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Proper time

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Proper time

In relativity, proper time along a timelike (or lightlike) world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and a Lorentz scalar.[1] The proper time interval between two events on a world line is the change in proper time. This is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line. The proper time between two events depends not only on the events but also the world line connecting them, and hence on the motion of the clock between the events. It is expressed as an integral over the world line. An accelerated clock will measure a smaller elapsed time between two events than that measured by a non-accelerated (inertial) clock between the same two events. The twin paradox is an example of this effect.






  • Mathematical formalism 1
    • In special relativity 1.1
    • In general relativity 1.2
  • Examples in special relativity 2
    • Example 1: The twin "paradox" 2.1
    • Example 2: The rotating disk 2.2
  • Examples in general relativity 3
    • Example 3: The rotating disk (again) 3.1
    • Example 4: The Schwarzschild solution — time on the Earth 3.2
  • See also 4
  • Footnotes 5
  • References 6

Mathematical formalism



In special relativity

Let the Minkowski metric be defined by

\eta_{\mu\nu} = \left ( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{matrix} \right ),

and define

(x^0, x^1, x^2, x^3) = (ct, x, y, z)

for arbitrary Lorentz frames.

Consider an infinitesimal interval

ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2 = \eta_{\mu\nu}dx^\mu dx^\nu,







ds^2 = c^2d\tau^2 - dx_\tau^2 - dy_\tau^2 - dz_\tau^2 = \eta_{\mu\nu}dx_\tau^\mu dx_\tau^\nu = c^2d\tau^2,

since in the instantaneous rest frame, the particle is at rest. Since the interval is assumed timelike (lightlike), one may take the square root of the above expression;[3]

ds = cd\tau,


d\tau = \frac{ds}{c}.


\Delta\tau = \int_P d\tau = \int \frac{ds}{c}.          (2)

Here P is the worldline from some initial event to some final event with the ordering of the events fixed by the requirement that the final event occurs later according to the clock (for a lightlike world line, if allowed, this differs, the result is zero according to definition (2) since in this case ds = 0[4]) than the initial event.

Using (1) and again the invariance of the interval, one may write[5]

\begin{align}\Delta\tau &= \int_P \frac{1}{c} \sqrt{\eta_{\mu\nu}dx^\mu dx^\nu}\\ &= \int_P \sqrt {dt^2 - {dx^2 \over c^2} - {dy^2 \over c^2} - {dz^2 \over c^2}}\\ &= \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\\ &= \int \sqrt {1 - \frac{v(t)^2}{c^2}} dt = \int \frac{dt}{\gamma(t)},\end{align}          (3)

where v(t) is the coordinate speed at coordinate time t, and x(t), y(t), and z(t) are space coordinates. It should be noted that the first expression is manifestly Lorentz invariant. They are all Lorentz invariant, since proper time and proper time intervals are coordinate-independent by definition.


\Delta\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.

If the motion is of the particle is constant, the expression simplifies to

\Delta \tau = \sqrt{\left(\Delta t\right)^2 - \frac{\left(\Delta x\right)^2}{c^2} - \frac{\left(\Delta y\right)^2}{c^2} - \frac{\left(\Delta z\right)^2}{c^2}},


In general relativity


\Delta\tau = \int_P \, d\tau = \int_P \frac{1}{c}\sqrt{g_{\mu\nu} \; dx^\mu \; dx^\nu}.








\Delta\tau = \int_P d\tau = \int_P \frac{1}{c}\sqrt{g_{00}} dx^0.


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 interval for A is then

\Delta \tau = \sqrt{(10\text{ years})^2} = 10\text{ 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 interval 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 interval 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 (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 \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, \;\, r\cos(\omega T),\;\, r\sin(\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

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 T_1 and T_2, the proper time experienced will be

\int_{T_1}^{T_2} d\tau = (T_2 - T_1) \sqrt{ 1 - \left ( \frac{r\omega}{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


Example 3: The rotating disk (again)




\theta = \arctan\left(\frac{y}{x}\right) - \omega t.


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}}.


d\tau = dt \sqrt{ 1 - \left (\frac{r \omega}{c} \right )^2 },

which is the same as in Example 2.


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.


Example 4: The Schwarzschild solution — time on the Earth

The Schwarzschild solution has an incremental proper time equation of

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\phi^2 - \frac{r^2}{c^2} \sin^2(\phi ) \, d\theta^2 },


M is the mass of the Earth,
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 and tidal deformation, 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 often 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 = 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 d\tau = 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

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

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


  1. ^ Zwiebach 2004, p. 25
  2. ^ Minkowski 1908, pp. 53–111
  3. ^ Zwiebach 2004, p. 25
  4. ^ Some authors include lightlike intervals in the definition, e. g. Lawden 2012, p. 150: Extract of page 150.

    Others leave it undefined, e.g. Kopeikin, Efroimsky & Kaplan 2011, p. 275: Extract of page 275.
    Yet others aren't clear on whether lightlike intervals are included: Landau & Lifshitz 1975, pp. 7–8
  5. ^ Foster & Nightingale 1978, p. 56
  6. ^ Foster & Nightingale 1978, p. 57
  7. ^ Landau & Lifshitz 1975, p. 251
  8. ^ Cook 2004, pp. 214–219


  • Cook, R. J. (2004). "Physical time and physical space in general relativity". Am. J. Phys. 72 (2): 214–219.  
