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4.1 Isotropy

Isotropy reduces the phase space of general relativity to be 2-dimensional since, up to SU(2)-gauge freedom, there is only one independent component in an isotropic connection and triad, respectively, which is not already determined by the symmetry. This is analogous to metric variables, where the scale factor a is the only free component in the spatial part of an isotropic metric
ds2 = - N (t)2dt2 + a(t)2((1 - kr2)-1dr2 + r2 d_O_2). (16)
The lapse function N (t) does not play a dynamical role and correspondingly does not appear in the Friedmann equation
( )2 a- + -k-= 8pG-a- 3Hmatter(a) (17) a a2 3
with the matter Hamiltonian H matter and the gravitational constant G, and the parameter k taking the discrete values zero or ±1 depending on the symmetry group or intrinsic spatial curvature.

Indeed, N (t) can simply be absorbed into the time coordinate by defining proper time t through dt = N (t)dt. This is not possible for the scale factor since it depends on time but multiplies space differentials in the line element. The scale factor can only be rescaled by an arbitrary constant, which can be normalized at least in the closed model where k = 1.

One can understand these different roles of metric components also from a Hamiltonian analysis of the Einstein-Hilbert action

integral S = --1--- dt d3xR[g] EH 16pG

specialized to isotropic metrics (16View Equation) whose Ricci scalar is

( ) ¨a a2 k a N R = 6 --2-+ --2-2-+ -2-- ----3 . N a N a a a N

The action then becomes

integral integral ( ) V0 3 3V0 aa2 S = ------ dt N a R = ----- dt N - --2-+ ka 16pG 8pG N

(with the spatial coordinate volume integral 3 V0 = S d x) after integrating by parts, from which one derives the momenta

@L 3V aa @L pa = --- = - ---0---, pN = ----= 0 @a 4pG N @N

illustrating the different roles of a and N. Since pN must vanish, N is not a degree of freedom but a Lagrange multiplier. It appears in the canonical action integral S = (16pG) -1 dt(apa - N H)) only as a factor of

2pG--p2a-- --3-- H = - 3 V0a - 8pG V0ak

such that variation with respect to N forces H, the Hamiltonian constraint, to be zero. In the presence of matter, H also contains the matter Hamiltonian, and its vanishing is equivalent to the Friedmann equation.

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