Some Subtle Aspects Of Relativity.

[Bombadil]

Remember, we are talking about our expectations, we are not talking about reality. This fact leads to the different frames of reference. The issue is that the observer may not be aware of the entire universe and/or it's impact upon his observations: i.e., if distant entities are to be ignored as insignificant, it is entirely possible that two observers may use a different coordinate system. More important, in comparing their coordinate systems, they may not agree that the sum of all momentums vanishes in the other's coordinate system. At the same time, in order to obtain valid results, they must both operate with the fundamental equation under the constraint that the sum of the momentum of the universe vanishes. It turns out that this means their coordinate systems will not be the same.

 

How I understand this is that if we have two different sets of elements where both are in a coordinate system that have the momentums vanish we want to compare the two sets and find a coordinate system in which the momentum of the combination of both sets of elements vanish.

 

If so how can we compare the two coordinate systems? Isn’t all that we can do is compare the elements in the coordinate systems? In which case isn’t it impossible to prove that the two systems are different (other then the number of elements in them). Due to not being able to prove if any element in one set is one of the elements in the other set, how are we even going to be able to determine if the two sets have a different value of 1/k.

 

I’m wondering at this point if the measure of the fundamental equation and the momentum of the elements is entirely defined by the arbitrary constants that were used to define the constants in your derivation of the Schrodinger equation in the first place, and in so doing defining the speed of the expanding sphere.

 

The issue is the form of the fundamental equation. If you can ignore the Dirac delta functions (and, since they are point interactions it is always possible to analyze the situation at a small enough scale to make them dynamically irrelevant) then the fundamental equation reduces to a collection of independent wave equations each propagating with a velocity of 1/K. That makes that propagation an expanding sphere. Everyone using that equation must see that same result. This is exactly the same conundrum presented by the Michelson-Morley results and the solution is exactly the same: their coordinate systems must display exactly the same transformations Lorentz and Fitzgerald hypothesized.

 

How is it that a scale transformation will cause the differentials to vanish? Are only some of the elements going to be scaled or does the scale of the measure have some effect on the value of the differentials?

 

I suspect that part of my problem is in understanding how the solution of the equation is the expansion of a sphere, how I understand it, any solution to the fundamental equation is describing a sphere expanding at a speed of 1/k and that this means that the term [imath]e^{it/k}[/imath] or one similar to it is multiplied through any solution to the fundamental equation, but I don’t understand how or why something like this comes about.

[Doctordick]

Hi Bombadil, I have been slow to respond to your post because I want to make some important points clear; points which seem to give everyone great difficulty. After reading over your questions, I think you are working under a subtle misunderstanding of what is going on here. I have deduced the validity of my fundamental equation under some specific definitions. In that deduction, I have made every effort to assure that there exists no explanation of anything which can not be interpreted in a way which obeys my equation. This is the fundamental issue of understanding itself. When you think you understand something, what it means is that you have discovered an interpretation of facts available to you which makes sense to you. What I am trying to drive home is that this is an issue of interpretation.

In essence, what I have proved is that it makes utterly no difference what your world view actually is; as long as that world view is internally consistent (that is, it is consistent with itself) there always exists an interpretation of every construct embedded in that world view which can be seen as a solution to that equation. Before we can even begin to analyze any aspect of that world view, we must first find the interpretation which obeys my equation. That is easy for me because I have used that interpretation for well over forty years already and I apologize for the fact that I often fail to pick up on the fact that others have not seen that interpretation. I think that I should put forth my perspective on the base issues of that interpretation.

First, I should point out that the fundamental equation is very much what any competent physicist would write down to represent a gas of massless quantized point entities which only interact on contact. The first term (sans constants which can be absorbed in the free evolution parameter K), the sum over momentum, yields the kinetic energy of those massless entities in quantum mechanical notation and the Dirac delta function interaction is exactly what one would write down as a “point contact interaction” which essentially yields the boundary conditions on the correct solutions. The right hand side of the equation is the total energy which would be the sum over the kinetic energy associated with the various components of momentum in the representation (the dimensionality of that representation, which is essentially another free parameter).

What is important here is that the fundamental picture is one of an uncorrelated gas. This is not something which serves up much of value. Essentially it can contain no objects and objects are one of the central aspects of anyone's world view. The fundamental difficulty is one of creating correlations within this picture. Objects, in your world view (presuming that world view includes quantum mechanics) are very much correlated collections of quantum mechanical entities. Correlated means that their association has to continue over time. A collection of elements of a simple gas, where the velocity of those elements is constant (1/K). cannot maintain that kind of correlation unless those elements happen to be moving in the same direction.

Let me define “an object” to be a collection of elements which maintain their internal relationships over a sufficient length of time that they can be accurately described as an entity unto itself. If this is the case, then all the elements which go to make up an object must be moving in the same direction (or, since momentum quantization is a possibility, in opposite directions). It should be clear that no other possibility exists. This implies that all “objects” are collections of elements which display “mass” (defined earlier by me to be momentum quantization in the tau direction; a totally fictional axis). Since we are looking for an interpretation which is consistent with our (presumed internally self consistent) world view, the dimensionality of the picture (which I have pointed out is a free parameter) must be four which yields a three dimensional picture after the quantized dimension is abstracted out by infinite uncertainty in position on the tau axis. (The uncertainty principle says that the uncertainty in momentum times the uncertainty in position must be equal to or greater than [math]\hbar/2[/math]. Thus, if there is no uncertainty in mass, the uncertainty in position on tau is infinite.)

The essential issue here expressed is that we are, for the most part, concerned with the behavior of three dimensional massive objects. This is to say that our world view must consist of collections of three dimensional objects which have the quality referred to as mass. Since I have proved that Schroedinger's equation (and thus classical mechanics) is a valid approximation of the behavior of all elements (so long as their motion is close to parallel to tau), we can conclude that our common Newtonian view of the universe is a roughly accurate solution to the fundamental equation; being absolutely a correct solution in the limit where the motion of the objects is entirely in the tau direction (such objects would of course appear to be at rest).

So, what you must understand is that our picture is totally consistent with the standard Newtonian world view so long as apparent motion (motion orthogonal to tau) is negligible relative to 1/K (another free parameter). The picture is absolutely accurate (a correct solution to the fundamental equation) so long as all significant objects are “at rest”. In order to talk about dynamic consequences of motion we need to design a clock. Note that everywhere where you see the parameter “c” in the following quoted document, it should be replaced with 1/K (that free parameter).

You can find a specific design of an ideal clock in the physicsforum thread, posts #64 and #66 (again the post is a bit long and required two parts. I apologize for the diagrams being url references and not images.

