Proper time - Aetherwisp

The proper time of an object is a measure of time that takes into account the time dilation that the object experiences due to its velocity with respect to another frame of reference. It is the time measured by a moving clock. Since time passes slower for things in motion, the proper time is always smaller than the time measured by a clock at rest (in some frame). In infinitesimal form it is defined as

d\tau=\sqrt{ 1- \frac{v^{2}}{c^{2}} }dt=\frac{dt}{\gamma}

where $v$ is the speed of the object as measured by an observer at rest, $c$ is the speed of light and $\gamma$ is the relativistic gamma. $dt$ is the time unit of a clock at rest.

Proper time is a relativistic invariant.

As usual, the concept of "at rest" is relative. That's kind of the whole point of "relativity", so be careful of how you measure $v$ . You might claim that the time of someone on a train is dilated and therefore their proper time is squished, but they might claim that your proper time is squished because from their perspective you are moving. It's the same deal of time dilation all over again (in fact, proper time is basically just infinitesimal time dilation).