TIME VARIES WITH RELATIVE SPEED
Gary D. Evans
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Experiment: Let a person sitting within a traincar measure the time it takes for a particle of light to travel between a held flashlight and a detector on the train's ceiling.
Let T_{1} = the time measured by the train rider performing the experiment.
Let T_{0} = the time measured by the observer of the same experiment standing outside the passing train.
Let v = the speed of train as measured by the outside observer as it passes by.
Let c = the speed of light

The lengths of the sides represented here are simply the (speed * time) = distances involved.
Pythagorean Theorm: for a right triangle, the length of the sides adjacent to the rightangle squared
and then added together will equal the length of the hypotenuse squared.
In this example: (c*T_{1})^{2} + (v*T_{0})^{2} = (c*T_{0})^{2}
Square the values: c^{2}T_{1}^{2} + v^{2}T_{0}^{2} = c^{2}T_{0}^{2}
Rearrange: c^{2}T_{1}^{2} = c^{2}T_{0}^{2}  v^{2}T_{0}^{2}
Divide by c^{2}: T_{1}^{2} = (c^{2}T_{0}^{2}  v^{2}T_{0}^{2}) / c^{2}
Simplify: T_{1}^{2} = T_{0}^{2} (1  v^{2}/c^{2})
Take the squareroots: T_{1} = T_{0}(1  v^{2}/c^{2})^{1/2}

CONCLUSION: As indicated by the final equation, as the speed of the train (v) approaches the speed of light (c), the value of the right side of the equation goes toward zero, meaning: as the speed of the train increases, the observer (wearing a T_{0} watch) sees the trainrider's T_{1} watch ticking ever more slowly. That is to say, the trainrider's rate of the passage of time is slowed relative to the outside observer's rate of time passage.