Proper time

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 tau to distinguish it from coordinate time represented by t or T.

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.

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.

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

In special relativity, proper time can be defined as

= 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 lambda, this can be written as

= int sqrt {left (frac{dt}{dlambda}right)^2 - frac{1}{c^2} left (left (frac{dx}{dlambda}right)^2 + left (frac{dy}{dlambda}right)^2 + left (frac{dz}{dlambda}right)^2 right) } ,dlambda.

In differential form it can be written as the path integral

tau = int_P sqrt {dt^2 - {dx^2 over c^2} - {dy^2 over c^2} - {dz^2 over 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

Delta tau = sqrt{Delta t^2 - {Delta x^2 over c^2} - {Delta y^2 over c^2} - {Delta z^2 over c^2}},

where Δ 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 x^mu and equipped with a corresponding metric tensor g_{munu}, the proper time tau experienced in moving between two events along a timelike path P is given by the line integral

tau = int_P , dtau


dtau = sqrt{dx_mu ; dx^mu} = sqrt{g_{munu} ; dx^mu ; dx^nu}.


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

ds^2 = g_{munu} , 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 tau . So for our purposes tau stackrel{mathrm{def}}{=} s .

Taking the square root of each side of the line element gives the above definition of dtau . After that, take the path integral of each side to get tau as described by the first equation.

Derivation for special relativity

In special relativity spacetime is mapped with a four-vector coordinate system 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:

g_{munu} = 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 dtau will always be positive definite for timelike paths.)

In special relativity, the proper time equation becomes

tau = int_P sqrt {dt^2 - {dx^2 over c^2} - {dy^2 over c^2} - {dz^2 over 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 x=y=z=0 for 10 years of coordinate time. The proper time for A is then

Delta tau = sqrt{(10text{ years})^2} = 10text{ 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

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 Delta T, the proper time experienced is

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 (dtau ) 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 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 (T, ;, rcos(omega T),;, rsin(omega T), ;, 0), where T is the current coordinate time. When r and omega are constant, dx = -r omega sin(omega T) ; dT and dy = r omega cos(omega T) ; dT. The incremental proper time formula then becomes

dtau = sqrt{dT^2 - (r omega /c)^2 sin^2(omega T); dT^2 - (r omega /c)^2 cos^2(omega T) ; dT^2} = dTsqrt{1 - left (frac{romega}{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 T_1 and T_2, the proper time experienced will be

int_{T_1}^{T_2} dtau = (T_2 - T_1) sqrt{ 1 - left (frac{romega}{c} right )^2}.

As v= 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

r=sqrt{x^2 + y^2}


theta = arctan(x/y) - omega t.

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

dtau = sqrt{left [1 - left (frac{r omega}{c} right )^2 right] dt^2 - frac{dr^2}{c^2} - frac{r^2, dtheta^2}{c^2} - frac{dz^2}{c^2} - 2 frac{r^2 omega , dt , dtheta}{c^2}}.

With r, θ, and z being constant over time, this simplifies to

dtau = 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

dtau = sqrt{left [1 - left (frac{R omega}{c} right )^2 right] dt^2 - left (frac{Romega}{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

dtau = 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} dtheta^2 - frac{r^2}{c^2} sin^2 theta ; dvarphi^2},


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.
varphi 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, M = 5.9742 × 1024 kg, meaning that m = 4.4354 × 10 −3 m. When standing on the north pole, we can assume dr = dtheta = dvarphi = 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 dtau = dt ,sqrt{1 - 2m/r}. Then using the polar radius of the Earth as the radial coordinate (or r = 6,356,752 meters), we find that

dtau = 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 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 dvarphi/dt of 2π divided by the sidereal period of the Earth's rotation, 86162.4 seconds. So dvarphi = 7.2923 times 10^{-5}, dt. The proper time equation then produces

dtau = 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.

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