An “analytical-metaphysical” take on Special Relativity!

[Doctordick]

Henri Poincareé once said, “One geometry can not be more true than another; it can only be more convenient.” The geometry I have used in all of my “analytical metaphysical” presentations has consistently been a Euclidean geometry. Even physicists, who hold that it is invalid if one's intention is to represent reality exactly, use it quite often so it still seems to qualify as “convenient”. I will show explicitly that my picture is not only totally consistent with special relativity but actually requires that special relativity be valid.

 

When I began the derivation of my fundamental equation, I was attempting to develop an analytical model of the concept I define to be an explanation. As all theories may be seen as epistemological constructs based upon ontological elements and it is the common philosophic position of metaphysics that ontological elements can not be proved, I took the collection of ontological elements standing behind any explanation to be “unknowns” and then attempted to set down the relationships those unknowns had to obey: the result was the derivation of my fundamental equation. The presentation of that proof may be found here; where the following relationship is both defined and derived

[math]\left\{\sum_i \vec{\alpha}_i \cdot \vec{\nabla}_i + \sum_{i \neq j}\beta_{ij}\delta(\vec x_i - \vec x_j)\right\}\vec{\Psi} = K\frac{\partial}{\partial t}\vec{\Psi} = iKm\vec{\Psi}[/math]

 

I have already given a specific proof that Schrödinger's equation is an approximate solution to my fundamental equation (including specific definitions of “energy”, “momentum” and “mass” which I define as specific terms in that equation without making any assumptions as to the applications to any specific explanation) and expanded that proof to show that a three dimensional representation of any explanation may be achieved via a three dimensional representation of that self same equation: i.e., it leads to a three dimensional form of Schrödinger equation.

 

At this moment, the fundamental equation has been shown to generate the standard mental model of reality (the standard world view) from a totally unknown universe given nothing but a totally undefined stream of data transcribed by a totally undefined process. During that development, I essentially defined "reality" to be a set of numbers (the numerical labels of those undefined ontological elements) and I decided to view those numbers as points plotted in a four dimensional Euclidean space: [imath](\hat{x},\hat{y},\hat{z},\hat{\tau})[/imath]. I refer to these points as elemental entities in the sense that they cannot be reduced: i.e., they constitute numerical references to those undefined ontological elements. When I did that I used the fact that my model must encompass any alteration in the data which was conceivable (that would be the transcription performed by the totally undefined process: i.e., our senses). In particular, I concerned myself with addition of a given constant to each and every number in a given observation. This was shown to be analogous to redefinition of the origin and I used that fact to obtain my fundamental equation.

 

During that process, I also showed that the Dirac delta function could be used as a rational representation of any rules that universe might have. This step provided another subtle alteration in the original numbers which was somewhat unexpected. Notice that multiplication of each of the original numbers representing any observation by any specific constant also has utterly no consequences within this model: i.e., the model is scale invariant. This scale invariance comes about purely because the delta function is non zero only when the argument is zero and zero has no scale. For the moment I will concern myself with the impact of scale invariance on objects (an object being defined to be a coherent collection of elemental entities which can be regarded as an entity unto itself).

 

I have pointed out a number of times that, if the data belonging to a given observation could be divided into two (or more) sets having negligible influence on one another, those sets could be examined independently of one another: i.e., these collections would end up being constrained by exactly the same relationship which constrained the original universe. This is to say that these subsets (or “objects”) could be analyzed as a universes unto themselves; however, there is a subtle problem here: the fundamental equation was constrained (see appendix 3 of the original proof) to be valid only in the rest frame of the universe. The central issue here is that the two collections of elemental entities either have significant influence on one another or they do not. If they do not have any significant influence on one another, the constraint that the equation is only valid in the rest frame of “the universe” cannot be a valid constraint as either object may be considered to be a universe unto itself: i.e., the rest frame of one collection of elemental entities may not be the same as the rest frame of the other. The solution to this problem lies with the scaling of the geometry between the two systems: there must exist a consistent way of converting a solution in one system to a solution in the other independent of any influence between the two.

 

Now, I have already shown that a given solution in the rest frame is easily transformed to a solution where the frame of reference is no longer at rest. Such a transformation is simply obtained via multiplication of [imath]\vec{\Psi}[/imath] by the simple function

[math]\prod_{j=1}^n e^{i\frac{Px_j}{n\hbar}}[/math].

 

This change in [imath]\vec{\Psi}[/imath] will simply add P/n to the momentum in the x direction of every elemental entity in the universe (the universe consisting of the elemental entities which make up that independent object). In other words, the transformation simply adds P to the momentum of the object and thus the object is no longer at rest in the rest frame used to solve for [imath]\vec{\Psi}[/imath]. Thus it is that we can always transform a solution in the rest frame of one object to a solution in the rest frame of the other (note that the transformation also requires a change in energy which is just as easily obtained).

 

The actual problem here is that the fundamental equation is no longer valid (we are simply no longer in the rest frame of the original object and our altered [imath]\vec{\Psi}[/imath] is thus no longer a solution to the correct equation). What we have is the fact that our mental model of reality must include the fundamental transformation such that all solutions will transform to valid solutions of the fundamental equation in the center of mass of any collection of data (so long as all outside influence can be ignored). This appears to imposes a major constraint on the character of the possible solutions [imath]\vec{\Psi}[/imath]. In reality,it does not as the scale invariant nature of our mental model provides a straight forward resolution of the difficulty.

 

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.

 

The velocity in our four dimensional “wave equation” is fixed by the value of K in our representation. (Notice that, in my derivation of Schrödinger's equation, I set [imath]c=\frac{1}{K\sqrt{2}}[/imath].) For the moment (since K is actually a totally open parameter) I will set this constant velocity to v_?.

 

In order to solve for the required transformation, consider uniform motion in the x direction (remember, we are still actually working in a four dimensional representation so x can be in any direction (though I will not really worry about tau as in the final analysis any dependence on tau will be integrated out anyway so tau is, in some sense special; particularly as it is a figment of our imagination created solely to allow representation of multiple occurrences of valid elemental entities). In the following picture, the tau axis is not shown. We just can't really show four orthogonal axes in a conventional picture. In this case, tau is simply another axis orthogonal to x and obeys exactly the same relationships as do the y and/or z axes: i.e., [imath]\tau'=\tau[/imath].

 

We need to have a formula for translating coordinate points in the first frame, [imath](x,y,z,\tau)[/imath], into the identical points represented in the second frame, which have to be [imath](x',y',z',\tau')[/imath] 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), [imath]\Psi[/imath], 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, [imath]\Psi(t)[/imath] 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 [imath]r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2}[/imath]. 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).

 

Thus the wave function is non zero only on the surface of a sphere expanding at a specific velocity (which I am calling v_? for the time being). What is important here is that this must be true in both frames (if it is not true in the primed frame, the non-zero portion of [imath]\Psi(t')[/imath] will not be on the surface of an expanding sphere). That is, both frames must yield exactly the same probability distribution; it is the two frames of reference which are different, not the probability of finding that elemental entity.

 

First, it is quite easy to show that the transformations in y, z and [imath]\tau[/imath] are trivial as they must always line up exactly with the same points on the unprimed axes (an entity not moving in one of those directions in the unprimed coordinate system can not be moving in those directions in the primed coordinate system): i.e., y'=y, z'=z and [imath]\tau'=\tau[/imath] (the scale of these coordinates must be identical). The only problems occur with the x axis and t. Note that, in my picture (though I can produce x, t diagrams) t is not an axis of my coordinate system; it is instead, a parameter of evolution, a distinctly different concept. It should be clear to the reader that there exists no way to guarantee that t in the primed coordinate system is identical to t used in the unprimed coordinate system (before we can discuss that issue one must first explain how time is to be determined). Nevertheless, it is fairly easy to show that the transformation from one coordinate system to the other can be no more complex than [imath]x'=\alpha x -\beta t[/imath] and [imath]t'=\gamma t -\delta x[/imath].

 

For those who don't believe that, consider the terms in a power series expansion of some supposed arbitrary function. The constant terms of that power series can be dropped as they move the primed origin at all times even t=0 where we have already defined it to be in exactly the same position as the unprimed coordinate system. Furthermore, all terms not linear in x or t will generate changes which will create different answers when we simply transform the origin (something both coordinate systems must allow). Thus it follows that the transformations just given are the most complex possible. Our problem becomes quite simple: if we can lay out four independent equations involving the coefficients alpha, beta, gamma and delta, we can solve for those elements.

 

The first thing I need to point out is that the position of the point, x'=0, being the origin of the primed coordinate system must be at x=vt in the unprimed coordinate system as that is the definition of the primed frame's movement in the unprimed coordinate system. That implies that [imath]\alpha(vt)-\beta t = 0[/imath]: i.e., that is exactly the transformation which yields the origin of the primed coordinate system which is, by definition x'=0. From that we can immediately deduce that [imath]vt=\frac{\beta}{\alpha}t[/imath] or, dividing by t, that [imath]v=\frac{\beta}{\alpha}[/imath]. This is first of those four equations we are looking for.

 

We now need to lay out three additional valid independent equations involving the unknown coefficients. We know that both coordinate systems must yield a spherical surface originally defined by [imath]x^2+y^2+z^2+\tau^2=v_?^2t^2[/imath]: i.e., that surface must transform exactly into the surface [imath]x'^2+y'^2+z'^2+\tau'^2=v_?^2t'^2[/imath] in the primed coordinate system. Simply performing the transformation defined above must yield exactly that result. When we use the proposed transformations perform the transformation (substitute the explicit forms for each primed coordinate) we get the following relationship: [imath](\alpha x -\beta t)^2+y^2+z^2+\tau^2=v_?^2(\gamma t -\delta x)^2[/imath] which expands algebraically directly into

[math]\alpha^2x^2-2\alpha x\beta t+\beta^2 t^2+y^2+z^2+\tau^2=v_?^2[\gamma^2t^2-2\gamma t \delta x+\delta^2 x^2][/math]

 

or, collecting terms related to the unprimed coordinates of interest, we get

[math](\alpha^2-v_?^2\delta^2)x^2+y^2+z^2+\tau^2=v_?^2t^2\left(\gamma^2-\frac{\beta^2}{v_?^2}\right)+2xt(\alpha \beta -v_?^2\gamma \delta)[/math]

 

which, as it must still yield that spherical surface as represented in the unprimed frame must be exactly [imath]x^2+y^2+z^2+\tau^2=v_?^2t^2[/imath]. This fact immediately yields three additional equations involving those four coefficients.

 

So we now have four equations in four unknowns:

[math]v=\frac{\beta}{\alpha}\quad ; \quad \alpha^2-v_?^2\delta^2=1 \quad ; \quad \gamma^2-\frac{\beta^2}{v_?^2} = 1\quad and \quad \alpha \beta -v_?^2\gamma \delta = 0 \quad which\quad is \quad\gamma=\frac{\alpha\beta}{v_?^2\delta}[/math].

 

You can then eliminate [imath]\beta[/imath] by substituting [imath]\beta=\alpha v[/imath] which is obtained from that first equation. This reduces the set to three equations in three unknowns:

[math]\alpha^2-v_?^2 \delta^2=1\quad ; \quad\gamma^2-\left(\frac{v}{v_?}\right)^2\alpha^2=1\quad and \quad\gamma=\frac{\alpha^2v}{v_?^2\delta}[/math]

 

Eliminating [imath]\alpha[/imath] via [imath]\alpha^2=1+v_?^2\delta^2[/imath] (obtained from the new first equation) reduces the set to two equations in two unknowns:

[math]\gamma^2-\left(\frac{v}{v_?}\right)^2(1+v_?^2\delta^2)=1\quad and \quad\gamma=\frac{(1+v_?^2\delta^2)v}{v_?^2\delta}[/math].

 

And finally, we can eliminate [imath]\gamma[/imath] via [imath]\gamma^2=\frac{(1+v_?^2\delta^2)^2}{v_?^2\delta^2}\left(\frac{v}{v_?}\right)^2[/imath] (obtained by squaring the right hand equation of the two above). We thus arrive at a single equation with one unknown, “[imath]\delta[/imath]”:

[math]\frac{(1+v_?^2\delta^2)^2}{v_?^2\delta^2}\left(\frac{v}{v_?}\right)^2 -\left(\frac{v}{v_?}\right)^2(1+v_?^2\delta^2)=1[/math].

 

If you multiply this equation through by [imath]\frac{v_?^2\delta^2}{(1+v_?^2\delta^2)}[/imath], you will obtain

[math](1+v_?^2\delta^2)\left(\frac{v}{v_?}\right)^2 -v_?^2\delta^2\left(\frac{v}{v_?}\right)^2=\frac{v_?^2\delta^2}{(1+v_?^2\delta^2)}[/math].

 

The left hand side clearly reduces to [imath](v/v_?)^2[/imath]. Thus if we multiply through by [imath](1+v_?^2\delta^2)[/imath] we obtain a very simple result:

[math]\left(\frac{v}{v_?}\right)^2 +v_?^2\delta^2\left(\frac{v}{v_?}\right)^2=v_?^2\delta^2\quad or \quad\left[1-\left(\frac{v}{v_?}\right)^2\right]v_?^2\delta^2=\left(\frac{v}{v_?}\right)^2[/math]

 

which is easily solved for [imath]\delta[/imath] (just divide through by the coefficient of [imath]\delta[/imath] and take the square root of both sides of the equation. The final result is:

[math]\delta= \left(\frac{v}{v_?}\right)\frac{1}{v_?\sqrt{1-\left(\frac{v}{v_?}\right)^2}}[/math].

 

Since [imath]\alpha^2=1+v_?^2\delta^2[/imath], we know that

[math]\alpha^2= 1+\left(\frac{v}{v_?}\right)^2\frac{1}{\left[1-\left(\frac{v}{v_?}\right)^2\right]}[/math].

 

Use “common denominators” to add the two terms above and you will discover that the square root of the result is:

[math]\alpha= \frac{1}{\sqrt{1-\left(\frac{v}{v_?}\right)^2}}[/math].

 

Finally, since [imath]\beta=\alpha v[/imath] and [imath]v_?^2\delta=\alpha v[/imath], it is quite obvious that [imath]\gamma=\alpha \frac{\beta}{v_?^2\delta}[/imath] clearly implies [imath]\gamma=\alpha[/imath].

