase space with the following properties.

(i) There exist Lagrangian submanifolds orthogonal with respect to the 2n-dim biconformal(Killing) metric.

(ii) The restriction of the Killing metric to each Lagrangian submanifold is non-degenerate, ifand only if the initial four-dim space we gauge is Euclidean or signature zero. In either ofthese cases the resulting configuration sub-manifold is necessarily Lorentzian [28].

Thus, it is possible to impose the conditions necessary to make biconformal space ametric phase space only in a restricted subclass of cases, and the configuration space metricmust be Lorentzian. In [28], it was found that with a suitable choice of gauge, the metric maybe written in coordinates αy as

δ= −αβ α β αβ( )( )

hy

y y y1

2 , (15)2 2

2

where the signature changing character of the metric is easily seen.In the metric above, equation (15), =α αy W is the Weyl vector of the space. This points to

another unique characteristic of flat biconformal space. The structures of the conformal group,treated as described above, give rise to a natural direction of time, given by the gauge field ofdilatations. The situation is reminiscent of previous studies. In 1979, Stelle and Westintroduced a special vector field to choose the local symmetry of the MacDowell–Mansouritheory. The vector breaks the de Sitter symmetry, eliminating the need for the Wigner–Inönucontraction. Recently, Westman and Zlosnik [31] have looked in depth at both the de Sitterand anti-de Sitter cases using a class of actions which extend that of Stelle and West byincluding derivative terms for the vector field and therefore lead to dynamical symmetrybreaking. In [32, 33] and Einstein-aether theory [34], there is also a special vector fieldintroduced into the action by hand that can make the Lorentzian metric Euclidean. Theseapproaches are distinct from that of the biconformal approach, where the vector necessary forspecifying the timelike direction occurs naturally from the underlying group structure. Wewill have more to say about this below, where we show explicitly that the Euclidean gaugetheory necessarily posesses a special vector, ω η η= −v dab

bc1

2. This vector gives the time

direction on two Lagrangian submanifolds, making them necessarily Lorentzian. The fullmanifold retains its original symmetry.

4. A brief note on gravitation

Notice that our development to this point was based solely on group quotients and gen-eralization of the resulting principal fiber bundle. We have arrived at the form of the

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

13

curvatures in terms of the Cartan connection, and Bianchi identities required for integrability,thereby describing certain classes of geometry with local symmetry. Within the biconformalquotient, the demand for orthogonal Lagrangian submanifolds with non-degenerate n-dimrestrictions of the Killing metric leads to the selection of certain Lorentzian submanifolds.Though our present concern has to do with the geometric background rather than withgravitational theories on those backgrounds, for completeness we briefly digress to specify theaction functionals for gravity. The main results of the current study, taken up again in the finalthree sections, concern only the homogeneous space, n

0(2 ).

We are guided in the choice of action functionals by the example of general relativity.Given the Riemannin geometries of section 2.3, we may write the Einstein–Hilbert action andproceed. More systematically, however, we may write the most general, even-parity actionlinear in the curvature and torsion. This turns out to be the tetradic Palatini action,

∫ ε= ∧ ∧S R e ePab c d

abcd, and, as noted above, a full variation of the connection,

ωδ δ( )e ,bb

a , implies vanishing torsion in addition to the Einstein equation. The latter, morerobust approach is what we follow for conformal gravity theories.

It is generally of interest to build the simplest class of actions possible, and we use thefollowing criteria.

(i) The pure-gravity action should be built from the available curvature tensor(s) and othertensors which occur in the geometric construction.

(ii) The action should be of lowest possible order ⩾1 in the curvatures.(iii) The action should be of even parity.

These are of sufficient generality not to bias our choice. It may also be a reasonableassumption to set certain tensor fields, for example, the spacetime torsion to zero. This cansignificantly change the available tensors, allowing a wider range of action functionals.

Notice that if we perform an infinitesmal conformal transformation to the curvatures,Ω Ω Ω Ωβ

α αβ( ), , , , they all mix with one another, since the conformal curvature is really a

single Lie-algebra-valued two form. However, the generalization to a curved manifold breaksthe non-vertical symmetries3, allowing these different components to become independenttensors under the remaining Weyl group. Thus, to find the available tensors, we apply aninfinitesmal transformation of the fiber symmetry. Tensors are those objects which transformlinearly and homogeneously under these transformations.

4.1. The auxiliary gauging and Weyl gravity

According to our rules above, an action for the auxiliary gauging is constructible from theavailable tensors, Ωe ,c

BA , together with the invariant metric and Levi-Civita tensors,

η ε,ab abcd . In n2 -dimensions, scale invariance requires n factors of the curvature, so it is the=p q( , ) (4, 2) case that is of interest here. Then the most general even parity, -invariant

possibility is uniquely determined (up to an overall multiple) to be

3 The reduction of the conformal symmetry here should not be confused with ‘symmetry breaking’. Until weperform the quotient of a group by a (symmetry) subgroup, there is neither a manifold nor any symmetry—thequotient defines the initial physical geometry. Every formulation of gravity must make some such initial definition ofthe geometry.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

14

∫∫

Ω Ω

Ω Ω Ω Ω

α

α

= ∧

= ∧ + ∧ + ∧( )S

T S

*

* 4 * 2 * ,

BA

AB

ba

ab c

c

auxiliary

where Ω BA is the full SO (4, 2) curvature two-form. This leads to a Weyl–Cartan geometry

(i.e., one having nontrivial dilatation and torsion). To achieve Weyl gravity on the bundle, we need to break the special conformal symmetry with our choice of the action(putting aside the question of whether this might be done dynamically). Since the curvaturehas already broken the translational symmetry, we expect both non-dynamical torsion andnon-dynamical special conformal curvature. Dropping the center term in Sauxiliary, we have the -invariant Weyl–Bach action [27],

∫ Ω Ω Ω Ωα β= ∧ + ∧( )S * * . (16)ba

ab

auxiliary

Various special cases of this action have been studied. With the absence of translations andspecial conformal transformations, Ω T S, ,b

a aa and Ω all become independent tensors under

the remaining symmetry, making the choice of α and β arbitrary. Bach, [27], examined theexceptional case β α= 2 , for which the dilatation Ω is nontrivial. Assuming a suitable metricdependence of the connection components, ω ωβ

αα( )f, , , metric variation leads to the fourth-

order Bach equation. In efforts to study superconformal gravity, two collaborations,[8–11, 35] set β = 0, and showed that the action reduces to the Weyl curvature squared. Boththese sets of investigations assumed vanishing torsion. Recently (with β α≠ 2 ) it has beenshown that when the full connection is varied independently, and the torsion set to zero onlyin the resulting field equations, Sauxiliary leads to the locally dilatationally invariantgeneralization of the vacuum Einstein equation [19].

In dimensions higher than four, our criteria lead to still higher order actions. Alter-natively, curvature-linear actions can be written in any dimension by introducing a suitablepower of a scalar field [30, 36]. This latter [36], gives the ϕ R2 action often used in tractorstudies.

4.2. Gravity in the biconformal gauging

The biconformal gauging, based on , also has tensorial basis forms ω ωαα( ), . Moreover,

each of the component curvatures, Ω Ω Ω Ωβα α

β( ), , , , becomes an independent tensor underthe Weyl group.

In the biconformal case, the volume form ω ω ω ω ω∧ ∧ … ∧ ∧ ∧ ∧αβ νρσ λ α β ν

ρ σ……e

ω… ∧ λ has zero conformal weight. Since both Ω βα and Ω also have zero conformal weight,

there exists a curvature-linear action in any dimension and for any p q( , ) [2]. The mostgeneral linear case is

∫ Ω Ω ω ω ω ω ω ωα β δ γ= + + ∧ ∧ ∧ … ∧ ∧ ∧ … ∧βα

βα α

β αμ νβρ σ μ ν

ρ σ……( )S e .