If you can follow that discussion, you should come to realize that the object which was designed to measure time does not do so (as I have said many times, “time” is not a measurable variable it is an evolution parameter created by the observer). What one cycle of such a clock does measure is the displacement of the clock in the tau direction during that cycle. What is fascinating about this result is the fact that, if the length of the clock contracts in the direction of motion by the factor [math]sin \theta[/math], such a clock still measures exactly the displacement in the tau direction during that cycle. The important point about this fact is that it has absolutely nothing to do with the value of K.

In the universe so described, everything moves at the velocity 1/K. Because of mass quantization (as I have defined mass in my deduction of Schroedinger's equation) motion in the tau direction is undetectable. Thus the apparent velocity of objects can vary from zero (motion is parallel to tau) to 1/K (motion is orthogonal to tau). Notice that when the motion is orthogonal to tau, the momentum in the tau direction must be zero: i.e., only massless entities can have a velocity of 1/K. Now, as I have said many times, time is not a measurable variable. It is an evolution parameter and is entirely open to what ever the interest of the observer (the solver of the fundamental equation), The object referred to above as an ideal clock is an example of the phenomena exactly behind any time measure dependent upon electromagnetic resonance. The point being that the entity defined measures tau displacement and not time.

What is important here is that the apparent velocity is set by [math]sin \theta[/math], the angle between the tau axis and the direction of motion in the four dimensional space. Since no ruler may be defined which measures tau displacements, what clocks measure and what rulers measure seem to be independent scales; however, the fact that the energy connected to momentum in the tau direction must be the same as the energy connected to momentum in any other direction sets the factor for rest energy (no apparent motion) to be [math]mc^2[/math] (again, see my derivation of Schroedinger's equation). This sets a connection between apparent time and and space measures (sets the apparent speed of light to be c).

The above sets the stage for the issue of relativity. The issue here is that it must be possible to interpret your world view as a solution to my fundamental equation. That world view is exactly the world view which you imagine to be the case. So we have two observers in this universe who share a world view (consisting of a collection of defined objects whose motion is being analyzed). They use exactly the same methods to define their scales of apparent space and time and they are examining exactly the same experiment. The only difference is that their frames of references are different; they are using frames of reference which are moving with respect to one another. We are concerned with the proper mechanism for translating the measures made by one observer into the corresponding measurements made by the other.

Now, if they are both using methods which yield the expectations consistent with their pasts (what they know) and what they know is essentially identical, then they are both working with solutions to my fundamental equation. The problem arises when they use different frames of reference. My equation is only valid in the frame of reference where the sum of the momentum of all the elements in the universe vanishes. If they are using frames of reference which are moving with respect to one another, the momentum of all the elements in the universe certainly does not vanish in both frames: i.e., one of them is wrong (the fundamental equation is not valid in their frame of reference). On the other hand, the fundamental equation must yield their expectations based on what they know or think they know. This suggests that one is as right as the other. There exists only one solution to this dilemma.

The fundamental equation says that they live in a gas of massless entities moving with a fixed velocity in a four dimensional universe. The existence of an element at some point (a spike in probability at that point) suggests that the probability of finding that same element somewhere in the future is given by an expanding shell propagating at a velocity of 1/K from that start point. The point being that both observers must see the proper solution for the evolution starting from a point as an expanding sphere. As I said, it is a simple high school algebra problem to figure out exactly what that constraint must be. Now, as I said earlier, K is a free evolution parameter established by the observer; however, since t is not a measurable variable, the general procedure is to express velocities with an ideal clock such as the one I have defined above. Both observers will express the evolution of this probability distribution as an expanding shell propagating at a velocity equal to c.

We can solve the problem as follows. First, choose the x axis to be the axis of uniform motion where the origins at t=t'=0 are the same: i.e., x'=0 is identical to the point x=vt. Then, as far as y, z, and tau are concerned, there is no reason for them to use different measures. Both observers can specify these positions via entities whose position does not change over time (our motion specifies only a change in their x specification). Thus we can express their expectations for that expanding shell (be aware that they are talking about the same expanding shell of probability) with the following equations.

 

[math]ct=\sqrt{x^2+y^2+z^2+\tau^2}[/math] and [math] ct'=\sqrt{{x'}^2+{y'}^2+{z'}^2+{\tau'}^2}[/math]

 

Where y'=y, z'=z and [math]\tau' =\tau[/math]. The only possibility for x' and t' are given by [math]x'=\alpha x-\beta t[/math] and [math]t'=\gamma t -\delta x[/math]. For uniform motion, [math]\alpha, \beta, \gamma[/math] and [math]\delta[/math] must be simple constants. There are rather clear reasons why the transformation can not be more complex than that. If you can not pick up on the problems which arise with other possibilities, let me know and I will explain it to you.

Figuring out what those four constants have to be is, as I said, simple high school algebra. First of all x'=0 requires [math]\alpha x - \beta t =0[/math] when x=vt or [math]\alpha vt=\beta t[/math] which implies that [math]v=\frac{\beta}{\alpha}[/math].

Essentially the above spheres must transform into exactly the form shown: i.e., the unprimed sphere must be exactly the primed sphere. We can immediately write down the following expression for the primed version above.

[math] c^2(\gamma t -\delta x)^2=(\alpha x -\beta t)^2+y^2+z^2+\tau^2[/math]

 

or, multiplying out and rearranging terms, one has

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

 

which must be identical to [math]c^2t^2=x^2+y^2+z^2+\tau^2[/math]. This yields four equations on four unknowns which must be valid.

[math]v=\frac{\beta}{\alpha}[/math] ; [math]\alpha^2-c^2\delta^2=1[/math] ; [math]\gamma^2 - \frac{\beta^2}{c^2}=1[/math] and [math]\alpha \beta -c^2\gamma\delta =0[/math].

 

The final expression may be solved for gamma yielding [math] \gamma=\frac{\alpha\beta}{c^2\delta}[/math] and the first may be solved for beta yielding [math]\beta = \alpha v[/math]. Substituting beta in the other three equations yields the following three equations on the remaining constants.

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

 

The first may be solved for alpha via [math]\alpha^2=1+c^2\delta^2[/math]. That result can be substituted into the remaining two equations yielding

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

 

The second of these two can be squared to yield an expression for the gamma squared in the first. Substitution yields a final equation on delta.

