Proper time

In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. It depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a shorter proper time between two events than a non-accelerated (inertial) clock between the same events. The twins paradox is an example of this.

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.

In terms of four-dimensional spacetime, proper time is analogous to arc length in three-dimensional (Euclidean) space.

By convention, proper time is usually represented by the Greek letter Failed to parse (Missing texvc executable; please see math/README to configure.): \tau

to distinguish it from coordinate time represented by Failed to parse (Missing texvc executable; please see math/README to configure.): t
or Failed to parse (Missing texvc executable; please see math/README to configure.): T

.

Mathematical formalism

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.

In special relativity

In special relativity, proper time can be defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau = \int \frac{dt}{\gamma} = \int \sqrt {1 - \frac{v(t)^2}{c^2}} dt = \int \sqrt {1 - \frac{1}{c^2} \left ( \left (\frac{dx}{dt}\right)^2 + \left (\frac{dy}{dt}\right)^2 + \left ( \frac{dz}{dt}\right)^2 \right) } dt ,

where v(t) is the coordinate speed at coordinate time t, and x, y and z are orthogonal spatial coordinates.

If t, x, y and z are all parameterised by a parameter Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda , this can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau = \int \sqrt {\left (\frac{dt}{d\lambda}\right)^2 - \frac{1}{c^2} \left ( \left (\frac{dx}{d\lambda}\right)^2 + \left (\frac{dy}{d\lambda}\right)^2 + \left ( \frac{dz}{d\lambda}\right)^2 \right) } d\lambda .

In differential form it can be written as the path integral

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau = \int_P \sqrt {dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2} ,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta \tau = \sqrt{\Delta t^2 - \Delta x^2/c^2 - \Delta y^2/c^2 - \Delta z^2/c^2} ,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta

means "the change in" between two events.

The special relativity equations are special cases of the general case that follows.

In general relativity

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 Failed to parse (Missing texvc executable; please see math/README to configure.): x^\mu

and equipped with a corresponding metric tensor Failed to parse (Missing texvc executable; please see math/README to configure.): g_{\mu\nu}

, the proper time Failed to parse (Missing texvc executable; please see math/README to configure.): \tau\

experienced in moving between two events along a timelike path P is given by the line integral

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau = \int_P \;\! d\tau

where

Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau = \sqrt{dx_\mu \; dx^\mu} = \sqrt{g_{\mu\nu} \; dx^\mu \; dx^\nu}

Derivation

For any spacetime, there is an incremental invariant interval ds between events with an incremental coordinate separation dx^μ of

Failed to parse (Missing texvc executable; please see math/README to configure.): ds^2 = g_{\mu\nu} \; dx^\mu \; dx^\nu .

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \tau\ . So for our purposes Failed to parse (Missing texvc executable; please see math/README to configure.): \tau\ \ \stackrel{\mathrm{def}}{=}\ s .

Taking the square root of each side of the line element gives the above definition of Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau\ . After that, take the path integral of each side to get Failed to parse (Missing texvc executable; please see math/README to configure.): \tau\

as described by the first equation.

Derivation for special relativity

In special relativity spacetime is mapped with a four-vector coordinate system Failed to parse (Missing texvc executable; please see math/README to configure.): x^\mu = (t,x,y,z)\,

where 

t is a temporal coordinate and

x, y, and z are orthogonal spatial coordinates.

This spacetime and mapping are described with the Minkowski metric:

Failed to parse (Missing texvc executable; please see math/README to configure.): g_{\mu\nu} = \left ( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & -\frac{1}{c^2} & 0 & 0 \\ 0 & 0 & -\frac{1}{c^2} & 0 \\ 0 & 0 & 0 & -\frac{1}{c^2} \end{matrix} \right ) .

(Note: The +--- metric signature is used in this article so that Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau\

will always be positive definite for timelike paths.)

In special relativity, the proper time equation becomes

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau = \int_P \sqrt {dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2} ,

as above.

Examples in special relativity

Example 1: The twin "paradox"

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 Failed to parse (Missing texvc executable; please see math/README to configure.): x=y=z=0

for 10 years of coordinate time.  The proper time for A is then

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta \tau = \sqrt{(10\; \mathrm{years})^2} = 10\; \mathrm{years}

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta \tau = \sqrt{(5\;\mathrm{years})^2 - (4.33\;\mathrm{years})^2} = \sqrt{6.25\;\mathrm{years}^2} = \sqrt{6.25\;} \mathrm{years}= 2.5 \; \mathrm{years}.

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta T , the proper time experienced is

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta \tau = \sqrt{\Delta T^2 - (v_x \Delta T/c)^2 - (v_y \Delta T/c)^2 - (v_z \Delta T/c)^2 } = \Delta T \sqrt{1 - v^2/c^2} ,

which is the SR time dilation formula.