 

At this point, we have solved the problem; from the above it is quite clear the only possible relationship which can exist between moving coordinate system (moving at constant velocity v) is given by;

[math]x'=\frac{1}{\sqrt{1-\left(\frac{v}{v_?}\right)^2}}[x-vt]\quad \quad t'= \frac{1}{\sqrt{1-\left(\frac{v}{v_?}\right)^2}}\left[t-\left(\frac{v}{v_?}\right)\left(\frac{x}{v_?}\right)\right][/math]

 

where the direction of the motion is defined to be along the x coordinate. All other coordinates (and note that the dimensionality can be carried to any level) map directly into one another: i.e., y'=y, z'=z, [imath]\tau'=\tau[/imath] etc. Just for convenience, one can define [imath]sine\theta = v/v_?[/imath] as this makes the square root in the above equations equal to [imath]cos(\theta)[/imath] yielding a simpler representation. If that constant velocity v_? were to be c, those would become exactly the standard relativistic transformations.

 

I did this derivation in detail for one very simple reason: most publications merely publish the results and imply that their truth is support for Einstein's theory of special relativity. I prefer to view it as nothing more than the result of requiring a very specific symmetry: namely that some specific velocity must be the same in any inertial coordinate system. These relations are exactly the standard Lorentz transformations Einstein's theory of special relativity was concocted to explain. The fact that my model requires them for internal consistency implies that my model actually requires any conceivable universe to satisfy the relations associated with special relativity.

 

One very specific cavil still remains: in the derivation above, I set the velocity of a free element (that is neglecting interactions implied by the Dirac term, [imath]\sum_{i \neq j} \delta(\vec{x}_i -\vec{x}_j[/imath]) equal to v_? whereas the actual velocity is related to K , what seems, on first examination, to be a free parameter.

In fact, if we multiply through by [imath]-\hbar c[/imath] (which clearly has utterly no impact on the solution as it multiplies every term) and make the following definitions directly related to constants already defined,

[math]m=\frac{q\hbar}{c}[/math] , [math]c=\frac{1}{K\sqrt{2}}[/math] and [math]V(x)= -\frac{\hbar c}{2q}G(x)[/math]

 

it turns out that the equation of interest (without the introduction of a single free parameter: please note that no parameters not defined in the derivation of the equation have been introduced) is exactly one of the most fundamental equations of modern physics.

[math]\left\{-\left(\frac{\hbar^2}{2m}\right)\frac{\partial^2}{\partial x^2}+ V(x)\right\}\vec{\phi}(x,t)=i\hbar\frac{\partial}{\partial t}\vec{\phi}(x,t)[/math]

K is actually not a free parameter because we have not yet defined the actual measure of t. At this moment, t is an evolution parameter and is free to have any relationship with distances desired: i.e., velocities are essentially not defined. In order to relate that parameter to ordinary human perceptions, we have to design a mechanism to measure that parameter (essentially for reference purposes): i.e., it is required that a standard “clock” be defined before one can compare velocities as seen by different observers. In order to do that, one has to understand a few of the dynamic constraints implied by the model I have presented. In the design of my clock, for simple convenience, I will continue to use v_? as the fixed velocity implied by my fundamental equation.

 

First, is the issue of “objects”. I have defined an object to be a coherent collection of elemental entities which can be regarded as an entity unto itself. This implies that the elemental entities going to make up that object must, for practical purposes, be traveling in the same direction. If that is not true, our “object” will not remain a coherent collection of elemental entities but will rather disperse. How that internal coherence is maintained is not the central issue at this moment; what is important is that “objects” so defined cannot exist unless such coherence can be maintained. So I will merely (at least for the moment) presume “objects” can exist. Our standard clock will be “an object” thus, if objects cannot exist, a standard clock can not exist.

 

To begin with, during the design stage, I want my standard clock to be “at rest” in my coordinate system. In the deduction of Schrödinger's equation, I ended up integrating over all tau dependence and defined momentum in the tau direction to be mass. Clearly the fact that my clock is to remain a coherent object requires that, if it is to be “at rest” the major components must have mass: i.e., the important component of the momentum of the majority of the underlying elemental entities must be in the tau direction.

 

I will define my standard clock to consist of two components: a mirror assembly and an oscillator. Both components are coherent macroscopic assemblies of elemental entities. The oscillator will have zero rest mass; therefore, every elemental entity which is part of the oscillator will have exactly zero momentum in the [imath]\tau[/imath] direction. The mirror assembly, on the other hand, will be massive: i.e., every elemental entity making up the mirror will have non zero mass. It follows that every event making up the mirror assembly must have significant momentum in the [imath]\tau[/imath] direction.

 

The probability of finding the collection those elemental entities in any specific positions is given by the magnitude of [imath]\vec{\Psi}[/imath] squared; thus, since that result cannot be a function of [imath]\tau[/imath], the macroscopic cross section of both structures perpendicular to [imath]\tau[/imath] must be uniform and their extension in the [imath]\tau[/imath] direction must be infinite. This being the case, a description of their three dimensional cross section completely describes their macroscopic shape. The standard “clock” will be defined to be the entity pictured below.

 

The clock is further defined by the following constraints: all elemental entities making up the mirror assembly have non negligible momentum in the [imath]\tau[/imath] direction (they are massive entities) and negligible momentum in the x, y, and z directions (this all being relative to the macroscopic scale of the clock). On the other hand, all elemental entities making up the oscillator will have exactly zero momentum in the [imath]\tau[/imath] direction (they are massless entities), non-negligible momentum in the y direction and negligible momentum in the x and z directions. Furthermore, the non-negligible momentum of the oscillator in the y direction will be negligible with respect to the momentum of the mirror assembly in the y direction. We are totally free to make these assertions as we are defining an object and, in the absence of contradiction, such an object could certainly exist.

 

It follows from the above that, in macroscopic terms, although every elemental entity has exactly the same velocity, the mirror assembly is essentially an object moving parallel to the [imath]\tau[/imath] axis while the oscillator is an object (a coherent massless entity) moving parallel to the y axis. Since the entire assembly is infinite and uniform in the [imath]\tau[/imath] direction, motion in the [imath]\tau[/imath] direction yields utterly no changes in the structure of any part of our clock.

 

If we now postulate that microscopic interactions (created by those Dirac delta interactions which we are essentially ignoring) between the mirror and oscillator are capable of reversing the sign of the oscillator's momentum upon contact with the mirror, the oscillator will bounce back and forth between the legs of the mirror assembly. Our standard clock will clearly have a period of 2L₀/v_?.

 

Since every event in the system described has non-negligible momentum only in the [imath](y,\tau)[/imath] plane, we can display all important dynamic phenomena while considering only a cross section in that plane. Thus let us examine our standard clock as it appears in that cross section, paying particular attention to the associated velocity vectors. Notice that although no constraint has been imposed on the sign of the momentum of elemental entities making up the mirror (they can be in either direction of [imath]\tau[/imath]) , each entity making up the mirror must have momentum either in the plus or minus [imath]\tau[/imath] direction. As the sum of all events must maintain a coherent whole (by definition, our object is coherent over the time and space considered) we need only focus on the collection of entities having the same sign. For the sake of this graphic representation, I choose that sign to be positive.

 

Anssi's video of a stationary clock. http://www.youtube.com/watch?v=jbNqSnVgUZs

 

It is interesting to note that T, the period of our standard rest clock, is identical to 1/v_? times the distance the mirror moves in the [imath]\tau[/imath] direction during one clock cycle. Although actual position in the [imath]\tau[/imath] direction is a meaningless concept (as the entire object is infinite and uniform in that direction), our standard clock appears to be measuring the implied displacement of the mirror over time in that direction: i.e., we can infer that the mirror has moved a distance 2L₀ in the [imath]\tau[/imath] direction during one complete cycle. This will turn out to be a very significant fact since the scale of the [imath]\tau[/imath] dimension is set by the form of the fundamental equation (setting the scale of any dimension sets the scale of all the others) .

 

Our mechanism is certainly analogous to a standard clock since it will keep time if we can count the number of times the oscillator bounces back and forth (v_? is a fixed velocity defined by our K and L₀ is a defined length). Furthermore, the image is clearly that of a massless object (?a coherent pulse of photons?) bouncing back and forth between two reflective surfaces of a massive mirror, which constitutes the common construction of an accurate clock under the conventional physics viewpoint (an electro-magnetic oscillation in a defined cavity).

 

Now consider an identical standard clock in a moving reference frame: i.e., identical to the clock just described except for the fact that I will allow the momentum of the mirror assembly to be non negligible in the y direction. I use the y direction only because it is convenient to the drawing: i.e., the movement of the massless pulse is in the same direction as the clock, an issue which makes a drawing in two dimensions easy. If anyone is concerned about the issue, I will assist in clarifying the problem later.

 

Since all objects are uniform and infinite in the [imath]\tau[/imath] direction, it is reasonable to suppress actually drawing the objects themselves and, instead, deal entirely with the various displacement vectors. These displacement vectors are essentially v_?t where t is no more than a parameter of evolution: i.e., its scale is totally immaterial. It should be clear that these vectors contain all relevant information needed to predict the time evolution of the device. The only issue of great importance here is that, anytime the displacement vectors lead to identical (x,y,z) coordinates (which, in the [imath](y,\tau)[/imath] plane which is being shown, means simply that two entities have identical y coordinates), microscopic interactions can occur between our macroscopic object anytime they lie on the same vertical line in these drawings (such a line specifies all points with the same y coordinate). This is important because all macroscopic objects are actually infinite and uniform in the [imath]\tau[/imath] direction, an issue which is no longer being explicitly shown in the drawing. Essentially, in the following drawings, x and z of every point in the picture is always identical so we need only concern our selves with a line at a y coordinate and the directions of the displacement vectors (essentially the angle [imath]\theta[/imath] they make with the tau axis).

 

Note that the length of the moving clock is shown to be L'. This has been done because we know that the symmetry discussed in the previous section must require the Lorentz contraction to be a valid on any macroscopic solution if interactions with the rest of the universe may be neglected (up to this point the model was scale invariant): i.e., when we solve the problem in the moving clocks system we want the length of the clock as seen by the observer in that moving frame to be L₀. We use the scale freedom in our model to set that length (as seen from the rest system) to be L'; then and only then can we seriously call the clocks identical. This will require [imath]L'=L_0\sqrt{1-sin^2(\theta)}[/imath] (the inverse of the relativistic transformation deduced earlier: i.e., in order to get the length of the moving clock in the primed coordinate system we have to multibly by [imath]\alpha[/imath]). Note that [imath]sin(\theta)[/imath] is exactly the apparent velocity of the moving clock divided by the velocity of the elemental entities, v_?, which actually has nothing to do with time. Since all velocities are v_?, it follows directly that d₁ + d₂ = S. Please note that everything so far is being graphed as seen in the frame of the rest clock: i.e., S=v_?T_m, where T_m is the period of the moving clock as seen from the rest frame.

 

Notice that the following geometric figure is embedded in the previous diagram.

 

Anssi's video of a moving clock. http://www.youtube.com/watch?v=-UDrWCIgmTk

 

Once again, since the triangles A and B are identical as are the triangles a and b, we discover that one clock cycle, rather surprisingly, measures exactly the length of time it takes the mirror to move the distance 2L₀ in the [imath]\tau[/imath] direction. Again, although our standard clock was originally designed to measure time, it appears that what is actually being measured here is inferred displacement in the [imath]\tau[/imath] direction. Once again, I assert that this is a very significant fact. One cycle of both the moving clock and the rest clock measure exactly an inferred displacement of 2L₀ in the [imath]\tau[/imath] direction; however, the time required for our moving clock to accomplish this feat is given by the length of the displacement vector S which is very clearly longer than 2L₀. It follows, from the fact that everything here moves at the velocity v_?, that S = v_?T_m or, the period of the moving clock (as seen in the rest observer's frame) is given by ,

[math]T_m=\frac{S_m}{v_?}= \frac{2L_0}{\sqrt{1-sin^2(\theta)}}\left(\frac{1}{v_?}\right)=\frac{T}{\sqrt{1-sin^2(\theta)}}[/math]

 

Which happens to be exactly the result expected from the standard Lorentz relation: i.e., the moving clock appears to run slow by exactly the factor [imath]\sqrt{1-sin^2(\theta)}[/imath]. T_m is larger than T (the period of our standard rest clock) which means the moving clock appears to run slow when viewed from the frame where the original clock was at rest, the frame being used to display events here. An observer in the moving frame will call this particular period T as it constitutes his standard of time (the period of a defined standard clock at rest in his frame).

 

At this point, it is very important to examine the reverse case: using the moving “clock” as his standard, what does an observer in the moving frame obtain for the period of the rest clock we started with? Before we can accomplish that result, we need to know exactly how the the moving observer determines the length of his clock.

 

There is a subtle point here (having to do with the central character of relativity) that I have never heard a professional physicist point out. If the clock has a device which will display the clocks reading, there will be utterly no argument about what the reading on a clock will be. All observers will agree as to what that reading is. What they will argue about is “when” and “where” the event of that reading is in his own personal coordinate system. Here “when” can be defined either by the evolution parameter I have defined or by the t coordinate in Einstein's space-time; either way, the two observers will not agree. The problem is intimately tied to the definition of simultaneity.

 

The moving observer will call the period of his standard clock T and the length of his clock L₀; however, there is a second point buried here which is practically never mentioned. When the moving observer goes to actually measure the distance between the mirrors of his standard clock (that resonance cavity mentioned earlier), the rest observer can watch him doing it. Suppose we request that the moving observer mark that length on a physical ruler. The observer will have to get the ruler, lay it aside the clock and mark the positions of the two mirrors on the ruler. The central point being “when” does he do this. Suppose, just for the sake of argument, the moving observer lines up his ruler to measure the distance and, when he is satisfied that everything is correct, he will arrange for flash bulbs to go off simultaneously (one at each end of the distance to be measured) in order to let us know "when" he actually performed the measurement.

 

It is important that the reader should understand that the outcome of that act clearly depends on exactly how he defines simultaneity. In our diagram of the dynamics of the circumstance, what value of y' for the left hand mirror will he see as being simultaneous with a given specific position of the right hand mirror? This is a serious issue which has to be examined very carefully.

 

First of all, the “moving observer” has no way to determine that he is moving so long as our statement that the interactions with the rest of the universe are negligible.

 

Just as an aside, everyone who reads this should realize that, in reality, this is a false statement. A photon from the farthest star is not really a negligible fact when the question “are we moving?” is being analyzed. In fact, people are today measuring our movement through the universe relative to the microwave background radiation. So the question above is actually rather unrealistic.

