Editing Proper time (section)

=== Example 1: The twin "paradox" ===

For a [[twin paradox]] scenario, let there be an observer ''A'' who moves between the ''A''-coordinates (0,0,0,0) and (10 years, 0, 0, 0) inertially. This means that ''A'' stays at <math>x = y = z = 0</math> for 10 years of ''A''-coordinate time. The proper time interval for ''A'' between the two events is then
<math display="block">\Delta \tau_A = \sqrt{(10\text{ years})^2} = 10\text{ years}.</math>

So 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 ''A''-coordinate time at 0.866''c'' to (5 years, 4.33 light-years, 0, 0). Once there, ''B'' accelerates, and travels in the other spatial direction for another 5 years of ''A''-coordinate time to (10 years, 0, 0, 0). For each leg of the trip, the proper time interval can be calculated using ''A''-coordinates, and is given by
<math display="block">\Delta \tau_{leg} = \sqrt{(\text{5 years})^2 - (\text{4.33 years})^2}
= \sqrt{6.25\;\mathrm{years}^2}
= \text{2.5 years}.</math>

So the total proper time for observer ''B'' to go from (0,0,0,0) to (5 years, 4.33 light-years, 0, 0) and then to (10 years, 0, 0, 0) is
<math display="block">\Delta \tau_B = 2 \Delta \tau_{leg} = \text{5 years}.</math>

Thus it is shown that the proper time equation incorporates the [[time dilation]] effect. In fact, for an object in a SR (special relativity) spacetime traveling with velocity <math>v</math> for a time <math>\Delta T</math>, the proper time interval experienced is
<math display="block">\Delta \tau
= \sqrt{\Delta T^2 - \left(\frac{v_x \Delta T}{c}\right)^2 - \left(\frac{v_y \Delta T}{c}\right)^2 - \left(\frac{v_z \Delta T}{c}\right)^2 }
= \Delta T \sqrt{1 - \frac{v^2}{c^2}}, </math>
which is the SR time dilation formula.