From configuration to dynamicsEmergence of Lorentz signature in classical field theory

# From configuration to dynamics Emergence of Lorentz signature in classical field theory

Shinji Mukohyama Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan    Jean-Philippe Uzan Institut d’Astrophysique de Paris, Université Pierre & Marie Curie - Paris VI, CNRS-UMR 7095, 98 bis, Bd Arago, 75014 Paris, France.
July 15, 2019
###### Abstract

The Lorentzian metric structure used in any field theory allows one to implement the relativistic notion of causality and to define a notion of time dimension. This article investigates the possibility that at the microscopic level the metric is Riemannian, i.e. locally Euclidean, and that the Lorentzian structure, that we usually consider as fundamental, is in fact an effective property that emerges in some regions of a 4-dimensional space with a positive definite metric. In such a model, there is no dynamics nor signature flip across some hypersurface; instead, all the fields develop a Lorentzian dynamics in these regions because they propagate in an effective metric. It is shown that one can construct a decent classical field theory for scalars, vectors and (Dirac) spinors in flat spacetime. It is then shown that gravity can be included but that the theory for the effective Lorentzian metric is not general relativity but of the covariant Galileon type. The constraints arising from stability, the equivalence principle and the constancy of fundamental constants are detailed and a phenomenological picture of the emergence of the Lorentzian metric is also given. The construction, while restricted to classical fields in this article, offers a new view on the notion of time.

###### pacs:
03.50.-z,04.20.Cv,04.50.Kd
preprint: IPMU13-0008

## I Introduction

When constructing a physical theory, there is a large freedom in the choice of the mathematical structures. The developments of theoretical physics taught us that some of these structures are well-suited to describe some classes of phenomena (e.g. the use of a vector field for electromagnetism, of spinors for some class of particles, the use of some symmetries, etc.). However, these choices can only be validated by the mathematical consistency of the theory and the agreement between the consequences of these structures and experiments. It may even be that different structures are possible to reproduce what we know about physics and one may choose one over the other on the basis of less well-defined criteria such as simplicity and economy.

At each step, some properties such as the topology of space topology (), the number of spatial dimensions or the numerical values of the free parameters that are the fundamental constants jpucte (), may remain a priori free in a given framework, or imposed in another framework (e.g. the number of space dimensions is fixed in string theory stringT ()).

Among all these structures, and in the framework of metric theories of gravitation, the signature of the metric is in principle arbitrary. Indeed, it seems that on the scales that have been probed so far there is the need for only one time dimension and three spatial dimensions. In special and general relativity, time and space are geometrically different because the geometry of spacetime is locally Minkowskian, i.e. it enjoys a Lorentzian metric with signature , i.e. the line element is . While the existence of two time directions may lead to confusion V4 (), it is not clear if there are theoretical obstacles to have more than one time directions, as even suggested by some framework V5 () (see the argument for such possibilities made by Ref. no2 () and detailed further in Ref. gibbons ()). Several models for the birth of the universe nothing () are based on a change of signature via an instanton in which a Riemannian and a Lorentzian manifolds are joined across a hypersurface which may be thought of as the origin of time. While there is no time in the Euclidean region, where the signature is , it flips to . Eddington even suggested G85 () that it can flip across some surface to . Signature flip also arises in brane-world scenarios P86 () (see Ref. gibbons () for a review of these possibilities) or in loop quantum cosmology lqc (). These discussions however let the problem of the origin of the time direction open zeh ().

In Newtonian theory, time is a fundamental concept. It is assumed to flow and is described by a real variable. It can be measured by good clocks and any observers shall, irrespective of their motion, agree on the time elapsed between two events vuibert (). The laws of dynamics describe the change of configurations of a system with time. In relativity, first the notions of space and time are set on the same footing and second, the notion of time is no more unique. One has to distinguish between a coordinate time, with no physical meaning, and the proper time that can be measured by an observer. Quantum mechanics offers another insight on time: there, while there may be operators or observables corresponding to spatial positions, time is not an observable, and thus not an operator rovelli0 (). As detailed in Ref. gibbons (), by an argument going back to Pauli, commutation relations like are incompatible with the spectrum of lying in the future lightcone and the notion of time is intimately related to the complex (Hilbert Space) structure of quantum mechanics gibbons ().

