Reviewing Special Relativity and Lorentz Trasformations, Relevance to Alternate Cosmologies

In the spirit of this thread, for anyone interested, here’s a great video series introducing SR:

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OK, after searching hard, I found the derivation stated nicely the way I want, I might have to rework it to make it clearer, but it required some nasty algebra tricks. UGH!

I was asking so I could understand, not skeptically questioning. Finding solutions to differential equations with one variable/dimension – not so bad, but with 4 variable x,y,z, t under the constraints of Maxwell’s equations, not so clear. I mean yeah:

\nabla \times (\nabla \times \vec{\Psi}) = \nabla (\nabla \cdot \vec{\Psi}) - \nabla^2\vec{\Psi}

which leads to

\nabla^2 E = \mu_0 \epsilon_0 \frac {\partial^2 E}{\partial t}

where we can put B or E in place of \Psi

but where does

\nabla \times (\nabla \times \vec{\Psi})

come from in the first place. I can dig it up, but I was asking in case anyone knew off the top of their head.

Since many Creationists are Engineers, and a few are Electrical Engineers, the question of derivation of relativity will invariably come up. Since most Electrical Engineers, especially the antenna and motor engineers are acquainted with Maxwell’s equations of electro magnetism (since Maxwell is one of the founding fathers of Electrical Engineering), I thought it would be good to show how Lorentzian relativity arises naturally out of Electromagnetism. But there are a few catches to all this as I pondered it.

At issue, since the days of Lorentz and Larmor is whether there is absolute time and maybe the clocks in moving frames slow down, or whether time actually flows differently depending on velocity. The Lorentz transformations that make Maxwell’s equations invariant, as I’ve clumsily tried to show (and finally had to provide a link that had been searching for on and off for two years), does not actually say whether time flows differently or if the clocks merely slow down!

I will argue the clocks slow down, and that there is absolute time, and I will use argument by contradiction using Lorentz transformations.

But I wanted to apply some rigor to show that the Lorentz transformations at the heart of Einstein’s Special Relativity weren’t pulled out of a vacuum (pun intended).

Is this not Sal just using PS as a handy notepad for his musings? Is that a legitimate use of the web site?

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If is boring, people will ignore it.

The reason I’m posting this is to showcase some of the forgotten work of Lorentz and others like Fizeau, Fresnel, Hoek, Stokes, Maxwell, Larmor, Miller and other fine physicists who worked on the Aether formulations of relativity.

The experiments on Fresnel drag and the clock slowing (misnamed time dilation) suggest fairly absolute cooridinate systems and that the Aether is related to the zeropoint energy, and variable zeropoint energy implies anisotropic variable light speed, exactly what the YEC/YCC are searching for.

With that in mind, based on the Lorentz transformations derived from Maxwell’s equations above:

\Delta t = \frac{1}{\sqrt(1-\frac{v^2}{c^2}}t_0 = \gamma \Delta t_0

But this needs some clarification from mainstream interpretation, so let me add subscripts

\Delta t_\text{Earth} = \gamma \Delta t_\text{Spaceship} or

\frac{\Delta t_\text{Earth}}{\gamma} = \Delta t_\text{Spaceship}

now there is subtlety that requires some time dilation(clock slowing) due to General Relativity (GR) when the space ship is accelerating and decelerating as well, but that can be accounted for, suffice to say the clock on the space ship will have fewer ticks than the clocks on Earth due to both SR and GR type effects.

No problem, so far, and the Hafele Keating experiments show these effects using cesium beam clocks flown on airplanes:

It confirmed that clocks slow down when they travel fast relative to the aether/Zero Point Enegy field.

BUT there is a subtle issue here – the clock that slowed down was the clock on the object that was subject to accelerations and decelerations! This would suggest that there could in principle be an object with a net velocity of zero, based on this integral

\LARGE v(t) = \int {a(t)}

but also the fact that we might infer what the zero velocity point is by firing space ships in opposite directions at various speeds. I suspect Earth is moving close to the zero velocity, but not exactly because of it’s orbit around the sun and the sun’s movement relative to the universal/cosomological frame. Magueijo and the VSL advocates were right to point out the Cosmos itself defines a frame of reference. Surely we can say one object is accelerating relative to another – i.e. You assume when you’re driving and step on the accelerator, that you’re the one accelerating, not the rest of the universe! This leads then to the inference from the above integral that there exists a zero velocity in principle, where the clock from that frame is the reference clock for all other clocks.

