Time Dilation

Lorentz Time Dilation

This one was requested by a reader, and I think that it's a neat proof of a fundamental property of nature anyway. The only two principles we need to begin with are the principle of relativity - that you can't determine who is at rest if two people are moving at a constant velocity with respect to one another - and the principle of the equivalence of the speed of light - that light travels at the same speed for everybody (no matter what velocity they are moving at)

We can prove this using a thought experiment and a little bit of algebra. Our thought experiment will require a light clock which works by having two mirrors a set distance apart and bouncing light between them. 

The light clock on the left is stationary to the observer, and the mirrors are a distance L apart. The light clock on the right is in constant motion to the observer with a velocity of v. The distance D is the distance the light travels from the bottom mirror to the top mirror. vt'/2 is the horizontal distance between the points where the light bounces off one of the mirrors in the light clock. Time in the left light clock is represented by t; time in the other light clock is represented by t'.

From the experiment, we can say:

t=2L/c

D=ct'

Using Pythagoras' Theorem and the diagram, we can say:

As we now have two values of D, we can say:

Sqaure each side and re-arrange:

Factorise:

Solve for the square of t':

Take the square root:

Substitute in t from the first formula from the thought experiment:

Thus we can say that at high velocities, time is dilated by a function of the velocity (usually represented by a lower-case gamma: γ)

QED

 

At low, everyday speeds the effect is very small. If we substitue v=0.0001c (0.01% of light speed), then t' will equal 1.0000005000000375 times t, which is only a small dilation of time.

However, at high speeds reached by high-energy particles, the effect can be very noticeable. If we substitute v=0.99c (99% of light speed), the t' will equal 7.0888120501 times t. Which means that if someone were moving at 99% the speed of light with repect to an observer, the observer will pass through time 7 times faster than the person in motion.

It is because we don't experience these kinds of velocities in our everyday lives that it took us so long to discover the effects of time-dilation.