Second postulate of Special Relativity - Finite Invariante Speed?

Question

@JohnRennie, the most decorated physics stack exchange user of all time, answered the question whether the first postulate of special relativity implies the second (Einstein's first postulate implies the second?), in the following way:

The first postulate is satisfied by Galilean relativity with an infinite speed of light, but this violates the second postulate. Therefore the second postulate does not follow from the first.

Of course experiment tells us that the speed of light isn't infinite, and if we combine the first postulate with a finite speed of light we find they are inconsistent unless further assumptions are made. This is where the second postulate comes in i.e. it is one way of reconciling the first postulate with a finite speed of light. The second postulate requires physical laws to be Lorentz covariant, which leads immediately to special relativity. https://physics.stackexchange.com/a/220821/288306

I'm struggling to understund why infinite light speed would violate the second postulate. Any idea's?

Answer 1

It possible that you can't define the 2nd postulate, since according to https://math.stackexchange.com/questions/650718/does-infinite-equal-infinite

$$ \infty \ne \infty $$

Regarding the 1st postulate, the brilliant book Application of Classical Physics by Thorne & Blanford (http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/), and absolute requirement for any physics graduate student, the measurement of the speed of light is discused (Section 24.2.3), without a clock.

They describe an apparatus with which you measure $\epsilon_0$ and $\mu_0$ to then infer:

$$ c = \frac 1 {\sqrt{\epsilon_0 \mu_0}} $$

which then, by the 1st postulate, must be invariant.

Of course, this opens up the classic can-O-worms: are we talking about the speed of light, or are we talking about the speed of causality?

For years I thought "who cares, what's the difference?", but then I saw Jefimenko's eq, where light is not really self propagating wave, rather at a point $(t, x)$, there are $\vec E$ and $\vec B$ fields caused by all $\rho(t',x'), \dot{\rho}(t',x'), \vec j(t',x'), \dot{\vec j}(t',x')$ under the condition:

$$ t' = t - \frac{|x-x'|} c $$

Now for what we would call a photon (or a classical short pulse), the point $(t, x)$ where there is support moves away from the source at $c$, which looks a lot like a wave traveling at the speed of light, but is described more like the speed of causality.

So just something to think about.