Notice that we now have three important properties of biconformal gravity that arise becauseof the doubled dimension: (1) the non-degenerate conformal Killing metric induces a non-degenerate metric on the manifold, (2) the dilatational structure equation generically gives asymplectic form, and (3) there exists a Weyl symmetric action functional linear in thecurvature, valid in any dimension.

There are a number of known results following from the linear action. In [2] torsion-constrained solutions are found which are consistent with scale-invariant general relativity.Subsequent work along the same lines by one of us (JTW) shows that the torsion-free

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

15

solutions are determined by the spacetime solder form, and reduce to describe spaces con-formal to Ricci-flat spacetimes on the corresponding spacetime submanifold. A super-symmetric version is presented in [37], and studies of Hamiltonian dynamics [38, 39] andquantum dynamics [40] support the idea that the models describe some type of relativisticphase space determined by the configuration space solution.

5. Homogeneous biconformal space in a conformally orthonormal, symplecticbasis

The central goal of the remainder of this manuscript is to examine properties of the homo-geneous manifold, n

0(2 ), which become evident in a conformally orthonormal basis, that is, a

basis which is orthonormal up to an overall conformal factor. Generically, the properties wediscuss will be inherited by the related gravity theories as well.

As noted above, biconformal space is immediately seen to possess several structures notseen in other gravitational gauge theories: a non-degenerate restriction of the Killing metric4,a symplectic form, and Kähler structure. In addition, the signature theorem in [28] shows thatif the original space has signature ±n or zero, the imposition of involution conditions leads toorthogonal Lagrangian submanifolds that have non-degenerate n-dim restrictions of theKilling metric. Further, constraining the momentum space to be as flat as permitted requiresthe restricted metrics to be Lorentzian. We strengthen these results in this section and thenext. Concerning ourselves only with elements of the geometry of the Euclidean = ±s n( )cases (as opposed to the additional restrictions of the field equations, involution conditions orother constraints), we show the presence of exactly such Lorentzian signature Lagrangiansubmanifolds without further assumptions.

We go on to study the transformation of the spin connection when we transform the basisof an eight-dim biconformal space to one adapted to the Lagrangian submanifolds. We showthat in addition to the Lorentzian metric, a Lorentzian connection emerges on the config-uration and momentum spaces and there are two new tensor fields. Finally, we examine thecurvature of these Lorentzian connections and find both a cosmological constant and cos-mological ‘dust’. While it is premature to make quantitative predictions, these new geometricfeatures provide novel, predictive candidates for dark energy and dark matter.

5.1. The biconformal quotient

We start with the biconformal gauging of section 3, specialized to the case of compactified,Euclidean 4 in a conformally orthonormal, symplectic basis. The Maurer–Cartan structureequations are

ω ω ω ω ωΔ= ∧ + ∧βα

βμ

μα

νβαμ

μνd 2 , (17)

ω ω ω ω ω= ∧ + ∧α ββα αd , (18)

ω ω ω ω ω= ∧ + ∧α αβ

β αd , (19)

ω ω ω= ∧ααd (20),

where the connection one-forms represent SO (4) rotations, translations, specialconformal transformations and dilatations respectively. The projection operator

4 There are non-degenerate restrictions in anti-de Sitter and de Sitter gravitational gauge theories.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

16

Δ δ δ δ δ≡ −γβαμ

γα

βμ αμ

γβ( )1

2in equation (17) gives that part of any ( )1

1-tensor antisymmetric

with respect to the original Euclidean metric, δαβ. As discussed in section 3.2.2, this group hasa non-degenerate, 15-dim Killing metric. We stress that the structure equations and Killingmetric—and hence their restrictions to the quotient manifold—are intrinsic to the conformalsymmetry.

The gauging begins with the quotient of this conformal group, SO (5, 1), by its Weylsubgroup, spanned by the connection forms ω β

α (here dual to SO (4) generators) and ω. Theco-tangent space of the quotient manifold is then spanned by the solder form, ωα, and the co-solder form, ωα, and the full conformal group becomes a principal fiber bundle with localWeyl symmetry over this eight-dim quotient manifold. The independence of ωα and ωα in thebiconformal gauging makes the two-form ω ω∧α

α non-degenerate, and equation (20)immediately shows that ω ω∧α

α is a symplectic form. The basis ω ωαα( ), is canonical.

The involution evident in equation (18) shows that the solder forms, ωα, span a sub-manifold, and from the simultaneous vanishing of the symplectic form this submanifold isLagrangian. Similarly, equation (19) shows that the ωβ span a Lagrangian submanifold.However, notice that neither of these submanifolds, spanned by either ωα or ωα alone, has an

induced metric, since by equation (14), ω ω ω ω= =α βα β, , 0. The orthonormal basis

will make the Killing metric block diagonal, guaranteeing that its restriction to the config-uration and momentum submanifolds have well-defined, non-degenerate metrics.

It was shown in [28] that it is consistent (for signatures ±n, 0 only) to impose involutionconditions and momentum flatness in this rotated basis in such a way that the new basis stillgives Lagrangian submanifolds. Moreover, the restriction of the Killing metric to these newsubmanifolds is necessarily Lorentzian. In what follows, we do not need the assumptions ofmomentum flatness or involution, and work only with intrinsic properties of n

0(2 ). This

section describes the new basis and resulting connection, while the next establishes that forinitial Euclidean signature, the principal results of [28] follow necessarily. Our results showthat the timelike directions in these models arise from intrinsically conformal structures.

We now change to a new canonical basis, adapted to the Lagrangian submanifolds.

5.2. The conformally-orthonormal Lagrangian basis

In [28] the ω ωαα( ), basis is rotated so that the metric, hAB becomes block diagonal

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢

⎤⎦⎥

δδ

⇒ =−

βα

βα [ ]h

h

h

0

0

0

0AB

ab

ab

while the symplectic form remains canonical. This makes the Lagrangian submanifoldsorthogonal with a non-degenerate restriction to the metric. Here we use the same basischange, but in addition define coefficients, αha to relate the orthogonal metric to oneconformally orthonormal on the submanifolds, η = α

αββh h hab a b , where ηab is conformal to

± ± ± ±diag( 1, 1, 1, 1). From [28] we know that hab is necessarily Lorentzian,η η= = − =ϕ ϕh e diag( 1, 1, 1, 1) eab ab ab

2 2 0 and we give a more general proof below. Noticethat the definition of ηab includes an unknown conformal factor. The required change of basisis then

⎜ ⎟⎛⎝

⎞⎠ω ω= +α

α αββh he

1

2(21)a a

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

17

⎜ ⎟⎛⎝

⎞⎠ω ω= −α

α αββh hf

1

2(22)a a

with inverse basis change

ω η= −α α ( )h e f1

2(23)a

a abb

ω η= +α α ( )h f e . (24)aa ab

b

Using (14), the Killing metric is easily checked to be

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

η

η

=−

=−

α βαβ

α βαβ

ϕ

ϕ

−

h h h

h h h

e

e e e f

f e f f

, ,

, ,

0

0

e 0

0,

a b ab

ab

a b

a b

a b

ab

ab

( )

( )

20

2 0

where =αβ αβh h( ), and δ=αββγ γ

αh h .The new basis is also canonical, as we see by transforming the dilatation equation (20), to

find ω = ∧d e faa. We refer to the =f 0a and =e 0a submanifolds as the configuration and

momentum submanifolds respectively.

