Lorentz transformation

Of course, four-vectors are always given _relative_ to somebody or something. For example, if I say: "I see a ping-pong ball 3 m in front of, 1.5 m to the right and 1 m up from my nose" (assume that 'in front of', 'to the right' and 'up' are actual directions!), it is meaningless unless you know where my nose is and where I am facing. If I turn my face 20 degrees to the right, I will have to modify those numbers, in a way that I can work out mathematically! This is a type of a Lorentz transformation.

Now, special relativity extends this to include the time as well! In special relativity, when you change the velocity you are moving at, you rotate in space-time! To return to the ping-pong example, if I say: "1 s from now, there will be a ping-pong ball 5 cm in front of my nose", if I start moving back at a velocity of 20 m/s, I will have to change both of those numbers _the instant_ I start moving.

Of course, the effects of special relativity can't in practice be detected when playing ping-pong, but in principle they are there. In astronomical scales, special relativity predicts many curious things, like time dilation and Lorentz contraction, all of which have been verified.

Let's get a bit more mathematical. We look at Lorentz transformations with 2 coordinates, t and x, t being the time. We use units of length and time where c, the speed of light is equal to 1. So when we say x=t, it means x=c*t. Now, as you can see, x=t describes the motion of a ray of light along x-axis. Remember, the speed of light must remain the same regardless of the velocity you are moving at. But at first this seems weird, because if you change your velocity from 0 to v, x=t becomes x=(1-v)t by common sense, so after changing your velocity the ray of light would be moving at velocity 1-v. So if the speed of light relative to you must stay the same regardless of your speed, we must extend common sense a bit.

Now, it turns out that Lorentz transformation is a linear transformation, which means that it can be represented with a matrix.

For example, the following matrix equation represents ordinary rotation through an angle phi in a plane:

(x')  =	 (a  b) (x)           (equation 1)
(y') 	 (-b a) (y)

a = cos(phi)
b = sin(phi)

Anyway, you can prove using simple trigonometry that x'²+y'² = x²+y² under equation 1.

In Lorentz transformations what stays constant is not t²+x² but rather t²-x²! (You should have no trouble proving that if changing the velocity is expressed by a transformation where t²-x² = constant, it means that the speed of light remains the same.) This leads (modulo flipping, again) to the following matrix equation:

(x')  =	 gamma * (1 v) (x)           (equation 2)
(t') 	         (v 1) (t)

gamma = 1/sqrt(1-v²)

If you want to read more, special relativity is explained more pedagogically for example in the book "Spacetime physics" by John Wheeler and Edwin Taylor.

Consider two inertial frames of reference, S and S'. Frame S' is moving at a constant velocity V with respect to S. Define a quantity Γ,

Γ=1/(1-V²/c²)^1/2

If the frame S' is moving at a constant speed V along the x-axis and the two frames coincide at t=0 then the transformation may be written as follows

x=Γ(x'+Vt')
y=y'
z=z'
t=Γ(t'+Vx'/c²)

This transformation agrees with the experimental observation that the speed of light is constant whether measured in the reference frame S or the moving frame S' (and as postulated by the theory of Special Relativity). It can also be shown that Maxwell's equations are covariant under the transformation. Therefore, Electromagnetism is consistent with the requirements of Special Relativity.

The above transformation reduce to the Galilean case when V is much less than c (Γ->1). However, several effects arise that are outside our daily experience when V is a significant fraction of c (Γ>1). See time dilation, length contraction and the twin paradox.

Well here's a small but interesting effect. The Lorentz transformation equations dont go over completely into the Galilean transformation equations as (v/c) tends to zero!
What happens is that gamma tends to 1, and this means that the transformations for all the spatial coordinates tend towards the respective galilean transformations. Look at the equation for time however:
t' = gamma(t-ux/c^2)
Notice that if x is very large then even if u/c is very small you would still get a substantial effect!
One way out is to say that we should take the limit c tending to inf. Thats not very satisfying however, because what we're actually concerned with is the ratio v/c . Conclusion: This peculiarity is just a fact of life!