Galilean Special Relativity as a Many-Aethers Theory

Einstein's special theory of relativity SR is based on the following postulates:

1. Laws of physics take the same form in all inertial systems.
2. The speed of light is the same in all inertial systems.

An inertial system is a Euclidean coordinate system in space together with a time coordinate and different inertial systems (with the same spatial orientation) are moving with constant velocity with respect to each other.  

Consider the following realisation of 1+2: Assume

• Inertial systems are connected by a Galilean transformation. 
• Newtonian mechanics.
• Maxwell's equations for electromagnetics in each inertial system.

Newtonian mechanics is Galilean invariant and thus satisfies 1. Assuming that identical Maxwell's equations are used in all inertial systems makes 1 true by definition, and then also 2.

Using formally identical Maxwell's equations in all inertial systems can be viewed as a "many-aethers"-theory with each Euclidean coordinate system representing an "aether" for propagation of light. The speed of light will then be the same in all systems. Compare with this post.

This is the set-up in Many-Minds Relativity as a "many-aethers"-theory. It should be compared with Einstein's "no-aether" SR, where Galilean transformation is replaced by Lorentz transformation. 

A Galilean transformation connects the space coordinates $x$ and $x^\prime$ of two inertial system moving with velocity $v$ with respect to each other by the simple transformation 

• $x^\prime = x-tv$

where $t$ is a time coordinate shared by both systems. A Galilean transformation has a direct simple realisation as a translation  with constant velocity, as the simplest possible motion. 

A Lorentz transformation mixes space coordinates with time and has no physical realisation. Even more disturbing: Newtonian mechanics is not Lorentz invariant and thus has to be given up in SR. Einstein paid his tribute in his "Newton, forgive me!"

How could then Einstein end up with his SR based on Lorentz transformation with all its mysteries and sacking of Newton, starting from the same basic postulates 1+2  as we saw could as well be satisfied by Galilean transformation without mystery and with Newton intact? 

The answer is hidden in Einstein's derivation of the Lorentz transformation from 1 + 2, which starts with light pulses initiated in two different inertial systems. By identifying the two light pulses to be one and the same taking different expressions in the two systems, Einstein then derived the Lorentz transformation. But the identification is unphysical in the sense that initialisation as coexistence at an initial time of a wave form in space necessarily takes different forms in different systems when space is mixed into time. It means that the identification of the pulses cannot be made and so Einstein's derivation of the Lorentz transformation from 1+2 is incorrect from physical point of view.

In his derivation of the Lorentz transformation Einstein relied on the concept of event  as something of unknown nature which has no extension in space, which can be labeled with a single space coordinate $x$ and time coordinate $t$. This made it possible for Einstein to view the launch of the  light pulses as one and the same event with different labels in the different systems and from that derive a connection between the labels in the form of Lorentz transformation.

An event labeled by $(x,t)$ carries unclear physics and can lead to misunderstanding of physics. The notion of particle as something without spatial extension but still physical presence, also can lead to misunderstanding.

Sum up: Einstein's SR is a "no-aether"-theory with strange physics in conflict with Newtonian mechanics. Many-Minds Relativity is a natural "many-aethers"-theory in harmony with Newtonian mechanics and electromagnetics.

PS To see the difference between one-pulse and two-pulse physics, consider two intertial systems which coincide at the light pulse launch. In the Galilean setting the launch physics will be the same in both systems which effectively means launch of two pulses with different translation speeds in two different "aethers", while in the Lorentzian setting they will be the same which is unphysical.