5.3. Properties of the structure equations in the new basis

We now explore the properties of the biconformal system in this adapted basis. Rewriting theremaining structure equations (17)–(19), in terms of ea and fa, we show some striking can-celations that lead to the emergence of a connection compatible with the Lorentzian metric,and two new tensors.

We begin with the exterior derivative of equation (21), using structure equations (18) and(19), and then using the basis change equations (23) and (24). Because equations (23) and(24) involve the sum and difference of ea and fb, separating by these new basis forms leads toa separation of symmetries. This leads to a cumbersome expansion, which reduces con-siderably and in significant ways, to

τ τ ωΘ η Ξ η η η η= ∧ − ∧ + ∧ + ∧ + ∧de e f d e d f f1

2

1

22 , (25)a b

cbad

dc bc

c dbae

ed

bcab c ab

bab

b

where we define projections Θ δ δ η η≡ −( )dbac

da

bc ac

bd1

2and Ξ δ δ η η≡ +( )cb

adca

bd ad

cb1

2that

separate symmetries with respect to the new metric ηab rather than δαβ. These give the

antisymmetric and symmetric parts, respectively, of a ( )1

1-tensor with respect to the new

orthonormal metric, ηab. Notice that these projections are independent of the conformal factoron ηab.

The significance of the reduction lies in how the symmetries separate between the dif-ferent subspaces. Just as the curvatures split into three parts, equation (25) and each of theremaining structure equations splits into three parts. Expanding these independent partsseparately allows us to see the Riemannian structure of the configuration and momentumspaces. It is useful to first define

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

18

τ α β≡ + , (26)ba

ba

ba

where α τΘ≡ba

cbad

dc and β τΞ≡b

acbad

dc . Then, to facilitate the split into ∧e ea b, ∧e fa

b and∧f fa b parts, we partition the spin connection and Weyl vector by submanifold, defining

α σ γ σ γ≡ + = +e f (27)ba

ba

ba

bca c

ba c

c,

β μ ρ μ ρ≡ + = +e f (28)ba

ba

ba

bca c

ba c

c,

ω ≡ +W We f . (29)aa a

a

We also split the exterior derivative, d d d= +x y( ) ( ), where coordinates αx and αy areused on the α

αe xe d=a a and ααf yf d=a a submanifolds, respectively. Using these, we

expand each of the structure equations into three -invariant parts. The complete set (withcurvatures included for completeness) is given in appendix A.

The simplifying features and notable properties include.

(i) The new connection: the first thing that is evident is that all occurences of the spinconnection ω β

α may be written in terms of the combination

τ ω≡ −α βα β α

αdh h h h (30)ba a

b ba

which, as we show below, transforms as a Lorentz spin connection. Although the basischange is not a gauge transformation, the change in the connection has a similarinhomogeneous form. Because αh a is a change of basis rather than local SO n( ) or localLorentz, the inhomogeneous term has no particular symmetry property, so τ b

a will haveboth symmetric and antisymmetric parts.

(ii) Separation of symmetric and antisymmetric parts: notice in equation (25) how theantisymmetric part of the new connection, α b

a , is associated with eb, while the symmetricpart, β b

a pairs with fc. This surprising correspondence puts the symmetric part into thecross-terms while leaving the connection of the configuration submanifold metriccompatible, up to the conformal factor.

(iii) Cancellation of the submanifold Weyl vector: the Weyl vector terms cancel on theconfiguration submanifold, while the fa terms add. The expansion of the dfa structureequation shows that the Weyl vector also drops out of the momentum submanifoldequations. Nonetheless, these submanifold equations are scale invariant because of theresidual metric derivative. Recognizing the combination of hd terms that arises as ηd ab,and recalling that η η= ϕeab ab

2 0 , we have η η δ ϕ− =d daccb b

a1

2. When the metric is

rescaled, this term changes with the same inhomogeneous term as the Weyl vector.(iv) Covariant derivative and a second Weyl-type connection: it is natural to define the

τcb-covariant derivative of the metric. Since α αη η+ = 0cb

ca ac

cb , it depends only on βc

a

and the Weyl vector,

τ τ ωη η η η η≡ + + −D d 2 (31)ab ab cbc

a acc

b ae

β ωη η η= + −d 2 2 . (32)ab cbc

a ab

This derivative allows us to express the structure of the biconformal space in terms of theLorentzian properties.

When all of the identifications and definitions are included, and carrying out similarcalculations for the remaining structure equations, the full set becomes

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

19

τ τ τ Δ η Δ η Δ Ξ= ∧ + ∧ − ∧ + ∧d e e f f f e2 , (33)ba

bc

ca

dbae

ecc d

ebac ed

c d fbae

defc

cd

α η η η= ∧ + ∧ + ∧de e d e D f1

2

1

2, (34)a c

ca

cbac b ab

b

α η η η= ∧ + ∧ − ∧df f d f D e1

2

1

2, (35)a a

bb

bcab c ab

b

ω = ∧d e f , (36)aa

with the complete -invariant separation in appendix A.

5.4. Gauge transformations and new tensors

The biconformal bundle now allows local Lorentz transformations and local dilatations on n

0(2 ). Under local Lorentz transformations, Λ c

a, the connection τ ba changes with an inho-

mogeneous term of the form Λ Λd¯bc

ca. Since this term lies in the Lie algebra of the Lorentz

group, it is antisymmetric with respect to ηab, Θ Λ Λ Λ Λ=( )d d¯ ¯dbac

ce

ed

be

ea and therefore only

changes the corresponding Θdbac-antisymmetric part of the connection, with the symmetric part

transforming homogeneously:

α α

β β

Λ Λ Λ Λ

Λ Λ

= −

=

d˜ ¯ ¯

˜ ¯ .

ba

ca

dc

bd

bc

ca

ba

ca

dc

bd

Having no inhomogeneous term, β ba is a Lorentz tensor. This new tensor field β b

a necessarilyincludes degrees of freedom from the original connection that cannot be present in α b

a , thetotal equaling the degrees of freedom present in τ b

a . As there is no obvious constraint on theconnection α b

a , we expect β ba to be highly constrained. Clearly, α d

c transforms as aLorentzian spin connection, and the addition of the tensor β b

a preserves this property, so τ ba

is a local Lorentz connection.Transformation of the connection under dilatations reveals another new tensor. The Weyl

vector transforms inhomogeneously in the usual way, ω ω= + fd˜ , but, as noted above, theexpression η ηdcb

ac1

2also transforms,

η η δ ϕ δ ϕ= = − fd d d d1

2˜ ˜ ˜ ( )cb

acba

ba

so that the combination

ω ϕ= +v d

is scale invariant. Notice the presence of two distinct scalars here. Obviously, givenη η δ ϕ=d dac

cb ba1

2we can choose a gauge function, ϕ= −f1 , such that η η =d 0ac

cb1

2. We also

have, ω =d 0, on the configuration submanifold, so that ω = fd 2, for some scalar f2 and thismight be gauged to zero instead. But while one or the other of ω or ϕd can be gauged to zero,their sum is gauge invariant. As we show below, it is the resulting vector v which determinesthe timelike directions.

Recall that certain involution relationships must be satisfied to ensure that spacetime andmomentum space are each submanifolds. The involution conditions in homogeneousbiconformal space are

μ= ∧ − ∧e v e0 (37)ba b

xa

( )

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

20

ρ= ∧ − ∧f u f0 , (38)ab

b y a( )

where ≡ + ≡ +v uv v u e fx y aa a

a( ) ( ) . These were imposed as constraints in [28], but areshown below to hold automatically in Euclidean cases.