 

However, if the observer ignores that fact and works from the perspective that he has absolutely no way of establishing his motion, he will presume he is at rest (with respect to the universe). This is essentially what he has done when he “postulates” that the speed of light is a constant (that is, when he presumes its speed is the same in every direction). The problem is that has no way to establish simultaneity if all observations involve that speed of light and he does not know his motion with respect to the universe. So he takes the short cut and presumes that the velocity is exactly the same in both directions.

 

So let us go back to my diagram above. Since he presumes he is at rest, (and postulates that speed of light is the same in all directions) he will be driven to make the assumption that reflection off the right mirror face is simultaneous with the position of the left face at the midpoint of displacement S: i.e., he will assume that the time for the oscillator to reach the right mirror is exactly equal to the time it takes for the oscillator to return to the left mirror (I have marked the point y', which he will take to be the simultaneous position of the left mirror, as point #1 in the above diagram). If the flashbulb on the right hand mirror goes off exactly as the oscillator reaches that mirror, he will set the flashbulb on the left hand mirror to go off at point #1.

 

The rest observer will totally disagree with the moving observer's measurement. The rest observer will say that the oscillator took considerably longer to get to the right mirror than it took to return to the left. From his perspective, the flash bulb on the left hand mirror should have been set to go off at the y' indicated by point #2. From the rest observer's perspective, the moving observer has first marked his ruler at point #1 and then waited a considerable fraction of time before marking the other end of the ruler. The markings are not being made simultaneously. The consequence is clearly that the distance between the two marks on the ruler (as seen by the rest observer) are exactly equal to the relativistically corrected distance between the two mirrors, L'₀.

 

This is an issue I have never heard any professional physicist discuss. The consequences are actually quite remarkable. The distance between the two points actually used by the moving observer to mark his ruler (as seen by the rest observer) is [imath]L'_m =L_0/cos(\theta)[/imath]. The point being that, anytime the moving observer goes to measure anything, he will perform exactly the same procedure: i.e., from the rest observer's perspective, he will measure some distance d and then numerically refer to the distance as being [imath]dcos(\theta)[/imath] in his coordinate system. Thus it is that when he goes to measure the rest observer's clock, which is L₀ in our graph (the graph is laid out in the rest observer's scale) he will call the distance [imath]L_0 cos(\theta)[/imath] in his coordinate system. Thus it is that he will hold forth that his clock is correct and the (so called) rest clock appears to be contracted by exactly the standard Lorentz contraction. The only reason I went through that, was because many people have difficulty comprehending how the moving observer (who' entire perspective appears to be Lorentz contracted) can see the rest observer as Lorentz contracted. (I strongly suspect that many trained physicists can't see it either; they are just throughly indoctrinated in conventional relativity. But that is only an opinion :lol:)

 

The significant issue here is that, when an interaction takes place, everyone (moving or not) sees the same event. The only difference in their interpretation of the event is that they use a different coordinate system to represent it. Since the drawing I have presented above is based upon the coordinate system at rest with the original “rest clock”, we need to be very careful as to exactly which locations (the exact events) the moving observer will use to establish his coordinate system.

 

Finally, in order for the moving observer to measure the period of the rest clock, he must receive two signals. Clearly, from his perspective the distance traveled by these two signals are very different. (Once again, I am presenting every specific description of the interactions in the rest frame of the original clock.) In order to assure that we examine the correct displacement vectors, I will use the concept of a half silvered mirror: i.e., we will start the clock with the oscillator at the right mirror headed to the left. After it has reflected off the left mirror and returned to the original position at the right mirror, it will penetrate that mirror and become the second signal to be received by the observer. At t=0, the oscillator, the right reflector and our observer will be defined to be at exactly the same point. It follows that the distance traveled by the first signal is exactly zero. (This is just a convenient place to set up the start circumstance.)

 

The observer will proceed to the right while the oscillator proceeds to the left. After moving a distance L₀, the oscillator will reflect off the left mirror and proceed to the right until it overtakes the observer. The second signal is the arrival of the oscillator. The following diagram lays out the paths of the observer and the second signal:

 

Again, as all velocities along our displacement vectors are v_?, d₁+d₂=S. It follows that, from the perspective of our rest frame, this is exactly [imath]S=2L_0+Ssin(\theta)[/imath] or, solving for S,

[math]S=\frac{2L_0}{1-sin(\theta)}[/math].

 

During this time the observer will have moved a distance of [imath]Ssin(\theta)[/imath]. He however will call this distance [imath]Ssin(\theta)cos(\theta)[/imath] (based upon his personal measurements of length and his perception of what he has measured) and will assume that the clock has receded from him by that distance. Since, as far as he is concerned, his standard clock is correctly measuring time, he will read the elapsed time between the received signals as [imath]cos(\theta)S/v_?[/imath]. He will therefore see the clock as receding from him at a rate given by

[math]\frac{\Delta y}{\Delta t}= v_? sin(\theta)[/math]

 

and everyone agrees as to the relative velocities. I need to point out that, as these two observers do not agree about either their distance measurements nor their time measurements one should find this agreement with regard to relative velocities somewhat surprising (it is not really a trivial issue).

 

Since the moving observer presumes he is at rest, we can conclude that he will subtract, from the elapsed time between signals, [imath]\Delta t [/imath], the time he presumes the signal took to reach him. He will assume the distance is equal to the distance the clock receded from him during one cycle: i.e., [imath]d=sin(\theta) v_?T_r[/imath]. Since he will presume the signal traveled at the rate v_?, he will subtract from [imath]\Delta t[/imath] the factor [imath]sin(\theta) T_r[/imath]. He will thus obtain the period of the rest clock as follows: from the moving observers perspective, the period of the rest clock T_r is given by [imath]\Delta t -sin(\theta) T_r[/imath] or, [imath]T_r=\Delta t/(1+sin(\theta))[/imath] but

[math]\Delta t = \frac{S}{v_?}cos(\theta) = \frac{2L_0}{v_?}\frac{cos(\theta)}{1-sin(\theta)}[/math].

 

Thus it follows that the observed period of the rest clock (as seen by the moving observer) is,

[math]T_r = \frac{2L_0 cos(\theta)}{v_?(1-sin(\theta)^2)}=\frac{2L_0}{v_?}\frac{1}{\sqrt{1-sin(\theta)^2}}[/math]

 

It follows that he will presume what we have called the rest clock is running slow by exactly the same factor which was predicted by the relativistic transformation.

 

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.

 

There is one other thing worth taking note of. When I set off to design a clock, my intention was to duplicate the common idea of a clock used in our world view. The fact that time measurements in coordinate systems moving with respect to one another had to obey the Lorentz relationships was nice, but the fact that they measured exactly the inferred motion of any object in the tau direction was quite surprising. Clearly (by looking at the associated differentials of standard relativity) this was equivalent to the fact that real clocks do in fact measure proper time.

 

At this point it seems quite rational to point out that no one in the history of the world has ever been able to create a real manufactured device which will measure time if one defines time by requiring interacting entities to exist at the same time as per their personal reckoning of time: i.e., interacting entities cannot carry clocks who's readings will indicate their ability to interact (think about the twin paradox). All so called clocks actually measure what a modern physicist calls proper time, commonly referred to as [imath]\tau[/imath]. Proper time (the actual reading on any real clock) is calculated via a line integral taken along the path of that clock of [imath]d\tau[/imath] which is given by

[math]d\tau=-\frac{i}{c}\sqrt{(dx)^2+(dy)^2+(dz)^2-c^2(dt)^2}[/math].

 

The only case in which a clock actually measures “time” is when dx, dy, and dz are, all three, identically zero along the path of that clock. Technically, if a so called clock undergoes any motion at all (and this would reasonably include thermal agitation) where the differentials do not perfectly vanish that clock does not, technically, measure time. Against this, it should be noted that clocks do indeed measure "proper time" exactly, even when in an arbitrarily accelerated frame! I have always found it rather strange that this fact was never pointed out to me during my graduate studies. Nor have I ever heard it proffered by any professional physicists, they all tend, rather to avoid the issue as unimportant. Not exactly the reaction I would expect from “exact scientists”.

 

The problem is very clear, physicists have defined measures for both space and time as if they were independent entities whereas they are not. Strictly speaking, if “same time” is to mean two entities can interact, then “time” is not a measurable thing it is, instead, a totally arbitrary evolution parameter definable only in the frame of a specific observer (from the observers perspective, time is what my clock measures, all other clocks are wrong).

 

You can define time as “what clocks measure” or you can define time as “an interaction can occur if two entities occupy the same position at the same time” but you cannot use both definitions at the same time as they are mutually contradictory.

 

In the final analysis, I hold that my explanation of the relativistic transformations is superior to Einstein's on many counts: first, it is entirely deduced from first principals, second, no postulate as to the constant velocity of light is required and third, it is not only totally consistent with quantum mechanics, but actually includes the deduction of the validity of quantum mechanics from the very beginning and the requirment of relativity flows directly from a proper deduction of quantum mechanics. In addition, all paths in my geometry are allowed whereas, it Einstein's geometry, there exist paths (paths outside the light cone) which can not be followed by any physical entity and must be specifically outlawed as possible paths: i.e., the basic geometry does not correspond one to one with the reality we seem to find ourselves in.

 

(Of course, I am a certified crack pot!) Perhaps I should change that to "a certified sloppy crack pot!" Thank you Anssi for all the valuable proof reading.

 

Have fun guys -- Dick

[Bombadil]

The actual problem here is that the fundamental equation is no longer valid (we are simply no longer in the rest frame of the original object and our altered [imath]\vec{\Psi}[/imath] is thus no longer a solution to the correct equation). What we have is the fact that our mental model of reality must include the fundamental symmetry that all solutions, in the absence of outside influence, must transform to valid solutions to the correct equation in the center of mass system of any collection of data. This symmetry appears to imposes a major constraint on the character of the possible solutions [imath]\vec{\Psi}[/imath]. In reality,it does not as the scale invariant nature of our mental model provides a straight forward resolution of the difficulty.

 

So would the new [math]\vec{\Psi}[/math] after being changed to a new frame still describe the behavior of the element if we had the correct solution for [math]\vec{\Psi}[/math] or would the new function no longer describing the correct behavior of the elements. It also seems to me that the new function would no longer expand in a spherical manner but would expand in an elliptical manner.

 

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.

 

So the fundamental equation without the Dirac delta function is just the equation of an expanding sphere, (is it actually an expanding wave and only the distance that it has expanded is of interest?) and the speed at which the wave is expanding is defined by k. So what happens if we don’t ignore the effect of the Dirac delta function? Will it still be an expanding wave function where the only change is that it behaves in such a way that the value of it at any other element is always zero?

 

Thus the wave function is non zero only on the surface of a sphere expanding at a specific velocity (which I am calling v_? for the time being). What is important here is that this must be true in both frames (if it is not true in the primed frame, the non-zero portion of [imath]\Psi(t')[/imath] will not be on the surface of an expanding sphere). That is, both frames must yield exactly the same probability distribution; it is the two frames of reference which are different, not the probability of finding that elemental entity.

 

So we are considering an expanding sphere for each set of elements that make up an object, and the probability for any object in the rest frame of some other object and the probability when all elements are considered or any other set of elements must be equivalent if the objects can truly be considered separately. But, what if they can‘t be considered separately? Will the Lorenz transformation still need to be applied, or will we no longer be able to apply the Lorenz transformation to the objects?

 

To begin with, during the design stage, I want my standard clock to be “at rest” in my coordinate system. In the deduction of Schrödinger's equation, I ended up integrating over all tau dependence and defined momentum in the tau direction to be mass. Clearly the fact that my clock is to remain a coherent object requires that, if it is to be “at rest” the major components must have mass: i.e., the important component of the momentum of the majority of the underlying elemental entities must be in the tau direction.

 

So is this due to the only possible function that describes the objects that make up your clock must have a constant derivative to [math]\tau[/math] as a result the object will have a positive mass?

 

I suspect that you may have overwhelmed some people with the length of this post (although I really don’t see any problem with it), you have answered quite a few questions and I have managed to read the whole thing. But I think that it is perhaps best if I don’t try and ask all of my questions at once, so when you answer this I will continue. Also I should have a P.M for you in a couple of days.

[Doctordick]

So would the new [math]\vec{\Psi}[/math] after being changed to a new frame still describe the behavior of the element if we had the correct solution for [math]\vec{\Psi}[/math] or would the new function no longer describing the correct behavior of the elements. It also seems to me that the new function would no longer expand in a spherical manner but would expand in an elliptical manner.

It seems to me that you have a misconception of the issues here. My fundamental equation was designed such that any flaw-free explanation could be cast in a manner such that it fulfilled that equation. The issue “correct” then falls to the question, “are the expectations yielded by [imath]\vec{\Psi}^\dagger \cdot \vec{\Psi}[/imath] consistent with your expectations”. The momentum adjustment provided by [imath]e^{ikx}[/imath] does not change the predicted expectations at all; it merely changes your perspective on the circumstance. Go back and look at my analysis of the moving clock.

 

Essentially, I am looking at the solution from exactly the same frame as was used to analyze the rest clock. The only difference is that oscillator's momentum to the right is higher than it's momentum to the left (there is a significant change in momentum when the oscillator reflects off the mirror). I think you need to understand “Doppler shift”. The analysis is not how the solution looks from a new frame but rather, the issue of changing the momentum of a specific solution.

So what happens if we don’t ignore the effect of the Dirac delta function?

Then you have to include all the interactions with the rest of the universe (or at least some aspect of it). Go look at my deduction of Schrödinger's equation. The solution is then a very complex thing and depends very intimately on “what you are trying to explain”. I made it quite clear that I was talking about a very specific circumstance here: “if the data belonging to a given observation could be divided into two (or more) sets having negligible influence on one another”. The “influence” is a direct consequence of the Dirac delta function.

 

If influence is allowed, all bets are off! This is in stark contrast to Einstein's theory of relativity which postulates that the speed of light is c. My presentation is based entirely on the lack of influence between the two collections of information being explained.

So we are considering an expanding sphere for each set of elements that make up an object

No, we are hypothesizing the structure of a specific object, ignoring how the structure of that object is maintained (essentially assuming that such an object as I describe, my clock, is a possibility). Since such objects appear to be possible in our universe, it seems reasonable that there could exist a solution to my fundamental equation which would explain such an object (most of the individual interactions would have to be approximate solutions to Schrödinger's equation, where rest mass of the elements are momentum in the tau direction). “We are totally free to make these assertions as we are defining an object and, in the absence of contradiction, such an object could certainly exist.”

But, what if they can‘t be considered separately? Will the Lorenz transformation still need to be applied, or will we no longer be able to apply the Lorenz transformation to the objects?

