Gravity Wave Speed: Deriving Constancy from Maxwell Eqs.

[Meir Achuz]

he also meant that "the equations of mechanics" would have to be modified

Why didn't he say what "he meant"?

 

[Sagittarius A-Star]

Why is the second postulate needed today?

I think the opposite direction would be nicer. Derive Minkowski spacetime from postulate 1 (the laws of physics are the same in all inertial frames) and postulate 2 (there exists an invariant speed) + assumed linearity. Then derive a theory of electromagnetism from assumed Coulomb’s law +SR.

There exist approaches in this direction (I did not read it because of the paywall):

ABSTRACT
Maxwell’s equations are obtained by generalizing the laws of electrostatics, which follow from Coulomb’s law and the principle of superposition, so that they are consistent with special relativity.

Source:
https://aapt.scitation.org/doi/abs/10.1119/1.14521

 

[PeterDonis]

Why didn't he say what "he meant"?

If "he" means "Einstein", he did. That's the entire point of his 1905 paper "On the Electrodynamics of Moving Bodies", which is where the phrase you quote appears: to show how "the equations of mechanics" must transform in order to be Lorentz invariant, and what the implications of this are.

 

[vanhees71]

Why is the second postulate needed today?

With the 1st postulate alone and the other symmetry assumptions you get Galilei-Newton spacetime or Einstein-Minkowski spacetime. To decide, which one is the correct (of rather the better) description of the spacetime structure you need empirical input. That empirical input at the time when Einstein wrote his paper were Maxwell's equations, summarizing all then known phenomena concerning electromagnetism including its application to optics, and from a purely kinematic point of view the one conclusion from the assumption that Maxwell's equations must look the same in any inertial reference frame is the "constancy of the speed of light", i.e., the independence of the phase velocity of electromagnetic waves in a vacuum of the velocity of the light source wrt. any inertial reference frame.

 

[vanhees71]

Why didn't he say what "he meant"?

But he did! In his 1905 paper he gave the modified equations of motion for point particles, though in a very deformed way from the modern point of view, introducing various kinds of "relativistic mass". This was clarified already in 1906 by Planck and finally by Minkowski in 1908, revealing the mathematical structure of Einstein's special-relativistic spacetime model.