6. Riemannian spacetime in Euclidean biconformal space

The principal result of [28] was to show that the flat biconformal space n0(2 ) arising from

any SO p q( , ) symmetric biconformal gauging can be identified with a metric phase spaceonly when the initial n-space is of signature ±n or zero. To make the identification, involutionof the Lagrangian submanifolds was imposed, and it was assumed that the momentum spaceis conformally flat. With these assumptions the Lagrangian configuration and momentumsubmanifolds of the signature ±n cases are necessarily Lorentzian.

Here we substantially strengthen this result, by considering only the Euclidean case. Weare able to show that further assumptions are unnecessary. The gauging always leads toLorentzian configuration and momentum submanifolds, the involution conditions are auto-matically satisfied by the structure equations, and both the configuration and momentumspaces are conformally flat. We make no assumptions beyond the choice of the quotient and the structures that follow from these groups.

Because this result shows the development of the Lorentzian metric on the Lagrangiansubmanifolds, we give details of the calculation.

6.1. Solution of the structure equations

A complete solution of the structure equations in the original basis, equations (17)–(20) isgiven in [41] and derived in [2], with a concise derivation presented in [38]. By choosing thegauge and coordinates α

β( )w s, appropriately, where Greek indices now refer to coordinatesand will do so for the remainder of this manuscript5, the solution may be given the form

ω Δ=βα

νβαμ

μνs wd2 , (39)

ω =α αwd , (40)

⎜ ⎟⎛⎝

⎞⎠ω δ= − −α α α β αβ

βs s s s wd d1

2, (41)2

ω = − ααs wd (42)

as is easily checked by direct substitution. Our first goal is to express this solution in theadapted basis and find the resulting metric.

From the original form of the Killing metric, equation (14), we find

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

δ

δ=

−

α β αβ

αβ

α β

βα

αβ

αβ

w w w s

s w s s k

d d d d

d d d d

, ,

, ,

0,

where we define δ≡ −αβ αβ α βk s s s22 . This shows that αwd and αsd do not span orthogonalsubspaces. We want to find the most general set of orthogonal Lagrangian submanifolds, andthe restriction of the Killing metric to them.

5 The connection forms could be written with distinct indices, for example as ω δ= ααwda a , but this is unnecessarily

cumbersome.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

21

Suppose we find linear combinations of the orginal basis κ λβα, that make the metric

block diagonal, with λ =α 0 and κ =β 0 giving Lagrangian submanifolds. Then any furthertransformation,

κ κ

λ λ

=

=

αβα β

α αβ

β

A

B

˜

˜

leaves these submanifolds unchanged and is therefore equivalent. Now suppose one of thelinear combinations is

λ α β

α β

= +

= +α α

ββ αμ

μ

αβ

β βμμ( )

A s C w

A s C w

d d

d d

˜ ˜

,

where = −C A C̃1 , and the constants are required to keep the transformation non-degenerate.Then λ α β= +α α αβ

βs wd dC spans the same subspace. A similar argument holds for κ β˜ , so ifwe can find a basis at all, there is also one of the form

λ

κ

α β

μ ν

= +

= +α α αβ

β

α α αββ

s C w

w B s

d d

d d .

Now check the symplectic condition,

κ λ μβ αμ δ νβ να∧ = ∧ + − ∧ + ∧αα αμ

α μμβ

αμαβ μ

βαβ

β α( ) ( )( ) w w w s s sC d d d d d dC B B .

To have κ λ∧ = ∧αα

ααw sd d , αβB and αβC must be symmetric and

αμνβ

αβ= = − ≡−B B C C1 ¯ .t 1

Replacing αβB in the basis, we look at orthogonality of the inner product, requiring

κ λ

μ αμβ

α β

αμ δβ

α αμ

=

= + − +

= − − −

αβ

α αμμ β βν

ν

βα αμ

μβ

w C s s C w

C k

d d d d

0 ,

1 ¯ ,

(2 1)1

( 1) ¯

with solution =αβα αμβ αμ αβ

−−C k( 1)

(2 1). Therefore, the basis

λ

κ

α α αμαμ

μ αμα

= + −−

= + −

α α αββ

α α αββ

s k w

w k s

d d

d d

( 1)

(2 1)2 1

satisfies the required properties and is equivalent to any other which does.The metric restrictions to the submanifolds are now immediate from the inner products:

κ κ

λ λ

αμα

ααμ

= −

= −−

α β αβ

α β αβ

k

k

,2 1

,2 1

.

2

2

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

22

This shows that the metric on the Lagrangian submanifolds is proportional to αβk , and we

normalize the proportionality to 1 by choosing μ = αα

+ k1

2

2

and β α≡ k , where = ±k 1. Thisputs the basis in the form

κ

λ

ββ β

ββ β

= + +

= + −

α α αββ

α α αββ

( )( )

( )

( )

kk w k k s

k s k k w

d d

d d

21 2

1

22 1 .

2 2

2 2

Now that we have established the metric

⎛⎝⎜

⎞⎠⎟δ= −αβ αβ α βk s

ss s

2,2

2

where δαβ is the Euclidean metric and δ= >αβα βs s s 02 , and have found one basis for the

submanifolds, we may form an orthonormal basis for each, setting η = α βαβh h kab a b .

ββ β= + +α

α αββ( )( )k

h k w k k se d d2

1 2 (43)a a 2 2

ββ β= − −α

α αββ( )( )h k s k k wf d d

1

22 1 . (44)a a

2 2

We see from the form δ= −αβ αβ α β( )k s s ss

2 22 that at any point αs 0, a rotation that takes

αss

1 0w

to a fixed direction n̂ will take αβk to − …s diag( 1, 1, , 1)2 so the orthonormal metricηab is Lorentzian. This is one of our central results. Since equations (39) through (42) providean exact, general solution to the structure equations, the induced configuration andmomentum spaces of Euclidean biconformal spaces are always Lorentzian, withoutrestrictions.

We now find the connection forms in the orthogonal basis and check the involutionconditions required to guarantee that the configuration and momentum subspaces areLagrangian submanifolds.

6.2. The connection in the adapted solution basis

We have defined τ ba in equation (30) with antisymmetric and symmetric parts α b

a and β ba ,

subdivided between the ea and fa subspaces, equations (27) and (28). All quantities may bewritten in terms of the new basis. We will make use of ≡ α

αs h sa a and δ δ≡ α βαβh hab a b . In

terms of these, the orthonormal metric is η δ= −( )s s sab ab s a b2 2

2 , where δ≡ >s s s 0aba b

2 .