If the two collections of data cannot be considered independently, then the rest frame of the collection is a definable thing and my equation is only valid in that rest frame: i.e., if the data cannot be considered independently, it can't be considered independently and no “transformation” can be defined.

So is this due to the only possible function that describes the objects that make up your clock must have a constant derivative to [math]\tau[/math] as a result the object will have a positive mass?

Mass is momentum in the tau direction and thus can be either positive or negative just as momentum in the x direction can be either positive or negative; however, the energy associated with that momentum can only be positive. Secondly, you seem to have lost sight of the fact that the structure of my object (my clock) is only bound by the fact that, “We are totally free to make these assertions as we are defining an object and, in the absence of contradiction, such an object could certainly exist.”

 

Have fun -- Dick

[Bombadil]

If influence is allowed, all bets are off! This is in stark contrast to Einstein's theory of relativity which postulates that the speed of light is c. My presentation is based entirely on the lack of influence between the two collections of information being explained.

 

Then c, that is the speed of light, is only a limiting speed when the objects can be considered separately, when they can’t be considered separately c is no longer a limiting factor?

 

No, we are hypothesizing the structure of a specific object, ignoring how the structure of that object is maintained (essentially assuming that such an object as I describe, my clock, is a possibility). Since such objects appear to be possible in our universe, it seems reasonable that there could exist a solution to my fundamental equation which would explain such an object (most of the individual interactions would have to be approximate solutions to Schrödinger's equation, where rest mass of the elements are momentum in the tau direction). “We are totally free to make these assertions as we are defining an object and, in the absence of contradiction, such an object could certainly exist.”

 

So only when a object can be considered to be separate from any other object of interest can we consider the Dirac delta function to have no effect so that we can consider an expanding sphere of the probability wave for the object?

 

Mass is momentum in the tau direction and thus can be either positive or negative just as momentum in the x direction can be either positive or negative; however, the energy associated with that momentum can only be positive. Secondly, you seem to have lost sight of the fact that the structure of my object (my clock) is only bound by the fact that, “We are totally free to make these assertions as we are defining an object and, in the absence of contradiction, such an object could certainly exist.”

 

I don’t understand what the relevance of the energy associated with the momentum having to be positive is, but will the energy have to be positive due to energy being associated with the derivative to t, which due to it being defined as nothing more then a evolution parameter must be positive.

 

It is interesting to note that T, the period of our standard rest clock, is identical to 1/v_? times the distance the mirror moves in the [imath]\tau[/imath] direction during one clock cycle. Although actual position in the [imath]\tau[/imath] direction is a meaningless concept (as the entire object is infinite and uniform in that direction), our standard clock appears to be measuring the implied displacement of the mirror over time in that direction: i.e., we can infer that the mirror has moved a distance 2L₀ in the [imath]\tau[/imath] direction during one complete cycle. This will turn out to be a very significant fact since the scale of the [imath]\tau[/imath] dimension is set by the form of the fundamental equation (setting the scale of any dimension sets the scale of all the others) .

 

Then this is due to the velocity of the expanding sphere having to be a constant so that when distance along one axis has been defined in order for the expanding sphere to expand at a constant speed in all directions the distance in all directions has to have been defined, this includes the [math]\tau[/math] direction. Will this define a measure along the t axis seeing as it is just an evolution parameter? Or will it define t only if we have defined the value of [math]v_?[/math].

 

Now consider an identical standard clock in a moving reference frame: i.e., identical to the clock just described except for the fact that I will allow the momentum of the mirror assembly to be non negligible in the y direction. I use the y direction only because it is convenient to the drawing: i.e., the movement of the massless pulse is in the same direction as the clock, an issue which makes a drawing in two dimensions easy. If anyone is concerned about the issue, I will assist in clarifying the problem later.

 

So then, can movement in any other direction be dealt with simply by considering the additional distance that the oscillator will have to move in the direction of movement?

 

Note that the length of the moving clock is shown to be L'. This has been done because we know that the symmetry discussed in the previous section must require the Lorentz contraction to be a valid on any macroscopic solution if interactions with the rest of the universe may be neglected (up to this point the model was scale invariant): i.e., when we solve the problem in the moving clocks system we want the length of the clock as seen by the observer in that moving frame to be L₀. We use the scale freedom in our model to set that length (as seen from the rest system) to be L'; then and only then can we seriously call the clocks identical. This will require [imath]L'=L_0\sqrt{1-sin^2(\theta)}[/imath] (the inverse of the relativistic transformation deduced earlier: i.e., in order to get the length of the moving clock in the primed coordinate system we have to multibly by [imath]\alpha[/imath]). Note that [imath]sin(\theta)[/imath] is exactly the apparent velocity of the moving clock divided by the velocity of the elemental entities, v_?, which actually has nothing to do with time. Since all velocities are v_?, it follows directly that d₁ + d₂ = S. Please note that everything so far is being graphed as seen in the frame of the rest clock: i.e., S=v_?T_m, where T_m is the period of the moving clock as seen from the rest frame.

 

So is the time dilation due to space dilating while distance for the moving frame is still defined in the same way although it has been scaled for an observer not in its rest frame.

 

Do we have to use the inverse transform here because we are moving in the opposite direction as to what we assumed to be transforming to in the first place? If so how can we tell the inverse Lorenz transform from the Lorenz transform aren’t both just moving from a rest frame to a frame that is not at rest with respect to it? It seems that it is a question of what frame it is that the measurements are in that we are transforming.

 

The rest observer will totally disagree with the moving observer's measurement. The rest observer will say that the oscillator took considerably longer to get to the right mirror than it took to return to the left. From his perspective, the flash bulb on the left hand mirror should have been set to go off at the y' indicated by point #2. From the rest observer's perspective, the moving observer has first marked his ruler at point #1 and then waited a considerable fraction of time before marking the other end of the ruler. The markings are not being made simultaneously. The consequence is clearly that the distance between the two marks on the ruler (as seen by the rest observer) are exactly equal to the relativistically corrected distance between the two mirrors, L'₀.

 

So then this is due to both observers using the assumption that the speed of the oscillator that their clock uses has a constant speed. And while it is measured to have a constant speed in any frame the comparison of it to a moving frame (that is if we measure the speed of the oscillator inside of a moving frame from our rest frame) it will not appear to be moving at the speed V_? in comparison to the moving observer. However in the moving frame they will measure the speed of the oscillator to be unchanged due to how time and distance have been defined.

 

Of course we could take the stance that time and distance have been scaled and as a result the oscillators measured speed is unchanged but that doesn’t seem to bring as much attention to the fact that we have circularly defined distance and time.

 

When you answer this I will continue.

[Doctordick]

Sorry about being so slow to answer. At 3:30 AM Saturday March 28 we got a telephone call that my wife's mother's house had been struck by lightning and was burning. At four o'clock we were packed and headed to the gulf coast. We had intended to be there for a couple of days but ended up staying two weeks dealing with the insurance and arranging for a place for her to stay until the house was rebuilt. Believe me, cleaning up after a major fire is no simple matter.

 

Then c, that is the speed of light, is only a limiting speed when the objects can be considered separately, when they can’t be considered separately c is no longer a limiting factor?

You seem to miss the central issue here. I have “defined” the past to be what you know; the the future to be what you do not know and the present to be a specific change in what you know. I then introduced the tau axis for the simple purpose of allowing the undefined data (the past) to be representable by a set of points in a Euclidean space. The parameter “t” is then no more than a reference parameter referring to a specific “present”. Through symmetry arguments I showed that any function [imath]\vec{\psi}[/imath] capable of producing your expectations (your explanation of that past) would have to obey my equation so long as the momentum of the entire universe was zero in that Euclidean reference space (momentum being defined in my presentation).

 

Now, I have shown Schrödinger's equation to be an excellent approximation to my equation so long as the velocity of objects (also defined in my presentation) are small with respect to “c” and Schrödinger's equation has been studied extensively for many many years. Anyone competent in physics would recognize my fundamental equation as an n body equation of interacting point elements lacking mass (analogous to a wave equation with quantized interactions).

 

It follows that the equation results in an undefined speed (undefined because the parameter “t” is not a measurable element) for “non-interacting” elements. Thus it is that any “object” is most definitely limited to travel at less than that speed (after integrating out the dependence on tau all that is left is the apparent velocity orthogonal to tau). When the elements cannot be considered separately then the collection certainly is limited to be less than or equal to c (equal to c when every element is moving orthogonal to tau).

So only when a object can be considered to be separate from any other object of interest can we consider the Dirac delta function to have no effect so that we can consider an expanding sphere of the probability wave for the object?

I am afraid it is a little worse than that. Not only must it be separate from any other object (the argument of the Dirac delta function cannot vanish for any other element: i.e., [imath]x_1 \neq x_i[/imath] for any i) but you must know the exact position of the element we are talking about (it is known to be at zero in both frames when t=0 by definition of the problem we are examining)

I don’t understand what the relevance of the energy associated with the momentum having to be positive is, but will the energy have to be positive due to energy being associated with the derivative to t, which due to it being defined as nothing more then a evolution parameter must be positive.

As I said, the equation is being related to the n body equation of interacting point elements lacking mass (the common physical meaning attached to such an equation). In such a representation, momentum in the positive x direction is taken to imply exactly the same energy and as does the same momentum in the negative x direction. In exactly the same vein, when I set momentum in the tau direction to be mass, the energy associated with positive mass (momentum in the positive tau direction) would be exactly the same as the energy associated with negative mass (momentum in the negative tau direction). In fact, energy of an element is, in my picture, no more or less than the magnitude of the momentum of that entity the same as one would expect in a universe consisting entirely of photons.

Will this define a measure along the t axis seeing as it is just an evolution parameter? Or will it define t only if we have defined the value of [math]v_?[/math].

There is no “t axis”; “t” is a simple parameter identified with a change in our knowledge. Defining “t” and defining [math]v_?[/math] are opposite sides of the same coin.

So then, can movement in any other direction be dealt with simply by considering the additional distance that the oscillator will have to move in the direction of movement?

Yes, it is no more than a problem in analytical geometry and exactly the same answer will be obtained.

So is the time dilation due to space dilating while distance for the moving frame is still defined in the same way although it has been scaled for an observer not in its rest frame.

Time does not “dilate”; neither does space “dilate”. The “apparent dilation” is entirely due to the mechanisms we use to establish our measures. Our explanation of the universe needs to be internally self consistent; that, and that alone, is the fundamental “?cause?” of the relativistic transformations.

Do we have to use the inverse transform here because we are moving in the opposite direction as to what we assumed to be transforming to in the first place?

Here I get the definite impression that you simply don't follow what I am talking about. The whole thing is the issue of observing physical phenomena with measures defined in a specific frame. If we look at Jones' measurement of a specific phenomena we have to transform our measures to his measure. If we look at a specific measurement performed by Jones and how that measurement looks in our frame, we use the inverse of that same transformation.

 

You have to think these things out carefully in detail. The idea that space and time change “physically” is little more than a rule of thumb for performing transformations between numerical results of performing experiments from different frames of reference. What actually happens has nothing to do with your frame of reference. It is the internal consistency of your explanation (which is almost always based on some frame of reference) which leads to these transformations.

So then this is due to both observers using the assumption that the speed of the oscillator that their clock uses has a constant speed.

Not really; it is the assumption that the speed of the oscillator that their clock uses is the same in all directions which generates the problems. It arises directly from the fact that there exists no self consistent way of defining simultaneity except when the distance between the events of interest is zero; essentially the concept “time” is meaningful only at the location of the entity defining it.

 

As I have said many times, if “existence at the same time” is what defines interactions, then “time is not a measurable variable”. If, on the other hand, “time is what clocks measure” is the definition of time, then time has nothing to do with whether or not things can interact (think about the twin paradox). Physicists problems arise because they use two conflicting definitions of time as if they are equivalent (they can only do this by defining what is called simultaneity, a concept which requires a preferred frame of reference).

 

Have fun -- Dick

[Bombadil]

You seem to miss the central issue here. I have “defined” the past to be what you know; the the future to be what you do not know and the present to be a specific change in what you know. I then introduced the tau axis for the simple purpose of allowing the undefined data (the past) to be representable by a set of points in a Euclidean space. The parameter “t” is then no more than a reference parameter referring to a specific “present”. Through symmetry arguments I showed that any function [imath]\vec{\psi}[/imath] capable of producing your expectations (your explanation of that past) would have to obey my equation so long as the momentum of the entire universe was zero in that Euclidean reference space (momentum being defined in my presentation).

 

So then t is nothing more then a way of telling if two or more elements can interact and is not an axis that is moved along. That is elements can interact if they try to occupy the same location at the same value of t.

 

Also didn’t you at the same time require mass to sum to zero in order for the fundamental equation to be valid which would also mean that the total energy must be zero. In both of these cases is a positive value the only possible value resulting in all elements having a zero mass and momentum? The energy must of course then be zero in order for equality to hold.

 

It follows that the equation results in an undefined speed (undefined because the parameter “t” is not a measurable element) for “non-interacting” elements. Thus it is that any “object” is most definitely limited to travel at less than that speed (after integrating out the dependence on tau all that is left is the apparent velocity orthogonal to tau). When the elements cannot be considered separately then the collection certainly is limited to be less than or equal to c (equal to c when every element is moving orthogonal to tau).

 

But wouldn’t even an element that is not interacting with other elements have to have interacted with other elements in order to be known which implies that at some point it had a defined speed and that if an element interacts at some future time that its speed can be defined during the time that it wasn’t interacting or does it only have a defined speed while it is interacting with other elements?

 

Either way if I’m understanding this right during the time that an element has an undefined speed (that is it is not interacting with any other elements) it can’t travel a distance that would suggest that it traveled faster then c during the time that its speed was undefined.

 

As I said, the equation is being related to the n body equation of interacting point elements lacking mass (the common physical meaning attached to such an equation). In such a representation, momentum in the positive x direction is taken to imply exactly the same energy and as does the same momentum in the negative x direction. In exactly the same vein, when I set momentum in the tau direction to be mass, the energy associated with positive mass (momentum in the positive tau direction) would be exactly the same as the energy associated with negative mass (momentum in the negative tau direction). In fact, energy of an element is, in my picture, no more or less than the magnitude of the momentum of that entity the same as one would expect in a universe consisting entirely of photons.

 

So the momentum associated with an element traveling in the positive direction is the same as that which is associated with an element traveling in the negative direction and weather the derivative is positive or negative has no effect on the value of the momentum?

