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5.10 Einstein-Rosen waves

One class of 1-dimensional models is given by cylindrically symmetric gravitational waves, with connections and triads
A = Ax(x)t3 dx + (A1(x)t1 + A2(x)t2)dz + (A3(x)t1 + A4(x)t2) df, (54) x -@- 1 2 @-- 3 4 -@- E = E (x)t3@x + (E (x)t1 + E (x)t2) @z + (E (x)t1 + E (x)t2)@f (55)
in cylindrical coordinates. This form is more special than (53View Equation), but still not simple enough for arbitrary A1, A2, A3 and A4. Einstein-Rosen waves [10934] are a special example of cylindrical waves subject to the polarization condition A2A4 + A1A3 = 0, and analogously for triad components. This is just what is needed to restrict the model to internally perpendicular connection components and is thus analogous to diagonalization in a homogeneous model.

5.10.1 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 /\i I is the case as a consequence of the Gauss constraint ijk j I e f Ipk = 0. With inhomogeneous fields, now, the Gauss constraint reads

x' 2 1 4 3 E + A1E - A2E + A3E - A4E = 0 (56)
or, after splitting off norms and internal directions
V~ -------- V~ -------- 2 2 2 2 Az := A 1 + A2 , Af := A3 + A 4 (57) A1t1 + A2t2 A3t1 + A4t2 /\Az := ------------ , /\Af := ------------ (58) Az Af
and analogously Ez, Ef, /\zE and /\fE,
Ex'+ (A Ez + A Ef) sin a = 0 (59) z f
with sina := -2 tr(/\Az /\zEt3). If Ex is not constant, a cannot be zero and thus connections and triads have different internal directions.

As a consequence, z E is not conjugate to Az, anymore, and instead the momentum of Az is z E cos a [52Jump To The Next Citation Point]. This seems to make a quantization more complicated since the momenta will be quantized to simple flux operators, but do not directly determine the geometry such as the volume integral V~ ---------- V = 4p dx |ExEzEf |. For this, one would need to know the angle a which depends on both connections and triads. Moreover, it would not be obvious how to obtain a discrete volume spectrum since then 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 Ef playing the role of momenta of A cosa z and A cos a f [75Jump To The Next Citation Point]. This seems to be undesirable, too, since now the connection variables are modified which play an important role for holonomies. 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 cos a = gKz, Af cos a = gKf, 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 5.14).

5.10.2 Representation

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

hzv(A) = exp(ignvKz(v)), hfv(A) = exp(igmvKf(v))

in vertices v with real m ,n > 0 v v. Cylindrical functions depend on finitely many of those holonomies, whose edges and vertices form a graph in the 1-dimensional manifold. Flux operators, i.e., quantized triad components, act simply by

x gl2Pke+(x) + ke-(x) E^ (x)Tg,k,m = ------------------Tg,k,m (60) integral 8p2 sum 2 E^zTg,k,m = gl-P nvTg,k,m (61) I 4p v (- I integral 2 sum E^fT = gl-P m T (62) I g,k,m 4p v g,k,m v (- I
on a spin network state
prod prod Tg,k,m(A) = rke(he) rmv(gKf(v))rnv (gKz(v))rkv (b(v)) e (- g ( v (- V(g) ) prod 1 prod igmvKf(v) ignvKz(v) ikvb(v) = exp 2ike integral Ax(x)dx e e e , (63) e (- g e v (- V (g)
which also depend on the gauge angle b determining the internal direction of E /\ 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 + b'.

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

g3/2l3P sum V~ ------------------- Vk,m,n = -- V~ -- mvnv|ke+(v) + ke-(v)|. (64) 4 p v
The labels m v and n 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 (one obtains one difference equation for each vertex).

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 (64View Equation). Now, k + e, say, and thus x E can be zero without volume eigenvalues in neighboring vertices having zero volume.


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