Solving for δab, we find δ η= + s sab s ab s a b1 22 2

. Similar relations hold for the inverses, η δ,ab ab

(see appendix B). In addition, we may invert the basis change to write the coordinate dif-ferentials,

β η

ββ η β

= −

= − + +

β β

α α ( )( )

( ) ( )w k h k

s h k k k

d e f

d e f1

21 1 .

aa ab

b

aab

ba

2 2

The known solution for the spin connection and Weyl form, equations (39) and (42)immediately become

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

23

ω Δ β η= −( )s k ke f2 (45)ba

dbac

cd de

e

ω β βη= − +k s se f , (46)aa ab

a b

where we easily expand the projection Δdbac in terms of the new metric. Substituting this

expansion to find τ ba , results in

τ β Θ η η η η= + + − − αα( )( ) ke f d2 s 2 s 2 s s s h h .b

adbac

cae

bd eae

e b dd dg

g ba

The antisymmetric part is then α τΘ Θ≡ = − ααdh hb

acbad

dc

cbad

dc with the remaining terms

cancelling identically. Furthermore, as described above, αh c is a purely αs -dependent rotationat each point. Therefore the remaining α

αh hddc term will lie totally in the subspace spanned

by αsd , giving the parts of α ba as

⎛⎝⎜⎜

⎞⎠⎟⎟σ β

βΘ η= − − ∂

∂α

βα β e

1 k

2h

sh h (47)b

acbad

ba c

cdd

2

⎛⎝⎜⎜

⎞⎠⎟⎟γ β

βΘ= − + ∂

∂α

βα β f

k

2h

sh h . (48)b

acbad

ba c

c

2

Recall that the value of k or β in these expressions is essentially a gauge choice andshould be physically irrelavant. If we choose β = 12 , we get either σ = 0b

a or γ = 0ba ,

depending on the sign of k.Continuing, we are particularly interested in the symmetric pieces of the connection since

they constitute a new feature of the theory. Applying the symmetric projection to τ ba , we

expand

β τΞ≡ .ba

cbad

dc

Using Ξ =μμ α

β αμμβ( )h h h h k kd dab

cdd

a cb

1

2(see appendix C) to express the derivative term in

terms of va, we find the independent parts

μ

ρ

βδ βγ δ δ η η η

βδ η βγ δ η δ η η η η

= − + + + +

= + + + +

+

−

( )( )

( )( )

k s s s s s s s

s k s s s s s s

e

f

2 ,

2 ,

ba

ba

c ba

c ca

bad

bc dad

b c dc

ba

ba cd

d ba cd

d bc ad

dac

bad ce

b d e c

where γ β≡ ±β± ( )k11

22 . Written in this form, the tensor character of μ b

a and ρ ba is not

evident, but since we have chosen ηab orthonormal (referred to later as the orthonormalgauge), ϕ = 0, and ω ωϕ= + =v d we have β βη+ = − +k s sv u e fe f a

a aba b( ) ( ) so that we

may equally well write

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟μ δ γ δ δ η η

βη= − + + ++v k v v v v v v e

2, (49)b

aba

c ba

c ca

bad

bc dad

b c dc

2

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟ρ δ γ δ δ η η

βη= + + + +−u k u u u u u u f

2, (50)b

aba c

ba c

bc a ac

bdd

bda c d

c2

which are manifestly tensorial. The involution conditions, equations (37) and (38), are easilyseen to be satisfied identically by equations (49) and (50). Therefore, the =f 0a and =e 0a

subspaces are Lagrangian submanifolds spanned respectively by ea and fa. There existcoordinates such that these basis forms may be written

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

24

= μμe xe d , (51)a a

= μμf yf d . (52)a a

To find such submanifold coordinates, the useful thing to note is that δ= ανμν

μα( ) k sd ds

s2

so that the basis may be written as

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

γ βδ

δ β γ δ

= + ≡

= − ≡

αα αβ β

αα

ααβ

βμ μμν

ν μμ

+

−( )

h k ws

sh x

h k ks

sw f y

e d d

f d d

,

,

a a a

a a a

2

2

with γ βδ= +α α αβ+

β( )x k ws

s2 and β γ δ= −μ μνν

−β( )y k w

s

s2 . This confirms the involution.

7. Curvature of the submanifolds

The nature of the configuration or momentum submanifold may be determined by restrictingthe structure equations by =f 0a or =e 0a , respectively. To aid in the interpretation of theresulting submanifold structure equations, we define the curvature of the antisymmetricconnection αb

a

α α α≡ − ∧R d , (53)ba

ba

bc

ca

= ∧ + ∧ + ∧R R Re e f e f f1

2

1

2. (54)bcd

a c db d

a cc

db

a cdc d

While all components of the overall Cartan curvature, Ω Ω Ω= ( )T S, , ,Aba a

a are zero on

n0(2 ), the curvature, R b

a , and in particular the curvatures ∧R e ebcda c d1

2and ∧R f fb

a cdc d

1

2on

the submanifolds, may or may not be. Here we examine this question, using the structureequations to find the Riemannian curvature of the connections, σb

a and γ ,ba of the Lorentzian

submanifolds.

7.1. Momentum space curvature

To see that the Lagrangian submanifold equations describe a Riemannian geometry, we set=e 0a in the structure equations (33)–(36) and choose the ϕ = 0 (orthonormal) gauge (or see

appendix A, equations (A.3a) and (A.3d), with the Cartan curvatures set to zero). Then,taking the Θdb

ac projection, we have

ρ ρ

γ

Θ η Δ= ∧ − ∧ + ∧

= − ∧

R f f f f

d f f

01

2,

0 . (55)

ba cd

c d bc

ca

dbac ac

cfeb

b a

y b ba

a( )

These are the structure equations of a Riemannian geometry with additional geometric terms,ρ ρ Θ η Δ− ∧ + ∧f fb

cc

adbac ac

cfeb

b a, reflecting the difference between the Cartan curvature ofthe spin connection and the Riemannian curvature of the Lorentzian connection, γ b

a. Thesymmetric projection is

ρ Ξ Δ η= − ∧=

kD f f

d u 0,

yba

dbac

ecdf eg

f g

y f

( )

( ) ( )

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

25

where γu ,f ba

( ) and ρ ba are given by equations (46), (48), (50), respectively. Rather than

computing R ba cd directly from γ b

a , which requires a complicated expression for the localrotation, αh a, we find it using the rest of equation (55).

Letting β = λe so that

⎧⎨⎩γ λλ

+ = == −−k k

k

cosh 1sinh 1

22

2

the curvature is

⎧⎨⎪⎩⎪

λΘ η η η

λΘ η η η∧ =

+ ∧ =

+ ∧ = −

( )( )

Rs s k

s s kf f

f f

f f

1

2

cosh 2 1

sinh 2 1.b

a cdc d

cbag cf cd fe

d e f g

cbag cf cd fe

d e f g

2

2

Now consider the symmetric equations. Notice that the Weyl vector has totally decoupled,with its equation showing that u f( ) is closed, a result which also follows from its definition.

For the symmetric projection, we find Ξ η Δ ∧ ≡f f 0dbac ac

cfeb

b a . Then, contraction of ρDaba c

with η η u uad cea e, together with =d u 0y f( ) ( ) shows that ua is covariantly constant, =D u 0y

a b( ) .

If we choose = −k 1 and λ = 0, the Riemann curvature of the momentum space van-ishes. This is a stronger result than in [28], since there only the Weyl curvature could be set tozero consistently. In this case, the Lagrangian submanifold becomes a vector space and thereis a natural interpretation as the co-tangent space of the configuration space. However, theorthonormal metric in this case, η=f f,a b ab, has the opposite sign from the metric of the

configuration space, η= −e e,a b ab. This reversal of sign of the metric together with the theunits, suggests that the physical (momentum) tangent space coordinates are related to thegeometrical ones by ∼α αp i y . This has been suggested previously [42] and explored in thecontext of quantization [40].

Leaving β and k unspecified, we see that in general momentum space has non-vanishingRiemannian curvature of the connection γ b

a , a situation suggested long ago for quantumgravity [43, 44]. We consider this further in section 7.3. Whatever the values of β and k, themomentum space is conformally flat. We see this from the decomposition of Riemanniancurvature into the Weyl curvature, C b

a , and Schouten tensor, a, given by

Θ= − ∧R C e2 .ba

ba

dbae

ed

The Schouten tensor, η≡ −− −( )eR Ra ab abb1

n 2

1

2(n 1)is algebraically equivalent to the

Ricci tensor, Rab. It is easy to prove that when the curvature two-form can be expressed as aprojection in the form Θ= − ∧R X e2b

adbae

ed , then Xa is the Schouten tensor, and the Weyl

curvature vanishes. Vanishing Weyl curvature implies conformal flatness.

