A first-order approach to conformal gravity

A first-order approach to conformal gravity

T.G. Złośnik111zlosnik@fzu.cz  and H.F. Westman222hans.westman@alefomega.com
Imperial College Theoretical Physics, Huxley Building, London, SW7 2AZ
Institute of Physics of the Czech Academy of Sciences, Na Slovance 1999/2, 182 21, Prague
Alef Omega, Inc., 1023 Walnut St., 80302 Boulder, Colorado, United States
Abstract

We investigate whether a spontaneously-broken gauge theory of the group may be a viable alternative to General Relativity. The basic ingredients of the theory are an gauge field and a Higgs field in the adjoint representation of the group with the Higgs field producing the symmetry breaking . The action for gravity is polynomial in and the field equations are first-order in derivatives of these fields. The new symmetry in the gravitational sector is interpreted in terms of an emergent local scale symmetry and the existence of ‘conformalized’ General Relativity and fourth-order Weyl conformal gravity as limits of the theory is demonstrated. Maximally symmetric spacetime solutions to the full theory are found and stability of the theory around these solutions is investigated; it is shown that regions of the theory’s parameter space describe perturbations identical to that of General Relativity coupled to a massive scalar field and a massless one-form field. The coupling of gravity to matter is considered and it is shown that Lagrangians for all fields are naturally gauge-invariant, polynomial in fields and yield first-order field equations; no auxiliary fields are introduced. Familiar Yang-Mills and Klein-Gordon type Lagrangians are recovered on-shell in the General-Relativistic limit of the theory. In this formalism, the General-Relativistic limit coincides with a spontaneous breaking of scale invariance and it is shown that this generates mass terms for Higgs and spinor fields.

1 Introduction

The prevailing classical theory of gravity remains Einstein’s General Relativity. The theory has enjoyed considerable success in accounting for data in gravitational experiments on scales of the solar system and below. Its success on larger scales is less clear. A considerable amount of evidence points towards the existence of a discrepancy between the properties of the universe as predicted by General Relativity in conjunction with known matter and what is actually observed [1]. The effects of this discrepancy can be incorporated into the framework of General Relativity via the introduction of additional matter described as a near-pressureless, perfect fluid (dark matter) but it is unclear whether this represents the effect of a genuinely new matter field or a manifestation of the breakdown of General Relativity. Additionally, there is evidence that the evolution of the very early universe was dominated by a scalar degree of freedom (the inflaton) that might not be formed from known matter and gravitational fields. Therefore, there exists an experimental motivation to consider alternative theories of gravitation [2].

In General Relativity the gravitational field is described solely by a field , the co-tetrad 333More usually gravity is discussed in terms of a metric tensor where is the invariant matrix of . We choose to phrase things in terms of as it is only with this field that one can couple gravity to fermionic fields.. The index denotes that the one-form is in the fundamental representation of the Lorentz group . The action for General Relativity - the Einstein-Hilbert action - possesses an invariance under local Lorentz transformations represented by matrices with [3]. This field is sufficient to describe the inherent dynamics of gravity and the coupling of gravity to known matter fields: scalar fields, gauge fields, and fermionic fields 444Fermionic fields are spinorial representations of the group which is the double-cover of and so the inclusion of fermions into gravitational theory implies that most accurately the local symmetry of General Relativity is that of .. In General Relativity the coupling of gravity to each of these fields requires use of the tetrad , where is the matrix inverse of . The coupling of gravity to all matter in General Relativity is hence non-polynomial and this contrasts with how other force fields in nature (the gauge fields of particle physics) couple to matter i.e. polynomial coupling within gauge-covariant derivatives of matter fields. Such is the success of the standard model of particle physics, it is conceivable that an alternative to General Relativity would share its Lagrangian structure of fermionic, gauge, and Higgs fields coupled to one another polynomially; in Appendix A we review how a promising approach towards this arises from regarding the local symmetry of General Relativity to be the symmetry broken phase of a theory with a gauge symmetry of a larger group than the Lorentz group.