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

 

Multiplying through by [math]\frac{c^2\delta^2}{1+c^2\delta^2}[/math] yields

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

or, by simple rearrangement,
 

[math]e^2\delta^2=(1+c^2\delta^2)\left(\frac{v}{c}\right)^2[/math]

 

which is easily solved for delta;
 

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

 

Since [math]\alpha^2=1+c^2\delta^2[/math]

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

 

adding via common denominators and taking the square root,

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

 

Since [math]\beta = \alpha v[/math] and [math]c^2\delta=\alpha v[/math], [math]\gamma=\alpha \frac{\beta}{c^2\delta}[/math] implies gamma is identical to alpha. It follows from the above that the only possible relationship between the measurements made by the observers moving with respect to one another is

[math]x' = \frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}[x-vt][/math] and [math]t' =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}\left[t-\left(\frac{vx}{c^2}\right)\right] [/math]

 

and, of course, the three other measurements y'=y, z'=z and [math]\tau' = \tau [/math].

These are exactly the relationships deduced from Einstein's axiom that the speed of light is a constant to all observers. I did this derivation in detail for one very simple reason: most people merely state the results and imply that their truth is support for Einstein's theory of special relativity. I prefer to see it as nothing more than the result of requiring that some specific velocity be observer independent (in this case, the implication of conservation of momentum for a free or none interacting element from the perspective of analysis under that free evolution operator).

Lorentz and Fitzgerald both showed that, if objects macroscopic structures were governed by Maxwell's equations then the contraction of the x coordinate above was an expected result. In the same vein, I have shown that if any objects structure is governed by my fundamental equation then the contraction of the x coordinate is an expected result. Since I have already proved that any and all explanations can be interpreted such that my fundamental equation is valid, the special relativistic transformation equations must be true and there is no need to pose the axiom that the speed of light is a constant to all observers.

If so how can we compare the two coordinate systems?

To reiterate, the issue here is the fact that there must exist an interpretation of your world view which obeys my fundamental equation. I am merely giving you that interpretation. How you compare the two coordinate systems is exactly how those coordinate systems are compared in your world view.

... how are we even going to be able to determine if the two sets have a different value of 1/k.

Again, K is a free evolution parameter used by the specific observer and is not a measurable thing. The fundamental equation, since mass is defined to be momentum in the tau direction, yields a maximum apparent velocity of 1/K for massless elements: i.e., conservation of momentum (in the absence of interactions) and a freely defined evolution parameter always yields a fixed maximum velocity. But as a “ideal clock” actually measures the rate of change in tau (which is also that same maximum velocity), any observer will use a clock to set that velocity will obtain the value “c” which is a direct consequence of his method of measuring change in x compared to his method of measuring change it tau.

I’m wondering at this point if the measure of the fundamental equation and the momentum of the elements is entirely defined by the arbitrary constants that were used to define the constants in your derivation of the Schrodinger equation in the first place, and in so doing defining the speed of the expanding sphere.

In a sense, this is absolutely correct: however, the real issue at the root of it all is the common scientific assumption that clocks measure time. That assumption is not consistent with any interpretation of my fundamental equation where time is no more than an observer's view of how things change (an evolution parameter).

How is it that a scale transformation will cause the differentials to vanish? ... but I don’t understand how or why something like this comes about.

You are simply looking at the wrong problem.

I hope this clarifies a few things -- Dick

 

Edited February 1, 2016 by Doctordick

[Kharakov]

Hi Dr. Dick,

 

I found your posts on 4d space (5d spacetime) while searching the internet for theories (or descriptions, I've seen your preference to not refer to your statements as theories) that describe a 4th dimension of space. I recently watched a program on TED.com (Brian Greene's lecture introducing string theory) that made me think of a few things (which aren't exactly in accordance with the string theory views on the matter, but may not oppose the various string theories as well).

 

Basically, it's some mathematical evidence that supports the view of a 4d space/ 5d spacetime. I actually called it a theory, because... well, I am not aware of the semantic issues as to why you do not call your ideas "theories" (this isn't an argument for using the term theory, I just think of an idea that is intended to describe reality as a theory or a hypothesis- please don't perceive this as a challenge to your non-usage of the term). Anyways, please forgive my longwindedness in this introduction, I will try and explain what I found.

 

 

The basic gist of the idea is this: that there is a 4th dimension of space and that looking at space as having a 4th dimension grants insight into the workings of physical reality. I arrived at what might be the same or at least similar conclusions to your own. Here are some mathematical examples of it (which are very simplistic):

 

Let’s start out with a demonstration of the theory, with an electron.

 

Electron mass: 510,998.910 eV/c2 ^[1,2]

Electron energy: 510,998.910 eV ^[1,2]

Planck constant in eV s: 4.13566733 * 10-15 eV s ^[1,2]

Speed of light in vacuum ©: 299792458 m/s ^[1,2]

 

Let’s find the frequency and wavelength of a photon with this energy.

f= E/h = 510,998.910 eV / 4.13566733 * 10-15 eV s = 1.23558998 * 1020 Hz

wavelength = c/f = 299792458 m/s / 1.23558998 * 1020 Hz = 2.42631021×10-12 m

 

Now we are going to accelerate the electron to .5 c (cos 60 * c) and calculate its de Broglie wavelength.

The equation for the de Broglie wavelength is:

[imath]\lambda = \frac{h}{mv} \sqrt{1-\frac {v^2}{c^2}}[/imath] ^[3,4]

 

[imath]

\lambda=4.20249256 \ \times \ 10^{-12} \ m

[/imath]

 

Wavelength, frequency, and energy are all vectors (they have direction) in 4d space. We are used to looking at them as scalar quantities, but that is not correct. The de Broglie wavelength is the wavelength component of the photon (or photons) on the standard 3 dimensional space axis (3dspace). The wavelength we calculated for the rest mass of an electron earlier is the wavelength component of the photon (electron) on the orthogonal dimension axis (orthogonal to 3dspace).

Let’s determine what the total wavelength of the photon (the electron) is.

To find the total wavelength, we have to invert the wavelengths, square them, add them together, take the square root of the result, and invert the result of this. We invert the wavelengths because the total wavelength is shorter than either of the components. We could just convert the wavelength to frequencies as well, square them, add them, and then take the square root of the results.

Here are the 2 equations we can use (depending on whether we have wavelength or frequency vectors to calculate the totals of):

 

[imath]

\lambda_{total} = \frac{1}{\sqrt{\frac{1}{\lambda_{3dpace}}^2 + \frac{1}{\lambda_{orthogonal}}^2}}

[/imath]

 

[imath]

f_{total}=\sqrt{f^2_{3dspace}+f^2_{orthogonal}}

[/imath]

 

We can also calculate the relativistic energy if we have both energy components (3dspace and orthogonal):

 

[imath]

E_{total}=\sqrt{e^2_{3dspace}+{e}^2_{orthogonal}}

[/imath]

 

Anyways, let’s keep going with our previous example and calculate the total wavelength of the photon (electron).

