A generalized setup for the dynamics of spatially homogeneous models

In this section we introduce a general setup which will serve as a starting point for quantization. Recall that we did not discuss the inclusion of matter fields in section 2.1.3. The case of ideal fluids was discussed by Jantzen [31]. For a canonical treatment of ideal fluids one might employ the formalism developed by Schutz [71, 72] and for the particular case of dust the one by Brown and Kuchaˇr [73]. The setup we will introduce should be general enough to cover the coupling of matter fields such as scalar fields and Yang-Mills fields to spatially homogeneous cosmological models. Most importantly it should emulate the main features of the minisuperspace models obtained via the symmetry reduction of general relativity to spatially homogeneous spacetimes. The setup that we will introduce in the following has the virtue of offering a quite general view on the dynamics of minisuperspace models and on the geometry of minisuperspace. It has, however, a major disadvantage: while the spatially homogeneous cosmological models admit a dynamical and a spacetime point of view, the model considered here offers only a dynamical point of view. Although the following setup is more general than the homogeneous cosmological model, we will nevertheless use the terminology that is common in the context of (quantum) cosmology.

We assume ad-dimensional minisuperspace_{M parametrized by the variables {q}0, . . . , qd−1}

with an action of the form

S = Z dt L = Z dt  1 2NGAB q˙ A_{− N}i_A iA
˙ qB_{− N}j_A jB
− NV  , (2.84)

where N =: N0 _{and N}i _{are Lagrange multipliers and the index i runs from 1 to d}

mc < d

(for cosmological models usually dmc ≤ 3). The vectors Ai = AiA(q)∂A are assumed to be

linearly independent in all points q_{∈ M. The (generalized) DeWitt metric}8 _{is defined via}

d_S2 :=_GABdqA⊗ dqB . (2.85)

The DeWitt metric constitutes a metric on minisuperspace. In general it has a Lorentzian signature (_{−, +, . . . , +). This signals the presence of a lightcone structure in M. The function} V : M → R is the minisuperspace potential, which is determined by the spatial three- curvature and the matter potentials. We adopt the terminology of Misner [42, 43] and Ryan [45] and refer to the configuration q = _{qA_{} ∈ M of the universe as the universe point.}

8_{Often one calls the metric on the gravitational part ofM the DeWitt metric. In this thesis, we will call}

2.1. SPATIALLY HOMOGENEOUS COSMOLOGICAL MODELS 41

Note that the coupling of phantom fields [24] changes the signature of the DeWitt metric. The volume element on minisuperspace is given by

?1 =p|G| dq0∧ dq1∧ . . . ∧ dqd−1 , (2.86)

where_{G is the determinant of the DeWitt metric, that is, G := det ({G}AB}).

In the following we want to go over to the Hamiltonian formulation via the Dirac- Bergmann algorithm. We will use thereby to the terminology of Sundermeyer [74]. Note that the first step of the algorithm has already been performed by identifying the lapse and shift functions as non–dynamical variables.

The DeWitt metric and the vector fieldsAiA both appear in the Legendre transform

qA, ˙qA
7→  qA, pA= 1 NGAB ˙q B − Ni AiB   . (2.87)

The Legendre transform provides a linear map from the velocity phase space TM to the momentum phase spaceT∗

M. Performing the Legendre transform at the level of the action we obtain S = Z dt ˙qA_p A− N0H0− NiHi+λ0P0+λiPi
. (2.88)

The variables Λ0 _{and Λ}i _{are Lagrange multipliers, which ensure that the momenta conjugate}

to N0 _{and N}i _{weakly vanish, that is, P}

0 ' 0 and Pi ' 0. These are the so called primary

constraints, which are usually not written explicitly and we will also omit these terms from now on. The Hamiltonian constraint takes the form

H0 =

1 2G

AB_p

ApB+V ' 0 , (2.89)

This equation provides some important information about the dynamics. It tells us if the momentum p = _{pA} is “timelike”, “spacelike” or “lightlike” as determined by the sign of

the potential. This already allows us to make some simple qualitative statements about the behavior of the solutions. For example, the trajectory of a recollapsing universe has to turn “spacelike” in a region around the turning point. Consequently, if _{V ≥ 0 for all q ∈ M, the} trajectory of the universe point will be “timelike” or “lightlike” and a recollapse is impossible. In addition we assume in (2.84) the presence ofdmc momentum constraints, which are linear

in the momenta. They have the form

We collect the constraints into a single vector by introducing the notation _Hµ ={H0,Hi},

where µ = 0, 1, . . . , dmc.