The question of whether time does actually “exist” has been widely debated in the context of classical physics barbour (), relativity time-RG () and quantum mechanics rovelli0 (). The debate on the nature of time has shifted with quantum gravity where the recovery of a classical notion of time is considered as a problem. In that case, the Schrödinger equation becomes the Wheeler-de Witt equation, of the form , so that the allowed states are those for which the Hamiltonian vanishes. Thus, it determines in which states the universe can be but does not give any evolution through time. We refer to Refs. rovelli (); time (); Ellis:2006sq () for general discussions on the nature of time. This has led to numerous works on the emergence of time in different versions of quantum gravity rovelli2 (); markopoulou (); time-emergence (); time-emergence2 (); time-emergence3 (); time-emergence0 () (and indeed the reverse opinion has been argued carroll ()). Also, the thermodynamical aspects of gravity, the existence of dualities between gauge theories and gravity theories maldacena (), and holography holo () have led to the idea that the metric itself may have to be thought of as the result of a coarse-graining of underlying more fundamental degrees of freedom V6 ().

The local Minkowski structure is an efficient way to implement the notion of causality in realistic theories and is today accepted as a central ingredient of the construction of the relativistic theory of fields. When gravity is included, the equivalence principle implies (this is not a theoretical requirement, but just an experimental fact, required at a given accuracy) that all the fields are universally coupled to the same Lorentzian metric. From the previous discussion, we may wonder whether the signature of this metric is only a convenient way to implement causality or whether it is just a property of an effective description of a microscopic theory in which there is no such notion.

This article proposes the view according to which the fundamental physical theory is intrinsically purely Euclidean so that its field equations determine a static 4-dimensional field configuration. The Lorentzian dynamics that we can observe in our universe has then to be thought of as an emergent property, that is as an illusion holding in a small patch of a Euclidean mathematical space. This is thus an attempt to go further than early proposals Greensite:1992np (); Girelli (); Goulart () and see to which extent this can be an open possibility. We emphasize that it is different from the models discussed above involving a signature change across a boundary or obtained by rotating to an Euclidean space. We consider it important to take the freedom to see how far one can go in such a direction. As we shall later discuss, if possible, such a setting may shed a new light on several theoretical issues from the nature of singularities to quantum gravity.

Our attitude is however more modest and we want to start by constructing a decent classical field theory under this hypothesis. Section II explains the basics of our mechanism and then describes the construction of the scalar, vector and spinor sectors in flat spacetime. We show that the whole standard model of particle physics can be constructed from a Euclidean theory, at the classical level. Section III addresses the more difficult question of gravity. While general relativity is not recovered in general, it shows that an extended -essence theory of gravity called covariant Galileon can be obtained. We then show in Section IV that the dynamics of scalar and vector in curved spacetime can also be obtained. We then discuss the experimental and theoretical constraint on our construction in § V and also propose a way to understand phenomenologically the emergence of the effective Lorentzian dynamics. It is however to be remembered that there is no dynamics at the fundamental level and that this illusion is restricted to a domain of a large Euclidean space.

## Ii Field theory in flat space

This section introduces the mechanism in the simple case of a flat space (§ II.1). It shows how scalars (§ II.2), vectors (§ II.3 and § II.4) and spinors (§ II.5) defined in Euclidean space can have an apparent Lorentzian dynamics. We finish by pointing out the properties and limits of this mechanism in § II.7, many of them being discussed in a more realistic version in the following sections.