 

[imath]

\lambda_{total}=\frac{1}{\sqrt{{\frac{1}{4.20249256 \ \times 10^{-12}m}}^2+{\frac{1}{2.42631021\ \ times 10^{-12}m}}^2}}

[/imath]

 

[imath]

\lambda=2.10124628 \ \times \ 10^{-12} \ m

[/imath]

 

Let’s calculate our total energy from the wavelength:

 

[imath]

E= \frac{hc}{\lambda}

[/imath]

 

[imath]

E_{total}=590050.717 \ eV

[/imath]

 

Let’s go ahead and check this against the result from the standard relativity energy equation ^[5,6,7,8]:

 

[imath]

E= \frac{e}{\sqrt{1-\frac {v^2}{c^2}}}

[/imath]

 

[imath]

E= \frac{510998.910 eV}{\sqrt{1-\frac {(.5c)^2}{c^2}}}

[/imath]

E = 590050.716 eV

It’s the same (except for slight error due to rounding).

We can also determine the angle of the photon from the 3dspace axis and all of its component vector magnitudes in a much easier manner, simple trigonometric functions. I wanted to show you the hard classical method first.

 

If we know the velocity of the photon along the 3dspace axis, we can do something really simple.

Since:

 

[imath]

v_{total}=\sqrt{v^2_{3dspace}+v^2_{orthogonal}}=c

[/imath]

 

We can take the inverse cosine of the velocity unit vector component along the 3dspace axis divided by the total velocity of the photon. In other words, the inverse cosine of (.5c)/c (the scalar is taken out) gives us the angle away from 3dspace. If we know the total relativistic energy of the photon (particle) we can use the angle to generate its 3dspace and orthogonal component vectors.

 

[imath]

\theta=\arccos{.5}=60

[/imath]

 

Say we know the relativistic energy of this particle, how do we find its rest mass energy (the orthogonal component of the photons energy)? Simple trigonometric calculation in 5d physics:

 

[imath]

e_{rest\ mass}=\sin\theta \times E_{total}

[/imath]

 

[imath]

e_{rest \ mass}=\sin 60 \ \times \ 590050.717 \ eV=510998.910 \ eV

[/imath]

 

Likewise if we know the photons rest mass energy component and its velocity, we can take the inverse cosine of the unit vector, get the angle, divide its orthogonal (rest mass) energy by the sin of the angle and get its relativistic energy.

 

[imath]

v=\sqrt{.96}c

[/imath]

 

[imath]

e_{orthogonal}=510998.910 \ eV

[/imath]

 

[imath]

\theta = \arccos{\sqrt{.96}}=11.5369590

[/imath]

 

[imath]

E_{total}=\frac{510998.910}{\sin{\theta}}= 2554994.55 \ eV

[/imath]

 

You already know the relativistic energy equation ^[5,6,7,8], so you can compare the above answer with the result you get from it.

 

Energy is a vector in space with 4 dimensions, rather than a scalar quantity. It has direction.

Anything with mass is comprised of photons with an energy component orthogonal (in a 4th dimension) to 3dspace. From this we can discover new ways of interacting with … photons.

 

Anyways, as to why these photons don't move away from 3dspace, well:

In the "Theory of Orthogonality", photons are described moving in a 4th dimension of space, but what wasn’t described is why they do not appear to recede from 3dspace.

It’s simple. Conservation of energy (in 3dspace) cannot be violated. Therefore, the photon “pulls” space into 3dspace from the 4th dimension as it “travels” in that direction. In the beginning, the first photons traveling this "way" produced tons of space (compared to the volume of the universe). Now, the volume is soooooo big that we don’t see as much expansion (but it still is going on, it’s just a smaller ratio to the total volume).

 

 

Bibliography for the above:

 

1. wdoubleUw<dot>physicstoday<dot>org/guide/fundconst.pdf CODATA Recommended Values of the Fundamental Physical Constants – 2006

2. "Planck constant in eV s". 2006 CODATA recommended values. NIST. Retrieved on 2007-08-08.

3. en<dot>wikipedia<dot>org/wiki/De_Broglie_hypothesis de Broglie hypothesis - Wikipedia, the free encyclopedia (The equation is not cited on this page, so must not be copyrighted: it is de Broglie’s equation, however)

4. L. de Broglie, Recherches sur la théorie des quanta (Researches on the quantum theory), Thesis (Paris), 1924; L. de Broglie, Ann. Phys. (Paris) 3, 22 (1925). Reprinted in Ann. Found. Louis de Broglie 17 (1992) p. 22.

5. Albert Einstein in letter to L Barnett (quote from L. B. Okun, “The Concept of Mass,” Phys. Today 42, 31, June 1989.)

6. Tolman, R. C. (1934). Relativity, Thermodynamics, and Cosmology. Oxford: Clarendon Press. LCCN 340-32023. Reissued (1987) New York: Dover ISBN 0-486-65383-8.

7. Larmor, J. (1897), "A dynamical theory of the electric and luminiferous medium — Part III: Relations with material media", Philosophical Transactions of the Royal Society 190: 205–300, doi:10.1098/rsta.1897.0020

8. Lorentz, Hendrik Antoon (1899), "Simplified theory of electrical and optical phenomena in moving systems", Proc. Acad. Science Amsterdam I: 427–443; and Lorentz, Hendrik Antoon (1904), "Electromagnetic phenomena in a system moving with any velocity less than that of light", Proc. Acad. Science Amsterdam IV: 669–678

[Bombadil]

After reading your post I think I have somewhat of a better idea of what the issue that is being brought up is. The way that I am understanding this is that, given two different reference frames (that is two separate observers, both using the same explanation but using a different set of values for the derivatives in the fundamental equation) the task is to find a transformation that will change a measurement made in one frame to the one arrived at if the object is measured in the other frame.

 

First, I should point out that the fundamental equation is very much what any competent physicist would write down to represent a gas of massless quantized point entities which only interact on contact. The first term (sans constants which can be absorbed in the free evolution parameter K), the sum over momentum, yields the kinetic energy of those massless entities in quantum mechanical notation and the Dirac delta function interaction is exactly what one would write down as a “point contact interaction” which essentially yields the boundary conditions on the correct solutions. The right hand side of the equation is the total energy which would be the sum over the kinetic energy associated with the various components of momentum in the representation (the dimensionality of that representation, which is essentially another free parameter).

