Editing Proper time (section)

=== 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>