 

Also the mass sums to zero in fact all elements have zero mass so that the momentum is the only source of energy in the equation. I have to wonder at this point how this fits with what we normally call mass and how it relates to what you have defined as mass? I wonder because if no element has mass then it seems that no object that is constructed of elements can have mass but nonzero mass is commonly used in Newtonian physics (in fact I can’t think of what you would do which would use a zero mass) and you have shown that Newtonian physics is an approximation to the fundamental equation so how are these related?

 

Again, as all velocities along our displacement vectors are v_?, d₁+d₂=S. It follows that, from the perspective of our rest frame, this is exactly [imath]S=2L_0+Ssin(\theta)[/imath] or, solving for S,

[math]S=\frac{2L_0}{1-sin(\theta)}[/math].

 

Here, are you still using the definition [imath] sin\theta = v/v_? [/imath] for the value of sine in the above equation, I think that you are? If so I’m not quite sure how you come to the choice of theta, I think how you do is that S is the total distance that the object has traveled which is the same as the distance that the oscillator has traveled which could be wrote out as [imath] \tau c[/imath] while the y axis is the distance that the object has traveled do to its velocity [imath] v_0 [/imath] which can be wrote out as [imath] \tau v_0 [/imath] using these we arrive at the same value of [imath] sin\theta [/imath] as you have previously defined. But is [imath] \tau [/imath] the proper thing to use here or should t be used. I think that [imath] \tau [/imath] as measured by the rest observer is the proper thing to use here although I think that this is equivalent to the value of t.

 

During this time the observer will have moved a distance of [imath]Ssin(\theta)[/imath]. He however will call this distance [imath]Ssin(\theta)cos(\theta)[/imath] (based upon his personal measurements of length and his perception of what he has measured) and will assume that the clock has receded from him by that distance. Since, as far as he is concerned, his standard clock is correctly measuring time, he will read the elapsed time between the received signals as [imath]cos(\theta)S/v_?[/imath]. He will therefore see the clock as receding from him at a rate given by

[math]\frac{\Delta y}{\Delta t}= v_? sin(\theta)[/math]

 

and everyone agrees as to the relative velocities. I need to point out that, as these two observers do not agree about either their distance measurements nor their time measurements one should find this agreement with regard to relative velocities somewhat surprising (it is not really a trivial issue).

 

Here I’m not sure I understand your use of [imath] cos \theta [/imath] instead of [imath] 1/cos /theta [/imath] as this seems to be how distance is scaled for a moving observer. Unless this is the transformation used to find the length of the observer in the rest frame in which place we would find the distance that the observer would measure between him and the clock by multiplying the distance in the rest frame by [imath] cos \theta [/imath].

 

Also, do you mean that the speed that the observer measures for the speed that the clock is receding from him is the same as what his speed moving away from the clock is measured to be in the rest frame?

[Doctordick]

Hi Bombadil,

 

After reading your post, I get the distinct feeling that you do not understand what I am doing here. I am not presuming anything about the transformations required. I am merely asserting that my fundamental equation must be the same in both reference frames. That is, that when the two observers (in the two different frames) examine the same phenomena, they obtain the same expectations. Initially, the phenomena I am examining is probability of an event at a specific point at a specific time. They are essentially solving the same problem. It follows that, no matter how they define their measure of x and t, the same actual events must be described (the actual events which take place have utterly nothing to do with the reference frame used to describe them).

 

That examination produces the standard relativistic transformations without actually defining how distance and time are measured and thus, without defining any value for [imath]v_?[/imath].

 

Once I describe the phenomena (as displayed in the drawings I show) distance can be defined anyway one chooses: i.e., I merely let the initial observer define the distances used in his four dimensional coordinate system. I can use the transformation equations to define the so called “moving observer's” distance measures from the initial observer's units. Time is a little different; I need to define a mechanism to measure time consistent with picture I have presented (a four dimensional Euclidean space with “massless” entities obeying a wave equation) where the tau component is to be integrated out as it is a fictional element. I have to design a "clock".

So then t is nothing more then a way of telling if two or more elements can interact and is not an axis that is moved along. That is elements can interact if they try to occupy the same location at the same value of t.

Essentially yes. Before you even begin to think about the relativistic transformations being discussed in this thread, you need to understand the fact that Schrödinger's equation is a valid approximation for the behavior of individual elements of any explanation. As I have said to Anssi, my deduction has nothing to do with reality as it is a tautological construct. It is only after the proof that Schrödinger's equation is a valid approximation for any possible explanation of anything that we can begin to relate my fundamental equation to the common concept of reality held by modern physicists (that is, the concepts of energy, momentum and mass).

Also didn’t you at the same time require mass to sum to zero in order for the fundamental equation to be valid which would also mean that the total energy must be zero.

In my representation, mass is defined to be momentum in the tau direction. This fact is what leads to the kinetic energy representation so long as [imath]E \approx mc^2[/imath] which brings up the need to define mass.

Notice that, if the term [imath]q^2[/imath] is moved to the right side of the equal sign, we may factor that side and obtain,

[math] \left\{\frac{\partial^2}{\partial x^2} + G(x)\right\}\vec{\Phi}(x,t)=\left\{\sqrt{2}K\frac{\partial}{\partial t}- iq\right\}\left\{\sqrt{2}K\frac{\partial}{\partial t}+iq\right\}\vec{\Phi}(x,t).[/math]

 

At this point, I will invoke a third approximation. I will concern myself only with cases where [imath]K\sqrt{2}\frac{\partial}{\partial t}\vec{\Phi} \approx -iq\vec{\Phi}[/imath] to a high degree of accuracy. In this case, the first term on the right may be replaced by -2iq and, after devision by 2q, we have ...

What is important here (and you should take careful note of) is the fact that if q is negative, the appropriate factor to set approximately valid is [imath]K\sqrt{2}\frac{\partial}{\partial t}\vec{\Phi} \approx +iq\vec{\Phi}[/imath]. In that case, the second term on the right may be replaced by +2iq = -2i(-q) (but -q is a positive number if q is negative) and thus it is that after division by 2|q| we get exactly the same thing we got in the original case.

 

What I am getting at here is that the energy is not a vector quantity but is rather related to the magnitude of the momentum. The energy of an entity with momentum in the opposite direction is still positive. Thus under my definition of mass [imath]mc^2[/imath] is always a positive quantity even when the momentum in the tau direction is opposite. So, no, the mass does not sum to zero even though the momentum in the tau direction of all the elements in the universe must sum to zero.

In both of these cases is a positive value the only possible value resulting in all elements having a zero mass and momentum? The energy must of course then be zero in order for equality to hold.

Wrong!

But wouldn’t even an element that is not interacting with other elements have to have interacted with other elements in order to be known which implies that at some point it had a defined speed and that if an element interacts at some future time that its speed can be defined during the time that it wasn’t interacting or does it only have a defined speed while it is interacting with other elements?

You are missing the entire import of the presentation. We have not yet defined a measure of either distance or time. Thus it is that speed is still undefined. However, the fundamental equation is essentially analogous to a “wave equation” when interactions are ignored. It is the fundamental equation itself which sets that fixed undefined “speed” [imath]v_?[/imath]. K is an arbitrary constant, however, once it is set, the magnitude of that undefined speed is set for all elemental entities in the universe (so long as they are not interacting). The interactions take place at a point and “a point” has a very very small existence (actually zero by definition) therefore, no matter how often interactions take place there is always a smaller scale such that we can talk about the speed when they are not interacting. Remember, our explanations are explaining a finite amount of information thus the number of required valid interactions can not be infinite.

 

When I say the speed is undefined, I mean that the definition of speed requires we know how to measure distance and how to measure time. No matter how distance is defined, the distance an entity moves in some fixed time “t” is equal to the speed times that time so it is quite reasonable to work with specified distances rather than the actual speed. Since I am working with a non-dispersive wave equation in four dimensions, I know that distances traveled in fixed times are exactly the same for all entities when displayed in that four dimensional space.

 

At any rate, since this equation specifies our expectations in a four dimensional universe where all probabilities dependence on tau must vanish, the speed of any element must be whatever that set speed happens to be but the “apparent” speed (in the three dimensional picture after the tau coordinate is integrated out) can have that value only when the element is moving orthogonal to tau (or it has no momentum in the tau direction: i.e., it is massless). An element moving in the tau direction will appear to be standing still and the “apparent” velocity of any massive entity is bounded by zero and that set speed [imath]v_?[/imath].

So the momentum associated with an element traveling in the positive direction is the same as that which is associated with an element traveling in the negative direction and weather the derivative is positive or negative has no effect on the value of the momentum?

Momentum is a vector quantity and direction is a significant issue. You are apparently confusing “magnitude” with the vector nature of momentum.

Here, are you still using the definition [imath] sin\theta = v/v_? [/imath] for the value of sine in the above equation, I think that you are?

The angle theta is the angle between the tau axis and the path of the entity being discussed. The velocity along that path is undefined but nonetheless fixed (by the form of the fundamental equation; which is essentially a wave equation). After integrating the tau dependence out, the apparent distance something moves (d = vt) is related to the actual distance it moves (which is [imath]v_?[/imath] in the four dimensional space) by exactly that same factor [imath]sin\theta[/imath]. This is no more than a simple way of expressing relationships in the diagrams I have drawn.

Here I’m not sure I understand your use of [imath] cos \theta [/imath] instead of [imath] 1/cos /theta [/imath] as this seems to be how distance is scaled for a moving observer.

I presume you meant to write [imath]1/cos\theta[/imath] instead of what you actually wrote. I am analyzing everything from the representation in my drawings. You, on the other hand, seem to ignore the drawings and apparently are trying to apply your understanding of the appropriate transformations. The whole issue here is that we are examining explicit events defined in my four dimensional space and obeying my fundamental equation in order to deduce the appropriate transformations. If you do not understand the drawings, you cannot deduce the transformations. And we cannot use something we have not deduced to be valid. What I am using is the fact that, if the moving observer's distance measures are going to obey the transformations deduced in part one of this presentation then I can use the inverse of that transformation to deduce the actual locations of points the moving observer uses to establish his distance measures. Remember, both observers are examining exactly the same phenomena.

Also, do you mean that the speed that the observer measures for the speed that the clock is receding from him is the same as what his speed moving away from the clock is measured to be in the rest frame?

Yes that is exactly what I am saying.

 

All I get from reading your response is that you do not understand my presentation. Try starting from the beginning and thinking about it one line at a time. You need to have every step clear in your head before going on. This is a logical deduction, not a zen thing that will eventually pop up in your mind. You need to understand the documentation.

 

Have fun -- Dick

[Bombadil]

After reading your post, I get the distinct feeling that you do not understand what I am doing here. I am not presuming anything about the transformations required. I am merely asserting that my fundamental equation must be the same in both reference frames. That is, that when the two observers (in the two different frames) examine the same phenomena, they obtain the same expectations. Initially, the phenomena I am examining is probability of an event at a specific point at a specific time. They are essentially solving the same problem. It follows that, no matter how they define their measure of x and t, the same actual events must be described (the actual events which take place have utterly nothing to do with the reference frame used to describe them).

 

So then in all of the analysis’s of the problems that you put forward there is no need for the actual transformations. All that is necessary is that the problem is analyzed in such a way that both observers, that is the rest and the moving observer, obtain the same result when observing a clock that they are at rest with.

 

Essentially yes. Before you even begin to think about the relativistic transformations being discussed in this thread, you need to understand the fact that Schrödinger's equation is a valid approximation for the behavior of individual elements of any explanation. As I have said to Anssi, my deduction has nothing to do with reality as it is a tautological construct. It is only after the proof that Schrödinger's equation is a valid approximation for any possible explanation of anything that we can begin to relate my fundamental equation to the common concept of reality held by modern physicists (that is, the concepts of energy, momentum and mass).

 

I suspect that part of the problem here is that while I can understand how the Schrödinger equation is arrived at I have very little idea as to what it suggests other then that it can be used to derive Newtonian mechanics. Before I take an in-depth look at the topic, which I plan to do when I have a sufficient understanding of the necessary math, do you know of any sites that might give me an idea of the kind of things that the Schrödinger equation implies? I am thinking of taking a closer look at Wikipedia as it looks like it explains it somewhat but I’m not sure if this is the best place to begin trying to get a better understanding of what the Schrödinger equation implies.

 

What I am getting at here is that the energy is not a vector quantity but is rather related to the magnitude of the momentum. The energy of an entity with momentum in the opposite direction is still positive. Thus under my definition of mass [imath]mc^2[/imath] is always a positive quantity even when the momentum in the tau direction is opposite. So, no, the mass does not sum to zero even though the momentum in the tau direction of all the elements in the universe must sum to zero.

 

Even with the sum of the mass being greater then zero will the sum of the mass operators [imath]-i\frac{\hbar}{c}\frac{\partial}{\partial \tau}[/imath] still sum to zero? (I am understanding these terms to be the differentials to [imath] \tau[/imath] on the left side of the fundamental equation) in order for the fundamental equation to remain valid while only the term

 

[imath]m=-i\frac{\hbar}{c}\int\vec{\Psi}^\dagger\cdot\frac{\partial}{\partial \tau}\vec{\Psi}dV.[/imath]

 

which represents the probability of what the actual mass is that all elements under consideration have where the integral is over all elements being considered?

 

In a similar way the energy operator [imath]i\hbar\frac{\partial}{\partial t}[/imath] must be zero however this does not imply that the energy defined by

 

[imath]E=i\hbar\int\vec{\Psi}^\dagger\cdot\frac{\partial}{\partial t}\vec{\Psi}dV.[/imath]

 

must also be zero. Again the integral is taken over all elements under consideration.

 

When I say the speed is undefined, I mean that the definition of speed requires we know how to measure distance and how to measure time. No matter how distance is defined, the distance an entity moves in some fixed time “t” is equal to the speed times that time so it is quite reasonable to work with specified distances rather than the actual speed. Since I am working with a non-dispersive wave equation in four dimensions, I know that distances traveled in fixed times are exactly the same for all entities when displayed in that four dimensional space.

 

So then the speed of an element that has zero movement along the [imath]\tau[/imath] axis will have a constant speed (this is entirely due to arbitrary constants that are part of the fundamental equation) What is important is that the speed is only defined if both time and distance are defined and that no mater what reference frame we are in we must observe the same events in other frames. That results in simultaneity as you have demonstrated it to be preserved.

 

For instance supposing that we use a unit rod as a way to measure distance, then in order to actually measure an object we must make measurements on both ends of the rod at what we consider to be simultaneous measurements, which requires that we define what we consider to be simultaneity something which must differ from one frame to anther due to the fact that your clock must behave the same way in any reference frames but that the observation of a clock in a different reference frame will differ. That is, events that appear simultaneous in one frame need not appear simultaneous in any other frames.

