Problem — The reference frame of minimum time

Problem

In which reference frame is the time interval between two events minimum? What is this time called? What condition must the two events satisfy for such a frame to exist?

Solution

The minimum time. The time interval between two events is minimum in the reference frame in which the two events occur at the same point in space (co-spatial events, $\Delta x' = 0$). In every other frame the time is dilated:

$\Delta t = \gamma\,\Delta\tau \geq \Delta\tau$

Name. This minimum time is the proper time $\Delta\tau$: it is the time measured by a single clock present at both events.

Condition for existence. Such a frame exists only if the two events are separated by a timelike interval, i.e. if they can be causally connected (a signal with $v < c$ can travel from one to the other). In terms of the spacetime invariant:

$\ev{(c\,\Delta t)^2 - (\Delta x)^2 > 0 \;\Rightarrow\; \text{timelike events}}$

If instead the interval is spacelike, $(c\,\Delta t)^2 - (\Delta x)^2 < 0$, no clock can be present at both events and there is no frame in which they are co-spatial: the proper time is not defined.

Links

Topics: Special relativity Concepts: Time dilation · Spacetime invariant Skills: Changing reference frame