Intuitive Special Relativity - The Lorentz Transformation
I am writing a series of posts and trying to explain Special Relativity in an intuitive way.
In this post I am going to deviate from that and go straight to the math. So if you are math-phobic this is not the post for you :(
You may want to read my first four posts leading up to this one:
Intuitive Special Relativity (A Tough Concept Simply Explained)
Intuitive Special Relativity - What Is Time?
Intuitive Special Relativity - Time Dilation
Intuitive Special Relativity - Time Is Relative
The Lorentz Transformation
This is the equation we are trying to derive.
This is also called the Lorentz factor or gamma. In this equation v is the velocity of an object as observed by a stationary observer and c is the velocity of light.
Looking at the equation one can see:
- Gamma increases as the velocity, v, increases.
- It does not matter if v is positive (moving to the right) or negative (moving to the left) you get the same value.
- When v is equal to the speed of light the value of gamma goes to infinity (in other words any particle with a non-zero rest mass cannot actually get to this velocity).
- When v = 0 then gamma simply equals one (i.e no effect).
Figure 1. A Simple Time Clock
(Image Credit: Me. with Earth and Rocket icons taken from Pixabay, CC0 creative commons license)
(I release this image to the Public Domain with no restrictions)
So how do do we arrive at this equation?
Looking at Figure 1 above we see that for the moving observer on the right hand side the light in the simple clock has to move along a pathway that it longer than the pathway for a stationary observer on the left side of the figure.
Lets calculate the length of that pathway.
Using the Pythagorean Theorem we know the hypotenuse of the triangle described by the light path is,
t0 is time as measured by the stationary observer,
t is time of the moving observer as measured by the stationary observer,
v is the velocity of the moving observer as measured by the stationary observer,
c is the velocity of light (the same for all observers).
expanding the above equation we get,
isolating the t0 we get,
factoring out the variable t we get,
dividing both sides by c2 we get,
or,
which is,
this can be also be expressed as,
taking the square root of both sides of the equation we get,
divide both sides by the square root factor we get close to the final most workable form,
and one can see that the time of the moving observer as measured by the stationary observer differs by a factor of,
which is the Lorentz factor as discussed at the start of the post.
Closing Words
It is somewhat impressive that a person can work out the time dilation factor, the length contraction factor and the relativistic mass factor all from simple triangle geometry and the assumption that the speed of light is constant for all observers.
Equations in this post were generated using Online LaTex Equation Editor
Another good read, thanks mate