Proper Time

For an object, a particle, or an observer, the invariant spacetime interval between two events for that object is never spacelike because the object cannot move faster than the speed of light.

\bgroup\color{black}$\displaystyle s^2=-c^2\Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2 >0 $\egroup

So the space time interval for an object can always be written as the time interval in the object's rest frame.

\bgroup\color{black}$\displaystyle s^2=-c^2\Delta \tau^2 $\egroup

So it is convenient to use the proper time \bgroup\color{black}$ \tau$\egroup to understand the object's motion. \bgroup\color{black}$ \tau$\egroup is a scalar quantity since it is directly related to the scalar spacetime interval.

As the length of a vector is the square root of the dot product of the vector with itself in Euclidean space, the proper time is essentially the square root of the dot product of a vector with itself in Minkowski space (up to the factor of \bgroup\color{black}$ c$\egroup and the negative sign for timelike vectors that is conventional).

We will use the proper time to help us define the velocity and momentum in Minkowski space.

Jim Branson 2012-10-21