7.2. Spacetime curvature and geometric curvature

The curvature on the configuration space takes the same basic form. Still in the orthonormalgauge, and separating the symmetric and antisymmetric parts as before, we again find aRiemannian geometry with additional geometric terms,

μ μσ Θ Δ η= − ∧ − ∧R e e0 ( ) , (56)ba

bc

ca

dbac

fcde

egg f

σ= − ∧d e e0 , (57)xa b

ba

( )

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

26

together with

μ Ξ Δ η= − ∧=

D e e

d v

0 ,

0 .

x ba

dbac

fcde

egg f

x

( )

( )

Looking first at all the Θcbad-antisymmetric terms and substituting in (49) for μ b

a , we findthat the Riemannian curvature is

γ Θ η= − + ∧+( ) ( )k s sR e e2 ,ba

dbac

ce c ed e2

so the Weyl curvature vanishes and the Schouten tensor is

γ η= − ++( )( )s s e1

2k 2 . (58)a ab a b

b2

The vanishing Weyl curvature tensor shows that the spacetime is conformally flat. This resultis discussed in detail below.

The equation, =d v 0x( ) shows that v is hypersurface orthogonal. Expanding theremaining equation with η= =d v D0, 0x x ab( ) ( ) and =D e 0x

a( ) , contractions involving ηab

and va quickly show that

=D v 0.ax

b( )

This, combined with =uD 0y a( ) and η= −u k va abb shows that the full covariant derivative

vanishes, =D v 0a b . This vector is therefore a covariantly constant, hypersurface orthogonal,unit timelike Killing vector of the spacetime submanifold.

7.3. Curvature invariant

Substituting β = λe as before, the components of the momentum and configuration curvaturesbecome

⎧⎨⎪⎩⎪

η ηλ Θ δ Θ δ η

λ Θ δ Θ δ η=

− + =

− + = −

( )( )

( )( )

Rs s k

s s k

cosh 2 1

sinh 2 1df eg b

a fgdbac

ef

ebac

df

fc f c

dbac

ef

ebac

df

fc f c

2

2

and

⎧⎨⎪⎩⎪

λ Θ δ Θ δ η

λ Θ δ Θ δ η=

− + =

− + = −

( )( )

( )( )

Rs s k

s s k

sinh 2 1

cosh 2 1.bde

adbac

ef

ebac

df

fc f c

dbac

ef

ebac

df

fc f c

2

2

Subtracting these

η η Θ δ Θ δ η− = − +( )( )R R k s s2df eg ba fg

bdea

dbac

ef

ebac

df

fc f c

so that the difference of the configuration and momentum curvatures is independent of thelinear combination of basis forms used. This coupling between the momentum andconfiguration space curvatures adds a sort of complementarity that goes beyond thesuggestion by Born [43, 44] that momentum space might also be curved. As we continuouslyvary β2, the curvature moves between momentum and configuration space but this differenceremains unchanged. We may even make one or the other Lagrangian submanifold flat.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

27

For the Einstein tensors,

η η η− = − + −( )G G k n n s s1

2( 3) ( 2) .ac bd y

cdab

xab a b( )

( )

7.4. Candidate dark matter

There is a surprising consequence of the tensor μ ba in the Lorentz structure equation. The

structure equations for the configuration Lagrangian submanifold above describe an ordinarycurved Lorentzian spacetime with certain extra terms from the conformal geometry that existeven in the absence of matter. We gain some insight into the nature of these additional termsfrom the metric and Einstein tensor. In coordinates, the metric takes the form

⎛⎝⎜

⎞⎠⎟δ= −αβ αβ α βh s

ss s

2,2

2

which is straightforwardly boosted to ηαβs2 0 at a point. Since the spacetime is conformally flat,gradients of the conformal factor must be in the time direction, αs , so we may rescale the time,

′ =t s td d2 to put the line element in the form

= − ′ + ′ + +( )s t s t x y zd d ( ) d d d ,2 2 2 2 2 2

that is, the vacuum solution is a spatially flat FRW cosmology. Putting the results in terms ofthe Einstein tensor and a coordinate basis, we expect an equation of the form κ=αβ αβG T˜ matter

where the Cartan Einstein tensor is modified to

≡ − − + − −αβ αβ α β αβG G n s s s n n s h˜ 3( 2)3

2( 2)( 3) , (59)2 2

where αβG is the familiar Einstein tensor. The new geometric terms may be thought of as acombination of a cosmological constant and a cosmological perfect fluid. With thisinterpretation, we may write the new cosmological terms as

κ ρ Λ= + + −αβ α β αβ αβ( )T p v v p h h ,cosm0 0 0

where κ ≡ − − − −α β αβT n s v v n n s h3( 2) ( 2)( 3)abcosm 2 3

22 . In =n 4-dimensions,

ρ Λ+ = −( )p p1

2 0 0 0, with the equation of state and the overall scale undetermined. If weassume an equation of state ρ=p w0 0, this becomes

ρ Λ+ =w1

2(1 3 ) .0

This relation alone does not account for the values suggested by the current Planck data:about 0.68 for the cosmological constant, 0.268 for the density of dark matter, and vanishingpressure, w = 0. However, these values are based on standard cosmology, while we have notyet included matter terms in equation (59). Moreover, the proportions of the three geometricterms in equation (59) may change when curvature is included. Such a change is suggested bythe form of known solutions in the original basis, where αβh is augmented by a Schoutenterm. If this modification also occurs in the adapted basis, the ratios above will be modified.We are currently examining such solutions.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

28

8. Discussion

Using the quotient method of gauging, we constructed the class of biconformal geometries.The construction starts with the conformal group of an SO p q( , )-symmetric pseudo-metricspace. The quotient by ≡ ×p q SO p q dilatations( , ) ( , ) gives the homogeneous manifold, n

02 . We show that this manifold is metric and symplectic (as well as Kähler with a different

metric). Generalizing the manifold and connection while maintaining the local invariance,we display the resulting biconformal spaces, n2 [15, 16, 21].

This class of locally symmetric manifolds becomes a model for gravity when we recallthe most general curvature-linear action [2].

It is shown in [28] that p q( , )n0(2 ) in any dimension = +n p q will have Lagrangian

submanifolds that are orthogonal with respect to the 2n-dim biconformal (Killing) metric andhave non-degenerate n-dim metric restrictions on those submanifolds only if the originalspace is Euclidean or signature zero ∈( ){ }p n0, ,n

2, and then the signature of the sub-

manifolds is severely limited → ±p p( 1). This leads in the two Euclidean cases to Lor-enztian configuration space, and hence the origin of time [28]. For the case of flat, eight-dimbiconformal space the Lagrangian submanifolds are necessarily Lorentzian.

Our investigation explores properties of the homogeneous manifold, n( , 0)n02 . Starting

with Euclidean symmetry, SO n( ), we clarify the emergence of Lorentzian signatureLagrangian submanifolds. We extend the results of [28], eliminating all but the group-theoretic assumptions. By writing the structure equations in an adapted basis, we reveal newfeatures of these geometries. We summarize our new findings below.

A new connection

There is a natural SO n( ) Cartan connection on n02 . Rewriting the biconformal structure

equations in an orthogonal, canonically conjugate, conformally orthonormal basis auto-matically introduces a Lorentzian connection and decouples the Weyl vector from the sub-manifolds. This structure emerges directly from the transformation of the structure equations,as detailed in points 1 through 4 in section 5.3.