### ii.1 Clock field

In order to understand the basics of our model, let us consider a -dimensional Riemannian manifold with a positive definite Euclidean metric in a Cartesian coordinate system. As a consequence, the theory we shall consider on this manifold does not have a natural concept of time. In order to make such a notion emerge locally, we introduce a scalar field and assume that its derivative has a non-vanishing vacuum expectation value (vev) in a region of the Riemannian space (see Fig. 1). To be more precise, we assume that in . It follows that we can set

 ∂μϕ=M2nμinM0 (1)

with a unit constant vector (). We have introduced a mass scale so that is dimensionless. By construction, its norm is constant and satisfies

 XE>0inM0. (2)

Now, under this assumption, one of the coordinates can be chosen as

 dt=nμdxμ. (3)

This accounts for choosing

 t≡ϕM2 (4)

up to a constant that can be set to zero without loss of generality. The metric of the 4-dimensional Riemannian space (with Euclidean geometry) can be rewritten as

 ds2E = δμνdxμdxν (5) = (nμdxμ)2+(δμν−nμnν)dxμdxν = dt2+δijdxidxj,

by introducing a set of three independent coordinates () on the hypersurfaces normal to . Note that the geometry on would not be Euclidean if were not constant. As we shall now discuss, the scalar field will be related to what we usually call “time”, so that we shall call such a scalar field a clock field.

### ii.2 Scalar field

The Euclidean configuration of a scalar field can be obtained by combining the usual action for a scalar field, with a kinetic term and a potential,

 −∫d4x[12δμν∂μχ∂νχ+V(χ)],

with a coupling to the clock field as

 ∫d4x(δμν∂μϕ∂νχ)2.

Let us consider the action obtained by the following combination

 Sχ = ∫d4x[−12δμν∂μχ∂νχ−V(χ) (6) +1M4(δμν∂μϕ∂νχ)2].

It is straightforward to conclude that since and, when restricted to , , the action (6) reduces to

 Sχ = ∫dtd3x[12(∂tχ)2−12δij∂iχ∂jχ−V] (7)

in . This can indeed be rewritten as

 Sχ = ∫dtd3x[−12ημν∂μχ∂νχ−V]. (8)

The action (6) thus describes, when restricted to , the dynamics of a scalar field propagating in a 4-dimensional Minkowski spacetime with metric . The apparent Lorentzian dynamics, with a preferred time direction, is thus the result of the coupling to the scalar clock field.

### ii.3 Vector field

Usually, the dynamics of a vector field is dictated by the action where is the Faraday tensor defined as and where the subscript indicates that the indices are raised with the Euclidean metric .

The standard action of the vector field can be extended to include a coupling to the clock field of the form so that the action for the vector field we consider is

 SA=14∫d4x[−FμνFμνE+4M4FμρEFνEρ∂μϕ∂νϕ]. (9)

Since and since in , it is easily concluded that this action can be rewritten as

 SA=14∫dtd3x[2δijF0iF0j−δikδjlFijFkl], (10)

or more simply as

 SA=−14∫dtd3xημαηνβFμαFνβ. (11)

Because of the coupling of the Faraday tensor to the clock field in the Euclidean theory, the vector field propagates effectively in a Minkowski metric and we recover the standard Maxwell action for a vector field. The generalization to an non-Abelian group is straightforward.

### ii.4 Charged scalar field

The construction of § II.2 can easily be generalized to a complex scalar field charged under a . Considering a complex scalar field , we add to the standard kinetic term a coupling to the clock field of the form , where . The Euclidean action is then chosen to be

 Sω = ∫d4x[−12δμν(Dμω)∗(Dνω)−U(|ω|2) (12) +1M4δμν|∂μϕDνω|2].

Following the same arguments as for the real scalar field , this action takes the form

 Sω = ∫dtd3x[−12ημν(Dμω)∗(Dνω)−U]. (13)

Again, the coupling to the clock field implies that the Euclidean dynamics leads to an effectively Minkowskian dynamics for .

### ii.5 Spinor fields

The next step is to include fermions in such a way that the standard Dirac dynamics emerges from an Euclidean action. Let us start by comparing the standard Dirac algebra in Minkowski spacetime (§ II.5.1) and that in Euclidean space (§ II.5.2) before we propose a choice of Euclidean action for the fermions (§ II.5.3).