A more long-standing motivation to look at alternatives to General Relativity than the above considerations is the observation that General Relativity possesses fixed scales i.e. the Planck length and, potentially, the length scale provided by a cosmological constant; these scales may be interpreted as a kind of ‘absolute structure’ in the theory. By comparison, the absolute time of Newton’s theory of gravity was shown in the context of General Relativity to be an approximate notion arising from the - ultimately - dynamical spacetime geometry in the solar system. Does the presence of absolute scales in General Relativity indicate that it might be a limit of a larger theory in which such scales has a dynamical origin? A model for the dynamical origin of scales was proposed by Zee [4] who introduced a scalar field such that the collection of scales in the gravitational sector should be due to this field having evolved to reach a fixed, non-zero value. A special case of such a theory is when a scalar field is coupled to gravity in a way such that field equations are invariant under the local field transformations and ; this is sometimes referred to as the conformal coupling of a scalar field to gravity. If the scalar field in this case is non-vanishing then it can be set to a constant by a transformation with , in which case the gravitational part of the theory reduces to General Relativity [5]; thus is ‘pure gauge’ whenever it is non-vanishing. More generally, the invariance of a theory’s field equations under these local transformations for the metric and other fields is referred to as local scale invariance and a number of gravitational theories possessing this symmetry have been widely studied [6, 7, 8, 9, 10, 11, 12].

In this paper we will describe a theory of gravitation where gravity possesses a local symmetry and with field content consisting of an gauge field and a Higgs field in the adjoint representation of the group which will be able to perform the symmetry breaking . We will show that this structure allows gravity to couple polynomially to matter in a way more akin to the gauge fields of particle physics and that a consequence of pursuing this similarity is the existence of a new field in gravity: the gravitational Higgs field ; we take this field to be genuinely dynamical in the sense that it is not subject to any Lagrangian constraints. This seems justified given the observation of the electroweak Higgs field which is in this sense unconstrained. It will be seen that the field may propagate, thus introducing new degrees of freedom into the gravitational sector, and the extent to which these new degrees of freedom may play a cosmological role is explored. Furthermore, it will be shown that this theory contains scale-invariant models of gravity as limiting cases and may allow for a dynamical explanation for the origin of scale in the gravitational sector.

The outline of the paper is as follows: In Section 2 we discuss the field content of the theory and present its action. In Section 3 we consider the limit of the theory that follows from constraining all of the degrees of freedom present in the Higgs field and show how fourth-order conformal gravity [13] and a conformally coupled scalar tensor theory arise. It is shown that in this limiting theory scale symmetry is broken not by the introduction of a new scalar field but by the dynamical alignment of two sets of frame fields. In this section we also discuss motivations for looking at the group as an approach to conformal gravity rather than other orthogonal groups such as or . In Section 4 we return to the full theory and show that there exist maximally symmetric solutions to the field equations i.e. solutions interpretable as spacetimes possessing ten Killing vectors each. In Section 5 we examine the nature of small perturbations around these solutions, establishing conditions for linear stability. To aid the reader, the flow chart Figure 1 summarizes the structure of these sections and some results therein. In Section 6 we discuss how one can couple the gravitational fields to matter fields in a simple and elegant fashion. All actions are polynomial and yield equations of motion that are first order partial differential equations that in the General-Relativistic limit of the theory reduce to the familiar second order equations for Yang-Mills fields and Higgs fields. It is suggested that Yang-Mills fields for a symmetry group are necessarily accompanied by scalar fields in the adjoint representation of . In Section 7 we discuss some potentially observable consequences of the model with a focus on open problems in cosmology. In Section 8 we discuss the relation of the work presented in this paper to previous approaches in the literature and in Section 9 we discuss the paper’s results and present conclusions.

gauge theoryMaximally symmetricsolutionsPerturbed actionaround maximallysymmetric solutionsConformal Einstein-Cartantheory

Conformalized General Relativityin first-order formConformalized General Relativityin second-order formGeneral Relativityin second-order formFourth-order Weyl gravityFourth-order conformal gravity

Constrain

Constrain

If

Solve for

Solve for

Gauge fix

Constrain

Figure 1: Diagram depicting known results of gravity and relation to other gravitational models. Dotted paths denote those taken with entirely unconstrained.