Oh, that’s the other thing, if we use the stars and the cosmos to define a frame, we can also define what objects are accelerating relative to this frame, and this will also define absolute velocities which imply privileged frames, and the relativisitic frames are only convenience conceptions.

Further comments will go into other experiments that involve Fresnel drag, which have bearing on re-interpretations of Michelson-Morely and the variable Zero Point hypothesis which could lead to possible solutions to the distant starlight problem of YEC/YCC.

PS
I just discovered this, Lorentz won the Nobel Prize in 1902:

https://www.nobelprize.org/images/lorentz-13546-content-portrait-mobile-tiny.jpglorentz-13546-content-portrait-mobile-tiny%5B1%5D

The so-called Lorentz transformation (1904) was based on the fact that electromagnetic forces between charges are subject to slight alterations due to their motion, resulting in a minute contraction in the size of moving bodies. It not only adequately explains the apparent absence of the relative motion of the Earth with respect to the ether, as indicated by the experiments of Michelson and Morley, but also paved the way for Einstein’s special theory of relativity.

Yes, the absence is only apparent, and Lorentz and Fresnel and new interferometry experiments suggest this.

I expect, but cannot tell for sure, that Sal’s excursions into physics are fully as bad as his attempts at biology or geology. Can any of the physicists here confirm?

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first things firdst. Is there a speed of light? if gravity waves go the same speed as light why is not the Speed of light just the speed of gravity waves? unlikely both go the same speed so they must be speeding THROUGH something else. Then go from there to anything can speed that fast THEN why say light is speeding vas opposed to a provocation in a light field is what is speeding?
This also Einstein probably messed up.

ERRATA:

The above claim is incorrect!!!

I finally found the correct way to demonstrate invariance. I had done the derivation from scratch once as a homework assignment, but that was 12 years ago. I lost my notes, but I found a comparable derivation on the net and have to adapt the notation and symbols and conventions. Using the symbol conventions above, what I need to show is that under the Lorentz transformation is that assuming:

\LARGE \frac{\partial E}{\partial^2x} = \mu_o \epsilon_0 \frac{\partial^2}{\partial t^2}

under the Lorentz transformation

\LARGE \frac{\partial E}{\partial^2x'} = \mu_o \epsilon_0 \frac{\partial^2 E}{\partial t'^2}

Apologies to the reader. I will put an edit in the original comment to point to this correction.

The issue is whether Maxwell’s equations must assume a reference frame for them to valid, the answer is “no”, but that does not necessarily mean there is no preferred frame, it could also mean t-prime is measuring time with a slowed down clock, not that time is actually moving slower.

So going from Maxwell’s equations, we can show they hold for a frame where clock-time is defined by t or a frame where clock-time is defined by t-prime. Clock time isn’t necessarily actual/reference time. I’ve suggested evidence that there is an actual preferred/reference clock rate, and that is where:

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

We can deduce if our clock is at or close to absolute zero velocity, v = 0, by accelerating clocks in opposite directions and measuring how many ticks they lose.

For example, if we accelerate one clock in one direction and let it fly for a while, and then accelerate another clock in exact opposite direction and the two clocks end up losing the same amount of clock ticks relative the the lab reference clock, then we know the lab reference clock is at v=0 in the absolute sense. If we get an unbalanced result where one clock loses more ticks than another, or even one clock even gains ticks, then one knows the lab reference clock is not at v = 0.

The reason I wanted to go through the derivation above is that my claim doesn’t violate Lorentz invariance of Maxwell’s equations. There is an absolute speed of light for a given segment of space, but the clocks that measure the speed of light are affected by their absolute speed. If that were not the case, we couldn’t have the results of the Hafele-Keating experiment which had clocks slow down after being accelerated to a certain velocity.

This was from a respected Physcis journal. The full paper is behind a paywall, and the language in the abstract apparently was deliberately obtuse since the results were favorable to the Aether interpretation by Krisher:

Reginald Cahill showed the significance of the paper along with other experiments including the Michelson-Morely experiment that had Fresnel Dragging (aka the one that had AIR as a medium for light to travel) and other experiments with Fresnel Dragging. Cahill comments here:

We combine data from two high precision NASA/JPL experiments: (i) the
one-way speed of light experiment using optical fibers: Krisher T.P., Maleki
L., Lutes G.F., Primas L.E., Logan R.T., Anderson J.D. and Will C.M., Phys.
Rev. D, vol 42, 731-734, 1990, and (ii) the spacecraft earth-flyby doppler shift
data: Anderson J.D., Campbell J.K., Ekelund J.E., Ellis J. and Jordan J.F.,
Phys. Rev. Lett., vol 100, 091102, 2008, to give the solar-system galactic
3-space average speed of 486km/s in the direction RA=4.29h
, Dec=-75.0◦. Turbulence effects (gravitational waves) are also evident. Data also reveals
the 30km/s orbital speed of the earth and the sun inflow component at 1AU
of 42km/s and also 615km/s near the sun, and for the first time, experimental
measurement of the 3-space 11.2km/s inflow of the earth. The NASA/JPL
data is in remarkable agreement with that determined in other light speed
anisotropy experiments, such as Michelson-Morley (1887), Miller (1933), DeWitte (1991), Torr and Kolen (1981), Cahill (2006), Munera (2007), Cahill
and Stokes (2008) and Cahill (2009).

The Kicher experiment was done by multiple researchers and published in a very respected journal Physical Review Letters D. The other experiment Cahill and Stokes (2008) I attempted to replicate but was unsuccessful. I still have the laser parts from Thor Labs in my basement!

As I mentioned, I wanted to show the Lorentz transformation of arises from something a tad more down to Earth, namely Maxwell’s equations which are the foundation of electrical engineering. Antenna Engineers and Motor designers are acquainted with these equations. The Lorentzian interpretation of the Lorentz transformation rather than the Einstein interpretation of the Lorentz transformation is friendly to variable speed of light.

I’ve been combing the net to find the derivation explicitly stated, and have had to cobble and rederive it piecemeal, and I’m getting closer to having it ready for my ID/Creation class.

Anyway, I said this earlier, this is one solution to Maxwell’s differential equations for a lightwave travelling through a vacuum, namely in terms of the ELECTRIC field:

\LARGE \nabla^2E = \mu_0 \epsilon_0 \frac{\partial^2E}{\partial t^2}

I’m going to change the form and amend the symbols as I will try to make this more rigorous. To emphasize the E is vector

\LARGE \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2E}{\partial t^2}

note that E is function of x,y,z,t, that is

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

but for brevity I simply state \vec{E}(x,y,z,t) as \vec{E}

I had to shake off a lot of cobb webbs upstairs to remember how to do the derivation I did 12 years ago for a homework assignment, and the way I did specifically isn’t on the net anywhere and I had to reconstruct my version of the derivation since I think it’s the most accessible to the un-initiated.

Expanding the Laplacian:

\LARGE \nabla^2E = \frac{\partial^2 \vec{E}}{\partial x^2}+\frac{\partial^2 \vec{E}}{\partial y^2}+\frac{\partial^2 \vec{E}}{\partial z^2} = \mu_0 \epsilon_0 \frac{\partial^2\vec{E}}{\partial t^2}

thus

\LARGE \frac{\partial^2 \vec{E}}{\partial x^2}+\frac{\partial^2 \vec{E}}{\partial y^2}+\frac{\partial^2 \vec{E}}{\partial z^2} = \mu_0 \epsilon_0 \frac{\partial^2\vec{E}}{\partial t^2}

noting that \mu_0 \epsilon_0 = \frac{1}{c^2}

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

subtracting \frac{1}{c^2} \frac{\partial^2\vec{E}}{\partial t^2} from both sides yield the familiar d’Alamberitan form:

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

another solution to Maxwell’s equation for a light wave travelling through a vacuum can be stated in term of the MAGNETIC field, where B is the vector of magnetic field intensity and direction, and it yields a d’Alambertian of similar form:

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

since one could express the the d’Alambertian with the letter E (for the Electric field) or B (for the Magnetic field) I’ll use \Psi as a generalization for either:

\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

This sets the stage now for a more rigorous and accessible demonstration of Invariance of Maxwell’s equations under the Lorentz Transformations at the heart of special relativity (be it the Lorentz Aether version or the Einstein version).

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!