#### ii.5.1 Dirac matrices in Minkowski spacetime

In a Minkowski spacetime with signature (), Dirac matrices are matrices satisfying the anti-commutation relation

 {γμ,γν}=−2ημν. (14)

For concreteness, throughout this section we shall adopt the following form of the Dirac matrices in Minkowski spacetime

 γ0 = σ0⊗σ1=(0σ0σ00), γi = iσi⊗σ2=(cc0σi−σi0), (15)

where is the unit matrix and () are Pauli matrices,

 σ1=(0110),σ2=(0−ii0),σ3=(100−1). (16)

While is Hermitian, are anti-Hermitian. One then defines by

 γ5≡−iγ0γ1γ2γ3=σ0⊗σ3=(σ000−σ0), (17)

which satisfies

 (γ5)2=1,{γ5,γμ}=0(μ=0,⋯,3). (18)

The matrices

 Sμν≡i4[γμ,γν] (19)

satisfy the algebra of Lorentz generators

 [Sμν,Sρσ]=i(ηνρSμσ−ημρSνσ−ηνσSμρ+ημσSνρ). (20)

Hence, the Lorentz transformation for a Dirac field is

 ψ→Λ12ψ,Λ12=exp[−i2ωμνSμν], (21)

where are real numbers. Concretely,

 S0i = −i2(σi00−σi), Sij = 123∑k=1ϵijk(σk00σk). (22)

While are Hermitian, are anti-Hermitian. As a consequence, is not unitary in general. In particular this means that

 ψ†→ψ†Λ†12≠ψ†Λ−112 (23)

and that is not a scalar under Lorentz transformation. However, it is easy to check that

 ¯ψ→¯ψΛ−112,¯ψ≡ψ†γ0 (24)

so that is a scalar under Lorentz transformations. This is the reason why the Dirac action in Minkowski spacetime is usually constructed as

 SMψ=∫d4x¯ψ(iγμ∂μ−m)ψ. (25)

#### ii.5.2 γ matrices in Euclidean space

In a -dimensional Euclidean space with metric , one can also define matrices according to

 γ0E≡iγ5,γiE≡γi (26)

so that they obey the anti-commutation relation

 {γμE,γνE}=−2δμν. (27)

Then, we can define

 γ5E≡γ0Eγ1Eγ2Eγ3E=γ0, (28)

which satisfies

 (γ5E)2=1,{γ5E,γμE}=0(μ=0,⋯,3). (29)

It follows that the matrices

 SμνE≡i4[γμE,γνE] (30)

satisfy the algebra of rotation generators

 [SμνE,SρσE]=i(δνρSμσE−δμρSνσE−δνσSμρE+δμσSνρE). (31)

Hence, the rotation for the Dirac field is

 ψ→ΛE,12ψ,ΛE,12=exp[−i2ωEμνSμνE], (32)

where are real numbers. Since all are Hermitian, is unitary. In particular, this implies that

 ¯ψ→¯ψΛ−1E,12,ψ†→ψ†Λ−1E,12, (33)

and that both and () are scalars under a transformation (see e.g. Refs. Wetterich:2010ni (); mehta ()). Note also that can be written as

 ¯ψ=ψ†γ5E. (34)

#### ii.5.3 Euclidean action and emergence of the Lorentzian Dirac action

As in the previous sections, we will need to couple the spinor field to the clock field in order for the spinor to have an apparent Lorentzian dynamics. Starting from the Euclidean Dirac action in flat space with the metric ,

 ∫dx4¯ψ(i2γμE↔∂μ−m)ψ,

and assuming that the clock field has derivative couplings to the Euclidean Dirac field of the form

 ∫dx4δμν(i¯ψγ5E↔∂μψ)∂νϕ,∫dx4δμν(i¯ψγρE↔∂μψ)∂ρϕ∂νϕ,

we can consider an Euclidean action for the Dirac spinor of the form

 Sψ = ∫dx4{¯ψ(i2γμE↔∂μ−m)ψ + 12M2δμν[(i¯ψγ5E↔∂μψ)−1M2(i¯ψγρE↔∂μψ)∂ρϕ]∂νϕ}.

As in the previous sections, the action reduces to

 Sψ=∫dx4¯ψ[i2γ0↔∂0+i2γi↔∂i−m]ψ. (36)

The coupling to the clock field implies that effectively propagates in an effective Lorentzian metric and we recover the standard Minkowskian Dirac action (25) with the usual algebra (14) for the -matrices.