2 Gravitational action

The group has a matrix representation as the set of all complex matrices 555We reserve as indices whereas other Greek letters will be used to denote spacetime indices. of unit-determinant that satisfy

(1)

where is the identity matrix. This group is the double cover of the orthogonal group which itself is the double cover of the conformal group of coordinate transformations which preserve a metric of signature up to an overall multiplicative function. In taking gravity to possess a local symmetry we imply that gravitational fields and fields coupling to gravity belong to representations of this group and that terms in the Lagrangian involving these fields are invariant under local transformations . The group is fifteen-dimensional whilst the Lorentz group associated with General Relativity is six-dimensional so it is necessary for spontaneously symmetry breaking to occur in order to recover a General-Relativistic limit of the theory. We will show that a suitable Higgs sector for the theory can break the symmetry down to ; this is a larger symmetry than General Relativity and we will show how it is related to local scale-invariance.

For calculational purposes it is rather convenient in the gravitational sector to work with representations of . The machinery to move from representations of to is discussed in detail in Section 6.3 when we discuss the coupling of gravity to fermions, which we take to be fields in the fundamental representation of . A group element of may be represented as a matrix with unit-determinant and satisfying

(2)

A field in the fundamental representation of is a six-component vector and a field in the adjoint representation takes of the form of an antisymmetric matrix . Indices are lowered and raised with and its matrix inverse. Under a local transformation, the field transforms as and we require that our Lagrangians are invariant under such transformations for any fields belonging to representations of .

The gravitational fields will be an gauge field and a spacetime scalar field in the adjoint representation. The field can always be put in the following ‘block-diagonal’ form by appropriate transformations:

(3)

If (labeling indices on their rows and columns by ) and (labeling indices on its rows and columns by ) then this form of will be invariant under the transformations and . If then - by implication - represent hyperbolic rotations/boosts in the plane whilst are transformations generated by the Lorentz group subgroup of . Thus with , and , the residual symmetry is .

Useful quantities are the curvature two-form and covariant derivative one-form of - - given as follows:

(4)
(5)

where is the exterior derivative on forms and multiplication of forms is always taken to be via the wedge product. Note that for differential forms the wedge product satisfies . We will look to consider the most general locally -invariant and diffeomorphism-invariant action polynomial in . We will take actions to be integrals of Lagrangian four-forms and our restriction to Lagrangians which are diffeomorphism-invariant and polynomial in fields is a simplifying principle. We may then build four-forms from and , thus guaranteeing that the Lagrangian is coordinate-independent. To further enforce local invariance, we will look to contract away all free indices for which we can in principle additionally use the scalar and the invariants and the completely antisymmetric symbol . Our action is as follows:

(6)

where:

(7)
(8)
(9)

The zero-form coefficients may in general depend on invariants built from and the group invariants , . In this paper, as a first approach to the theory, we will take these coefficients to be constant numbers but it is conceivable that functional dependences on such invariants cannot be consistently neglected. Though the action contains terms quadratic in and quartic in , the wedge product structure guarantees that components of these fields appear at most linearly in the action. The generation of higher-order partial derivatives in the equations of motion is made impossible by use of a polynomial Lagrangian and Bianchi identities and ‘’. Therefore, as the Lagrangian is at most linear in derivatives of any component, the equations of motion are at most first order in derivatives. The action may look very unfamiliar and so in the next section we will initially look at a simpler theory emerges when we ‘freeze’ all the degrees of freedom in the field . Finally, we note that though we will use in the following sections, we may alternatively (and equivalently) use an entirely antisymmetric field , where the two are related via

(10)

3 Conformal Einstein-Cartan theory

We first discuss a theory which emerges when the degrees of freedom of in the model (6) are completely frozen by means of constraints imposed at the level of the action. Recall that the theory recovered by the same process of freezing the Higgs degree of freedom for the gauge theories resulted in the Einstein-Cartan theory, which in the absence of matter is equivalent to General Relativity (see Appendix A). We shall show that the corresponding theory for gravity based on the gauge group is in some senses a straightforward scale-invariant generalization of the Einstein-Cartan theory.

