Galilean Relativity as Many-Minds Relativity

Many-Minds Relativity MMR explores an alternative to Einstein's special theory of relativity SR. The basic setting in both MMR and SR concerns observations made in different spatial Euclidean coordinate systems moving with constant velocity with respect to each other.

In MMR observers share a common time $t$ (Universal Common Time UTC as in the GPS system), while space coordinates in a system $X^\prime$ with space coordinates $x^\prime$ moving with velocity $v$ respect to a system $X$ with coordinates $x$, are connected by the Galilean transformation

• $x^\prime = x- tv$

stating that the origin of $X^\prime$ moves in $X$ with velocity $v$. 

In SR there is no common time and space-time coordinates are connected by the Lorentz transformation mixing space into time in violation of Newtonian mechanics. SR leads to many strange effects such as "time dilation" and "space contraction" as effects of motion with constant velocity, which are deeply paradoxical. 

The basic assumption of MMR are (see also this post):

1. All observers share a common time set by synchronised cesium clocks according to the SI standard.  
2. The same Maxwell's equations for light propagation is used in all systems. 
3. The length scale in each system is set according to the SI standard with the same speed of light in all systems in compatibility with 2.
4. All observers share the laws of Newtonian mechanics.

A2 as Assumption 2 means that each system carries its own "aether" as the coordinate system used to express Maxwell's equations, in accordance with this post. MMR is a "many-aethers"-theory with as many "aethers" as spatial coordinate systems. A2 means that the speed of light is the same in all systems, by SI standard.

A3 means that the same length scale is used in all systems, by SI standard. 

A1 + A3 means that time and length scale is the same in all systems. There is no time dilation or space contraction in MMR. 

In classical Newtonian/Galilean relativity there is "one-unique-aether", while in MMR there are many  (non-unique) aethers. 

The reason Newtonian/Galilean relativity was replaced by SR was the Michelson-Morley experiment MM, which showed to be in conflict with a "one-unique-aether". On the other hand, the null result of  MM is in harmony with MMR, since each arm in the MM experiment carries its own coordinate system.

MMR is thus a "many-observer"-theory, where different observers use different coordinate systems moving with to respect to each other connected by the Galilean transformation, while they share time and length scale and the same Maxwell's equations expressed the same way in their respective systems.

Recall that SR shares the Lorentz transformation with a "Lorentz unique-aether"-theory, including time dilation and space contraction, with the difference that Lorentz considered these effects to be more illusion than reality. Time dilation and space contraction present a veritable night-mare to science with a wealth of "paradoxes" to handle, and so each possibility of getting out of the trauma must be tried. 

But different observers of MMR may not agree on everything, such as Doppler shifts because Doppler shifts depend on the motion of source and receiver, and the observers are moving with respect to each other. 

To see this consider a (1d) situation where at a specific time $X$ and $X^\prime$ with observers $O$ and $O^\prime$ in their respective coordinate origin overlap, while moving with velocity $w$ with respect to each other.  Consider the perceptions of a signal sent from a source at frequency 1, assuming the speed of light is normalised to 1, moving with velocity $v$ with respect to $X$. The frequency $f$ recorded by $O$ using $X$ will then be 

• $f=\frac{1}{1+v}$

since the source is moving with speed $v$ (to the right say) and the receiver is fixed. On the other hand $X$ would attribute the following frequency to an observation made by $O^\prime$:

• $\bar f =\frac{1-w}{1+v}$ for $X^\prime$, 

because $X$ sees the the receiver of $O^\prime$ moving with speed $w$ (to the left say). But the frequency $f^\prime$ de facto observed by $O^\prime$ will be

• $f^\prime  =\frac{1}{1+v+w}$ 

since $v+w$ is the velocity of the source with respect to a fixed $O^\prime$. Computing we find that 

• $\bar f - f^\prime =-\frac{w^2+vw}{(1+v)(1+v+w)}$,

which is second order in $v$ and $w$ compared to the speed of light = 1. Further aspects are given in  the MMR book.

The observers $O$ and $O^\prime$ will thus share the same time and length scale, but will have different perceptions depending on mutual motion, which can differ up the square of motion speed vs speed of light. With a speed of light of $3\times 10^8$ meter/second the difference would be of size $10^{-6}$ for velocities up to $300$ km/second. In short MMR appears to give a consistent description which can be shared by all observers up to second order accuracy in speeds. For observers on Earth $w$ may be up $1$ km/second and for observers in space up to $10$ km/second, far away from $300$ km/second.

We recall that the addition of two velocities $v$ and $w$ from composite Doppler shift in MMR reads:

• $v+w+vw$,     

  since

• $\frac{1}{1+w}\frac{1}{1+v}=\frac{1}{1+v+w+vw}$.

Altogether, MMR is compatible with the MM null result, and thus offers an alternative to SR which is compatible with Newtonian mechanics and electromagnetics (up to second order in relative light velocity).

The idea of a "many-aethers"-theory can be found in the work by the English mathematician Ebenezer Cunningham, but it was swamped by Einsteins "no-aether"-radicalism.

One can argue that the idea of a "many-aethers"-theory connects post-modern multiculturalism, while Einstein's "no-aether"-theory would correspond to a "no-culture-at-all".

PS1 Lee Smolin writes in Einstein's Lonely Path:

• Special relativity was the result of 10 years of intellectual struggle, yet Einstein had convinced himself it was wrong within two years of publishing it. He rejected his own theory, even before most physicists had come to accept it, for reasons that only he cared about. For another 10 years, as others in the world of physics slowly absorbed special relativity, Einstein pursued a lonely path away from it.

Einstein thus gave up SR in 1907. It is high time for physicists of today to do the same thing.

PS2 The physics community views Einstein with mixed feelings: Sometimes as a God but more often as a crackpot. One may argue that "once champ always champ", or "once crackpot always crackpot". What is your view?

PS3 Of course there is no twin paradox to resolve in MMR, since all twins age at the same rate according to the same clocks, independent of inertial motion. Of course acceleration or gravity forces or temperature, pressure and friction may affect the workings of a mechanical clock and thus clock rate (all clocks are mechanical), but certainly not pure translation with constant velocity, against the core of SR.