 

Also couldn’t we define velocity by defining the oscillator to have a unit velocity then by defining either distance or time define the remaining one by properly considering the requirements for events to happen simultaneously? While it seems that we could do this in order for us to define the remaining measurement we would have to use what we consider to be a unit length or a unit of time so that we would have to come to the same conclusions. Although it seems that it would perhaps be impractical to use an oscillator to define velocity for most proposes.

[Doctordick]

So then in all of the analysis’s of the problems that you put forward there is no need for the actual transformations. All that is necessary is that the problem is analyzed in such a way that both observers, that is the rest and the moving observer, obtain the same result when observing a clock that they are at rest with.

You seem to have missed the entire exercise. When the two observers go to analyze any dynamic (means changing) collection of data referenced via positions in their personal reference frames (think “do physics”), they first need to establish their units of measure in that frame of reference. That means that they must establish both a unit of measure for the coordinates of their frame of reference (a measure of distance) and a unit of measure for time.

 

Now, the original problem was to obtain a flaw-free explanation of “any dynamic collection of data” in the universe. That means that “both” observers are using exactly the same explanation! Whatever specific phenomena those observers use to establish their units of measure, that phenomena must be an excellent example of that “flaw-free explanation”: i.e., by excellent I mean that the nature of the phenomena behind these measures is presumed to be well understood. The interesting thing here is that, due to the “rules” being enforced by a Dirac delta function, that “flaw-free” solution must be scale invariant. That means that, if the two observers are dealing with “well understood bodies of information which are totally independent of the linear motion of their personal frame of reference”, there exists no such “well understood phenomena”: i.e., a unique linear measure can not be established (the problem is scale invariant and therefore the solution must be scale invariant).

 

The way out of that problem is the fact of that very scale invariance itself. We have already established that the only possibility which leaves the form of the fundamental equation identical in both frames (under the assumption that a solution does exist) is exactly the transformation deduced by Lorentz and Fitzgerald. That alone is actually insufficient as it does not eliminate the possibility that “no flaw-free solution exists”: i.e., it is possible that the only flaw-free solution demands one unique frame of reference and, if you are not using that frame, your solution is flawed. If true, that possibility would pretty well destroy the field of physics as an exact science.

 

Thus I finish the essay by presuming that there does exist phenomena capable of establishing linear measure in those coordinate systems which is in accordance with the Lorentz-Fitzgerald solution. Note that I am not presuming that “all phenomena” are in accordance with that solution; however, when I am given that specific established linear measure, the approximate validity of Schrödinger's equation as a solution to my fundamental equation allows me to design objects in an arbitrary “rest frame”. This empowers me to design a “clock” and thus define time in exactly the same way in two non-accelerating frames moving with respect to one another. I then show explicitly that those clocks yield exactly the time relationship required. At this point, I have proved that, not only is the standard relativistic transformation the only possibility but that it is indeed totally consistent with my fundamental equation.

 

Since via Schrödinger's equation we have a guarantee that objects can exist we can actually move (via acceleration) a specific measuring rod (no matter how arbitrarily that distance is defined) from one frame into the other which is exactly what the marks on that rod in Paris were all about.

I suspect that part of the problem here is that while I can understand how the Schrödinger equation is arrived at I have very little idea as to what it suggests other then that it can be used to derive Newtonian mechanics.

The first issue is that Schrödinger's equation is an equation for determining probabilities for the results of one's measurements. We have, at this point, defined momentum, mass and energy in terms of the various components of the fundamental equation. From that perspective, as a function of “position x”, Schrödinger's equation amounts to a statement as to how one half the square of the momentum divided by mass differs from the energy; the difference being exactly what we have called “the potential energy”. For practical purposes, that is a simple “conservation of energy” equation.

 

Newton's fundamental equation is F=ma and his concept of momentum is p=mv. Another fundamental concept in Newtonian mechanics is kinetic energy (kinetic means “associated with motion) and, finally, the concept of “potential energy” was introduced in order to maintain “conservation of energy”. Thus the equation of conservation of energy in Newtonian mechanics is

[math]\frac{1}{2}mv^2+V(x) = E.[/math]

 

Since velocity is defined to be the time derivative of position, that equation essentially defines the behavior of the normally referred to as Newtonian mechanics. There is an important relationship embedded in the above; Newton's kinetic energy is exactly the square of the momentum divided by 2m. Thus it is that Newton's energy conservation equation is essentially identical to Schrödinger's energy conservation equation (exactly the same terms are related in exactly the same way). As an aside, since acceleration is (again by definition) the time derivative of velocity, Newton's force can be seen as identical to the time derivative of momentum so all of Newton's concepts are also defined in Schrödinger's approach and, since both are essentially using the same “conservation of energy” equation, they both yield essentially identical solutions. The only real difference is that, in Schrödinger's picture, the actual measurements are probabilistic while, in Newton's picture, the actual measurements are discrete and established. Thus, whenever the actual variables in the solutions to Schrödinger's can be seen as established discrete values, the results are essentially, for practical purposes, identical to Newton's. (I say “essentially” because there are problems which can be handled in Schrödinger's picture which are not handleable in Newton's picture; however, even in that case the solution can generally be cast into Newton's picture.)

 

The only aspect of that fact that I am using is that the behavior of “objects” (collections of fundamental elements which can be seen as physical structures) will essentially obey common sense behavior of standard objects. In particular, the structure I call a mirror assembly can be seen as a classical object in a four dimensional Euclidean space which will obey simple conservation of momentum in that geometry.

Even with the sum of the mass being greater then zero will the sum of the mass operators [imath]-i\frac{\hbar}{c}\frac{\partial}{\partial \tau}[/imath] still sum to zero?

This would be true except for one subtle fact. The tau dimension is totally fictional and, in the final analysis, the actual distribution of our points in the tau direction can not have any effect on our final “flaw-free” solution. Thus, in a sense, it is impossible for any actual valid data to establish a unique frame of reference in that direction. In a sense, although velocity in the tau direction is defined, real motion in the tau direction is essentially undefinable. The fact that one can not define a rest frame with regard to motion in the tau direction, seems to me to imply that this data need not be so constrained.

 

I personally think that this is the source of the philosophical problem concerning the lack of universal symmetry between the fundamental elements and the anti-elements of those same elements. Something worth thinking about anyway. (Actually, I have not asserted that I have solved every problem in physics; I have only solved a great many.)

For instance supposing that we use a unit rod as a way to measure distance, then in order to actually measure an object we must make measurements on both ends of the rod at what we consider to be simultaneous measurements, which requires that we define what we consider to be simultaneity something which must differ from one frame to anther due to the fact that your clock must behave the same way in any reference frames but that the observation of a clock in a different reference frame will differ. That is, events that appear simultaneous in one frame need not appear simultaneous in any other frames.

I think you are overlooking a much more fundamental issue here. Everyone is defining “simultaneity” via the assumption that the speed of light is the same in both directions and that is the basis of the problem here. All physicists in the universe could just do all their calculation in the rest frame of the background radiation. Then they would all agree and there would be no problems with identifying “simultaneity”.

 

The real problem with that solution is that it makes the actual calculations so difficult that no physicist would even consider such a solution. Think about some poor guy doing an experiment in his laboratory calculating the energy levels in hydrogen who is required to use a measuring rod at rest with respect to background radiation. His standard measuring rod would change with the rotation of the earth (since he must use a different speed of light in opposite directions which varies because he is moving). Now the final result would be exactly the same (Maxwell's equations, in fact, insure that) but the intermediate calculations would be horrific.

 

I'll stop here because I don't think your last paragraph is well thought out.

 

Have fun -- Dick

[Bombadil]

Now, the original problem was to obtain a flaw-free explanation of “any dynamic collection of data” in the universe. That means that “both” observers are using exactly the same explanation! Whatever specific phenomena those observers use to establish their units of measure, that phenomena must be an excellent example of that “flaw-free explanation”: i.e., by excellent I mean that the nature of the phenomena behind these measures is presumed to be well understood. The interesting thing here is that, due to the “rules” being enforced by a Dirac delta function, that “flaw-free” solution must be scale invariant. That means that, if the two observers are dealing with “well understood bodies of information which are totally independent of the linear motion of their personal frame of reference”, there exists no such “well understood phenomena”: i.e., a unique linear measure can not be established (the problem is scale invariant and therefore the solution must be scale invariant).

 

Are you referring to an explanation that is capable of explaining any set of data because it seems that if two different observers are using different sets of data to arrive at an explanation while their expectations may be the same for some subsets of the date that overlaps between them the actual explanations may differ. In fact if they did not give the same expectations one of them would seem to have to be flawed or at least incomplete.

 

Or perhaps you are only referring to the phenomena that is presumed to be well understood in which case since it is a well understood phenomena we can presume that any observers that considers it to be a well understood phenomena must use the same explanation of it.

 

I assume that by, “by excellent I mean that the nature of the phenomena behind these measures is presumed to be well understood.” that you mean that the object must behave the same over any particular measurement. That is if something is measured at one time and then at a latter time then the measurements will agree or if a particular event is measured at one time the same event will have the same measurements at a latter time. And so by using such a property to define a measure a unit measure will always have the same properties (they will be considered to be the same length) in any frame.

 

Now since any flaw-free explanation must be scale invariant there must exist frames in which two rods defined to be a unit of measurement by any particular event that is the same property of the explanation was used to define the units of measure. The unit measures will differ when compared with each other. In fact this goes even further in that if the two observers are using any well understood property of their explanation to define their measures, any two events that are scale to each other will appear to be the same event. So there is no way to tell one frame from a frame that is scaled up or down from the other.

 

The problem is that if we try to compare the two different observers measurements we will come up with different measurements and so will come up with different events when measuring any particular event. That is the observers will not agree on what they see.

 

The way out of that problem is the fact of that very scale invariance itself. We have already established that the only possibility which leaves the form of the fundamental equation identical in both frames (under the assumption that a solution does exist) is exactly the transformation deduced by Lorentz and Fitzgerald. That alone is actually insufficient as it does not eliminate the possibility that “no flaw-free solution exists”: i.e., it is possible that the only flaw-free solution demands one unique frame of reference and, if you are not using that frame, your solution is flawed. If true, that possibility would pretty well destroy the field of physics as an exact science.

 

So the solution is that if the scale of the flaw-free explanations differ by exactly the Lorentz transformation then both observers can measure the same events and while their measurements will not agree they will agree on what happened if the measurements are transformed into any particular reference frame.

 

This does not however prove that the explanation obeys the Lorenz transformation only that if it does then there is no frame that is proffered over any other frame and that measurements in one frame can be changed to any other frame.

 

Thus I finish the essay by presuming that there does exist phenomena capable of establishing linear measure in those coordinate systems which is in accordance with the Lorentz-Fitzgerald solution. Note that I am not presuming that “all phenomena” are in accordance with that solution; however, when I am given that specific established linear measure, the approximate validity of Schrödinger's equation as a solution to my fundamental equation allows me to design objects in an arbitrary “rest frame”. This empowers me to design a “clock” and thus define time in exactly the same way in two non-accelerating frames moving with respect to one another. I then show explicitly that those clocks yield exactly the time relationship required. At this point, I have proved that, not only is the standard relativistic transformation the only possibility but that it is indeed totally consistent with my fundamental equation.

 

So this is what the reason for using an element that has zero movement in the [imath]\tau[/imath] direction is for. It gives a very convenient way to define time or distance assuming that one has already been defined, because such elements will have the same speed no matter what frame they are in after speed has been defined. Its speed is in fact scale invariant, that is, no matter what the scale of the equation is after speed has been defined it’s speed will be the same in any frame. And so it can be used in any reference frame and the same result will be arrived at. However the speed being invariant is not necessary, all that is necessary is that objects in different frames behave in the same way independent of the scale.

 

Now since Schrödinger’s equation is a solution to the fundamental equation this allows you to define a particular object in a rest frame to be a unit of distance. The question then becomes the problem of moving such an object from one frame into any other frame. Now when you say,

 

Now, I have already shown that a given solution in the rest frame is easily transformed to a solution where the frame of reference is no longer at rest. Such a transformation is simply obtained via multiplication of [imath]\vec{\Psi}[/imath] by the simple function

[math]\prod_{j=1}^n e^{i\frac{Px_j}{n}}[/math].

 

This change in [imath]\vec{\Psi}[/imath] will simply add P/n to the momentum in the x direction of every elemental entity in the universe (the universe consisting of the elemental entities which make up that independent object). In other words, the transformation simply adds P to the momentum of the object and thus the object is no longer at rest in the rest frame used to solve for [imath]\vec{\Psi}[/imath]. Thus it is that we can always transform a solution in the rest frame of one object to a solution in the rest frame of the other (note that the transformation also requires a change in energy which is just as easily obtained).

 

This shows that there exists a way on moving from one frame to any other frame that is adding momentum to the fundamental equation. Now when we compare this to the Schrödinger equation, adding momentum in this way results in adding velocity to a object to change it to a new reference frame.

 

The only aspect of that fact that I am using is that the behavior of “objects” (collections of fundamental elements which can be seen as physical structures) will essentially obey common sense behavior of standard objects. In particular, the structure I call a mirror assembly can be seen as a classical object in a four dimensional Euclidean space which will obey simple conservation of momentum in that geometry.

 

So all that is being used is that energy and momentum will behave like they do in Newtonian physics and that it is possible to form objects?

 

This would be true except for one subtle fact. The tau dimension is totally fictional and, in the final analysis, the actual distribution of our points in the tau direction can not have any effect on our final “flaw-free” solution. Thus, in a sense, it is impossible for any actual valid data to establish a unique frame of reference in that direction. In a sense, although velocity in the tau direction is defined, real motion in the tau direction is essentially undefinable. The fact that one can not define a rest frame with regard to motion in the tau direction, seems to me to imply that this data need not be so constrained.

 

So in effect since movement along the [imath]\tau[/imath] direction has no effect in the finale analysis of the problem and any difference from zero of the terms [imath]-i\frac{\hbar}{c}\frac{\partial}{\partial \tau}[/imath] would just be integrated out and so would make no difference in the final analysis of the problem. So there is no reason to assume that there is any reason that the mass operators will sum to zero as it will have no effect in the final analysis of the problem.

 

I think you are overlooking a much more fundamental issue here. Everyone is defining “simultaneity” via the assumption that the speed of light is the same in both directions and that is the basis of the problem here. All physicists in the universe could just do all their calculation in the rest frame of the background radiation. Then they would all agree and there would be no problems with identifying “simultaneity”.

