# Temporal reparametrization

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*

and taking — > 0.

*dt*