In order to emulate the dynamics of spatially homogeneous cosmological models we require the Dirac-Bergmann algorithm to stop at this point. Thus the secondary constraints_Hµ ' 0

should be first class, that is, they should be preserved in time. This is equivalent to the requirement that the constraint algebra closes:

{Hµ,Hν} = Cµνλ Hλ ' 0 . (2.91)

Note that the structure functions_Cλ

µν =Cµνλ (q, p) are functions of all the phase-space variables

in general. The demand for closure of the constraint algebra will imply certain conditions on the DeWitt metric d_S2, the potential_{V and most importantly the vector fields A}i which we

will work out in the following. We readily find that

{Hi,Hj} = − [Ai, Aj]ApA=− AiB∇BAjA− AjB∇BAiA
pA , (2.92)

where _{∇ denotes the Levi-Civita connection compatible with dS}2. Hence the requirement (2.91) demands that the vector fields Ai form a closed Lie sub-algebra of vector fields, that is,

LAiAj = [Ai, Aj] =−C

k

ijAk. Following chapter 14 in the textbook [75], we can draw a clear

geometrical picture of the situation. Because of the closure condition, the vector fields Ai

define what is called an integrable distribution. This yields a foliation of the minisuperspace M into dmc–dimensional sub-manifolds Σ.9 These submanifolds are the so called integral

manifolds of the vector fields Ai, that is, their tangent space at any point q is spanned by

the vectors Ai|q. The integral manifolds are then also called the leaves of the foliation.

For a cosmological minisuperspace model, the leaves are usually spacelike and, in the case of the vacuum Bianchi models, the leaves of the foliation can be identified with the inner automorphism group.

We can construct the components of the induced metric on the leaves via_Gij :=AiAAjBGAB.

We define the dual of Ai as Bi = BAidqA, where BAi := GABGijAjB. By construction the

duality relation AjABAi =δji is satisfied. We can then write

{Hi,Hj} = − [Ai, Aj]yBkHk=CijkHk . (2.93)

Furthermore, the matrix

PAB :=BAiAiB (2.94) 9_Ad

mc– dimensional submanifold is also called submanifold of codimensiond−dmcin the mathematically

2.1. SPATIALLY HOMOGENEOUS COSMOLOGICAL MODELS 43

satisfies PACPCB = PAB and thus it acts as projection operator TM → T Σ (and T∗M →

T∗_{Σ). The projector P}

AB maps the momenta into span{Hi} and it is called the tangential

projection operator. The normal projection operator can be defined as

¯

PAB :=δAB− PAB . (2.95)

The projector ¯PAB is the complement of PAB, that is, PABP¯BC = ¯PABPBC = 0. By

construction the projectors are orthogonal, i.e.,

PAB _=PBA _{and P¯}AB _{= ¯P}BA _{. (2.96)}

The normal projector ¯PAB provides a mapTM → N (Σ) where N (Σ) is the so called normal

space, which is the space of all vector fields in_{N (Σ) that are normal to the leaves Σ. For the} traces of the projection operators, it holds thatPAA=dmc and ¯PAA=d− dmc. Furthermore,

the projectors allow for a decomposition of the DeWitt metric into two parts according to

GAB = ¯PACP¯BDGCD+PACPBDGCD . (2.97)

Note thatPACPBDGCD =BAiBBjGij is the induced metric on the leaves. The decomposition

of the inverse metric _GAB _{proceeds analogously.}

Let us now consider the Poisson brackets_{H0,Hi}. By direct calculation, we find that

{H0,Hi} = 1 2 LAidS 2
AB pApB+LAiV =_−∇(AAiB)pApB+AiA∂AV , (2.98) where (_LAidS 2₎AB _{= −2∇}(A_A

iB) is the Lie derivative of the (inverse) DeWitt metric with

respect to the vector field Ai. Our considerations so far lead us to impose the following

conditions:

• The vector fields Ai define a distribution onM.