 

But isn’t it also true that if we where to try and define simultaneity in any other way then the results would still have to differ from one frame to another. Unless of course, we define simultaneity to only exist in a particular frame and use it as the rest frame and do all of our calculations in that frame which is not a practical solution to the problem.

[Doctordick]

In fact if they did not give the same expectations one of them would seem to have to be flawed or at least incomplete.

That is the central issue of special relativity: how does one transfer the measurements made in one inertial frame of reference to those made in a second inertial frame of reference moving with respect to the first. It is the actual measurements of the actual phenomena which is of significance. Central to the whole question is the idea that physics (the explanation of reality which is physics) must allow us to explain how that is to be accomplished. The explanation (the physics itself) is not to be changed. Any “change” would clearly be seen as a flaw.

I assume that by, “by excellent I mean that the nature of the phenomena behind these measures is presumed to be well understood.” that you mean that the object must behave the same over any particular measurement. That is if something is measured at one time and then at a latter time then the measurements will agree or if a particular event is measured at one time the same event will have the same measurements at a latter time. And so by using such a property to define a measure a unit measure will always have the same properties (they will be considered to be the same length) in any frame.

No I do not mean any such thing. That is why I put it the way I did! How these measures are to be established is of utterly no consequence so long as the the procedure used is not confused by either party; that they will agree that the procedure being used is the “correct” procedure as per their explanation of reality (their “physics”). My presentation is much more open to alternate possibilities than is your statement.

Now since any flaw-free explanation must be scale invariant there must exist frames in which two rods defined to be a unit of measurement by any particular event that is the same property of the explanation was used to define the units of measure.

That sentence just doesn't make any sense to me. The flaw-free explanation is scale invariant when the entire universe is included. If you have a solution (an explanation) and that explanation includes a “scale” obtained by some procedure (internal to that universe) and you change the scale of the entire universe, the scale used in that explanation changes in exactly the same way. That is scale invariance. The problem arises when the two observers are leaving out different pieces of the universe (which is exactly what they are doing when they each propose their frame of reference is inertially “at rest”): i.e., they are presuming that the motion of the far away portions of the universe are of no significance to their physics (their explanation). The consequence of that fact is that, to quote you, “the observers will not agree on what they see” if their personal universes are scale invariant. The obvious answer is that they can not be scale invariant. Their “Physics” must establish a mechanism which “explains” the transformation required.

So the solution is that if the scale of the flaw-free explanations differ by exactly the Lorentz transformation then both observers can measure the same events and while their measurements will not agree they will agree on what happened if the measurements are transformed into any particular reference frame.

Let us say that their explanations of what happened will agree if those explanation include such a transformation.

This does not however prove that the explanation obeys the Lorenz transformation only that if it does then there is no frame that is proffered over any other frame and that measurements in one frame can be changed to any other frame.

I think you have the horse on entirely the wrong side of the cart here. You should have said, “This does not however prove that the explanation obeys the Lorenz transformation; only that, if it does not, the explanation is flawed in that “the rest of the universe can not be omitted”. It is entirely possible that the proper “physics” (the flaw-free explanation requires information about the rest of the universe).

 

We already know that the background microwave radiation sets up a unique frame of reference wherein the common inertial frames used in physics are not at rest. People today commonly speak about the motion of our solar system or our galaxy through the universe so there is very definitely a “preferred” inertial frame. What they don't talk about is the possibility that the “laws of physics” could be a function of that motion. I am afraid my presentation leaves that issue open. Einsteinian relativity presumes such a thing is impossible; however, no such independence of that frame has ever actually been proved. As a matter of fact, when and if we get into general relativity, the issue rises again in a somewhat different form.

So this is what the reason for using an element that has zero movement in the [imath]\tau[/imath] direction is for. It gives a very convenient way to define time or distance assuming that one has already been defined, because such elements will have the same speed no matter what frame they are in after speed has been defined. Its speed is in fact scale invariant, that is, no matter what the scale of the equation is after speed has been defined it’s speed will be the same in any frame. And so it can be used in any reference frame and the same result will be arrived at. However the speed being invariant is not necessary, all that is necessary is that objects in different frames behave in the same way independent of the scale.

Once again, I find your statement rather askew of what I am saying. I think everything would be clearer if you just left this paragraph out. I certainly would not agree with it.

Now since Schrödinger’s equation is a solution to the fundamental equation this allows you to define a particular object in a rest frame to be a unit of distance.

Again I find your statement confused. Schrödinger's equation being an approximate solution to my equation implies that “objects” (the structure of which physics explains through Schrödinger's equation) can also be an approximate solution to my equation: i.e., what we call “physical objects” can exist and will obey standard Newtonian rules.

The question then becomes the problem of moving such an object from one frame into any other frame.

The question then becomes the problem of examining such an object from two different frames. There is no problem of moving such an object: movement from one inertial frame to another occurs all the time in Newtonian mechanics.

This shows that there exists a way on moving from one frame to any other frame that is adding momentum to the fundamental equation. Now when we compare this to the Schrödinger equation, adding momentum in this way results in adding velocity to a object to change it to a new reference frame.

Yes, but the solution is only flaw-free in the original inertial frame: i.e., these are your expectations if you take the position that the moving frame is “wrong”.

So all that is being used is that energy and momentum will behave like they do in Newtonian physics and that it is possible to form objects?

Yes!

So in effect since movement along the [imath]\tau[/imath] direction has no effect in the finale analysis of the problem and any difference from zero of the terms [imath]-i\frac{\hbar}{c}\frac{\partial}{\partial \tau}[/imath] would just be integrated out and so would make no difference in the final analysis of the problem. So there is no reason to assume that there is any reason that the mass operators will sum to zero as it will have no effect in the final analysis of the problem.

Essentially yes.

But isn’t it also true that if we where to try and define simultaneity in any other way then the results would still have to differ from one frame to another. Unless of course, we define simultaneity to only exist in a particular frame and use it as the rest frame and do all of our calculations in that frame which is not a practical solution to the problem.

Yes, I would have to agree with that; however, you need to define “a practical solution”. Remember, we are talking about an explanation which yields our expectations. An explanation is flaw-free if it does indeed yield our expectations. If, as the basis of our explanation, we are going to presume that there is nothing in our “local” measurements which can identify that preferred frame, then our explanation must include exactly the relativistic transformations we have deduced.

 

What you have to comprehend is the fact that our solutions (the problems physics can solve exactly) can not be “general many body problems” and are essentially restricted to interactions between two bodies, the things our physics can explain are totally limited to phenomena which obey the transformations we have deduced. Essentially, this can not be taken to imply our expectations are correct. We may perhaps discover events which contradict a physics deduced under such a constraint. In particular, the assumption that such transformations of physics to relativistic velocities compared to the rest frame of the background microwave radiation will have no consequences is an assumption. One which I suspect is in error and I will explain that problem when I get into general relativity.

 

Have fun -- Dick

[modest]

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.

 

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 [imath]\mathbf{R}^{1,3}[/imath] 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.

 

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.

 

We need to have a formula for translating coordinate points in the first frame, [imath](x,y,z,\tau)[/imath], into the identical points represented in the second frame, which have to be [imath](x',y',z',\tau')[/imath] 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), [imath]\Psi[/imath], 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, [imath]\Psi(t)[/imath] 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 [imath]r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2}[/imath]. 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 [imath]r=\sqrt{x^2+y^2+z^2+\tau^2}[/imath] 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 [imath]r=v_?t=\sqrt{(x'+vt')^2+y'^2+z'^2+\tau'^2}[/imath] 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:

 

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,

 

[math]\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} [/math]

 

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

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

 

Only experiment can answer the question which of the two possibilities, [imath]\kappa = 0[/imath] or [imath]\kappa <= 0[/imath], 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 [imath]\kappa < 0[/imath].

 

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

 

~modest

[Doctordick]

I've avoided examining your work probably because it seems like a rather involved undertaking.

Actually, my work is quite simple; what makes it involved is the great number of things which flow from it.

My knee jerk reaction and my first question: if your fundamental equation can be used to derive a [imath]\mathbf{R}^{1,3}[/imath] 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.

I am not. In fact, I have essentially spent most of Sunday trying to find an exchange I had with Erasmus00 concerning exactly this issue which you really should read. That is one of the problems with the structure of these forums. There is essentially no way to find a specific discussion even when you know exactly what was being said: the search function does not produce posts but only threads. There is no organization of the exchanges what so ever. I try to keep a database on where my posts are and what they are about but I have let it get out of date. It takes time to keep it up to date. If I knew the bash shell a little better, I could probably write a bash script to do it for me but I am too busy (or maybe too old) to get competent in bash.

 

At any rate, he and I have discussed the fact that my picture and Einstein's are isomorphic with regard to any specific observations. I managed to convince him of the truth of that statement but he still didn't see any value in it and dropped out of the discussion.

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.

You should really look at the derivation of my fundamental equation as, without understanding the necessity of that result the whole thing seems to be a bit “off the wall”. My original web site no longer exists but that derivation has been put on WiKipedia. Sans the interaction term, that is nothing more than a standard four dimensional wave equation.

I understand and agree r = v_?t.

Then we see eye to eye insofar as that paragraph is concerned.

It almost feels like you've asserted the principle of relativity. It would help me to see how you rule out the possibility of [imath]r=v_?t=\sqrt{(x'+vt')^2+y'^2+z'^2+\tau'^2}[/imath] where v_? would be constant relative to the center of the sphere but not relative to S'.

I would comment that, to anyone who understands relativity and why Maxwell's equations require these transformations (and you are certainly a member of that set), asserting that the fundamental equation is a wave equation with constant velocity is exactly equivalent to asserting the principal of relativity (or rather, the problem relativity was created to solve). That is why I suggest you closely follow my deduction of that fundamental equation; the issue of relativity is embedded in the equation itself.

 

The equation [imath]r=v_?t=\sqrt{(x'+vt')^2+y'^2+z'^2+\tau'^2}[/imath] is invalid because because it requires the moving observer (the observer in the primed frame) to know his frame is moving. In my derivation of my fundamental equation, the equation is only valid in the rest frame of the universe. Since it is not talking about “correct” answers but rather about “expectations” the issue arises as to what happens if the two observers ignore the data which sets their frames of reference apart (for example, rotation of the earth, movement of the earth through the galaxy or possibly the microwave background radiation).

 

In that case, they would each presume their reference frame to be at rest with respect to the universe and clearly any physical phenomena which did not depend upon the validity of their frames being “true” rest frames of the universe would be erroneous (their expressions of the fundamental equations would be different). There is a contradiction here which can only be removed by taking advantage of the scale invariance of that equation.

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

No, it is not an issue for two explicit reasons. First, you need to understand the deduction of the fundamental equation. The constant K, which is essentially the inverse of v_?, is a free parameter of the deduction. Infinite v_? would essentially amount to the case K=0. That implies that the magnitude of the momentum of the individual entities making up the universe vanishes identically. The only conclusion which can be drawn from that is that “no change whatsoever can occur”; that demands an exactly static universe.

 

And, secondly, the fundamental equation is essentially scale invariant. Since t is a free evolution parameter, the actual change per unit time is set by two issues, the actual scale used for distance (which is defined by some specific solution to the equation) and the scale used for time (which is simply picked to describe some convenient repeating phenomena). That is, your explanation of the universe itself (as expressed by [imath]\vec{\Psi}[/imath]) sets the explicit value of v_? (whatever it is, it is a finite constant set by your physics). So why don't we just call it “c” for convenience?

 

Have fun -- Dick

[Bombadil]

No I do not mean any such thing. That is why I put it the way I did! How these measures are to be established is of utterly no consequence so long as the the procedure used is not confused by either party; that they will agree that the procedure being used is the “correct” procedure as per their explanation of reality (their “physics”). My presentation is much more open to alternate possibilities than is your statement.

 

So all that is necessary to insure that measurements in one frame will agree when transformed to a new frame with the measurements made in the new frame is that the same procedure is used in both frames to arrive at their units of measure. It doesn’t matter how the units used are defined as long as they are made in the same way since in order for a flaw-free explanation to exist in both frames their laws of physics (their explanations) must agree when their measurements are transformed to a single frame.

 

Now in the case in which the measurements that the observers make do not follow the Lorenz transformation, the only possibility is that one or both of the observers are using a flawed explanation.

 

I’m somewhat puzzled though by exactly what the scale invariance is, it seems that it is stating that no matter how you define distance and time that the same explanation will be a valid explanation and as long as you use the same units it doesn’t matter what they are. Or in other words it doesn’t matter how we define length along any axis as along as it is consistent with the explanation. I’m wondering if this is the case because it seems that the scale invariance vanishes when we define a system of units.

 

That sentence just doesn't make any sense to me. The flaw-free explanation is scale invariant when the entire universe is included. If you have a solution (an explanation) and that explanation includes a “scale” obtained by some procedure (internal to that universe) and you change the scale of the entire universe, the scale used in that explanation changes in exactly the same way. That is scale invariance. The problem arises when the two observers are leaving out different pieces of the universe (which is exactly what they are doing when they each propose their frame of reference is inertially “at rest”): i.e., they are presuming that the motion of the far away portions of the universe are of no significance to their physics (their explanation). The consequence of that fact is that, to quote you, “the observers will not agree on what they see” if their personal universes are scale invariant. The obvious answer is that they can not be scale invariant. Their “Physics” must establish a mechanism which “explains” the transformation required.

 

So since both observers are leaving out different parts of the universe when they define the units of there explanations they will not agree on there explanations (that is the units defined using one explanation will not be consistent with the other) but in order for both explanations to be flaw-free they must differ by nothing more then the scale (this corresponds to the scale of objects in their explanation) and in order for them to see the same event and agree on what happened in any particular frame that scale must be the same as what is defined by the Lorenz transformation.

 

Now the actual scaling of the fundamental equation is just a consequence of the transformation used to change the explanation in one frame to a new frame while still keeping the fundamental equation valid.

 

I think you have the horse on entirely the wrong side of the cart here. You should have said, “This does not however prove that the explanation obeys the Lorenz transformation; only that, if it does not, the explanation is flawed in that “the rest of the universe can not be omitted”. It is entirely possible that the proper “physics” (the flaw-free explanation requires information about the rest of the universe).

 

But isn’t the issue of if any flaw-free explanations can obey the Lorenz transformation still open or is the fact that if they do, observers in different frames will agree on the measurements when transformed to the same reference frame enough to insure that two different flaw-free explanations will obey the Lorenz transformation. Basically I’m asking how do we know that the measurements in one frame can be transformed to any other frame even if both explanations are flaw-free. It seems that this may be equivalent to asking if all flaw-free explanations differ only by a scale.