Assuming the block-diagonal form of from (3), degrees of freedom in can be characterized in terms of three -invariant quantities:

(11)
(12)
(13)

If it is enforced that and , this implies that two out of are zero. For example, we can choose (the label ‘3’ is completely arbitrary at this point) from . Subsequently, the condition takes the form:

(14)

Hence we can choose . Now, if we further require we have the condition:

(15)

Therefore and so we have the breaking of the original symmetry down to . If we then add on the following four-form Lagrange-multiplier constraints to the theory:

(16)

then the constraints will be enforced via the field equations obtained by varying . We may instead enforce the constraints at the level of the action, and so may be assumed to take the following form at the level of the action:

(17)

where implies an equality that holds in a specified gauge, we use indices as indices in the fundamental representation of , we use the convention , and the quantity is a constant. Given the application of these constraints, there are no longer any degrees of freedom for left in the action (6); the action is now a functional only of , a general ansatz for which is given in this gauge by:

(18)

The one-form field is the Lorentz-group spin connection, while the one-form is a connection for the group . The ‘off-diagonal’ components look much less familiar; they transform homogeneously under the remnant symmetry and appear in the -covariant derivative of as follows:

(19)

Now we write down the total constrained form of (6) in the ‘preferred gauge’, making use of the following results:

(20)

where is the curvature two-form; the action then becomes:

(21)

where we have used the result . To make further progress, we make the following general ansatz for in an arbitrarily chosen gauge:

(22)

where components in the column vector (22) are in components in an arbitrary null basis of the space of vectors. If we transform from this basis to another basis with an transformation represented by a matrix

(23)

Then in the new basis we have

(24)
(25)

where . Thus, transformations cause dilations of opposing weight in and . Applying the ansatz (22) to (21) yields:

(26)

where

As expected, the action is manifestly locally Lorentz invariant and invariant under local gauge transformations , , . The action additionally possesses invariance under local dilations of and of opposite weight:

(27)

where is entirely independent gauge transformation parameter . By appearance, the theory (26) resembles the Einstein-Cartan theory but instead has a pair of frame fields with a local scale symmetry under opposite rescalings due to them always appearing in the combination in the Lagrangian. Note that this is only possible because of the choice of the group which led to a remnant internal symmetry in the gravitational sector which in turn allowed for the definition of two frame fields in terms of a null basis in the space of vectors. If, instead, we had looked at theories with a local gauge symmetry for the de Sitter groups and , an adjoint Higgs field could break the symmetry instead to ; one could proceed to choose an arbitrary basis for the vector space, so defining frame fields in the manner that were defined in (22). It can be shown that these frame fields would always appear in the Lagrangian in the combination and no local scale symmetry is present.

We will refer to the theory (26) as Conformal Einstein-Cartan theory. Note that if there exist solutions when then the first three terms in (26) become terms familiar from the Einstein-Cartan theory: Palatini, cosmological, and Holst terms respectively [3]. The terms proportional to the coefficient vanish in this limit and so represent new behaviour, whilst the final terms quadratic in curvature are boundary terms and do not contribute to the equations of motion.

3.1 General-Relativistic limit

By conducting small variations of (26) one may straightforwardly obtain the equations of motion. Remarkably, solutions to these equations of motion exist for the ansatz i.e.

