Editing Proper time (section)

== Examples in general relativity ==

The difference between SR and general relativity (GR) is that in GR one 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 line integral form of the proper time equation must always be used.

=== Example 3: The rotating disk (again) ===

An appropriate [[Polar coordinate system#Converting between polar and Cartesian coordinates|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
<math display="block">r= \sqrt{x^2 + y^2}</math>
and
<math display="block">\theta = \arctan\left(\frac{y}{x}\right) - \omega t.</math>

The ''t'' and ''z'' coordinates remain unchanged. In this new coordinate system, the incremental proper time equation is
<math display="block">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}}.</math>

With ''r'', ''θ'', and ''z'' being constant over time, this simplifies to
<math display="block">d\tau = dt \sqrt{ 1 - \left (\frac{r \omega}{c} \right )^2 },</math>
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 {{math|1=''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
<math display="block">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. </math>

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.<ref>{{harvnb|Cook|2004|pp=214–219}}</ref>

=== Example 4: The Schwarzschild solution – time on the Earth ===

The [[Schwarzschild solution]] has an incremental proper time equation of
<math display="block"> 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
}, </math>
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),
*''ɸ'' is a co-latitudinal coordinate, the angular separation from the [[north pole]] in [[radian]]s.
*''θ'' is a longitudinal coordinate, analogous to the longitude on the Earth's surface but independent of the Earth's [[rotation]]. This is also given in radians.
*''m'' is the [[geometrized]] mass of the Earth, ''m'' = ''GM''/''c''<sup>2</sup>,
**''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.

For the [[Earth]], {{math|1=''M'' = {{val|5.9742e24|u=kg}}}}, meaning that {{math|1=''m'' = {{val|4.4354e-3|u=m}}}}. When standing on the north pole, we can assume <math>dr = d\theta = d\phi = 0 </math> (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 <math display="inline">d\tau = dt \,\sqrt{1 - 2m/r}</math>. Then using the polar radius of the Earth as the radial coordinate (or <math>r = \text{6,356,752 metres}</math>), we find that
<math display="block">d\tau = \sqrt{\left ( 1 - 1.3908 \times 10^{-9} \right ) \;dt^2} = \left (1 - 6.9540 \times 10^{-10} \right ) \,dt.</math>

At the [[equator]], the radius of the Earth is {{math|1=''r'' = {{val|6378137|u=metres}}}}. In addition, the rotation of the Earth needs to be taken into account. This imparts on an observer an angular velocity of <math> d\theta / dt</math> of 2''π'' divided by the [[sidereal time|sidereal period]] of the Earth's rotation, 86162.4 seconds. So <math>d\theta = 7.2923 \times 10^{-5} \, dt</math>. The proper time equation then produces
<math display="block">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.</math>

From a non-relativistic point of view this should have been the same as the previous result. This example demonstrates how the proper time equation is used, even though the Earth rotates and hence is not spherically symmetric as assumed by the Schwarzschild solution. To describe the effects of rotation more accurately the [[Kerr metric]] may be used.