Example 2: The rotating disk

An observer rotating around another, inertial observer is in an accelerated frame of reference. For such an observer, the incremental (Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau\ ) 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \omega

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 Failed to parse (Missing texvc executable; please see math/README to configure.): (T, \;\, r\cos(\omega T),\;\, r\sin(\omega T), \;\, 0)

, where Failed to parse (Missing texvc executable; please see math/README to configure.): T

is the current coordinate time.  When r and Failed to parse (Missing texvc executable; please see math/README to configure.): \omega
are constant, Failed to parse (Missing texvc executable; please see math/README to configure.): dx = -r \omega \sin(\omega T) \; dT
and Failed to parse (Missing texvc executable; please see math/README to configure.): dy = r \omega \cos(\omega T) \; dT

. The incremental proper time formula then becomes

Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau = \sqrt{dT^2 - (r \omega /c)^2 \sin^2(\omega T)\; dT^2 - (r \omega /c)^2 \cos^2(\omega T) \; dT^2} = dT\sqrt{1 - \left ( \frac{r\omega}{c} \right )^2} .

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 Failed to parse (Missing texvc executable; please see math/README to configure.): T_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): T_2

, the proper time experienced will be

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{T_1}^{T_2} d\tau = (T_2 - T_1) \sqrt{ 1 - \left ( \frac{r\omega}{c} \right )^2} .

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.

Examples in general relativity

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.

Example 3: The rotating disk (again)

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 Failed to parse (Missing texvc executable; please see math/README to configure.): r=\sqrt{x^2 + y^2}

and Failed to parse (Missing texvc executable; please see math/README to configure.): \theta = \arctan(x/y) - \omega t

. The t and z coordinates remain unchanged. In this new coordinate system, the incremental proper time equation is

Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau = \sqrt{\left [1 - \left (\frac{r \omega}{c} \right )^2 \right] dt^2 - \frac{dr^2}{c^2} - \frac{r^2\, d\theta^2}{c^2} - \frac{dz^2}{c^2} - 2 \frac{r^2 \omega \, dt \, d\theta}{c^2}}

With r, θ, and z being constant over time, this simplifies to Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau = dt \sqrt{ 1 - \left (\frac{r \omega}{c} \right )^2 } , 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

Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau = \sqrt{\left [1 - \left (\frac{R \omega}{c} \right )^2 \right] dt^2 - \left (\frac{R\omega}{c} \right ) ^2 \,dt^2 + 2 \left ( \frac{R \omega}{c} \right ) ^2 \,dt^2} = dt .

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.

Example 4: The Schwarzschild solution - Time on the Earth

The Schwarzschild solution has an incremental proper time equation of

Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau = \sqrt{\left( 1 - \frac{2m}{r} \right ) dt^2 - \frac{1}{c^2}\left ( 1 - \frac{2m}{r} \right )^{-1} dr^2 - \frac{r^2}{c^2} d\theta^2 - \frac{r^2}{c^2} \sin^2 \theta \; d\phi^2} ,

where

t is time as calibrated with a clock distant from and at inertial rest with respect to the Earth,

r is a radial coordinate (which is effectively the distance from the Earth's center),

θ is the latitudinal coordinate, being the angular separation from the north pole in radians.

Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\

is a longitudinal coordinate, analogous to the latitude on the Earth's surface but independent of the Earth's rotation.  This is also given in radians.

m is the geometrized mass of a central massive object, being m=MG/c²,

M is the mass of the object,

G is the gravitational constant.

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:

• Due to its rotation, the Earth is an oblate spheroid instead of being a true sphere. This results in the gravitational field also being oblate instead of spherical.
• In GR, a rotating object also drags spacetime along with itself. This is described by the Kerr solution. However, the amount of frame dragging that occurs for the Earth is so small that it can be ignored.

For the Earth, Failed to parse (Missing texvc executable; please see math/README to configure.): M=5.9742 \times 10^{24}

kg, meaning that Failed to parse (Missing texvc executable; please see math/README to configure.): m = 4.4354 \times 10^{-3}
m.  When standing on the north pole, we can assume Failed to parse (Missing texvc executable; please see math/README to configure.): dr = d\theta\ = d\phi\ = 0 
(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 Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau = dt \sqrt{1 - 2m/r}

. Then using the polar radius of the Earth as the radial coordinate (or Failed to parse (Missing texvc executable; please see math/README to configure.): r\ = 6,356,752

meters), we find that

Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau = \sqrt{\left ( 1 - 1.3908 \times 10^{-9} \right ) \;dt^2} = \left (1 - 6.9540 \times 10^{-10} \right ) dt .

At the equator, the radius of the Earth is Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \ d\phi/dt
of Failed to parse (Missing texvc executable; please see math/README to configure.): \ 2\pi
divided by the sidereal period of the Earth's rotation, Failed to parse (Missing texvc executable; please see math/README to configure.): \ 86162.4
seconds. So Failed to parse (Missing texvc executable; please see math/README to configure.): d\phi = 7.2923 \times 10^{-5} dt

. The proper time equation then produces

Failed to parse (Missing texvc executable; please see math/README to configure.): d\tau = \sqrt{\left ( 1 - 1.3908 \times 10^{-9} \right ) dt^2 - 2.4069 \times 10^{-12} dt^2} = \left( 1 - 6.9660 \times 10^{-10}\right ) \, dt .

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.

See also

• Special relativity
• General relativity
• Lorentz transformation
• Four-vector
• Minkowski space
• proper length
• proper acceleration

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