• The DeWitt metric GAB, the potential V and the vector fields Ai satisfy

AiA∂AV = 2λiV and ∇(AAiB)+λiGAB

_¯

PACP¯BD = 0 , (2.99)

where λi is defined via

λi :=−

1 d− dmc∇

(A_A

From now on, we will always assume that these conditions are satisfied. Note that, as a particular case, Ai might be a conformal Killing vector field of both the metric and the

potential. The conditions imply that the constraint algebra closes, that is, _{Hµ,Hν} =

Cλ

µνHλ ' 0, where the structure functions are given by

C0 0i= 2λi , C0ij =− ∇ (A AiB)+λiGAB
PAC + 2 ¯PAC
pCBBj Ck ij =− AiA∇AAjB− AjA∇AAiB
BBk . (2.101)

The calculation in this section shows that the conditions are both necessary and sufficient for the closure of the constraint algebra. Note that _C0

0i and Cijk are functions of only q while

the _C0ij are functions of q and p.

Equations of motion: The evaluation of the Poisson bracketsqA, H and {pA, H} gives

2_{× d equations of motion. In the momentum phase space, the equations of motion constitute} the dynamical system

˙ qA_=N GAB_p B+NiAiA ˙ pA=− NGCDΓBACpBpD − N∂AV − Ni∂AAiBpB . (2.102)

The equations can now be solved after fixing the gauge N and Ni _{and setting up initial}

conditions _{qin, pin} obeying the constraints

Hµ(qin, pin) = 0 . (2.103)

The time preservation of the constraints is implied by the closure of the constraint algebra. In configuration space, the equations of motion read

(∂t− ∂tlogN ) ˙qA− NiAiA
+ ΓABCq˙ B_+Ni GCD∇AAiD
˙ qC − Ni_A iC
+N_GAB_∂ BV = 0 , (2.104)

where_∇B is the Levi-Civita connection that is compatible with the DeWitt metric and ΓABC

are its coefficients. If we gauge Ni _{= 0 and assume that N = N (q) we obtain}

˙

2.1. SPATIALLY HOMOGENEOUS COSMOLOGICAL MODELS 45

Note that the equations of motion in configuration phase space are manifestly covariant. After choosing the gauge N = 1 we get

¨ qA_{+ Γ}A BCq˙ B_q˙C ₊ GAB_∂ BV = 0 , (2.106)

This is of course the geodesic equation on (_{M, dS}2) for a parametrized curve with tangent vector ˙q plus a conservative force term. Note that the precise form of the solution curves q(t) depends on the gauge Ni_.

Transformations of the constraint algebra: We consider the following transformation of the lapse and shift functions:

N _{7→ e}N = Ω2_{N , (2.107)}

Ni

7→ eNi _=Nj_L

ji , (2.108)

where Ω : _{M 7→ R}+ _{is sufficiently smooth and {L}

ji} ∈ GL(dmc, R).10 The full Hamiltonian

is invariant under this transformation, that is,

H = Nµ

Hµ7→ H = eNµHeµ (2.109)

where the Hamiltonian and momentum constraints transform like

H0 7→ eH0 = Ω−2H0 = 1 2Ge AB_p ApB+ eV , (2.110) Hi 7→ eHi = eAiApA , (2.111)

with e_GAB = Ω2GAB, eV = Ω−2V and eAiA =LijAjA. In the context of Quantum Cosmology,

we will refer to the transformation (2.107) as a conformal transformation. Furthermore, we introduce the obvious notation

d e_S2 = Ω2d_S2 . (2.112) The transformation (2.108) can also be understood as a change of the basis on the leaves of the foliation. The full transformation made up of (2.107) and (2.108) induces a transformation of the structure functions according to

{ eHµ, eHν} = eCµνλ Heλ . (2.113) 10_{More generally, one could also consider transformations of the form N}i

