The reparametrization of time is a passive transformation, corresponding to a coordinate change involving the real line R. Given an initial temporal coordinate a ^ t ^ b over the interval I = [a, b], and a final temporal coordinate a ^t ^ b over interval I = [a, b], we will assume that t is an invertible function of t, mapping points in I to points in I, viz. t ^ t(t), such that a = t(a), b = t(b). If we wanted to, we could choose to make the new time parameter t run in the opposite direction, that is, a = t(b), b = t(a), but there is no advantage in this so we shall not make this choice.

If we believe in the continuity of time, then we require t to be a continuous function of t. Additionally, since we want to deal with differentiable functions, we will require t to be a differentiable function of t over the open interval (a, b).

dt

All of these conditions means that we must have — > 0 over the interval (a, b).

dt

If this condition failed to hold anywhere over that interval then that would mean that the new time parameter t appeared to stop for some value of the old time parameter. Whilst we are prepared to accept that our new time parameter t could vary non-uniformly with the old (Absolute) time parameter t in the way Newton commented on above, we should not be prepared to use a clock that appears to stop at some point.

In the following analysis, we shall use the inverse transformation, writing t = t(t) dt