### ii.6 Massive point particle

The dynamics of massive object is usually derived from an action defined from the length of their worldline. In order to recover a proper dynamics, we start from the Euclidean action for a point particle

 12∫(N−1δμνdxμdτdxνdτ−Nm2)dτ

to which we add the coupling to the clock field of the form

 ∫N−1∂μϕ∂νϕdxμdτdxνdτdτ

The Euclidean action for a point particle is thus given by

 Spp = 12∫[N−1(δμν−2M4∂μϕ∂νϕ)dxμdτdxνdτ (37) −Nm2]dτ.

The equation of motion is thus simply given by the geodesic equation for the effective metric . It is obvious that in this effective metric reduces to the Minkowski metric .

### ii.7 Discussion

This section has provided the general construction of a mechanism that allows for scalar, vector, and spinor fields to actually propagate in an effective Lorentzian metric even though the underlying theory is purely Euclidean and written in terms of the Euclidean metric . This general construction assumes the existence of a scalar field , called clock field, that couples to all fields (scalar, vector and spinor fields). In particular, this implies that we can construct the whole standard model of particle physics.

Let us now discuss some properties and limitations of such a construction.

1. It requires that the clock field satisfies in a region of the Euclidean space. It follows that the effective Lorentzian description is local and holds in . The properties of this model when is not constant will be discussed in § V.3 below. As we shall see in the next section the clock field should enjoy a shift symmetry in order for the system to exhibit the time translation symmetry after the emergence of time. In , both the shift symmetry and the translational symmetry along the direction of are spontaneously broken, but a combination of them remains unbroken and is responsible for the existence of a conserved quantity that reduces in to the usual notion of energy.

2. It is limited to classical field theory in flat space. The extension to curved space is discussed in § III and § IV below and the quantum aspects are left for future investigations.

3. The origin of the effective Lorentzian dynamics in can be intuitively understood for scalars and vectors. For scalars, the action (6) is equivalent to the coupling to the effective metric

 ^gμν=δμν−2M4δμαδνβ∂αϕ∂βϕ. (38)

For vectors, one could have simply used a coupling to and a Lagrangian of the form since the extra term quartic in compared to the action (9) is of the form and does not contribute (note that it reduces to in ). Hence, the apparent Lorentzian dynamics for scalars and vectors boils down to the fact that . Massive point particles also propagate in this metric.

4. This interpretation cannot be extended to spinors mostly because of the -matrices, at least straightforwardly.

5. It is however important to realize that despite this, when restricted to all fields propagate in the same effective Minkowski metric so that the equivalence principle is safe in first approximation.

6. The couplings to the clock field have been tuned in order to recover the exact Minkowski actions. For instance, the action (6) for a scalar field could have been chosen as

 Sχ = ∫d4x[−κχ2δμν∂μχ∂νχ−V(χ) (39) +αχ2M4(δμν∂μϕ∂νχ)2].

In such a case, a Lorentzian signature is recovered only if . In the case where these constants are not tuned, different fields can have different lightcones. This will be discussed in Section V.

7. In the bosonic sector, since the theory is invariant under the Euclidean parity () as well as the field parity (), both and invariances in the emergent Lorentzian theory are ensured. Without the field parity invariance, the invariance would be spontaneously broken by a non-vanishing vacuum expectation value (vev) of the derivative of the clock field. This explains the reason why we have included only quadratic terms in in the actions for scalars and vectors.

8. In the fermionic sector, let us first remark that one could have constructed independent Euclidean -matrices, explicitly given by

 1,γ5E,γμE,γ5EγμE,SμνE. (40)

From the Dirac spinor , we can thus construct bilinear combinations that transform as scalars under rotations. Among them, Hermitian bilinears that do not include more than one derivative acting on spinors are the following ten possibilities

 ¯ψψ,¯ψγ5Eψ,i¯ψγμE↔∂μψ,¯ψγ5EγμE↔∂μψ,(¯ψγμEψ)∂μϕ,(i¯ψγ5EγμEψ)∂μϕ,δμν(i¯ψ↔∂μψ)∂νϕ,δμν(i¯ψγ5E↔∂μψ)∂νϕ,δμν(i¯ψγρE↔∂μψ)∂ρϕ∂νϕ,δμν(¯ψγ5EγρE↔∂μψ)∂ρϕ∂νϕ.