7→ Ni_{+ ∆N}i_{. We leave this}

The constraint algebra of the transformed system closes and the new structure functions are related to the old ones by

e C0 0i=Lij C0j0 − 2AjA∂Alog Ω
= 2eλ_i , e C0ij = Ω −2_L ilC0lk − G AB ∂ALik
pB  L−1
k j , e Ck ij =LilLjnClnm+LjlAlA(∂ALim)− LilAlA(∂ALjm)  L−1
m k _. (2.114)

Note that if we only consider rescalings of the lapse function then λi transforms similar to a

Weyl vector field (see appendix B.3):

λi 7→ eλi =λi− AiA∂Alog Ω . (2.115)

The transformation law suggests that we can interpret the one–formλiBias a Weyl one–form

on the leaves of the foliation.

For completeness and later use, we remark that we can define the 2-form F :=_FijBi∧Bj,

where

Fij := 2AiAAjB∇[AλB] and λA:=BAiλi . (2.116)

Note that by definition ¯PABλB = 0. The tensor F is in certain aspects analogous to

the Faraday 2-form in electrodynamics. Its most notable feature is its invariance under rescalings of the lapse. Moreover, it transforms covariantly under transformations of the shift functions. To be more precise, it transforms like _Fij 7→ eFij = LikLjlFkl under the

transformation (2.108). A certain simplification arises if F = 0. This is in particular true if all λi = 0 (or all eλi = 0 after a rescaling of the lapse) which is usually the case for spatially

homogeneous cosmological models. If we, however, perform a rescaling of the lapse, which satisfies AiA∂AΩ 6= 0, then eλi = −AiA∂Alog Ω is non-zero. The condition F = 0 seems

to be important in the context of quantization. In fact, this condition is equivalent to the statement that the Weyl structure on the leaves is integrable, as we will show later in section 2.2.7.

In addition, we have encountered the objects

KAB

i :=∇

(A_A

iB)+λiGAB (2.117)

in our calculations. We find that they transform as

KiAB 7→ eK AB

i = Ω

−2

2.1. SPATIALLY HOMOGENEOUS COSMOLOGICAL MODELS 47

when we rescale the lapse. This also implies that the scalars Ki := GABKiAB are invariant

under rescalings of the lapse function. However, neither KAB

i nor Ki transform covariantly

under transformations of the shift. The KAB

i transform as

KAB

i 7→ eKiAB =LijKjAB+Aj(A∂B)Lij . (2.119)

Nevertheless, since AiCP¯CA = 0 the object KiCDP¯CAP¯DB transforms covariantly under

transformations of the shift. Therefore, the condition KCD

i P¯CAP¯DB = 0 has a covariant

meaning. Furthermore, we remark that the structure functions_Ck

ij can be rewritten as

Ck

ij = 2A[iAAj]B∇ABBk = 2AiAAjB∇[ABB]k . (2.120)

Our considerations so far might be important in the context of canonical quantization. The transformations (2.107) and (2.108) certainly leave the physics invariant. We will later demand that the same holds true for the quantized version of the system. We continue the discussion on the geometry of minisuperspace in section 2.2.7. But before closing this section let us have brief look at the notion of symmetries.

Symmetries: A phase-space functionf (q, p) is a constant of motion if it weakly commutes with all constraints

{Hµ, f} ' 0 . (2.121)

We consider first functions of the form f (q, p) = ξA_p

A with ξ = ξA∂A ∈ T M. The

discussion proceeds analogously to the discussion of the momentum constraints. The condition (2.121) is satisfied if

• The vector field ξ satisfies [ξ, Ai] ∈ span{Ai}. In the mathematical literature on

foliations, these vector fields are called basic or also foliate. The set of all basic vector fields forms a Lie algebra [76].

• ξA_∂

AV = 2λξV and ∇(AξB)+λξGAB

_¯

PACP¯BD = 0.

The scalar λξ is completely analogous to the one defined in (2.100). Note that the above

conditions are satisfied by the momentum constraints_Hi as well (the associated constant of

motion, however, is constrained to be zero). An important example of such a symmetry are the generators of the special automorphism group in the case of (vacuum) Bianchi models.

