Proper time

Time along a worldline

Proper time is the time measured by a clock that travels along a particular worldline. If you carry a wristwatch from one event to another, the elapsed time on your watch is the proper time along your path.

Proper time is invariant: all observers agree on how much time that clock records, even if they assign different coordinate times and distances to the trip.

Proper time from the interval

For motion in one spatial dimension, the spacetime interval for a timelike path satisfies

$c^2d au^2=c^2dt^2-dx^2$

Using velocity $v=dx/dt$, this becomes

$d au=dtsqrt{1-v^2/c^2}$

or

d au=rac{dt}{gamma}

where

gamma=rac{1}{sqrt{1-v^2/c^2}}

For a finite trip,

au=int d au=int rac{dt}{gamma}

Straight paths maximize proper time

In Euclidean geometry, a straight line between two points minimizes distance. In Minkowski spacetime, an inertial straight worldline between two timelike-separated events maximizes proper time.

This surprising fact explains why an inertial twin ages more than an accelerating twin who leaves and returns. The traveling twin follows a different worldline with less proper time.

Proper time and time dilation

For an object moving at constant speed $v$ relative to an inertial frame, the coordinate time interval is related to proper time by

$Delta t=gammaDelta au$

Equivalently,

Delta au=rac{Delta t}{gamma}

The moving clock records less time than the coordinate time measured by clocks in the inertial frame it moves through.

Four-velocity normalization

Four-velocity is defined as

U^mu=rac{dx^mu}{d au}

Because proper time is invariant, four-velocity is a genuine four-vector. Its invariant magnitude is

$U^mu U_mu=-c^2$

This is analogous to saying all massive particles move through spacetime with a fixed four-speed magnitude $c$.

Accelerated motion

Special relativity can handle accelerated worldlines, even though inertial frames are its basic reference frames. For an accelerated clock, proper time is still found by integrating along the path:

$au=int sqrt{1-v(t)^2/c^2},dt$

Acceleration matters because it changes the worldline, not because acceleration directly slows clocks in special relativity independent of velocity history.

Clock hypothesis

The clock hypothesis states that an ideal clock's instantaneous rate depends only on its instantaneous velocity, not on its acceleration. Real clocks can be damaged by acceleration, but an ideal clock measures proper time along its worldline.

The big idea

Proper time is the invariant time measured by a clock along its own worldline. It is calculated from the spacetime interval and depends on the path through spacetime. Different paths between the same events can have different proper times, which is the geometric heart of relativistic aging effects.