Fundamentals of Time and Relativity

Proper Time

The concept of proper time (from Latin, meaning own time) was introduced by Hermann Minkowski in 1908, and is an important feature of Minkowski space. Proper time is measured directly when two events both occur at the position of a certain clock. Then the time interval between the two events that this stationary clock measures, is called its proper time.

More generally, proper time along a time-like worldline, from some initial event $P$ to some final event $Q$ in Minkowski space, is defined as the time as measured by a clock following that line. Let $d τ$ be an infinitesimal timelike interval on $P Q$ . The proper time interval is then defined as a line integral along the worldline

4.9

Δ τ : = \int_{P}^{Q} d τ = \frac{1}{c} \int_{P}^{Q} \sqrt{g_{μ ν} d x^{μ} d x^{ν}}

Since the Minkowski inner product is independent of coordinates, proper time is an invariant time coordinate by definition.

By substituting the spatial coordinate speed $v^{i} = d x^{i} / d t$ , the proper time interval in a particular inertial frame can be written in the useful form

4.10

Δ τ = \int d t \sqrt{1 - \frac{v^{2} (t)}{c^{2}}}

The integral runs from the initial time $t (P)$ to the final time $t (Q)$ as seen from the stationary reference frame. If the motion of the particle is constant, expression (4.10) simplifies to $Δ τ = Δ t \sqrt{1 - {(v / c)}^{2}}$ .

The square root is ubiquitous in relativity theory and has its own symbol

4.11

γ (v) : = {(\sqrt{1 - \frac{v^{2}}{c^{2}}})}^{- 1}

called the Lorentz factor or stretch factor; the inverse may be called the shrink factor.

The time of a moving object runs slower as seen by a stationary observer; this is the relativistic effect of time dilation.
The shrink factor also describes Lorentz contraction, the measured shortening of a moving object along its direction of motion.