 

So, will the Dirac delta function only have an effect on the equation by changing the wave function in such a way that it will cause it to equal 0 at any other element, or will only invalid elements have any effect on elements in an explanation?.

So, what is the link then between energy on the right side and the mass and momentum on the left side? Does the fundamental equation minus the Dirac delta function give the link between mass, momentum and energy?

 

Let me define “an object” to be a collection of elements which maintain their internal relationships over a sufficient length of time that they can be accurately described as an entity unto itself. If this is the case, then all the elements which go to make up an object must be moving in the same direction (or, since momentum quantization is a possibility, in opposite directions). It should be clear that no other possibility exists. This implies that all “objects” are collections of elements which display “mass” (defined earlier by me to be momentum quantization in the tau direction; a totally fictional axis). Since we are looking for an interpretation which is consistent with our (presumed internally self consistent) world view, the dimensionality of the picture (which I have pointed out is a free parameter) must be four which yields a three dimensional picture after the quantized dimension is abstracted out by infinite uncertainty in position on the tau axis. (The uncertainty principle says that the uncertainty in momentum times the uncertainty in position must be equal to or greater than \hbar/2. Thus, if there is no uncertainty in mass, the uncertainty in position on tau is infinite.)

 

I’m not quite sure of what you mean by (or, since momentum quantization is a possibility, in opposite directions) but it sounds like you have defined an object so that it is any set of points fixed under rotations and shifts in that it retains any internal relationship ( that is it has a fixed mass, momentum and energy)?

How do we know that the uncertainty principle can be applied in the way that you are applying it (that is to mass and the [imath]\tau[/imath] axis) I’m not trying to go into a long discussion about how it is derived but what is necessary for us to know that the uncertainty principle takes affect on the [imath]\tau[/imath] axis?

 

Now, if they are both using methods which yield the expectations consistent with their pasts (what they know) and what they know is essentially identical, then they are both working with solutions to my fundamental equation. The problem arises when they use different frames of reference. My equation is only valid in the frame of reference where the sum of the momentum of all the elements in the universe vanishes. If they are using frames of reference which are moving with respect to one another, the momentum of all the elements in the universe certainly does not vanish in both frames: i.e., one of them is wrong (the fundamental equation is not valid in their frame of reference). On the other hand, the fundamental equation must yield their expectations based on what they know or think they know. This suggests that one is as right as the other. There exists only one solution to this dilemma.

 

Do they to have to be working with essentially the same past or is it enough that their expectations for the area of the objects of interest are the same or is all that is necessary is that they define distances in the same way?

If I am understanding this right there is no need for the momentum in either frame to vanish. That is, we can change between any two frames of reference. There is no need to use a frame where the momentum vanishes, all that is required is that they both use the same expectations.

 

So, what you must understand is that our picture is totally consistent with the standard Newtonian world view so long as apparent motion (motion orthogonal to tau) is negligible relative to 1/K (another free parameter). The picture is absolutely accurate (a correct solution to the fundamental equation) so long as all significant objects are “at rest”. In order to talk about dynamic consequences of motion we need to design a clock. Note that everywhere where you see the parameter “c” in the following quoted document, it should be replaced with 1/K (that free parameter).

 

I’m somewhat puzzled by your referring to the value of c as 1/k, when in your derivation of the Schrödinger equation you defined it equal to the value of [imath]\frac{1}{k\sqrt{2}}[/imath]. Also without assuming that an object moving at such a speed has zero mass I cant understand how we can conclude that it has zero movement along the [imath]\tau[/imath] axis although it seems that there should be a relatively strata forward way of showing this, or did you show this when you derived the Schrödinger equation?

 

Where y'=y, z'=z and [imath]\tau' =\tau[/imath]. The only possibility for x' and t' are given by [imath]x'=\alpha x-\beta[/imath] t and [imath]t'=\gamma t -\delta x[/imath]. For uniform motion, [imath]\alpha, \beta, \gamma[/imath] and [imath]\delta[/imath] must be simple constants. There are rather clear reasons why the transformation can not be more complex than that. If you can not pick up on the problems which arise with other possibilities, let me know and I will explain it to you.

 

The first two y'=y, z'=z seem quite reasonable seeing as neither of these have any difference between the two frames. The next two seem backwards to me in that it seems that the [imath]\tau[/imath] and t should be reversed, in that we cant measure the change in t so any change in it would only seem to scale the equation along all of the axis’s.

 

Now the form of the transformation seems reasonable in that the fundamental equation being scale invariant seems to me to mean that we can simply scale any axis and the behavior of the fundamental equation will be unchanged. Also it seems that we can define any location as the origin of the x or t axis so any function that would be used must be shift invariant. While this suggests to me that this transformation will be sufficient, I don’t see how we can know that this is the only possible transformation unless we can show that the fundamental equation is not invariant under any other transformations that can leave the expanding sphere invariant. Although this being of the form of a change in coordinate bases makes this appear to be a quite reasonable choice.

[Doctordick]

Let me put a slightly different spin on the situation as you are to see it. First of all you already have a world view and we are discussing events which can take place in that world view.

The way that I am understanding this is that, given two different reference frames (that is two separate observers, both using the same explanation but using a different set of values for the derivatives in the fundamental equation) the task is to find a transformation that will change a measurement made in one frame to the one arrived at if the object is measured in the other frame.

The phrase in parenthesis should be replaced with, “two separate observers with flaw free explanations of their world views”. That they are even aware of my fundamental equation is of no consequence here. The issue is that either of their world views, if they are indeed flaw free, can be translated into my representation which requires their measurements be consistent with my fundamental equation (and that quantum gas picture quantized in the tau direction which I have put forward).

... or will only invalid elements have any effect on elements in an explanation?.

Invalid elements have no bearing on the problem as they obey exactly the same rules obeyed by the valid elements. The only point of interest here has to do with the transformation between the actual measurements made by the two observers.

So, what is the link then between energy on the right side and the mass and momentum on the left side? Does the fundamental equation minus the Dirac delta function give the link between mass, momentum and energy?

Seen as a quantum gas with “point” interactions and my definition of mass, momentum and energy, my fundamental equation certainly does provide exactly such a link.

