Time dialation

    Einstein based the special theory of relativity on two postulates.
The first being that the laws of physics are the same in any inertial frame.
The second being that the speed of light is measured to be the same in any inertial frame. Take these two light clocks for example...

The guy standing next to the stationary clock measures the speed of light to be c.

A guy riding on the moving clock measures the speed of light to be c.

On the surface, these simple postulates seem both logical and convenient, but they lead to a plethora of paradoxes. Take for example a person in a spaceship travelling at half the speed of light, if they turn on the headlights for their spaceship, the light travels away from them at the speed of light. 

But how does this appear to a stationary observer watching the same ship?

Classical Newtonian physics predicts that if light has a velocity of c compared to the spaceship, and the spaceship has velocity .5c compared to the observer, then the light must be travelling at  1.5c compared to the stationary observer.This however violates the second postulate, that the speed of light is measured to be the same by the people in the spaceship AND the stationary observer. How is this possible?

In a deft side step, Albert invents this half baked theory of time dilation.

What is time dilation you say?

In the simple form it means that time is travelling slower for the people on the spaceship than it is for the stationary observer. (A more through approach reveals that this too produces a paradox, but lets just wave our hands and say 'OKAY Albert, what ever you say must be true.')

Well, if time is travelling at different rates for different people, how do we compare them to one another? 

BAM! We have this convenient formula.

The t' represents the magnitude of a unit of time, lets call it a second, compared to the stationary observer's time, t. The denominator has terms v, for the velocity of the travelling thing, and c for the speed of light. The denominator can also be represented by multiplying t by a term we refer to as gamma. Interesting to note is that in the limit where v approaches c, t' approaches infinity. As such, it appears all things happen simultaneously to things travelling at the speed of light.

One might ask, how do we use this? Well, the nice people over at activphysics

provided us with an example using the light clocks from above. If told that the round trip time for the moving clock is 7.45 us, measured from the stationary frame, and that the stationary clock takes 6.67 us, we can determine the value that the coefficient of t must take in the above equation. The value that expression takes is also known as the Lorentz Factor. 

A simple division results that the Lorentz factor for the moving clock is 1.117, meaning that a single second in the stationary frame lasts 1.117 seconds in the moving frame. Solving this expression for velocity results in a v of, .445c.