More on the free motion: The Kasner solution

2.6 More on the free motion: The Kasner solution

The free motion between two bounces is a straight line in the space of the scale factors. In terms of the original metric components, it takes the form of the Kasner solution with dilaton. Indeed, the free motion is given by

μ μ 0 μ β = q x + β 0,

where the “velocities” $q μ$ are subject to

( ) ∑ ∑ 2 (qi)2 − qi + q2φ = 0, i i

since the motion is lightlike by the Hamiltonian constraint. The proper time $√ -- 0 dT = − g dx$ is then $0 T = B exp(− Kx )$ , with $∑ i K = iq$ and for some constant $B$ (we assume, as before, that the singularity is at $T = 0+$ ). Redefining then

μ pμ = ∑q---- iqi

yields the celebrated Kasner solution

2 2 ∑ 2pi ( i)2 ds = − dT + T dx , (2.53 ) i φ = − pφlnT + A, (2.54 )

subject to the constraints

where $A$ is a constant of integration and where the coordinates $xi$ have been suitably rescaled (if necessary).