 

The question then becomes the problem of examining such an object from two different frames. There is no problem of moving such an object: movement from one inertial frame to another occurs all the time in Newtonian mechanics.

 

This still seems to leave open the question can an object in one frame be moved to any other frame. That is are there frames that an object in one frame can’t be moved into from some other frame. Or can an object in any frame be moved to any other frame.

 

Yes, but the solution is only flaw-free in the original inertial frame: i.e., these are your expectations if you take the position that the moving frame is “wrong”.

 

Isn’t it still an open issue though that is if we can call the moving frame wrong or not it may be that there are no wrong frames only incomplete explanations or it may be as you seem to be suggesting that there is in fact a unique frame that is the only frame in which the fundamental equation is valid in.

[Doctordick]

I am sorry Bombadil, but I very much get the impression that some very important aspects of my presentation are just missing their mark. It seems to me that you are just attacking the problem from the wrong direction. I suspect there are two issues in play here. First, I suspect that your understanding of mathematics does not include much experience solving difficult differential equations. The single most important issue there is that no general solution to a many body problem has ever been found. The validity of my fundamental equation can not be judged through its solutions because those solutions simply are not available to us. Its validity rests entirely on my deduction and nothing else.

 

If you followed my deduction of the fundamental equation, you would understand that it is no more than a proof that any explanation of anything can be interpreted in a way which obeys that equation. That is, any ontological basis may be translated into a set of points in that Euclidean space I have set up. As Anssi has realized, what we are really talking about is a general epistemological construct consistent with that hypothetical ontological basis. Persistence is the central issue of any epistemological construct. Those points which make up a given present (defined to be new knowledge added to the past, “what we knew or thought we knew” ) are simply presumed to be a new distribution of the previous present. Each time slice is presumed to consist of exactly the same elements which made up the previous slice (indicated by the use of the same index i ); however, the presumed persistence says that they existed between those time slices. This is the central issue behind the dual identification: identification via the index “i” with the further numerical label x_i being plotted. That continuation which presumes existence between “presents” yields an epistemological construct which can model absolutely any explanation which can be conceived.

 

The epistemological construct is no more than a vast collection of point entities moving around in a Euclidean space. That epistemological construct is clearly flaw-free because it is clearly consistent with the fundamental equation I deduced: i.e., point entities traveling in straight lines except for point interactions (scattering) caused by the Dirac delta function interactions. But it is also a solution of utterly no use as I still can not solve the many body problem so represented. (Though I have shown that absolutely any "physical" representation of the universe can be represented by such a thing. See my geometric proof.)

 

But what I have done is to show that Schrödinger's Equation is an approximation to my equation given some rather simple and straight forward assumptions (assumptions totally consistent with the assumptions used in all applications of Schrödinger's Equation). What this leads me to is my definitions of mass, momentum, energy and (even more significant when it comes to constructing a mental picture consistent with my equation) “objects”. This gives me the power to transform this “flaw-free” epistemological construct into my personal world view (that solution to the problem provided to me by “squirrel thought” the correctness of which I can not check).

 

That any conceivable explanation can be so interpreted is the central issue; not what that explanation is to the person who dreamt it up. When you attempt to give me your explanation of anything, I make an attempt to understand what you are telling me. If that explanation is flaw-free (and I would certainly presume it is or I wouldn't bother trying to understand it as trying to understand a flawed explanation is clearly a waste of time) then your communication of your explanation can be interpreted in such a way that it obeys my equation. That is the first step! What you must remember is that your world view includes the meanings of the words you use and the mechanism you use to convey those words (sounds or electronic signals or even your own senses).

 

Your world view is the whole magillah. I likewise possess a world view and my interpretations of the communications you put forth are just another piece of my world view. There exists no evidence at all that there is any “real” correlation between your explanation (your method of keeping track of your expectations) and my explanation of what I think I know. What I know of “other people” is no more than the explanation I have managed to construct in my own mind (my mind itself being such a construction). We are each perceiving the others view through our own solution to the problem; our own personal explanation of our experiences.

 

Are any of those explanations flaw-free? Well of course not; neither of us is all knowing and our abilities to actually analyze our own world views are so far below the requirements of such a feat that it is ridiculous to even suggest attempting such a feat. But, what I do know (as I have proved it) is that any “flaw-free” explanation must be a solution to my fundamental equation. I can use this to examine certain aspects of the problem of understanding.

 

The fact that any solution of Schrödinger's equation is an approximate solution to my equation implies that I have some facts about my world view (and yours) which can not be countered. First, that the universe can always be seen as a collection of energy quantized massless entities interacting via point scattering (the Dirac delta function); essentially a gas of such point “noumena” (to accord Anssi a little credit). And, second, that if we constrain our examinations to entities which are momentum quantized in the tau direction (collections of noumena which are mass quantized) we are presented with the possibility that collections of these “noumena” can remain in close proximity for extended times (collections I have dubbed “objects”). Furthermore, these objects will approximately obey Newton's mechanics.

 

Well, where does that get us? You and I have already reached (even before we had our first conversation) the fact that one huge portion of our world-view is constructed of objects which obey Newton's equations and, through advanced concepts of quantum mechanics (essentially Schrödinger's equation plus the many extensions which have been proposed) chemical theory, and thus the field of biology. I am not saying that any of these fields are flawless; merely that they are all connected by a logical thread in agreement with our experiences with the universe we find ourselves in.

 

Thus it behooves us to speak of such things in terms of my fundamental equation. This thread was dedicated to the simple issue that my fundamental equation is valid “only in the rest frame of the universe”. If it is possible for two observers to define personal rest frames which are moving with respect to one another (something held, in our scientific community, as easily arranged) a very specific constraint exists on the transformations of measurements between those two frames. That constraint is exactly the constraint commonly held as “special relativity”. What you were supposed to do with this thread was to follow the logic of the deduction of that constraint; not go off in wild directions as to the interpretations which can be attached to that fact.

Now in the case in which the measurements that the observers make do not follow the Lorenz transformation, the only possibility is that one or both of the observers are using a flawed explanation.

If that transformation is not part of their explanation then their explanation is certainly flawed.

Now the actual scaling of the fundamental equation is just a consequence of the transformation used to change the explanation in one frame to a new frame while still keeping the fundamental equation valid.

Again, you seem to have the horse on the wrong side of the cart. Any flaw-free explanation of anything can be interpreted as a solution to that fundamental equation (which is only valid in the rest frame of the universe). It isn't a question of “keeping the fundamental equation valid”, it is a question of so interpreting their explanations.

Basically I’m asking how do we know that the measurements in one frame can be transformed to any other frame even if both explanations are flaw-free.

The method of performing the transformation is part of the explanation! If it does not conform to the relativistic transformation, the explanation can be simply declared flawed.

This still seems to leave open the question can an object in one frame be moved to any other frame. That is are there frames that an object in one frame can’t be moved into from some other frame.

Tell me, does your world view include the possibility of two inertial frames moving with respect to one another? And does your world view include the possibility that an object originally in one frame can be moved to the other: i.e., can acceleration exist?

... or it may be as you seem to be suggesting that there is in fact a unique frame that is the only frame in which the fundamental equation is valid in.

The fundamental equation is valid only in the rest frame of the universe. That does not require a unique frame for two reasons; first I defined “time” to be what we know and “the present” to be “a change in what we know”; it follows that “what we know” is continually changing and thus it is entirely possible that the “rest frame of the universe” may change. And secondly, whenever an attempt to explain any specific phenomena is undertaken, great quantities of information concerning the universe are commonly ignored (we work out the problem in a mental environment which clearly presumes a “universe” consisting of considerably less than what is defined to be “The Universe”).

 

What I am getting at here is that my proof that Schrödinger's equation is an approximation to my fundamental equation allows me to talk about phenomena actually described by Newtonian mechanics as possible solutions to my fundamental equation (at least roughly approximate solutions). It is this fact which allows me to do thought experiments with flaw-free explanations. It should be clear to you that my fundamental equation is not itself solvable.

 

Have fun – Dick

[modest]

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.

You should really look at the derivation of my fundamental equation as, without understanding the necessity of that result the whole thing seems to be a bit “off the wall”. My original web site no longer exists but that derivation has been put on WiKipedia. Sans the interaction term, that is nothing more than a standard four dimensional wave equation.

I agree that's the next step for me. While the derivation in the OP seems sound to me, it is firmly based on the character of something I know nothing about. As I say, your post seems to take as a postulate that some certain wave has a fixed speed regardless of frame of reference. So long as that speed is finite I agree the Lorentz transformations are the only logical conclusion.

 

Before I get to your fundamental equation I might have some questions about this metric you're using. My confusion is right now all in my head and I'd have to work at putting it down in a post... if you think this is the wrong thread for discussing that metric then let me know.

 

The equation [imath]r=v_?t=\sqrt{(x'+vt')^2+y'^2+z'^2+\tau'^2}[/imath] is invalid because because it requires the moving observer (the observer in the primed frame) to know his frame is moving.

 

Well, I certainly agree :) But, in another thread you're saying that the moving train frame can decide to use the fixed frame of the platform in defining simultaneity. By your own objection, this would require the person in the train to know they are moving and thus not be a valid approach.

 

 

Actually, If you can indulge me, I might get this conversation on the metric started.

 

Defining tau loosely as what clocks measure and x as what a ruler measures and the metric [imath]dt^2=dx^2+d\tau^2[/imath] where c=1 and y & z are omitted we might plot something like:

S is not moving in x while S' is. The red lines are light emitted from S and detected at S'. Taking things slowly, I'll just ask one question: what is the change in tau between the detection events for S'? Is there enough information (as I've said nothing about time) to answer this question?

 

I appreciate your help on this. I'd really like to understand this alternative view of relativity and it's not coming to me intuitively.

 

~modest

[Doctordick]

My confusion is right now all in my head and I'd have to work at putting it down in a post... if you think this is the wrong thread for discussing that metric then let me know.

No, it is a fine place to discuss your difficulties. Please go and read the opening post to this thread. You will find almost exactly your diagram in that opening post (with some subtle but very important differences).

But, in another thread you're saying that the moving train frame can decide to use the fixed frame of the platform in defining simultaneity. By your own objection, this would require the person in the train to know they are moving and thus not be a valid approach.

Then perhaps we are confusing the meaning of the term “know” here. English is not really a very exact means of expressing ideas. If you substitute the word “presume” for the word “know” perhaps the confusion would kind of dissipate a bit. What we are really talking about here is the ability to perform a valid change in our frame of reference.

 

It is Einstein's contention that there exists no preferred frame of reference (insofar as the laws of physics are concerned). That is an assumption and cannot be proved (that is why he “postulated” that the speed of light is a constant). It is nice in that we need not worry about how we are moving relative to “a preferred frame of reference” however, we can still always do all our physics in “a preferred frame of reference” if we wish (that is, in fact, the central issue of relativity itself).

Defining tau loosely as what clocks measure and x as what a ruler measures and the metric [imath]dt^2=dx^2+d\tau^2[/imath] where c=1 and y & z are omitted we might plot something like:

S is not moving in x while S' is. The red lines are light emitted from S and detected at S'. Taking things slowly, I'll just ask one question: what is the change in tau between the detection events for S'? Is there enough information (as I've said nothing about time) to answer this question?

Your diagram is fine except for one very important issue: all three entities being referred to are momentum quantized in the tau direction (even the photon which just happens to have a tau momentum of zero) thus their position in the tau direction has an uncertainty of infinity. It is thus an error to presume that their path (and thus their interactions) can be explicitly represented in your diagram you have displayed. There is nothing wrong with the coordinate system you have laid out; the problem is that you have to very very careful in deducing the positions where interactions can occur.

 

Time which I define to be an interaction parameter (things can interact if they are in the same place at the same time) is not a measurable thing but rather a hypothesized parameter used to divide the future from the past (something which can not be defined except at the point of interaction). Since both interacting elements positions in the tau direction have an uncertainty of infinity, so does that point of interaction.

 

The only factor which we have to offset the difficulty of establishing when and where an interaction can occur is the fact of my fundamental equation which is a wave equation with a velocity of v_?: i.e., the distance between interactions is explicitly v_?t where t is the time change since the previous interactions. For convenience, if we ignoring the momentum quantization and examining only the vectors which show the actual direction of the momentum we need to talk in terms of interactions themselves. We can start with the assumption that S and S' interacted at the origin of your coordinate system. This a good reason to show the vectors for the momentum of S and S' as originating there and establishes t=0 for both entities.

 

Now let us describe the phenomena you are attempting to show in your diagram. The next interaction of significance occurs after a delay of one unit of distance in the coordinate system (v_?t=1): after waiting one unit of time (or one unit of tau as measured on his clock: i.e., he defines time so that c=unity) S emits a photon towards S'. Then, after waiting another unit of time (essentially identical to the first period) S emits a second photon towards S'. The difficult issue is to establish exactly the position in your diagram where those two photons interact with S'. Those interaction points are very definitely not at the end of the red lines you show.

 

Interaction can only occur when the length of the path of both S' and the photons (plus their delay on S) are exactly the same. Since you have shown S' to be at the angle of 45 degrees, that point is easy to calculate (actually it is not difficult for any angle but this one I can do in my head). The point where the first photon interacts with S' (in the rest coordinate system) will be when x+1 (essentially the age of the photon) is equal to x divided by the cosine of 45 degrees which is [imath]x\sqrt{2}[/imath] or

[math]x=\frac{1}{\sqrt{2}-1}=\frac{1}{.414}=2.4155[/math].

 

Which is, of course, the point at which S' is at the point x=2.4155 (remember, both the photon and S' are smeared out in the tau direction). Since S' is moving at an angle of 45 degrees with respect to S and their clocks agreed at the origin the clock on S' will read exactly 2.4155 at the moment of receipt of the photon (assuming S' also defines time such that c=unity).

 

The second photon will be intercepted at a distance where x+2 is equal to [imath]x\sqrt{2}[/imath] or

[math]x=\frac{2}{\sqrt{2}-1}=\frac{2}{.414}=4.8309[/math].

 

Thus an observer on S' will actually see the period of the clock on S (which is being used to time the photons being emitted) as having a period of 2.4155 times the period of his clock. Now all you have to do is figure out the Doppler effect the S' observer attaches to that timing (essentially, the delay he attributes to the fact that S is moving away from him).

 

You really need to follow the second part of my opening post on this thread. The details of this kind of calculations are worked out in detail.

 

I hope I have cleared up the significant issues here. Looking forward to hearing from you again.

 

Have fun -- Dick