The first part is easy.
The Lorentz Transformations part requires some graphics which I'll have to make to do it properly. I can give you the Monarch Notes text version which may suffice.
First one must understand just what Einstein said in Special Relativity.
He started with two postulates (i.e., assumptions). I'm paraphrasing here.
- There is no absolute reference frame
- Within any given media and inertial reference frame, the speed of light is the same.
So what do those postulates mean?
The first is simple.
There's no universal piece of graph paper on which we all exist.
That is, there's no 0,0 point that is universal.
Consider the following.
You're sitting in a train car looking out the window.
Next to you is another train car.
You see it move towards your rear.
Is your train car moving forward or is the other car moving backwards?
As long as you stay in your train car, you can't tell.
If you were in train station, you could tell. The train station would be the absolute reference frame. Einstein assumed it didn't exist, so all you can say is that something is moving
relative to you.
The second has a few caveats that often are not mentioned.
The first is the same media. i.e., air, water, space, etc.
The fact that light travels at a different speed in water/glass and air is why refraction occurs and how a microscope and telescope work.
The phrase "inertial reference frame" simply means that there is no acceleration going on, things are moving at a constant speed.
So if we assume these two postulates are correct, let's consider the following experiment.
You have a train car in which Observer A stand. In the car is a light source adjacent to A and 10 meters away is a target with a light detector.
Adjacent to the train tracks stands Observer B.
The train moves at a constant speed towards B, and when A is right next to B, the light source is turned on.
Both Observers measure the time it takes and the distance the light travels to reach the detector.
What do they observe?
Well A see's the light travel 10 meters.
How far does B observe the light to travel?
10 meters, plus the distance the train car traveled before the light hit the detector.
OK, what does this mean?
Well recall the second postulate. Both have to agree on the velocity of light.
Velocity, by definition, is distance/time.
So if the velocity is the same, then A and B must disagree on how much time it took for the light to travel, the distance it traveled, or "when" it reached the detector.
So one, or more, of the following must be true.
A) A and B didn't agree "when" the light hit the detector. In other words, the events were not "simultaneous". The light arrived at the detector, from the observer's perspective, more than once. First for A and later for B.
B)
From B's perspective, the length of the train car was reduced by the exact same distance the train traveled during the experiment. A observed no change in the length of the train car.
C)
From B's perspective, the flow of time in the train was reduced enough to make the velocity match the longer distance. A observed no change in the flow of time in the car.
If one accepts the postulates, then one of these, or some combination of those three options, must occur. There is no other logical possibility.
Agreed?
If not, let me know your questions/issues.
It is often said that Einstein "proved" the speed of light is a constant. That's not technically true. He "assumed it" and developed Special Relativity based on that assumption.
Now we have good experimental evidence to support the predictions of time dilation made by Special Relativity, so the theory is accepted as being "true".
However, that isn't; the same as "proof".