I’m not quite sure of what you mean by (or, since momentum quantization is a possibility, in opposite directions)

That implies to me that you do not understand the Heisenberg uncertainty principle. If momentum in the tau direction is quantized the uncertainty is zero. It follows that the uncertainty in tau is infinite: i.e., any flaw free explanation must yield exactly the same expectations for any explicit value of tau. Thus tau can not be an observable variable in your world view. Or, more accurately, tau can not be an observable in my interpretation of your world view: i.e., it is not a measurable element in the transformations being discussed. The elements which go to make up an object must appear to remain in association. That means their individual apparent motions must be essentially the same. The component in the tau direction may be in either direction because actual position in the tau direction is unknowable.

... but it sounds like you have defined an object so that it is any set of points fixed under rotations and shifts in that it retains any internal relationship ( that is it has a fixed mass, momentum and energy)?

There is nothing in my definition of “an object” which requires a fixed mass, momentum or energy. Essentially, I have merely defined an object to be a collection of fundamental elements which stay associated with one another over times of interest to the observer. This is only possible if those fundamental elements are essentially going in the same direction (the fundamental equation is describing a quantum gas) in our four dimensional space. We are not particularly concerned with microscopic deviations, we are concerned with the long term average values. What phenomena behind the existence of these objects is not being discussed; we are only presuming that there exist solutions which correspond to “objects”. Since the ordinary concept of objects consists of macroscopic collections of elements which “stay associated with one another over times of interest to the observer”, I would suggest that there must be solutions with this property as, our world view contains such “objects” and I have already proved that any world view can be interpreted in such a way to make my fundamental equation valid. If I knew all the correct solutions, that collection would have to include “objects”.

How do we know that the uncertainty principle can be applied in the way that you are applying it (that is to mass and the [imath]\tau[/imath] axis) I’m not trying to go into a long discussion about how it is derived but what is necessary for us to know that the uncertainty principle takes affect on the [imath]\tau[/imath] axis?

You can look at it from two very different directions. Either we are in a universe where mass, as I have defined it, is quantized (in which case any flaw free explanation must yield exactly the same expectations for any explicit value of tau) and the uncertainty in tau is infinite; or you can comprehend that tau, as I have introduced it, is an invalid ontological element and thus must be perfectly uncertain which is just the other side of the same coin. The central point here is that elemental ontological elements specified by the indices x, y and z are uncertain only on a microscopic level (microscopic being defined as small enough to bring in such issues).

Do they to have to be working with essentially the same past or is it enough that their expectations for the area of the objects of interest are the same or is all that is necessary is that they define distances in the same way?

It's your world view; you have to answer these questions. When you get your answers (that flaw free explanation) explain it to me. As I learn your explanation, I will translate it into a form which obeys my equation.

If I am understanding this right there is no need for the momentum in either frame to vanish. That is, we can change between any two frames of reference. There is no need to use a frame where the momentum vanishes, all that is required is that they both use the same expectations.

Both observers can see the measuring instruments used by the other. If they both are using flaw free explanations, they must agree that the readings on the instruments are correct as those are elements of the facts which constitute their pasts.

I’m somewhat puzzled by your referring to the value of c as 1/k, when in your derivation of the Schrödinger equation you defined it equal to the value of [imath]\frac{1}{k\sqrt{2}}[/imath].

My reference to c as 1/K is little more than a statement of the inverse nature of the relationship; you have to realize that K is an open parameter having to do with the unmeasurable evolution parameter t. The consequences of the choice of an actual value is another issue to be discussed later. In my derivation of Schrödinger's equation, I am concerned with the actual value. The factor [imath]\sqrt{2}[/imath] arose from the definition of the alpha and beta operators. Please note my multiplication of the entire equation by [imath]\hbar[/imath] is rather strange from a mathematical perspective. None of this has any real consequences when it comes to relativity.

Also without assuming that an object moving at such a speed has zero mass I cant understand how we can conclude that it has zero movement along the [imath]\tau[/imath] axis although it seems that there should be a relatively strata forward way of showing this, or did you show this when you derived the Schrödinger equation?

In the quantum gas, the evolution parameter gives all elements exactly the same velocity through the four dimensional space. It is only when the momentum of the elements in the tau direction is quantized that the existence of this dimension vanishes from one's world view; in that case, the apparent velocity of all elements consists of the component orthogonal to tau. Only when that velocity is perfectly orthogonal to tau (that is, the velocity in the tau direction is zero) is the apparent velocity equal to that velocity set by the fundamental equation. But that requires the momentum in the tau direction to also be zero: i.e., the mass of the element must be zero.

The first two y'=y, z'=z seem quite reasonable seeing as neither of these have any difference between the two frames. The next two seem backwards to me in that it seems that the [imath]\tau[/imath] and t should be reversed, in that we cant measure the change in t so any change in it would only seem to scale the equation along all of the axis’s.

Let's go back to the underlying world view of a quantum gas in a four dimensional space (which is the interpretation which is to be made on each of the two referenced observers explanations). From that perspective, what makes tau different from y or z. The direction of the motion between the two frames of reference has been defined to be along the x axis. No comment was made concerning motion in the tau direction so why would you presume that [imath]\tau' \neq \tau[/imath]. The only place where tau comes into play is when I interpret the observers world view into mine and that is the axis upon which the momentum is quantized and no more.

Now the form of the transformation seems reasonable in that the fundamental equation being scale invariant seems to me to mean that we can simply scale any axis and the behavior of the fundamental equation will be unchanged.

It isn't the behavior of the fundamental equation which concerns us; what is important, is that a change in scale has no impact on the collection of possible solutions. The actual scale is set by the observers via the collection of solutions which go to make up their world view, a subtly different statement. At the same time, both observers are working with exactly the same collection of solutions, scaled of course in the parameters x and t they use.

Also it seems that we can define any location as the origin of the x or t axis so any function that would be used must be shift invariant. While this suggests to me that this transformation will be sufficient, I don’t see how we can know that this is the only possible transformation unless we can show that the fundamental equation is not invariant under any other transformations that can leave the expanding sphere invariant. Although this being of the form of a change in coordinate bases makes this appear to be a quite reasonable choice.

I was hoping I would not have to go through the following analysis; however, it seems necessary. To begin with the most general transformation possible (given that v is set) is [imath]x'=f(x,t)[/imath] and [imath]t'=g(x,t)[/imath] (they cannot depend upon y, z or tau as we are free to move the origin in those directions and the result cannot change).

 

We can make a power series expansion of these functions, obtaining

[math]x'= \sum_{i=0}^\infty \sum_{j=0}^\infty a_{ij}x^i t^j =f(x,t) [/math]

 

plus a similar expression for g(x,t). Now let us look at the effect of the various terms. First, i=0 and j=0 simply add a constant. Since we defined the origins at the start point to be the same point, such a constant would invalidate that representation. The result for i=1 and j=0 plus the term i=0 and j=1 is exactly the same as the representation I presented. It follows that all I need show is that all of the terms for i or j greater than 1 (or a₁₁)must invalidate our representation.