In full analogy to the vacuum Bianchi models, we introduce the following terminology: We refer to the group generated by all vector fields ξ which satisfy the above conditions as

the symmetry group of the system. The subgroup generated by the vector fields Ai will be

called the inner symmetry group. The group which is generated by all vector fields that are not in the span_{Ai} will be called the outer symmetry group.

Let us next consider phase-space functions of the form f (q, p) = XAB_p

BpA with X =

XAB_∂

A⊗ ∂B ∈ T M ⊗ T M being a symmetric 2₀
–tensor field XAB = XBA. We compute

the commutators with the secondary constraints as follows

{f, H0} = 2pA  1 2∇ (A XBC)pBpC− XAB∂BV  , {f, Hi} = (LAiX) AB pApB . (2.122)

We find that the condition (2.121) is satisfied if

• (LAiX) AB_P¯ ACP¯BD = 0, • XAB_∂ BV = λXAV, • ∇(A_XBC)_{+ 2λ} X(AGBC)
_¯ PADP¯BEP¯CF = 0.

The vector field λXA∂A is the vector-field analogue of the scalar (2.100) and the last two

conditions are in some sense equivalent to (2.99). As a particular sub-case, X might be a conformal Killing tensor field of the DeWitt metric and the potential. Another example is given by X = ξ _{⊗ ξ where ξ generates a symmetry. The reader can easily convince} herself/himself that this tensor satisfies all requirements.

2.1.6

Hamilton-Jacobi formalism

The Hamilton-Jacobi formalism might be considered as the formulation of classical mechanics which is closest to quantum mechanics. In fact, it was Schr¨odinger’s actual starting point for deriving his famous wave equation [77]. In this sense the current section will serve as a preparation for quantization, which is to be understood in this thesis as a procedure that tries to reverse the eikonal approximation. More precisely, the Wheeler-DeWitt equation and the quantum momentum constraints should be constructed such that the Hamilton-Jacobi equation is recovered in the semi-classical limit. We write now the action in the form

S0[q, p, qin, pin] = Z γ dt ˙qA_p A− NµHµ  , (2.123)

whereγ is a path with starting point qin, pin and end point q∈ M, p ∈ Tq∗M. The physical

2.1. SPATIALLY HOMOGENEOUS COSMOLOGICAL MODELS 49

for physical trajectories. On-shell, the action therefore becomes

S0[q, p, qin, pin] ˙= Z γ dτ := Z γ (∂AS0) dqA= Z γ pAdqA . (2.124)

The 1-form dτ is the differential of the so called WKB time. It can now be shown that (on-shell) S0 satisfies the Hamilton-Jacobi equations

H0 q, p = ∂AS0dqA
= 1 2G AB_(∂ AS0)(∂BS0) +V = 0 , Hi q, p = ∂AS0dqA
= AiA∂AS0 = 0 , (2.125)

where initial conditions are to be given on some suitable hypersurface Σ_{⊂ M. The initial} momenta ∂AS0|Σ are then also subject to the constraints Hµ(qA, ∂AS0)|Σ = 0. Note that

after solving the equations S0 provides a scalar field on some submanifold ofM. Moreover,

ifS0 is a solution to the Hamilton-Jacobi equations then−S0 is also a solution. By switching

to the Hamilton-Jacobi formalism, we shift the problem of solving equations of motion to the problem of solving one non-linear first order partial differential equation for the so called Hamilton-Jacobi action. Moreover, the formalism so far is completely independent of the gauge Nµ_{. After solving the Hamilton-Jacobi equation, we obtain the physical momenta}

pA =∂AS0 by construction. The momentum constraints AiA∂AS0 = AiApA = 0 imply that

S0 is constant on the leaves of the foliation spanned by the vector fields Ai.

