### 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
where the “velocities” are subject to

since the motion is lightlike by the Hamiltonian constraint. The proper time is then
, with and for some constant (we assume, as before, that the
singularity is at ). Redefining then

yields the celebrated Kasner solution

subject to the constraints
where is a constant of integration and where the coordinates have been suitably rescaled (if
necessary).