Specifically, we showed that all occurences of the SO (4) spin connection ω βα may be

written in terms of the new connection, τ ω≡ −βαα β α

αdh h h hba a

b ba, which has both symmetric

and antisymmetric parts. These symmetric and antisymmetric parts separate automatically inthe structure equations, with only the Lorentz part of the connection, α τΘ=b

adbac

cd

describing the evolution of the configuration submanifold solder form. The spacetime andmomentum space connections are metric compatible, up to a conformal factor.

The Weyl vector terms drop out of the submanifold basis equations. The submanifoldequations remain scale invariant because of the residual metric derivative, η η δ ϕ=d dac

cb ba1

2.

When the metric is rescaled, this term changes with the negative of the inhomogeneous termacquired by the Weyl vector.

Two new tensors

It is especially striking how the Weyl vector and the symmetric piece of the connection arepushed from the basis submanifolds into the mixed basis equations. These extra degrees offreedom are embodied in two new Lorentz tensors.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

29

The factor δ ϕdba which replaces the Weyl vector in the basis equations allows us to form

a scale-invariant one-form,

ω ϕ= +v d .

It is ultimately this vector which determines the time direction.We showed that the symmetric part of the spin connection, β b

a , despite being a piece ofthe connection, transforms as a tensor. The solution of the structure equations shows that thetwo tensors, v and β b

a are related, with β ba constructed cubically, purely from v and the

metric. Although the presence of β ba changes the form of the momentum space curvature, we

find the same signature changing metric as found in [28]. Rather than imposing vanishingmomentum space curvature as in [28], we make use of a complete solution of theMaurer–Cartan equations to derive the metric. The integrability of the Lagrangian sub-manifolds, the Lorentzian metric and connection, and the possibility of a flat momentumspace are all now seen as direct consequences of the structure equations, without assumptions.

Riemannian spacetime and momentum space

The configuration and momentum submanifolds have vanishing dilatational curvature,making them gauge equivalent to Riemannian geometries. Together with the signature changefrom the original Euclidean space to these Lorentzian manifolds, we arrive at a suitable arenafor general relativity in which time is constructed covariantly from a scale-invariant Killingfield. This field is provided automatically from the group structure.

Effective cosmological fluid and cosmological constant

Though we work in the homogeneous space, n02 , so that there are no Cartan curvatures,

there is a net Riemannian curvature remaining on the spacetime submanifold. We show this todescribe a conformally flat spacetime with the deviation from flatness provided by additionalgeometric terms of the form

ρ Λ≡ − + =αβ αβ α β αβG G v v h˜ 00

that is, a background dust and a cosmological constant. The values ρ = −n s3( 2)02 and

Λ = − −n n s( 2)( 3)3

22 give, in the absence of physical sources, the relation

ρ Λ+ =w(2 3 ) 0 for an equation of state ρ=p w0 0. An examination of more realistic

cosmological models involving matter fields and curved biconformal spaces, n2 , isunderway. Nonetheless, the current homogeneous model demonstrates that just as theMacDowell–Mansouri treatment of de Sitter gravity introduces a cosmological constant,conformally based gravitation can introduce other geometric contributions to curvature.

Appendix A. Subparts of the structure equations

Here we write the structure equations, including Cartan curvature. We expand the config-uration, mixed and momentum terms separately. Note that the ∧f fa b part of the dea equationand the ∧e ea b part of the dfa equation are set to zero. These are the involution conditions,which guarantee that the configuration and momentum subspaces are integrable submanifoldsby the Frobenius theorem.

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

30

In the conformal-orthonormal basis, we have η η δ ϕ= =ϕ ϕ− ( )g gd d de e 2abbc

abbc c

a2 2 . Thestructure equations in the conformal-orthonormal basis are

τ τ τ Ω

α

α

ω Ω

Δ η Δ η Δ Ξ

η η η

η η η

= ∧ + ∧ − ∧ + ∧ +

= ∧ + ∧ + ∧ +

= ∧ + ∧ − ∧ +

= ∧ +

d e e f f f e

de e d e D f T

df f d f D e S

d e f

2 ,

1

2

1

2,

1

2

1

2,

.

ba

bc

ca

dbae

ecc d

ebac ed

c d fbae

defc

cd

ba

a cca

cbac b ab

ba

a ab

bbc

ab c abb

a

aa

Then defining

μ μ μ σ σ μ

ρ ρ ρ σ σ ρ

μ μ μ γ γ μ

ρ ρ ρ γ γ ρ

≡ − ∧ − ∧

≡ − ∧ − ∧

≡ − ∧ − ∧

≡ − ∧ − ∧

D d

D d

D d

D d

,

,

,

xb

a xb

ab

cc

ab

cc

a

xb

a xb

ab

cc

ab

cc

a

yb

a yb

ab

cc

ab

cc

a

yb

a yb

ab

cc

ab

cc

a

( ) ( )

( ) ( )

( ) ( )

( ) ( )

the separation of the structure equations into independent parts gives:

Configuration space:

σ σ σ μ μ μΩ Δ η∧ = − ∧ + − ∧ − ∧k ae e d D e e1

2, (A.1 )bcd

a c d xb

ab

cc

a xb

ab

cc

aebac

cdd e( ) ( )

σ η η∧ = − ∧ + ∧T be e d e e d e1

2

1

2, (A.1 )bc

a b cx

a bb

a ac xcb

b( )

( )

⎜ ⎟⎛⎝

⎞⎠μη δ η η∧ = ∧ − +S k W ce e e e d

1

2

1

2, (A.1 )abc

b cab

cc

bcb

dd

cex be( )

Ω ∧ = ( )W de e d e1

2. (A.1 )ab

a bx a

a( )

Cross-term:

σ γ γ σ σ γ

ρ μ ρ μ μ ρ

Ω

Δ Ξ

∧ = + − ∧ − ∧

+ + − ∧ − ∧

− ∧ a

f e d d

D D

f e

,

2 , (A.2 )

b da c

cd y

ba x

ba

bc

ca

bc

ca

xba y

ba

bc

ca

bc

ca

dbac

cefd

fe

( ) ( )

( ) ( )

γ η η= − + ∧T f e d e e d e1

2cab

bc y a b

ba ac y

cbb( ) ( )

⎜ ⎟⎛⎝

⎞⎠μη η η− ∧ + ∧ − ∧k W bf f e d f

1

2, (A.2 )ac

cb

b d cd bd x

cd b( )

σ η η∧ = − ∧ − ∧S f e d f f d f1

2a cb

bc x

a ab

bcb x

ac b( ) ( )

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

31

⎜ ⎟⎛⎝

⎞⎠ρη η η+ ∧ + ∧ + ∧k W ce f e d e

1

2, (A.2 )ab

cc

b cc

b bc ycd

d( )

Ω ∧ = + − ∧( ) ( )W W df e d e d f e f . (A.2 )ba

ab

y aa

xa

aa

a( ) ( )

Momentum space:

γ γ γ ρ ρ ρΩ η∧ = − ∧ + − ∧ + Δ ∧ af f d D f f1

2k (A.3 )b

a cdc d

yb

ab

cc

ab

ab

cc

aebac ed

c d( )

γ η η∧ = − ∧ − ∧S bf f d f f d f1

2

1

2(A.3 )a

bcb c

ya a

bb

cb yac b

( ) ( )

⎜ ⎟⎛⎝

⎞⎠ρη η η∧ = − ∧ − ∧ − ∧T k W cf f f f f d f

1

2

1

2(A.3 )abc

b cac

cb

bb

b cbd y

cd b( )

Ω ∧ = ( )W df f d f1

2(A.3 )bc

b cy a

a( )

Appendix B. Coordinate to orthonormal basis

The Euclidean and Lorentzian metric components are related in the orthonormal basis by:

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

η δ

η δ δ δ

δ η

δ η η η

= −

= −

= +

= +( )( )

ss

s s

s ss s

ss s

s s s

2,

1 2,

12 ,

2 ,

ab ab a b

ab ab ac bdc d

ab ab a b

ab ab acc

add

22

2 2

2

2

where δ= >αβα βs s s 02 .