The first two of the left column correspond to the standard mass and kinetic terms while six among the eight others describe possible couplings to the clock field. Among these six couplings, we have only used the two which were sufficent as an existence proof of our mechanism for Dirac spinors, namely and . It has to be remarked that the second term is not CPT invariant after the clock field has a vev. Hence, unless the coefficient of this term is exactly the value shown in (II.5.3), the CPT invariance is violated. We also need to emphasize that we have been able to construct Dirac spinor but that we also need to construct Majorana and Weyl spinors. This is an open problem at the moment.

9. The mass scale is related to and is arbitrary. It is important to realize that it does not appear in the final expressions of the effective Lorentzian actions.

10. may not be constant if (curved space) and/or if is not strictly constant in . This will be discussed in § III and § V.

11. The configuration of the clock field is not arbitrary but should be determined by solving the equation of motion. Since the action for the clock field enjoys a shift symmetry, its equation of motion takes the form of a current conservation. This will be addressed in § III, where we will show that can be a solution, e.g. with .

## Iii Gravitation and curved space

So far, our description has been restricted to the classical dynamics of standard fields in flat spacetime. The first natural generalisation we must consider is the way to include gravity, i.e. a theory that will mimic or be close to general relativity.

For this purpose, we now consider a general -dimensional Riemannian111We use the term Riemannian for a curved spacetime with a positive definite metric and Lorentzian for a curved spacetime with a Lorentz signature. We keep the terms Euclidean and Minkowskian for the analog in flat space. However, for simplicity, we use the same subscript E for the Riemannian and Euclidean cases. manifold with a positive definite metric . Again, the theory we shall consider on this manifold does not have a microscopic concept of time. As previously, we introduce a clock field and assume it enjoys a shift symmetry that, as we have already seen, is necessary for the system to exhibit the time translation symmetry after the emergence of time.

### iii.1 Generic couplings to the clock field

In order to minimize the number of physical degrees of freedom, we demand that the equation of motion for is a second-order differential equation. Hence, the action for is restricted to the Riemannian version of the Horndeski theory Horndeski:1974wa () with shift symmetry. Equivalently, it is given by the shift-symmetric generalized Galileon Deffayet:2011gz () as

 Sg=∫dx4√gE(L2+L3+L4+L5), (41)

where the Lagrangians are explicitly given by

 L2 = K(XE), L3 = −G3(XE)∇2Eϕ, L4 = G4(XE)RE−2G′4(XE)[(∇2Eϕ)2−(∇Eμ∇Eνϕ)2], L5 = −g5GμνE∂μϕ∂νϕ+~G5(XE)GμνE∇Eμ∇Eνϕ (42) +13~G′5(XE)[(∇2Eϕ)3−3(∇2Eϕ)(∇Eμ∇Eνϕ)2 +2(∇Eμ∇Eνϕ)3].

Here, , and are the covariant derivative associated with the Riemannian metric , its Ricci scalar and Einstein tensor. The coefficient is a constant and , and are arbitrary functions of and a prime refers to a derivative with respect to that is defined as

 XE ≡ gμνE∂μϕ∂νϕ. (43)

We use the following short-hand notations

 ∇2Eϕ ≡ gμνE∇Eμ∇Eνϕ, (44) (∇Eμ∇Eνϕ)2 ≡ gνρEgσμE(∇Eμ∇Eνϕ)(∇Eρ∇Eσϕ), (∇Eμ∇Eνϕ)3 ≡ gνρEgσαEgβμE(∇Eμ∇Eνϕ)(∇Eρ∇Eσϕ)(∇Eα∇Eβϕ),

where is the inverse of .

For the effective equations, i.e. once the Lorentzian structure and the notion of time have emerged, we would like to ensure that the system is invariant not only under time translation but also under CPT 222As we have seen in the previous section, this requirement is not obviously fulfilled for spinors without fine-tuning. An additional mechanism is needed to naturally ensure the CPT invariance for spinors. In the present article we shall thus focus on the bosonic sector..

