# Reviewing Special Relativity and Lorentz Trasformations, Relevance to Alternate Cosmologies

The motivation for the Lorentz transformation was rooted both in experiment (the most important reason) and some conceptual challenges with Maxwell’s equations for electrodynamics. From a conceptual standpoint, it was desirable (for reasons I’ve since forgotten) that Maxwell’s equations in one frame (let us call this the “rest” frame) that yield:

\LARGE \frac{\partial^2 \vec{\Psi}}{\partial x^2}+\frac{\partial^2 \vec{\Psi}}{\partial y^2}+\frac{\partial^2 \vec{\Psi}}{\partial z^2} - \frac{1}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial t^2}=0

will have a comparable form in a frame moving with relative velocity to the rest frame which is described with prime coordinates (x’, y’, z’, t’) instead of (x,y,z,t):

\LARGE \frac{\partial^2 \vec{\Psi}}{\partial x'^2}+\frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2} - \frac{1}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial t'^2}=0

But as I mentioned, there is both a conceptually deduced and experimentally deduced (such as the Hafele Keating experiment) that suggests there really is a true rest frame where V = 0 because of the integration of acceleration over all time that would yield:

\LARGE v = \int a(t) = 0

and I described how one can conduct and experiment in principle where one can determine’s one’s absolute speed relative to the Aether. The Lorentz transformations apply to the Aether from of relativity, btw, but instead of time dilation we have clock slowing, and clock slowing based on absolute speed is indicated experimentally.

However to create a transformation that preserves the form or Maxwell’s equations such as:

\LARGE \frac{\partial^2 \vec{\Psi}}{\partial x'^2}+\frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2} - \frac{1}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial t'^2}=0

required the genius of Lorentz to accomplish. The transformation entailed doing the following using the chain rule on multiple varialbles on the d’Alamberitan above. I’ll elaborate it in brutal detail in order to make the idea more accessible:

Consdier that going from the rest frame to a moving frame, we have a change of coordinages from the unprimed to the primed system. For example, the unprimed coordinates in the x dimension are related the primed coordinates according to the chain rule for multiple variables:

\Large \frac{\partial^2 \vec{\Psi}}{\partial x^2} = \frac{\partial}{\partial x} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{x} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{x} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{x} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{x}\right]

I’m elaborating this in brutal detail to show the genius that Lorentz must have had find a transformation that would make Maxwell’s equations invariant! The above equation could be expanded in even more detail for the second derivative, but for brevity, I’ll spare the reader and keep it in that “simpler” form. Doing the same for all the dimensions the d’Alambertian in primed coordinates would be (gasp!):

\LARGE \frac{\partial^2 \vec{\Psi}}{\partial x^2}+\frac{\partial^2 \vec{\Psi}}{\partial y^2}+\frac{\partial^2 \vec{\Psi}}{\partial z^2} - \frac{1}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial t^2} =

\Large \frac{\partial}{\partial x} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{x} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{x} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{x} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{x}\right]

\Large + \frac{\partial}{\partial y} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{y} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{y} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{y} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{y}\right]

\Large + \frac{\partial}{\partial z} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{z} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{z} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{z} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{z}\right]

\Large - \frac{1}{c^2}\frac{\partial}{\partial t} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{t} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{t} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{t} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{t}\right]

=0

Lorentz searched for a transformation to reduce the above monstrosity to:

\LARGE \frac{\partial^2 \vec{\Psi}}{\partial x'^2}+\frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2} - \frac{1}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial t'^2}=0

and his solution contributed to him getting the Nobel prize. The above will not be solve by the Galilean transformation, so he had to concoct his own transformation. One way to make the above monstrosity manageable is to hyothesize the following 16 relations which can be substituted into the above monstrosity:

[for the x’ dimension]
\large \frac{\partial x'}{\partial x} = \gamma

\large \frac{\partial x'}{\partial y} = 0

\large \frac{\partial x'}{\partial z} = 0

\large \frac{\partial x'}{\partial t} = -\gamma V

[for the y’ dimension]
\large \frac{\partial y'}{\partial x} = 0

\large \frac{\partial y'}{\partial y} = 1

\large \frac{\partial y'}{\partial z} = 0

\large \frac{\partial y'}{\partial t} = 0

[for the z’ dimension]
\large \frac{\partial z'}{\partial x} = 0

\large \frac{\partial z'}{\partial y} = 0

\large \frac{\partial z'}{\partial z} = 1

\large \frac{\partial z'}{\partial t} = 0

[for the t’ prime dimension]

\large \frac{\partial t'}{\partial x} = -\gamma \frac{V}{c^2}

\large \frac{\partial t'}{\partial y} = 0

\large \frac{\partial z'}{\partial z} = 0

\large \frac{\partial z'}{\partial t} = \gamma

Hypothesizing the above 16 relations greatly reduces the nastiness of the above monstrosity to something more manageable which I cover in the next comment. How Lorenztz came up with that hypothesis is testament to his genius.

