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4.1 Isotropy

Isotropy reduces the phase space of general relativity to two-dimensions since, up to SU(2)-gauge freedom, there is only one independent component in an isotropic connection and triad, 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)2 dt2 + a (t)2((1 − kr2)−1dr2 + r2 dΩ2) . (16 )
The lapse function N (t) does not play a dynamical role and correspondingly does not appear in the Friedmann equation
( ˙a )2 k 8πG -- + -2-= ----a− 3Hmatter(a) (17 ) a a 3
where Hmatter is the matter Hamiltonian, G is the gravitational constant, and the parameter k takes 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 τ through dτ = 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 also understand these different roles of metric components from a Hamiltonian analysis of the Einstein–Hilbert action

∫ --1--- 3 SEH = 16πG dt d xR [g]

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

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

The action then becomes

∫ ∫ ( 2 ) S = -V0--- dt N a3R = 3V0-- dt N − a˙a--+ ka 16πG 8πG N 2

(with the spatial coordinate volume ∫ V0 = d3x Σ) after integrating by parts, from which one derives the momenta

∂L 3V0 a˙a ∂L pa = --- = − -------- , pN = --˙-= 0 ∂ ˙a 4πG 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 ∫ S = (16 πG )−1 dt(˙apa − N H )) only as a factor of

2πG p2a 3 HADM = − -------- − -----V0ak, (18 ) 3 V0a 8πG
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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