Note also that the Hamilton-Jacobi equations (2.125) are completely independent of any external time parameter. We can, however, define a time vector field by

∂τ :=

∂ ∂τ :=G

AB_(∂

BS0)∂A ∈ T M . (2.126)

The momentum constraintsAiA∂AS0 = 0 ensure thatS0 does not change in the direction Ai

which implies that∂τ does not have components proportional to the vectors Ai; in formulas:

PA

B(∂τ)B = 0 . (2.127)

Furthermore, we obtain the Lie bracket

[∂τ, Ai] =−2λi∂τ + 2(∂AS0)∇(AAiB)BBjAj . (2.128)

This implies that the algebra formed by_{∂τ, Ai} closes.11 If we want to obtain the physical

11_{This might be important in the context of semi-classical Quantum Cosmology since ∂}

τ appears on the

trajectories in _{M we have to solve the equation} 1 N ˙q = ∂ ∂τ +n i_A i , (2.129)

which requires to fix a gauge N and ni _:= Ni

N. The solution q(t) is then an integral curve

of the vector field N ∂ ∂τ +n

i_A

i
. We note that the flow of this vector field depends on the

gaugeni_{. We conclude that while different gauges for N only yield different parametrizations}

of the solution curves, different gaugesni_{, that is, different foliations, certainly yield different}

trajectories in_{M. A relation between WKB time τ and the coordinate time t can be obtained} from the equation

dτ := dS0 =pAdqA=GAB  ˙qA N − n i_A iA  ˙ qB_{dt . (2.130)}

Let us now turn to a discussion of symmetries at the level of the Hamilton-Jacobi formulation. Let f (q, p) = ξA_p

A be a symmetry of the system as discussed in section 2.1.5

of this thesis. We denote the flow generated by the vector ξ by

Φξ : M × R → M (q, T ) _{7→ Φ}T ξ(q) = exp (T ξ) q . (2.131) Furthermore, we define e ST_{(q) :=S Φ}T ξ(q)
. (2.132)

One can now show the following:

If ξ generates a symmetry and S solves the Hamilton-Jacobi equations (2.125) then ˜ST

is a solution as well.

It is sufficient to prove the infinitesimal version of this statement. We first expand ˜ST

aroundT = 0. Using the fact that S solves (2.125) and that ξ generates a symmetry one can show that

1 2G

AB_(∂

AS)(∂˜ BS) +˜ V = O(T2) and AiA∂AS =e O(T2). (2.133) The terms linear inT vanish identically and hence we can conclude the proof. Recall that the subalgebra of vector fields _{Ai} generate symmetries as well. The momentum constraints

AiA∂AS = 0, however, ensure that ˜S(Φ(q)) = S(q) if the flow Φ was generated by this

subalgebra.

Recall that an important example of the symmetries in this discussion are the automorphisms in the case of the vacuum Bianchi models. In that case ξ could be taken as a generator of a

2.1. SPATIALLY HOMOGENEOUS COSMOLOGICAL MODELS 51

constant special automorphism, that is, for a fixedT the flow is given by ΦT

ξ (hij) =hklLkiLlj,

where_{Li

j} is a constant element of SAut(g).

The Van Vleck factor

An important feature of any quantum theory of gravity should be that it contains the classical theory in some specific limits. In the Wheeler-DeWitt approach, one of the steps that leads to the recovery a classical limit usually involves the WKB approximation (see [10] or section of this 2.2.3 thesis). In the semi-classical limit, the wave functions are then of the form

Ψ_≈√DeiS _(2.134)

whereS is a solution to the Hamilton-Jacobi equation and D is the so called van Vleck factor (mostly referred to as van Vleck determinant). The following considerations might therefore be considered a first step into the realm of Quantum Cosmology.

Suppose that we have a solution S to the Hamilton-Jacobi equation (2.125). In the following, we want to derive a notion of a momentum flux density D in _{M for the given} solutionS . We define the momentum flux

JA_:=D

GAB_∂

BS0 (2.135)

and demand that it is conserved, that is,

∇AJA= 0 . (2.136)

The conservation equation yields a linear transport equation forD which reads

∂τD =−(S)D (2.137)

We remark that the definition (2.135) and the conservation law (2.136) correspond in some sense to Fick’s first and second law, respectively.

We can set up initial conditions for D at some point qin ∈ M and then evolve it along

the streamlines γ of the momentum flux by

D(q) = D(qin) exp  − Z γ d_{τ S}  . (2.138)

reference density.

Furthermore, note that the definition of D depends on the metric d_S2 _{and therefore on}

In document Quantum Fate of Singularities in Anisotropic Cosmological Models (Page 40-53)