Continuing, this is the monstrostity of the transformed d’Alambertian:

\Large \frac{\partial}{\partial x} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{x} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{x} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{x} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{x}\right]

\Large + \frac{\partial}{\partial y} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{y} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{y} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{y} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{y}\right]

\Large + \frac{\partial}{\partial z} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{z} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{z} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{z} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{z}\right]

\Large - \frac{1}{c^2}\frac{\partial}{\partial t} \left[\frac{\partial \vec{\Psi}}{\partial x'} \frac{\partial x'}{t} + \frac{\partial \vec{\Psi}}{\partial y'} \frac{\partial y'}{t} + \frac{\partial \vec{\Psi}}{\partial z'} \frac{\partial z'}{t} + \frac{\partial \vec{\Psi}}{\partial t'} \frac{\partial t'}{t}\right] = 0

by subsituting the above 16 relations hypothesized by Lorentz, this monstrosity becomes:

\Large \frac{\partial}{\partial x} \left[\frac{\partial \vec{\Psi}}{\partial x'} (\gamma) + \frac{\partial \vec{\Psi}}{\partial y'} (0) + \frac{\partial \vec{\Psi}}{\partial z'} (0) + \frac{\partial \vec{\Psi}}{\partial t'} (-\gamma \frac{V}{c^2})\right]

\Large + \frac{\partial}{\partial y} \left[\frac{\partial \vec{\Psi}}{\partial x'} (0) + \frac{\partial \vec{\Psi}}{\partial y'} (1)+ \frac{\partial \vec{\Psi}}{\partial z'} (0) + \frac{\partial \vec{\Psi}}{\partial t'} (0)\right]

\Large + \frac{\partial}{\partial z} \left[\frac{\partial \vec{\Psi}}{\partial x'} (0) + \frac{\partial \vec{\Psi}}{\partial y'} (0) + \frac{\partial \vec{\Psi}}{\partial z'} (1) + \frac{\partial \vec{\Psi}}{\partial t'} (0) \right]

\Large - \frac{1}{c^2}\frac{\partial}{\partial t} \left[\frac{\partial \vec{\Psi}}{\partial x'} (-\gamma V) + \frac{\partial \vec{\Psi}}{\partial y'} (0) + \frac{\partial \vec{\Psi}}{\partial z'} (0) + \frac{\partial \vec{\Psi}}{\partial t'} (\gamma)\right] = 0

which mercifully reduces to:

\Large \frac{\partial}{\partial x} \left[\gamma\frac{\partial \vec{\Psi}}{\partial x'} -\gamma \frac{V}{c^2}\frac{\partial \vec{\Psi}}{\partial t'} \right]

\Large + \frac{\partial}{\partial y} \left[\frac{\partial \vec{\Psi}}{\partial y'}\right]

\Large + \frac{\partial}{\partial z} \left[\frac{\partial \vec{\Psi}}{\partial z'}\right]

\Large - \frac{1}{c^2}\frac{\partial}{\partial t} \left[-\gamma V\frac{\partial \vec{\Psi}}{\partial x'} + \gamma\frac{\partial \vec{\Psi}}{\partial t'} \right] = 0

which makes stating the 2nd order derivative easier, which I’ll do in the next comment. But again, how Lorentz conceived of this transformation to reduce this monstrosity toward the desired result of invariance of Maxwell’s equations is sheer genius.

