Simple derivation of Lorentz contraction (time dilation formula) in special relativity

The general time dilation formula in special relativity can be proven with high school algebra. Here’s how:

Consider an object moving at a speed v relative to an observer. Define a plane passing through the observer parallel to the object’s motion, and assume there is a mirror along that plane. Then we can define two time intervals t and s as follows:

t: the time in the object’s coordinates for light to a make a one way trip to the mirror

s: the same time, but in the observer’s coordinates.

In the time s, in the observer coordinates, the object moves a distance vs. In the object’s own coordinates, the distance to the mirror is ct.

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by right angle geometry

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which is the Lorentz / Einstein formula. In other words, the object’s time is slowed by a factor of over time in the observer coordinate system.

To get a sense of the magnitude of , suppose you are going at half the speed of light.

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Which means time slows down by 15%. If you are going at 99% of the speed of light, then

So, going at 0.99 times the speed of light will slow time down by 7 times versus the observers coordinates.

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