History of relativity: A mathematical addendum

From yesterday’s post, the Galilean transformations have as an implicit postulate absolute space and absolute time, so measuring rods and clocks will perform the same way as seen by any observer regardless of the magnitude of their relative velocity (recall that we are restricting ourselves to inertial reference frames, ie. non-accelerating frames). We define an event to be a point in a set of spacetime coordinates $(x, y, z, t)$ by one observer and $(x', y', z', t')$ by another observer in uniform translatory motion with respect to the other. Then, for a particle moving along the x-axis in the positive direction we have the following transformation (note that the equation can be generalized to moving in any direction, but we can then just rotate our coordinate system to have the motion again be in the x-axis since we assume that the laws of physics are rotation invariant. The simpler case of restricting ourselves to the x-axis provides maximum clarity of the physics involved):

$x' = x - vt \\ y' = y \\ z' = z \\ t' = t$

We fairly easily see that if we are a stationary observer and an object moves passed us with a velocity v, we can simply apply the Galilean transformation to be in a reference frame moving parallel with the object so that it will appear stationary in our coordinate system.  Then, if it ejects an object moving with velocity u, we can perform the same trick and have a Galilean transformation with $x'' = x' - (u + v)t'$ given us a total velocity of $w = u + v$ which we call the velocity-addition formula. Now, presented without a derivation (as this is merely a brief historical overview, I’ll cover the derivation in a separate dedicated post) we have the Lorentz transformation:

$x' = \displaystyle\frac{x - vt}{\sqrt{1 - \frac{v^2}{c^2}}} \\ y' = y \\ z' = z \\ t' = \frac{t - v\frac{x}{c^2}}{\sqrt{1 - \frac{v^2}{c^2}}}$

We see that for v << c the terms that have a $\frac{1}{c^2}$ vanish and the Galilean transformation is recaptured. Moreover, we can guess intuitively that velocities should not add in the normal way.  Say you were traveling with a velocity 0.8c in a spaceship and then fired off a probe with velocity of 0.8c forward. You would naively expect the probe to have a velocity of 1.6c relative to a stationary observer (on a nearby space station say) but that would violate the law of the constancy of the speed of light. Moreover you can immediately see how the Lorentz transformation would break down for v > c to give non-physical answers (complex numbers). The actual velocity-addition formula is

$\displaystyle w = \frac{u + v}{1 + \frac{uv}{c^2}}$

presented here without derivation. You can immediately see how the same criteria of u, v << c will reduce the Lorentz invariant version to the Galilean invariant version. So we see that mathematically Newtonian mechanics is a limiting case of relativistic mechanics for velocities sufficiently small compared to the speed of light which explains why they are still extremely useful for things like bridges and buildings and vehicles, etc.