I realized I had to review a little basic calculus to show claim I made above, so I’ll elaborate it a little more since I had to re-learn it myself over the past week. If

\Large \vec{\Psi} = \vec{\Psi}(x,y,z,t)

then if we change coordinates from (x,y,z,t) to (x’,y’,z’,t’) the chain rule for the first derivative is:

\Large \frac{\partial \vec{\Psi}}{\partial x} = \frac{\partial \vec{\Psi}}{\partial x'}\frac{\partial x'}{\partial x}+\frac{\partial \vec{\Psi}}{\partial y'}\frac{\partial y'}{\partial x}+\frac{\partial \vec{\Psi}}{\partial z'}\frac{\partial z'}{\partial x}+\frac{\partial \vec{\Psi}}{\partial t'}\frac{\partial t'}{\partial x}

extending the idea to the second derivative in the z’ coordinates
\frac{\partial}{\partial z} \left[ \frac{\partial \vec{\Psi}}{\partial z'}\right] = \frac{\partial}{\partial x'}\left[\frac{\partial \vec{\Psi}}{\partial z'}\right]\frac{\partial x'}{\partial z} + \frac{\partial}{\partial y'}\left[\frac{\partial \vec{\Psi}}{\partial z'}\right]\frac{\partial y'}{\partial z}+ \frac{\partial}{\partial z'}\left[\frac{\partial \vec{\Psi}}{\partial z'}\right]\frac{\partial z'}{\partial z} + \frac{\partial}{\partial t'}\left[\frac{\partial \vec{\Psi}}{\partial z'}\right]\frac{\partial t'}{\partial z}

but taking from the above 16 relations hypothesized by Lorentz, specifically in our case this reduces to:

\frac{\partial}{\partial z} \left[ \frac{\partial \vec{\Psi}}{\partial z'}\right] = \frac{\partial}{\partial x'}\left[\frac{\partial \vec{\Psi}}{\partial z'}\right](0) + \frac{\partial}{\partial y'}\left[\frac{\partial \vec{\Psi}}{\partial z'}\right] (0)+ \frac{\partial}{\partial z'}\left[\frac{\partial \vec{\Psi}}{\partial z'}\right](1) + \frac{\partial}{\partial t'}\left[\frac{\partial \vec{\Psi}}{\partial z'}\right] (0)

thus

\Large \frac{\partial}{\partial z} \left[ \frac{\partial \vec{\Psi}}{\partial z'}\right] = \frac{\partial}{\partial z'}\left[\frac{\partial \vec{\Psi}}{\partial z'}\right] = \frac{\partial^2 \vec{\Psi}}{\partial z'^2}

similarly

\Large \frac{\partial}{\partial y} \left[ \frac{\partial \vec{\Psi}}{\partial y'}\right] = \frac{\partial}{\partial y'}\left[\frac{\partial \vec{\Psi}}{\partial y'}\right] = \frac{\partial^2 \vec{\Psi}}{\partial y'^2}

I’ll use this fact for the next comment.

The reduce transformed d’Alambertian monstrosity:

\Large \frac{\partial}{\partial x} \left[\gamma\frac{\partial \vec{\Psi}}{\partial x'} -\gamma \frac{V}{c^2}\frac{\partial \vec{\Psi}}{\partial t'} \right]

\Large + \frac{\partial}{\partial y} \left[\frac{\partial \vec{\Psi}}{\partial y'}\right]

\Large + \frac{\partial}{\partial z} \left[\frac{\partial \vec{\Psi}}{\partial z'}\right]

\Large - \frac{1}{c^2}\frac{\partial}{\partial t} \left[-\gamma V\frac{\partial \vec{\Psi}}{\partial x'} + \gamma\frac{\partial \vec{\Psi}}{\partial t'} \right] = 0

using the above results for the second derivatives with respect to y’ and z’ reduces to

\Large \frac{\partial}{\partial x} \left[\gamma\frac{\partial \vec{\Psi}}{\partial x'} -\gamma \frac{V}{c^2}\frac{\partial \vec{\Psi}}{\partial t'} \right] +\frac{\partial^2\vec{\Psi}}{\partial y'^2}+\frac{\partial^2\vec{\Psi}}{\partial z'^2}

\Large - \frac{1}{c^2}\frac{\partial}{\partial t} \left[-\gamma V\frac{\partial \vec{\Psi}}{\partial x'} + \gamma\frac{\partial \vec{\Psi}}{\partial t'} \right] = 0

rearranging for convenience

\Large \frac{\partial^2\vec{\Psi}}{\partial y'^2}+\frac{\partial^2\vec{\Psi}}{\partial z'^2}

+\Large \frac{\partial}{\partial x} \left[\gamma\frac{\partial \vec{\Psi}}{\partial x'} -\gamma \frac{V}{c^2}\frac{\partial \vec{\Psi}}{\partial t'} \right]

\Large - \frac{1}{c^2}\frac{\partial}{\partial t} \left[-\gamma V\frac{\partial \vec{\Psi}}{\partial x'} + \gamma\frac{\partial \vec{\Psi}}{\partial t'} \right] = 0

but unfortuantely we’re not out of the woods, as we have to execute the 2nd derivatives with respect to x’ and t’ and then do some algebra which I’ll do in the next comment which shows Lorentz invariance that is applicable to either the Einsteininan Special Relativity or the Lorentz Aether Theory of Special Relativity.