 

If there is a term in there proportional to x or t to a higher power than one (or proportional to xt), the resulting transformation becomes different for different x or t: i.e., moving the origin after performing the transformation yields a different result for the same point. That yields a coordinate system inconsistent with the postulates of a Euclidean coordinate system. I hope this clarifies your difficulties. Essentially, shifting the origin must not invalidate the transformation; simple scaling is the only possibility.

 

Have fun -- Dick

[Bombadil]

You can look at it from two very different directions. Either we are in a universe where mass, as I have defined it, is quantized (in which case any flaw free explanation must yield exactly the same expectations for any explicit value of tau) and the uncertainty in tau is infinite; or you can comprehend that tau, as I have introduced it, is an invalid ontological element and thus must be perfectly uncertain which is just the other side of the same coin. The central point here is that elemental ontological elements specified by the indices x, y and z are uncertain only on a microscopic level (microscopic being defined as small enough to bring in such issues).

 

The former seems to require that we know that we have a flaw free explanation (something that while it is a requirement of using the explanation it seems to me to be something that we cannot prove due to having finite knowledge). So the second one is, I think the one that I would think in terms more of. But, do you mean to call the tau axis an invalid object? I would think that an object was part of the explanation while the tau axis is something that you have used in defining an explanation. Although it seems that any object that makes the tau axis observable must be an invalid object.

 

It's your world view; you have to answer these questions. When you get your answers (that flaw free explanation) explain it to me. As I learn your explanation, I will translate it into a form which obeys my equation.

 

I take it then that these questions only mean something if we have a flaw free explanation to begin with while what we are interested in is what all flaw free explanations have in common?

 

My reference to c as 1/K is little more than a statement of the inverse nature of the relationship; you have to realize that K is an open parameter having to do with the unmeasurable evolution parameter t. The consequences of the choice of an actual value is another issue to be discussed later. In my derivation of Schrödinger's equation, I am concerned with the actual value. The factor [imath]\sqrt{2}[/imath] arose from the definition of the alpha and beta operators. Please note my multiplication of the entire equation by \hbar is rather strange from a mathematical perspective. None of this has any real consequences when it comes to relativity.

 

I noticed that it seems that it would be factored out. I don’t see why it is even in the equation at all. It seems that it shouldn’t have any effect on the equation at all except perhaps in how mass, momentum and energy are being defined.

 

Let's go back to the underlying world view of a quantum gas in a four dimensional space (which is the interpretation which is to be made on each of the two referenced observers explanations). From that perspective, what makes tau different from y or z. The direction of the motion between the two frames of reference has been defined to be along the x axis. No comment was made concerning motion in the tau direction so why would you presume that [imath]\tau' \neq \tau[/imath]. The only place where tau comes into play is when I interpret the observers world view into mine and that is the axis upon which the momentum is quantized and no more.

 

Didn’t we assume motion in the tau direction when we assumed that it is not a mass less element? In fact didn’t we assume that the elements speed along the tau axis changed when it changed the value of v as you have defined it? Why isn’t a scaling of the t axis going to be an unobservable scaling and have no effect or does this have some effect on the energy of the equation?

 

If there is a term in there proportional to x or t to a higher power than one (or proportional to xt), the resulting transformation becomes different for different x or t: i.e., moving the origin after performing the transformation yields a different result for the same point. That yields a coordinate system inconsistent with the postulates of a Euclidean coordinate system. I hope this clarifies your difficulties. Essentially, shifting the origin must not invalidate the transformation; simple scaling is the only possibility.

 

Then is there other possibilities that are not Euclidean, but we only chose a Euclidean system because we know that there is a flaw free explanation that is in a Euclidean geometry and it makes things simpler by working with such a geometry?

[Doctordick]

The former seems to require that we know that we have a flaw free explanation (something that while it is a requirement of using the explanation it seems to me to be something that we cannot prove due to having finite knowledge).

First, there is no need to prove our explanation is flaw free. If the explanation is flawed (i.e., it can be disproved with known information) it simply is of no interest to us. I think one point you may be confusing is that I am not concerned with producing explanations; I am concerned with constraints those explanations (if they are indeed flaw free) must obey, a very different issue.

 

Another point you may be missing is the fact that I use the term “flaw free” because we have finite knowledge: i.e., I do not use the term “correct” as future information is required to prove an explanation is “incorrect”. When I say an explanation is “flaw free” I mean that there is no known information which contradicts that explanation. If it is flawed and we don't know it, it must be due to improper analysis and not due to the finite character of our knowledge.

But, do you mean to call the tau axis an invalid object?

Of course I do. By construction it is a complete figment of my imagination. By definition, the valid ontological elements are those elements labeled by the numerical indices [math]x_i[/math] and tau is not a member of that set. Tau was introduced for the simple purpose that it allowed me to represent the collection of numerical indices, [math]x_i[/math], as points on an imaginary x axis. Without introducing tau, the fact that the collection of ontological elements referred to as a specific “present” (i.e., a specific change in what we know) could contain a given ontological element more than once would be lost. Remember, the ontological elements are totally unknown; “what” they are is part and parcel of our explanation: i.e., we don't know we are talking about the same element here until we have an explanation and, at no point do we actually "know" our explanation is correct. The best we can do is know it is "flaw free" at the moment. Another thing we know is that there always exists an interpretation of "what is known" which obeys my fundamental equation thus the constraints deduced from that equation are absolutely universal.

Although it seems that any object that makes the tau axis observable must be an invalid object.

I don't understand your definition of the word “object” here. My definition of an "object” is any collection of ontological elements which may be considered as unchanging over a elapse of time of interest to us.

I take it then that these questions only mean something if we have a flaw free explanation to begin with while what we are interested in is what all flaw free explanations have in common?

What we are interested in are the constraints that all flaw free explanation must obey. I don't know that I would use the phrase “have in common” as that seems to imply something more than what I am saying, but they certainly must “have in common” the fact that they can be interpreted in a manner which must obey my fundamental equation. I am not even saying that interpretation is correct; all I am saying is that, given that the information available to me (the valid but undefined ontological elements of that explanation), such an interpretation is flaw free.

I noticed that it seems that it would be factored out. I don’t see why it is even in the equation at all. It seems that it shouldn’t have any effect on the equation at all except perhaps in how mass, momentum and energy are being defined.