For this reason, we require that besides the shift symmetry () the theory also enjoys a symmetry () for the clock field action. With these symmetries, the action reduces to

 Sg = ∫dx4√gE{G4(XE)RE−g5GμνE∂μϕ∂νϕ+K(XE) (45) −2G′4(XE)[(∇2Eϕ)2−(∇Eμ∇Eνϕ)2]},

since only , and the first term of can contribute.

It is easy to show that the constant in the action (45) can be absorbed into the redefinition of up to a boundary term. Hence, by setting , hereafter we consider the Riemannian gravity action of the form

 Sg = ∫dx4√gE{G4(XE)RE+K(XE) (46) −2G′4(XE)[(∇2Eϕ)2−(∇Eμ∇Eνϕ)2]}.

### iii.2 Action for the gravitational sector

Following the logic developed in Section II, we restrict our analysis to a region in which so that we can define a preferred direction, that we shall call , defined as in Eq. (4),

 t≡ϕM2, (47)

that is chosen as one of coordinates of the 4-dimensional Riemannian manifold. We refer to such a coordinate choice (47) as unitary gauge.

#### iii.2.1 Decomposition of the Riemannian metric

One can then introduce a set of three other independent coordinates () so that the Riemannian metric is decomposed as

 gEμνdxμdxν=N2Edt2+γij(dxi+Nidt)(dxj+Njdt), (48)

where the lapse is given, thanks to Eq. (47), by

 NE ≡ 1√gttE=M2√XE. (49)

The 3-metric is given by

 γij≡gEij, (50)

and is its inverse. To finish, the shift vector is given by

 Ni ≡γijgEtj. (51)

One can then easily check that the inverse Riemannian metric is given by

 gttE = 1N2E, gtiE = gitE=−NiN2E, gijE = γij+NiNjN2E. (52)

#### iii.2.2 Riemannian geometrical quantities

With the decomposition (48), it is straightforward to show that the Einstein-Hilbert term reduces to

 √gERE = NE√γ(−KijEKEij+K2E+R(3)) (53) −2∂i(√γγij∂jNE)−2∂t(√γKE) +2∂i(√γNiKE),

in terms of the extrinsic curvature of the constant- hypersurface, , defined by

 KEij≡12NE(∂tγij−DiNj−DjNi), (54)

where is the spatial covariant derivative compatible with , and is its Ricci scalar. We have used the notations , , and .

#### iii.2.3 Riemannian action in M0

With the use of the quantities introduced above, the Riemannian action (46) takes the form

 Sg = ∫dtdx3NE√γ{−G4(KijEKEij−K2E) (55) +G4R(3)+Lϕ},

where the Lagrangian is given by

 Lϕ = −2(∂E⊥∂E⊥+D2)G4−2G′4[(∇2Eϕ)2 (56) −(∇μE∇νEϕ)(∇Eμ∇Eνϕ)]+K(XE),

in which the 3-dimensional Laplacian is defined as usual as . The perpendicular derivative is defined in terms of the unit vector normal to the constant hypersurfaces, , as

 ∂E⊥≡nμE∂μ≡1NE(∂t−Ni∂i), (57)

with .

In order to further simplify , note that in terms of the Christoffel symbols for the metric , . Its components are explicitly given by

 ϕE;ij ≡ ∇Ei∇Ejϕ=√XEKEij, ϕE;⊥i ≡ ϕE;i⊥≡nμE∇Eμ∇Eiϕ=12√XE∂ilnXE, ϕE;⊥⊥ ≡ nμEnνE∇Eμ∇Eνϕ=12√XE∂E⊥lnXE. (58)

It implies that the term appearing in Eq. (56) takes the form

 (γijγkl−γikγjl)ϕE;ijϕE;kl+2γij(ϕE;⊥⊥ϕE;ij−ϕE;⊥iϕE;⊥j)

and thus reduces to

 −XE(KijEKEij−K2E)+KE∂E⊥XE−12XE(DilnXE)2.

Inserting this into Eq. (56), it follows that takes the form

 Lϕ=2G′4XE(KijEKEij−K2E)+K(XE)+ΔϕNE√γ, (59)

where the last term is given by

 Δϕ = −2∂t(√γ∂E⊥G4)+2∂