I’m going to skip a couple steps by simply stating the 2nd derivatives which I’ll later derive in subsequent comments.

This was the above result of the original d’Alambertian in transformed coordinates:

\Large \frac{\partial^2\vec{\Psi}}{\partial y'^2}+\frac{\partial^2\vec{\Psi}}{\partial z'^2}

+\Large \frac{\partial}{\partial x} \left[\gamma\frac{\partial \vec{\Psi}}{\partial x'} -\gamma \frac{V}{c^2}\frac{\partial \vec{\Psi}}{\partial t'} \right]

\Large - \frac{1}{c^2}\frac{\partial}{\partial t} \left[-\gamma V\frac{\partial \vec{\Psi}}{\partial x'} + \gamma\frac{\partial \vec{\Psi}}{\partial t'} \right] = 0

executing the 2nd derivatives with respect to x’ and t’ (which I’ll show later how I can do it):

\frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2}+ \gamma^2 \frac{\partial^2 \vec{\Psi}}{\partial x'^2}-\gamma^2\frac{V^2}{c^2}\frac{\partial^2 \vec{\Psi}}{\partial x'^2} - \frac{\gamma^2}{c^2}\frac{\partial^2 \vec{\Psi}}{\partial t'^2}+\frac{\gamma^2 V^2}{c^4}\frac{\partial^2 \vec{\Psi}}{\partial t'^2} -\frac{\gamma^2 2V}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial x' \partial t'}+\frac{\gamma^2 2V}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial x' \partial t'}=

\frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2}+ \left[ \gamma^2 -\gamma^2\frac{V^2}{c^2}\right] \frac{\partial^2 \vec{\Psi}}{\partial x'^2} + \left[ \frac{\gamma^2 V^2}{c^4}-\frac{\gamma^2}{c^2}\right] \frac{\partial^2 \vec{\Psi}}{\partial t'^2} +\left[ \frac{\gamma^2 2V}{c^2} - \frac{\gamma^2 2V}{c^2}\right] \frac{\partial^2 \vec{\Psi}}{\partial x' \partial t'}=

\frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2}+ \left[ \gamma^2 -\gamma^2\frac{V^2}{c^2}\right] \frac{\partial^2 \vec{\Psi}}{\partial x'^2} + \left[ \frac{\gamma^2 V^2}{c^4}-\frac{\gamma^2}{c^2}\right] \frac{\partial^2 \vec{\Psi}}{\partial t'^2} + 0=

changing the form slightly
\Large \frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2}+ \left[ \gamma^2 -\gamma^2\frac{V^2}{c^2}\right] \frac{\partial^2 \vec{\Psi}}{\partial x'^2} - \left[ \frac{\gamma^2}{c^2} -\frac{\gamma^2 V^2}{c^4}\right] \frac{\partial^2 \vec{\Psi}}{\partial t'^2}=

changing the form sligthly by factoring
\Large \frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2}+ \gamma^2 \left[ 1 -\frac{V^2}{c^2}\right] \frac{\partial^2 \vec{\Psi}}{\partial x'^2} - \frac{\gamma^2}{c^2} \left[ 1 -\frac{V^2}{c^2}\right] \frac{\partial^2 \vec{\Psi}}{\partial t'^2}=

noting that \frac{1}{\gamma^2}=1-\frac{V^2}{c^2} under the Lorentz transformation (which yield the 16 relations mentioned above) we can further change the form to

\Large \frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2}+ \gamma^2 \left[ \frac{1}{\gamma^2}\right] \frac{\partial^2 \vec{\Psi}}{\partial x'^2} - \frac{\gamma^2}{c^2} \left[\frac{1}{\gamma^2}\right] \frac{\partial^2 \vec{\Psi}}{\partial t'^2}=

which reduces to

\Large \frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2}+ \frac{\partial^2 \vec{\Psi}}{\partial x'^2} - \frac{1}{c^2} \frac{\partial^2 \vec{\Psi}}{\partial t'^2}=

slightly re-arranging

\huge \frac{\partial^2 \vec{\Psi}}{\partial x'^2}+\frac{\partial^2 \vec{\Psi}}{\partial y'^2}+\frac{\partial^2 \vec{\Psi}}{\partial z'^2} - \frac{1}{c^2}\frac{\partial^2 \vec{\Psi}}{\partial t'^2} = 0

which shows the invariance of Maxwell’s equations under the Lorentz transformations, which is what Lorentz tried to establish for his Aether theory.

