Editing Proper time (section)

=== 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 (<math>d\tau</math>) 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 <math>\omega</math> and who is at a distance of ''r'' from the center of the disk with the center of the disk at {{math|1=''x'' = ''y'' = ''z'' = 0}}. The path of observer ''C'' is given by <math>(T, \, r\cos(\omega T), \, r\sin(\omega T), \, 0)</math>, where <math>T </math> is the current coordinate time. When ''r'' and <math>\omega</math> are constant, <math>dx = -r \omega \sin(\omega T) \, dT</math> and <math>dy = r \omega \cos(\omega T) \, dT</math>. The incremental proper time formula then becomes
<math display="block">d\tau
= \sqrt{dT^2 - \left(\frac{r \omega}{c}\right)^2 \sin^2(\omega T)\; dT^2 - \left(\frac{r \omega}{c}\right)^2 \cos^2(\omega T) \; dT^2}
= dT\sqrt{1 - \left ( \frac{r\omega}{c} \right )^2}.</math>

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 <math>T_1</math> and <math>T_2</math>, the proper time experienced will be
<math display="block">\int_{T_1}^{T_2} d\tau
= (T_2 - T_1) \sqrt{ 1 - \left ( \frac{r\omega}{c} \right )^2}
= \Delta T \sqrt{1 - v^2/c^2},</math>
as <math> v = r \omega </math> for a rotating observer. This result is the same as for the linear motion example, and shows the general application of the integral form of the proper time formula.