(28)

We now show that - as suggested above - such solutions constitute a General-Relativistic limit of the theory. In this limit, we find from combining the and field equations that

(29)

and the field disappears from the system of field equations. The remaining field equations are equivalent to those obtained from the following action:

(30)

Then, varying with respect to and solving for , eliminating it from the action, we recover the following, second-order action:

(31)

where , is the Ricci scalar according to the Christoffel symbols and

Thus we see in the second-order formalism a kinetic term for emerges. The definition of (28) implies that the invariance of the action is under the transformation and hence this is the action of a scalar field conformally coupled to gravity. We may utilize the scale gauge freedom in the theory to locally rescale : if we assume that the action is an integration over regions where then a convenient gauge choice is , in which case he action reduces to:

(32)

This is the Einstein-Hilbert action of General Relativity. Therefore the equations of motion obtained in the case are equivalent666We note that due to the presence of second-order partial derivatives of the metric tensor in the Einstein-Hilbert action it is necessary to introduce an additional topological term - the Gibbons-Hawking term - for the variational principle to work out correctly. to those of General Relativity777For a different approach to General Relativity and scale invariance see [14, 15, 16, 17].. Because of this, we refer to the theory (31) as conformalized General Relativity; if we instead had begun from the Einstein-Hilbert action (32) we could recover (31) via a local conformal rescaling of using .

We must now ask how a theory like General Relativity, with its absolute scales, can emerge from a theory where it is not clear that there is an inbuilt scale. As discussed in the introduction, frequently in the literature an additional ‘Higgs’ scalar is introduced alongside with scales in the gravitational sector to be due to dynamically reaching constancy; in this context, local scale invariance can be retained via the conformal coupling of to gravity and this is identical to the manner in which combine to yield the locally scale-invariant action (31). The scalar would have dimensions of length or mass and would this set a specific scale at each point in spacetime. However, locally in regions where , as in the case of the field , we may then readily impose a gauge in which the theory would no longer be manifestly scale-invariant. To some this introduction of scale invariance and then its immediate elimination might seem a bit contrived.

To that end we wish to point out that the breaking of scale invariance in our proposed model is not aided by the introduction by any additional fundamental Higgs fields but is a feature of a specific subclass of solutions including the General-Relativistic solutions characterized by the condition . At an extreme, as the equations of motion following from the action (6) are polynomial and each term is at least cubic in then there exist solutions to the unconstrained theory where and the entire -invariance is retained; for such solutions no notion of scale arises from the gravitational sector. It is conceivable that there may exist solutions to the full theory describing regions where and other regions where these fields are non-vanishing in a particular gauge and where . In this sense then, the theory may be a candidate for the dynamical origin of scale.

We may characterize the condition in a different fashion. The General-Relativistic limit is the limit in which a preferred basis of the vector space dynamically emerges. This basis is defined by two independent possibilities:

(33)

or

(34)

where recall the definition of from (18). Meanwhile, the ansatz (22) can be written equivalently as follows:

(35)

where and ; here and should be regarded as an arbitrary choice of basis of the two-dimensional vector space. For example in the case of the condition (33) holding, we have - using (35) and (33) - that

(36)

Thus, if we choose the preferred gauge as our basis then in this gauge we have from (36) that . If, alternatively, the condition (34) holds then and in the preferred gauge . Choosing the preferred gauge corresponds to choosing the scale gauge in which .

In some respects this is similar to the breaking of rotational invariance for a ferromagnet: although a preferred direction appears as a property of low-temperature solutions this direction makes no appearance in the fundamental equations of motion. Similarly, a preferred direction in the space - - appears for General-Relativistic solutions but its existence is a property of some solutions rather than a basic constituent of the theory. Neither can such a preferred vector be defined uniquely outside the General-Relativistic limit.

