Long Post Ahead!
The mathematics is actually really simple. The most complex thing to do is handle some square roots and second exponents but otherwise it's just dividing, multiplying, adding and substracting... you don't even need to use calculus (or differential mathematics, whatever you want to call it), basic algebra will do.
Anyway, as it was asked, the easiest way to derive equations for time dilation and Fitzgerald-Lorenz-contraction is as follows. I think showing how these two are derived is actually more informative in this context, since Lorenz-transformations actually are only useful if you want to know what co-ordinates a certain known point in space and time has in another, moving co-ordinate system. Anyway... Gunnery control, charge photon beam cannons, commence plasma core insertion! Let us begin.
We have a ship with a mirror fixed to it's side some distance from the ship itself, so that it will reflect photons emitted from the ship back towards the ship. The ship travels at, let's say, velocity of
v (arbitrary value, assign what you want to it) in relation to an observer. As it passes an arbitrary point (we'll call it A), the ship emits a photon perpendicularly to it's direction of velocity, ie. directly towards the mirror.
In the ship's reference frame, the photon simply travels to the mirror, reflects back and is observed back in the ship.
However, in the reference frame of an observer, the photon will move substantially more - this is a grossly exaggerated drawing and in fact the ship here would exceed the speed of light, but this will do for now. Observe:
On the upper level, there's how the situation appears to the ship itself, and on the lower level it shows how it looks like to the observer.
Note how there's two right-angled triangles there in the image? Let's concentrate on the first one of them - the one which has it's sides marked by points A, B and the point where the photon hits the mirror.
Now, the distance between B and the hit point is the distance the photon has travelled in ship's co-ordinates. Let's mark this length with
d'.
On the other hand, in the observer's reference frame the photon has moved from point A to the mirror hitpoint. Since this is also the distance that the photon has traveled (in observer's co-ordinates), we'll mark this length as
d without the '-mark. For convenience, we'll mark everything that happens in ship's reference frame with that mark and everything that happens in observer's frame without it.
Because we can assume that speed of light must be same for both observers - those inside the ship and the one outside, static in this case - we can form two simple equations. Since velocity is the distance divided by the time it took to travel that distance,
c = d / t and
c = d'/t'and subsequently,
d = c t
d' = c t'.
In these equations,
t is again the time that passes for the observer, and
t' is the time that passes in the ship. Since
c must be constant for both, these must be different. Now we just need to determine the difference between
d and
d', and we can determine the difference between
t and
t'...
At this point, we get back to the right-angled triangle. We now know the lengths of two edges - one is
d and the other is
d'. As it stands, we also do know the length of the third side, since we know the ship's velocity v. This third side of the triangle we will mark as dx, since that's convenient and we can say that the ship is moving along X axis of the observer's co-ordinates.
Now then... in observer's co-ordinates, the distance the ship moves between points A and B is obviously
dx = v tBut (this is important!) since the time in ship is different, they will measure the distance between A and B as
dx' = v t'. We'll get back to this later, but for now we need to first define the actual difference between
t and
t'.
This goes as follows. We now have all three sides of the triangle in our knowledge, in observer's reference frame.
side 1 =
d = c tside 2 =
d' = c t'side 3 =
dx = v tand from the Pythagoran theorem we can deduce that
d = Sqrt ( d'² + dx² )which we can now solve by inserting the above lengths into this equation:
(c t) = Sqrt ( [c t']² + [v t]² ) || (...)²
(c t)² = (c t')² + (v t)²c² t² = c² t'² + v² t²c² t'² = c² t² - v² t²c² t'² = (c² - v²) t²t'² = (c² - v²) / c² * t² Sqrt (...)
t' = Sqrt(c² - v²) / c * tt' = Sqrt( 1 - v²/ c² ) * t...and there's the equation for time dilation.
Now, back to the distance part - this defines the Fitzgerald-Lorenz-contraction of the axis aligned to the direction of velocity. We know that in the observer's reference frame, the ship moves the distance
dx = v t.
But since for the ship the time passed is different, the same distance will be measured differently by the ship crew. Specifically,
dx' = v t'.
Now, to define the difference between perceived distances, we simply insert the equation of time dilation to the latter one. Like this:
dx' = v Sqrt( 1 - v²/ c² ) * tWe can write this as
dx' = Sqrt( 1 - v²/ c² ) * v tand since
dx = v t, we can insert it into the equation:
dx' = Sqrt( 1 - v²/ c² ) * dxAnd there we got the equation for Fizgerald-Lorenz contraction.
In fact, the term
Sqrt( 1 - v²/ c² ) is often marked with the greek letter gamma for convenience, since it pops up all the time in calculatiosn that take relativity into account properly. It allows such easy markings as
t' = γ t and
x' = γ x.
Lorenz transformations are actually a way to bind co-ordinate systems with a relative velocity to each other, so that you can insert an XYZT co-ordinate values into the transformation equations and out comes X'Y'Z'T' values for another co-ordinate system, which is moving in relation to the other. They are a bit different from these ones, but basically once you get the idea clear for time dilation and Fitzgerald-Lorenz-contraction, you should have no problem with them. Basically, you first have two co-ordinates with their origos aligned, and the other origo is moving along the other one's X axis at velocity v. With Lorenz co-ordinate transformations you can simply transform one system's co-ordinates into another systems co-ordinates, very much similar to Galilei-transformations allow you to do with objects moving at subrelativistic speeds.