Appendix C. Symmetric projection of the derivative of the solder form

For the calculation of the symmetric pieces of the connection, we need to express the sym-metric part, Ξ α

αh hdcbad

dc, in terms of the metric. Expanding the metric derivatives,

η

η η η

η η

η η

Ξ

=

= +

= +

= +

=

αμμβ

αμμ β

α μμ β μ β

α μβ μ

αβ

αβ

μμ

αβ

μμ

αβ

μμ

( )( )

( )( )( )

k k k h h

h h h h h h

h h h h h h

h h h h h h h h

h h h h

d d

d d

d d

d d

d2

a bab

c dcd a b

aba b

ab

c db cd

aba

bb

cb cd

ab da

bc

cb

cb

abcd

da

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

32

so that we can write Ξ μμ( )h hdcb

add

c explicitly,

Ξ

δ

δ δ δ δ δ

δ δ δ δ δ

β

βδ η δ η η η η η

β

βδ η δ η η η η

=

= −

= − +

= − +

= −−

+ + +

−+

+ + +

μμ α

β αμμβ

αβ αμ

μβ μ β

αβ

βα νρ

ραν

β βν αμ

μ ν

νν

( )

( )( ) ( )( ) ( )

( )

( )

h h h h k k

h h k s s s

h hs

s s s s

ss s s h s

ks s s s s s

k ks s s s s s

d d

d

d

d

e

f

1

21

22

1

1

1

22

1

22 .

cbad

dc a

b

ab

ab

ba cd

dac

b bc ad

d c

ba cd

d bc ad

dac

baf ce

b e f cff

ba cd

d bc ad

dac

baf ce

b e f c

2

2

2

2

2

References

[1] MacDowell S W and Mansouri F 1977 Unified geometric theory of gravity and supergravity Phys.Rev. Lett. 38 739–42

[2] Wehner A and Wheeler J T 1999 Conformal actions in any dimension Nucl. Phys. B 557 380–406[3] Cartan E 1910 Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du

second ordre Ann. Sci. École Norm. Sup. (3) 27 109–92[4] Sharpe R W 1997 Differential geometry Graduate Texts in Mathematics vol 166 (New York:

Springer)[5] Yang C-N and Mills R L 1954 Conservation of isotopic spin and isotopic gauge invariance Phys.

Rev. 96 191–5[6] Utiyama R 1956 Invariant theoretical interpretation of interaction Phys. Rev. 101 1597–607[7] Kibble T W B 1961 Lorentz invariance and the gravitational field J. Math. Phys. 2 212–21[8] Romao J C, Ferber A and Freund P G O 1977 Unified superconformal gauge theories Nucl. Phys.

B126 429[9] Crispim-Romao J 1978 Conformal and superconformal gravity and nonlinear representations Nucl.

Phys. B145 535[10] Kaku M, Townsend P K and van Nieuwenhuizen P 1977 Gauge theory of the conformal and

superconformal group Phys. Lett. B69 304–8[11] Kaku M, Townsend P K and van Nieuwenhuizen P 1977 Superconformal unified field theory

Phys. Rev. Lett. 39 1109[12] Ne’eman Y and Regge T 1978 Gravity and supergravity as gauge theories on a group manifold

Phys. Lett. B 74 54–56[13] Ne’eman Y and Regge T 1978 Gauge theory of gravity and supergravity on a group manifold

La Rivista del Nuovo Cimento (1978–1999) 1 1–43[14] Kobayashi S 1969 Foundations of Differential Geometry (New York: Wiley)[15] Ivanov E A and Niederle J 1982 Gauge formulation of gravitation theories. I. The Poincaré, de

Sitter, and conformal cases Phys. Rev. D 25 976–87[16] Ivanov E A and Niederle J 1982 Gauge formulation of gravitation theories. II. The special

conformal case Phys. Rev. D 25 988–94[17] Wise D K 2010 MacDowell–Mansouri gravity and Cartan geometry Class. Quantum Grav.

27 155010[18] Gryb S and Mercati F 2013 2+1 gravity on the conformal sphere Phys. Rev. D 87 064006[19] Wheeler J T 2014 Weyl gravity as general relativity Phys. Rev. D 90 025027[20] Arkani-Hamed N et al 2012 Scattering amplitudes and the positive grassmannian arXiv:1212.5605

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

33

[21] Wheeler J T 1998 New conformal gauging and the electromagnetic theory of Weyl J. Math. Phys.39 199–328

[22] Weyl H 1918 Gravitation and Elektrizität, Sitzurgsberichte de Königlich Preußischen Akademieder Wissenschaften pp 465–80

[23] Weyl H 1918 Math. Z. 2 384[24] Weyl H 1919 Ann. Phys., Lpz. 4 101[25] Weyl H 1921 Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen

Auffassung Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen pp 99–112[26] Weyl H 1923 Raum, Zeit, Materie (Berlin: Springer)[27] Bach R 1920 Math. Z. 9 110[28] Spencer J A and Wheeler J T 2011 The existence of time Int. J. Geom. Method Mod. Phys. 8

273–301[29] Čap A and Slovák J 2009 Mathematical surveys and monographs Parabolic Geometries. I.

Background and General Theory vol 154 (Providence, RI: American Mathematical Society)pp 280–90

[30] Dirac P A M 1936 Wave equations in conformal space Ann. Math. 37 429–42[31] Westman H F and Zlosnik T G 2014 Exploring Cartan gravity with dynamical symmetry breaking

arXiv:1302:1103v2[gr-qc][32] Fernando J, Barbero G and Villasenor E J S 2003 Lorentz violations and Euclidean signature

metrics Phys. Rev. D 68 087501[33] Fernando J and Barbero G 1996 From Euclidean to Lorentzian general relativity: the real way

Phys. Rev. D54 1492–9[34] Jacobson T 2007 Einstein-aether gravity: a status report PoSQG-PH 020 arXiv:0801.1547[gr-qc][35] Kaku M, Townsend P K and van Nieuwenhuizen P 1978 Properties of conformal supergravity

Phys. Rev. D 17 3179[36] Dirac P A M 1973 Long range forces and broken symmetries Proc. R. Soc. A 333 403–18[37] Anderson L B and Wheeler J T 2004 Biconformal supergravity and the AdS/CFT conjecture Nucl.

Phys. B686 285–309[38] Anderson L B and Wheeler J T 2007 Yang–Mills gravity in biconformal space Class. Quantum

Grav. 24 475–96[39] Wheeler J T 2007 Gauging Newtonʼs law Can. J. Phys. 85 307–44[40] Anderson L B and Wheeler J T 2006 Quantum mechanics as a measurement theory on

biconformal space Int. J. Geom. Math. Mod. Phys. 3 315–40[41] Wheeler J T 1998 New conformal gauging and the electromagnetic theory of Weyl J. Math. Phys.

39 299–328[42] Wheeler J T 1996 Why quantum mechanics is complex arXiv:hep-th/9708088[43] Born M 1938 A suggestion for unifying quantum theory and relativity Proc. R. Soc. A 165 291[44] Freidel L and Livine E 2006 3d quantum gravity and effective noncommutative quantum field

theory Phys. Rev. Lett. 96 221301

Class. Quantum Grav. 31 (2014) 215001 J S Hazboun and J T Wheeler

34