3.2 Relation to Weyl and fourth-order conformal gravity

In the previous section we considered a theory recovered by ‘freezing’ all the degrees of freedom in to take a specific form; the resulting theory possessed a local invariance as well as independent rescaling invariances under , , . An interesting property of this theory is that beginning from the field equations for the set we may solve algebraically for any one of and eliminate it from the action. In Appendix C, this is explicitly illustrated for the following action:

(37)

where are constants. Solving algebraically for the field from its own equation of motion and inserting this solution back into the action, one obtains:

(38)

where is the Hodge dual operator built from the field , is the Ricci one-form and

(39)

where is the Ricci scalar. We now consider the effect of placing an additional constraint on this theory. Recall that in the Einstein-Cartan theory, the equation was the equation of motion for and allowed one to solve for and eliminate it from the action principle. Now consider the following generalisation of this equation:

(40)

This equation is invariant under a more restricted group of symmetry transformations of the Conformal Einstein-Cartan theory: that when . This equation is not the equation of motion for one would get by varying with respect to for this theory but we may enforce it via a Lagrangian constraint. Doing so, we may now use (40) to solve for and substitute this solution into the action (38). The resulting action is a functional only of (appearing via ) and (via ):

(41)

This is the action for fourth-order Weyl gravity which, as the name suggests, yields field equations containing fourth-derivatives of fields. If we further constrain then we recover the action:

(42)

The quantity is the Weyl two-form, related to the Weyl tensor via:

(43)

The action (42) is thus proportional to that of fourth-order conformal gravity plus a boundary term quadratic in . Hence, fourth-order conformal gravity can be recovered from the original gauge theory via the implementation of a number of constraints. Indeed, this was the result found by Kaku, Townsend, and Van Nieuwenhuizen [18]. Their approach was essentially the same as the steps discussed in this section i.e. fourth-order conformal gravity was recovered by implicitly constraining the symmetry breaking fields that break and explicitly constraining the spin-connection . For the particular action they considered (specifically the action (37) with ) vanishes automatically from the action and did not need to be constrained to vanish. The relation of this approach of recovering Weyl gravity from a gauge theory of gravity to Cartan’s conception of geometry has recently been discussed in detail888The link between Cartan geometry and conformal physics has previously been investigated in the case of spacetime dimensions [19]. [20].

It has become somewhat common lore that fourth-order Weyl gravity is the gauge theory of the conformal group [21, 22] . However, if one is looking to cast gravity as a gauge theory akin to those of particle physics, why completely freeze all the degrees of freedom in the symmetry breaking fields? The analogue in electroweak theory would be an insistence that for the electroweak Higgs were fixed to be a constant - this would force a non-vanishing expectation value for much as a non-vanishing expectation value for was achieved in the above approach. The discovery of the Higgs boson demonstrates that in that case it would be incorrect to apply such constraints; should gravity be any different? Even allowing this, why then further constrain to take a solution that would not generally follow from ’s equation of motion [23]?

4 Vacuum solutions of the full theory

We now ‘un-freeze’ the field . Its classical evolution will now be dictated entirely by its own equations of motion in conjunction with those of other fields. We will demonstrate that there exist simple solutions to the theory in which the field has a non-vanishing, constant expectation value and one may interpret the accompanying spacetime geometry as being de Sitter or anti de Sitter space. We will look for solutions where takes the following form:

(44)

where recall that are indices and the gauge-fixing condition has been imposed. We will focus on searching for solutions where

(45)

As detailed in the previous section, this form of (even if were not constant) breaks the original symmetry of the theory down to if the signature of is ; we will assume this to be the case. Clearly, the existence of solutions satisfying this condition does not indicate that they are dynamically favoured. In this paper our analysis will be limited to establishing linear stability of them with respect to small perturbations.

Now we turn to the form of the connection . Given the above symmetry breaking, a general ansatz for this field is:

(46)

For the remainder of this section we will remain in the gauge implied by the form (44) and so, subsequently, should be taken to mean an equality that holds given this gauge condition. It follows from (44) and (46) that

(47)

and we make an ansatz for such that :

(48)

Note that as in Section 3.1, we are effectively restricting ourselves to the case where contains a single independent frame field; this seems a reasonable assumption given that we are searching for solutions interpretable in terms of a single spacetime metric. The form of the ansatz (48) is invariant under a local rescaling , . The curvature two-form then takes the following form: