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modest · 05-10-2009

Quote:

Originally Posted by Doctordick

Could I interest you in going through my presentation line by line?

Sorry for the delay.

I've been telling Michael Mooney for some time that he should not pass judgment on relativity without first understanding it. Taking my own advice I would like to withhold any bias or opinion on your formalism until I understand it which is not yet the case. To be honest: for as much time as I've spent on Hypography, I've avoided examining your work probably because it seems like a rather involved undertaking.

Quote:

Originally Posted by Doctordick

I will show explicitly that my picture is not only totally consistent with special relativity but actually requires that special relativity be valid.

My knee jerk reaction and my first question: if your fundamental equation can be used to derive a $\mathbf{R}^{1,3}$ space-time then would I be correct that you would not object to using it—at the very least as a matter of convenience. I realize this question sidesteps the point of your presentation, but I'd just like to be sure you're not rejecting the utility of Minkowski.

Quote:

Originally Posted by Doctordick

It turns out that we are quite lucky in that the consequences of the above symmetry have already been completely worked out long ago by others. Notice that, if one ignores the Dirac delta function (as it has no spacial extension) my fundamental equation is a simple linear wave equation in four dimensions with wave solutions of fixed velocity. The constraint spoken of above is exactly the same constraint placed on the conventional Euclidean mental model of the universe by the fixed speed of light in Maxwell's equations. As we all know, if we constrain ourselves to linear scale changes, it turns out that there exists one very simple (and unique) relativistic transformation which maintains a given fixed velocity for all reference frames moving with constant velocity with respect to one another.

Taking as a postulate (simply because I have no idea how you derived these things) that some wave propagates at a fixed and finite speed for multiple inertial frames then I have no doubt the Lorentz transformations can and must be derived.

Quote:

Originally Posted by Doctordick

We need to have a formula for translating coordinate points in the first frame, $(x,y,z,\tau)$ , into the identical points represented in the second frame, which have to be $(x',y',z',\tau')$ in a way which continues the validity of the fundamental equation. In order to do that, I will use the fact that the fundamental equation is (sans interactions) a wave equation where the wave velocity, v_? is constant; thus, we can use an opening circumstance where (at t=0), $\Psi$ , the wave function of an object consisting of a single element (i.e., all interactions with the rest of the universe are being ignored), consists of a spike at the origin in both frames and is zero elsewhere (that means we are starting with the origins of both frames of reference exactly aligned origins). Anyone familiar with wave equations understands that the solution here is quite simple, $\Psi(t)$ is thereafter a spike at r=tv_? (where r is the radius of a four dimensional sphere centered on the origin) and zero elsewhere from then on. (Think of a flashbulb going off at the moment the origins of the two coordinate systems are exactly in the same point and then picture the sphere of light expanding at the speed of light.) The fact that our case is a four dimensional sphere is only of passing significance here, as we are still speaking of uniform radial expansion: i.e., the radius to that pulse of probability must be given by $r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2}$ . Please notice that this means that once a scale is set for one coordinate, it is likewise set for all the others (otherwise we wouldn't have a sphere).

Inertial frames of reference are Euclidean in both classic and relativistic mechanics and there being 4 proposed spatial dimensions then I can certainly accept $r=\sqrt{x^2+y^2+z^2+\tau^2}$ would make a 3-sphere in S. If time and velocity are typical (Newtonian) and the probability wave/sphere expands at v_? then I understand and agree r = v_?t.

The question I guess I have is how your fundamental equation insists that the origin of S' remain the origin of our 3-sphere as time progresses. In other words: why isn't the sphere moving in S'? It almost feels like you've asserted the principle of relativity. It would help me to see how you rule out the possibility of $r=v_?t=\sqrt{(x'+vt')^2+y'^2+z'^2+\tau'^2}$ where v_? would be constant relative to the center of the sphere but not relative to S'. That case of course giving x' = x-vt and t' = t.

And... speaking of Galilean transformations:

Quote:

Originally Posted by Doctordick

At this point, we have deduced the fact that observers from coordinate systems moving with respect to one another will totally agree with the transformations implied by the standard relativistic relationships. Virtually the only difference lies with the actual limiting velocity. Is v_? required to be c?

Before we can actually answer that question, we need to know exactly where the number “c” came from. The speed of light is c, thus it is, in our analysis, the apparent speed of that massless oscillator (what we have called v_?, the propagation velocity given to the changes of that probability function). It should be clear that the actual value of this velocity requires not only the definition of a clock (which we have done) but also a specification of a standard unit of time. It is here where the difference between my analysis and the conventional approach show up. In my approach, t is a free evolution parameter having absolutely nothing to do with actual physics of the issue. The velocity v_? can have any value one wishes (defining the standard unit of time or length is an open issue); however, once those units are defined, v_? will be exactly the apparent velocity of a massless entity and the limiting maximum velocity of any physical object in those specific units and that is exactly the underlying definition of c.

Where c approaches infinity the Lorentz transformations approach the Galilean transformations,

$\lim_{c \to \infty}\begin{cases} x' = \dfrac{x-vt}{\sqrt{1-\frac{v^2}{c^2}}} \ \Longrightarrow & x' = x-vt \\ \\ t' = \dfrac{t-\frac{vx}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}} \ \Longrightarrow & t'=t \end{cases}$

Usually it is left to observation to show that the invariant speed is finite. Wikipedia goes as far as saying experiment is necessary,

Quote:

Originally Posted by wikipedia—Lorentz Transformations

If $\kappa = 0$ then we get the Galilean-Newtonian kinematics with the Galilean transformation, where time is absolute, $t' = t$ , and the relative velocity v of two inertial frames is not limited...

Only experiment can answer the question which of the two possibilities, $\kappa = 0$ or $\kappa <= 0$ , is realized in our world. The experiments measuring the speed of light, first performed by a Danish physicist Ole Rømer, show that it is finite, and the Michelson–Morley experiment showed that it is an absolute speed, and thus that $\kappa < 0$ .

But, your post seems to leave open the possibility of an infinite v_?. Is this not an issue?

~modest

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