In contrast, coordinate time can be applied to events that occur a distance from an observer. In special relativity, coordinate time is reckoned relative only to inertial observers, whereas proper time can be measured by accelerated observers too.
By convention, proper time is usually represented by the Greek letter to distinguish it from coordinate time represented by or .
A Euclidean geometrical analogy is that coordinate time is like distance measured with a straight vertical ruler, whereas proper time is like distance measured with a tape measure. If the tape measure is taut and vertical it measures the same as the ruler, but if the tape measure is not taut, or taut but not vertical, it will not measure the same as the ruler.
The formal definition of proper time involves describing the path through spacetime that represents a clock, observer, or test particle, and the metric structure of that spacetime. Proper time is the pseudo-Riemannian arc length of world lines in four-dimensional spacetime.
From the mathematical point of view, coordinate time is assumed to be predefined and we require an expression for proper time as a function of coordinate time. (From the experimental point of view, proper time is what is measured experimentally and then coordinate time is calculated from the proper time of some inertial clocks.)
In special relativity, proper time can be defined as
If t, x, y and z are all parameterised by a parameter , this can be written as
In differential form it can be written as the path integral
where P is the path of the clock in spacetime.
To make things even easier, inertial motion in special relativity is where the spatial coordinates change at a constant rate with respect to the temporal coordinate. This further simplifies the proper time equation to
where Δ means "the change in" between two events.
The special relativity equations are special cases of the general case that follows.
Using tensor calculus, proper time is more rigorously defined as follows: Given a spacetime which is a pseudo-Riemannian manifold mapped with a coordinate system and equipped with a corresponding metric tensor , the proper time experienced in moving between two events along a timelike path P is given by the line integral
For any spacetime, there is an incremental invariant interval ds between events with an incremental coordinate separation dxμ of
This is referred to as the line element of the spacetime. s may be spacelike, lightlike, or timelike. Spacelike paths cannot be physically traveled (as they require moving faster than light). Lightlike paths can only be followed by light beams, for which there is no passage of proper time. Only timelike paths can be traveled by massive objects, in which case the invariant interval becomes the proper time . So for our purposes .
Taking the square root of each side of the line element gives the above definition of . After that, take the path integral of each side to get as described by the first equation.
In special relativity, the proper time equation becomes
For a twin "paradox" scenario, let there be an observer A who moves between the coordinates (0,0,0,0) and (10 years, 0, 0, 0) inertially. This means that A stays at for 10 years of coordinate time. The proper time for A is then
So we find that being "at rest" in a special relativity coordinate system means that proper time and coordinate time are the same.
Let there now be another observer B who travels in the x direction from (0,0,0,0) for 5 years of coordinate time at 0.866c to (5 years, 4.33 light-years, 0, 0). Once there, B accelerates, and travels in the other spatial direction for 5 years to (10 years, 0, 0, 0). For each leg of the trip, the proper time is
So the total proper time for observer B to go from (0,0,0,0) to (5 years, 4.33 light-years, 0, 0) to (10 years, 0, 0, 0) is 5 years. Thus it is shown that the proper time equation incorporates the time dilation effect. In fact, for an object in a SR spacetime traveling with a velocity of v for a time , the proper time experienced is
which is the SR time dilation formula.
An observer rotating around another inertial observer is in an accelerated frame of reference. For such an observer, the incremental () form of the proper time equation is needed, along with a parameterized description of the path being taken, as shown below.
Let there be an observer C on a disk rotating in the xy plane at a coordinate angular rate of and who is at a distance of r from the center of the disk with the center of the disk at x=y=z=0. The path of observer C is given by , where is the current coordinate time. When r and are constant, and . The incremental proper time formula then becomes
So for an observer rotating at a constant distance of r from a given point in spacetime at a constant angular rate of ω between coordinate times and , the proper time experienced will be
As v=rω for a rotating observer, this result is as expected given the time dilation formula above, and shows the general application of the integral form of the proper time formula.
The difference between SR and general relativity (GR) is that in GR you can use any metric which is a solution of the Einstein field equations, not just the Minkowski metric. Because inertial motion in curved spacetimes lacks the simple expression it has in SR, the path integral form of the proper time equation must always be used.
An appropriate coordinate conversion done against the Minkowski metric creates coordinates where an object on a rotating disk stays in the same spatial coordinate position. The new coordinates are
The t and z coordinates remain unchanged. In this new coordinate system, the incremental proper time equation is
With r, θ, and z being constant over time, this simplifies to
which is the same as in Example 2.
Now let there be an object off of the rotating disk and at inertial rest with respect to the center of the disk and at a distance of R from it. This object has a coordinate motion described by dθ = -ω dt, which describes the inertially at-rest object of counter-rotating in the view of the rotating observer. Now the proper time equation becomes
So for the inertial at-rest observer, coordinate time and proper time are once again found to pass at the same rate, as expected and required for the internal self-consistency of relativity theory.
The Schwarzschild solution has an incremental proper time equation of
To demonstrate the use of the proper time relationship, several sub-examples involving the Earth will be used here. The use of the Schwarzschild solution for the Earth is not entirely correct for the following reasons:
For the Earth, M = 5.9742 × 1024 kg, meaning that m = 4.4354 × 10 −3 m. When standing on the north pole, we can assume (meaning that we are neither moving up or down or along the surface of the Earth). In this case, the Schwarzschild solution proper time equation becomes . Then using the polar radius of the Earth as the radial coordinate (or meters), we find that
At the equator, the radius of the Earth is r = 6,378,137 meters. In addition, the rotation of the Earth needs to be taken into account. This imparts on an observer an angular velocity of of 2π divided by the sidereal period of the Earth's rotation, 86162.4 seconds. So . The proper time equation then produces
This should have been the same as the previous result, but as noted above the Earth is not spherical as assumed by the Schwarzschild solution. Even so this demonstrates how the proper time equation is used.