Which at least shows the hypothetical Lorentz transformation will preserve the form Maxwell’s equations while the Galilean transformations will not. Again the hypothetical Lorentz tranformations for his Aether theory is:

x' = \gamma (x - Vt)

t' = \gamma (t- \frac{Vx}{c^2})

y' = y

z' = z

taking the 16 partial derivatives of the above transformation of x’,y’,z’,t with respect to x,y,z,t result in the 16 relations I described above. How Lorentz conceived of this system, I don’t know, but I know for sure he was a genius, and he rightly sits at Einstein’s right hand both in the photo above and figuratively speaking.

The demonstration of the Invariance of Maxwell’s equations was tedious, but I did it to show that even using freshman calculus of partial derivatives and the chain rule, one could see the validity of the fundamental aspects of relativity, most especially clock slowing (what is alternatively called time dilation). BUT there is an issue that has always bothered me, namely the ambiguity as to what “V” means in the Lorentz transofrmation. Is V some absolute number relative to a reference velocity which I claim is

V_{ref} = \int a(t) = 0

I remember this being bothersome even in sophomore/junior level physics.

If we hypotehtically had two twin space craft in space one Turtle and the other Rabbit.
Let’s accelerate Rabbit to about 86.66% the speed of light relative to turtle, then the gamma factor will be about 2. Let him then decelerate to zero velocity (relative to Turtle) and start flying back to Turtle at about 86.66% the speed of light. Granted, as Rabbit accelerates, his clocks should experience GR-type clock slowing, which we assume can be accounted for somehow, but the point is has slowed Rabbit’s clock during the acceleration and deceleration phases.

For completeness, let me show the “clock slowing” equation.

For one frame, say the primed frame, let some arbitrary elapsed time on the clock in that frame be:

let \large \Delta t' = t'_2 - t'_1

how much time would elapse on the clock in the other frame according to the Lorentz transformation? I’m assuming x does not change over time because the space craft is travelling along the same axis, so I won’t use x_1 or x_2 but simply x.

\Delta t' = t'_2 - t'_1 = \gamma (t_2 - \frac{Vx}{c^2}) - \gamma (t_1 -\frac{Vx}{c^2}) =

\gamma (t_2-t_1) = \gamma \Delta t

or simply

\Delta t' = \gamma \Delta t

for a V = 86.66% the speed of light, \gamma = 2

Clearly one clock is ticking twice as slow as the other. The problem is deciding whether to assign Turtle’s clock to t or t’. Well experiments, using the above convention, we should assign it to Turtle’s clock to t’ because Rabbit is the spacecraft undergoing acceleration. There is a change of state in Rabbit because of the acceleration of Rabbit, not a change of state in Turtle. Rabbit can surely sense it is being accelerated!

Thus, when Turtle and Rabbit have a constant relative velocity of 86.66% the speed of light (hence are in inertial frames), using the above relation:

\Delta t_{Turtle} = \gamma \Delta t_{Rabbit}

or

\Large \frac{1}{\gamma} \Delta t_{Turtle} = \Delta t_{Rabbit}

if \gamma = 2, and 10 days elapsed for Turtle then

the clock on Rabbit (after GR-type corrections) would indicate 5 days time. I suppose, in principle, one could have Rabbit accelerate such that the clock slowing effect due to GR type effects would create an accumulated clock time of 5 days. In any case, one clock was definitely ticking slower than another.

Clearly we can’t say both clocks slowed down because of the relative velocity between them, as that would be a logical contradiction. Only one clock slows down. This is confirmed by experiment! How does this not suggest a preferred frame for accounting for the history of phenomenon? Extending this example with further thought experiments will show, there will be absurdities if one does not implicitly assume there exists an absolute velocity V = 0. That is to say, Lorentz invariance will lead to contradictions if there is not ultimately in principle:

V_{ref} = \int a(t) = 0

And that is what I meant by suggesting there is ambiguity as to what “V” in the Lorentz transformation really means. I claim V means V relative to some reference absolute V = 0, otherwise one gets absurdities in that transformation.

Hence, there is a preferred reference frame that gets the physics right. Thankfully, it takes a huge deviation from that preferred frame before relativistic effects like clock slowing make a mess of things.

Er, several comments by me, PdotdQ, and r_speir are missing? What happened. They were good comments too!