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5.11 Einstein–Rosen waves

One class of one-dimensional models is given by cylindrically-symmetric gravitational waves, with connections and triads
A = Ax (x)τ3dx + (A1(x )τ1 + A2 (x)τ2)dz + (A3 (x)τ1 + A4(x)τ2)dϕ (60 ) x -∂- 1 2 ∂-- 3 4 -∂- E = E (x)τ3∂x + (E (x)τ1 + E (x )τ2) ∂z + (E (x )τ1 + E (x)τ2)∂ϕ (61 )
in cylindrical coordinates. This form is more restricted than Equation (59View Equation), but still not simple enough for arbitrary A1, A2, A3 and A4. Einstein–Rosen waves [15941] are a special example of cylindrical waves subject to the polarization condition E2E4 + E1E3 = 0 for triad components. Since this is a quadratic condition for momenta, the polarization imposed on connection components takes a different form, which is analyzed in [34Jump To The Next Citation Point]. It turns out that this allows simplifications in the quantization analogous to models with internally-perpendicular connection components and is thus similar to diagonalization in a homogeneous model.

5.11.1 Canonical variables

A difference to homogeneous models, however, is that the internal directions of a connection and a triad do not need to be identical, which in homogeneous models with internal directions ΛiI is the case as a consequence of the Gauss constraint εijkφj pI= 0 I k. With inhomogeneous fields, now, the Gauss constraint reads

x′ 2 1 4 3 E + A1E − A2E + A3E − A4E = 0 (62 )
or, after splitting off norms and internal directions,
∘ -------- ∘ -------- A := A2 + A2 , A := A2 + A2 (63 ) z 1 2 ϕ 3 4 A A1τ1-+-A2τ2- A A3τ1-+-A4τ2- Λz := Az , Λϕ := A ϕ (64 )
and analogously Ez, E ϕ, ΛzE and ϕ ΛE,
Ex ′ + AzEz sin α + A ϕE ϕsinα¯= 0 (65 )
with A z sinα := − 2tr(Λ z ΛE τ3) and A ϕ sin ¯α := − 2tr(ΛϕΛ Eτ3). If x E is not constant, α and ¯α cannot both be zero and thus connections and triads have different internal directions.

As a consequence, Ez is not conjugate to Az anymore, and instead the momentum of Az is Ez cos α [60Jump To The Next Citation Point]. This seems to make quantization more complicated since the momenta will be quantized to simple flux operators, but do not directly determine the geometry, such as the volume ∫ ∘ --x--z--ϕ- V = 4π dx |E E E |. For this, one would need to know the angle α, which depends on both connections and triads. Moreover, it would not be obvious how to obtain a discrete volume spectrum since volume does not depend only on fluxes.

It turns out that there is a simple canonical transformation, which allows one to work with canonical variables Ez and Eϕ playing the role of momenta of Az cos α and A ϕ cos ¯α [109Jump To The Next Citation Point3435]. This seems to be undesirable, too, since now the connection variables that play an important role for holonomies are modified. That these canonical variables are very natural, however, follows after one considers the structure of spin connections and extrinsic curvature tensors in this model. The new canonical variables are then simply given by Az cosα = γKz, Aϕ cos ¯α = γK ϕ, i.e., proportional to extrinsic curvature components. Thus, in the inhomogeneous model we simply replace connection components with extrinsic curvature in homogeneous directions (note that Ax remains unchanged) while momenta remain elementary triad components. This is part of a broader scheme, which is also important for the Hamiltonian constraint operator (Section 5.15).

5.11.2 Representation

With the polarization condition the kinematics of the quantum theory simplify. Relevant holonomies are given by 1 he(A ) = exp(2i∫e Ax(x)dx ) along edges in the one-dimensional manifold and

z ϕ hv(A ) = exp(iγνvKz (v)), hv(A ) = exp(iγμvK ϕ (v ))

in vertices v with real μv,νv ≥ 0. Cylindrical functions depend on finitely many of these holonomies, whose edges and vertices form a graph in the one-dimensional manifold. Flux operators, i.e., quantized triad components, act simply by

x γ ℓ2Pke+(x) + ke−(x) Eˆ (x)Tg,k,μ = ------------------Tg,k,μ (66 ) ∫ 8π2∑ 2 EˆzTg,k,μ = γ-ℓP νvTg,k,μ (67 ) ℐ 4π v∈ ℐ ∫ 2∑ EˆϕT = γ-ℓP μ T (68 ) ℐ g,k,μ 4π v g,k,μ v∈ ℐ
on a spin network state
∏ ∏ Tg,k,μ(A) = ρke(he) ρ μv(γK ϕ(v))ρ νv(γKz (v))ρkv(β(v)) e∈g ( v∈V(g) ) ∏ 1 ∏ iγμvK ϕ(v) iγνvKz(v) ikvβ(v) = exp 2ike ∫Ax (x)dx e e e , (69 ) e∈g e v∈V (g)
which also depend on the gauge angle β determining the internal direction of E Λ z. If we solve the Gauss constraint at the quantum level, the labels kv will be such that a gauge invariant spin network only depends on the gauge-invariant combination Ax + β′.

Since triad components now have simple quantizations, one can directly combine them to get the volume operator and its spectrum

γ3∕2ℓ3P∑ ∘ ------------------- Vk,μ,ν = --√--- μv νv|ke+ (v) + ke− (v)|. (70 ) 4 π v
The labels μ v and ν v are always non-negative, and the local orientation is given through the sign of edge labels ke.

Commutators between holonomies and the volume operator will technically be similar to homogeneous models, except that there are more possibilities to combine different edges. Accordingly, one can easily compute all matrix elements of composite operators such as the Hamiltonian constraint. The result is only more cumbersome because there are more terms to keep track of. Again as in diagonal homogeneous cases, the triad representation exists and one can formulate the constraint equation there. Now, however, one has infinitely many coupled difference equations for the wave function since the lapse function is inhomogeneous, providing one difference equation for each vertex. With refinement, only μv and νv can possibly be scale dependent but not ke, which gives the representation for a holonomy along an inhomogeneous direction.

There are obvious differences to cases considered previously owing to inhomogeneity. For instance, each edge label can take positive or negative values, or go through zero during evolution, corresponding to the fact that a spatial slice does not need to intersect the classical singularity everywhere. Also the structure of coefficients of the difference equations, though qualitatively similar to homogeneous models, is changed crucially in inhomogeneous models, mainly due to the volume eigenvalues (70View Equation). Now, a single edge label ke+, say, and thus x E can be zero without volume eigenvalues in neighboring vertices having zero volume.


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