You are exactly correct. When I first started graduate school, our physics library had a copy of Gammow's “Mr Tompkins in Wonderland” which I read when I was first studying quantum mechanics (when I was a student, quantum was a graduate subject and I had never seen it as an undergraduate). At the moment, you can read an updated version of the book on the web called “The New World of Mr. Tompkins”.

 

I found most of it quite delightful; however, the chapter, “The quantum Safari” (chapter nine in the above link) bothered me quite a little. It made some rather stupid assumption which were simply contrary to quantum mechanics. I was moved to try and construct what Mr. Tompkins would actually see if [imath]\hbar[/imath] were actually a large number and failed quite miserably. The problem was that [imath]\hbar[/imath] comes into almost everything (size of atom's electron orbitals, consequential bond lengths, etc). I was totally unable to find a logical structure from which I could start (which did not depend upon the presumed value of [imath]\hbar[/imath]. I finally came to the conclusion that [imath]\hbar[/imath] was circularly defined. I brought the issue up to my physics advisor and was told that I "needed to study my physics a little better; [imath]\hbar[/imath] was certainly NOT circularly defined". What you have just noticed above is that, in my work, [imath]\hbar[/imath] is most certainly circularly defined and I take it as proof that my advisor was wrong. We can perhaps get deeper into that issue later.

Didn’t we assume motion in the tau direction when we assumed that it is not a mass less element?

You are confusing two different motions here. One is the motion of an element in that quantum gas and the other is the motion of an object in the world view of the two observers. The evolution parameter (the unmeasurable t associated with my equation) is a universal parameter attached to my interpretation of what is going on. Each observer will, because of the character of his world view, consider time to be defined by his personal clocks as tau is not an observable axis in his world view. If you go back to my definition of mass, being the momentum in the tau direction, you will find it is the rest mass of the entity of interest: i.e., the two observers (when they go to examine the situation under my interpretation) will agree as to the rest mass of the various elements making up the universe. In fact they will agree as to the momentum in any direction except x. It is only the momentum of these elements in the x direction (the partial with respect to x of their expectations) upon which they will disagree. What is important is that they will both (using my interpretation of the details) see the change in a specific expectation to be defined by a specific expanding sphere.

 

As his explanation (his expectations) are not to depend upon his absolute motion my fundamental equation must be as applicable under the assumption he is standing still as it is for the other frame (where the other observer is standing still). It is this fact which requires that the expanding sphere (in the quantum gas picture) must be spherical in both interpretations.

In fact didn’t we assume that the elements speed along the tau axis changed when it changed the value of v as you have defined it? Why isn’t a scaling of the t axis going to be an unobservable scaling and have no effect or does this have some effect on the energy of the equation?

Neither of the observers are actually using the unmeasurable evolution parameter. They are both using the dynamic properties of a thing they call a clock to establish what they call time (the evolution parameter under the assumption they are standing still).

Then is there other possibilities that are not Euclidean, but we only chose a Euclidean system because we know that there is a flaw free explanation that is in a Euclidean geometry and it makes things simpler by working with such a geometry?

Not really, there is something else going on here which seems to be missed by a lot of people who have great interest in “non Euclidean” geometries. If the sole purpose of your geometry is to visually display independent variables (which is exactly what we are doing here), then Euclidean geometry is the only geometry which can serve the purpose. All non-Euclidean geometries presume some relationship between the coordinates (how one coordinate behaves at a specific point is dependent upon the value of another coordinate). It is that relationship which defines the character of the coordinate system. If all coordinates are orthogonal for all values of the coordinates (essentially, no coordinates can cross one another) then the coordinate system is essentially Euclidean (there always exists a detailed local scaling which will make such a coordinate system Euclidean).

 

Have fun -- Dick

Edited February 1, 2016 by Doctordick

[Bombadil]

Another point you may be missing is the fact that I use the term “flaw free” because we have finite knowledge: i.e., I do not use the term “correct” as future information is required to prove an explanation is “incorrect”. When I say an explanation is “flaw free” I mean that there is no known information which contradicts that explanation. If it is flawed and we don't know it, it must be due to improper analysis and not due to the finite character of our knowledge.

 

So if any constraint for an explanation that we derive is not obeyed by an explanation then it must be due to an incorrect analysis of the known information? So is it possible to make an explanation for any set of data or is any possible list of known information constrained in some way so that it must obey an explanation? I suspect that any set of data can form an explanation, if in no other way then by adding invalid objects until such an explanation exists.

 

I don't understand your definition of the word “object” here. My definition of an "object” is any collection of ontological elements which may be considered as unchanging over a elapse of time of interest to us.

 

Perhaps this was a bad choice in wording. I should have called it any element of the explanation in this case though I was referring to any element or set of elements so I probably phrased this wrong. So to rephrase, “any element that behaves in such a way that it makes the tau axis a visible part of the explanation would be an invalid element“.

 

I suspect that I might still not be thinking of an object in the same way as how you are describing it. I am thinking of an object as any set of points that if we take them to be the entire set of elements in the fundamental equation all of the derivatives have the same value.

 

You are exactly correct. When I first started graduate school, our physics library had a copy of Gammow's “Mr Tompkins in Wonderland” which I read when I was first studying quantum mechanics (when I was a student, quantum was a graduate subject and I had never seen it as an undergraduate). At the moment, you can read an updated version of the book on the web called “The New World of Mr. Tompkins”.

 

 

I have looked at that link unfortunately it is like many of the books in the google book search only a preview so many of the pages (in fact most of the book) are missing including the entire quantum safari.

 

So is the way that you are using [imath]\hbar[/imath] equivalent to how it is used in quantum mechanics or are you using it in a slightly different way?

 

You are confusing two different motions here. One is the motion of an element in that quantum gas and the other is the motion of an object in the world view of the two observers. The evolution parameter (the unmeasurable t associated with my equation) is a universal parameter attached to my interpretation of what is going on. Each observer will, because of the character of his world view, consider time to be defined by his personal clocks as tau is not an observable axis in his world view. If you go back to my definition of mass, being the momentum in the tau direction, you will find it is the rest mass of the entity of interest: i.e., the two observers (when they go to examine the situation under my interpretation) will agree as to the rest mass of the various elements making up the universe. In fact they will agree as to the momentum in any direction except x. It is only the momentum of these elements in the x direction (the partial with respect to x of their expectations) upon which they will disagree. What is important is that they will both (using my interpretation of the details) see the change in a specific expectation to be defined by a specific expanding sphere.

 

So is the change in x as given by the equation [imath]x'=\alpha x-\beta t[/imath] an un-measurable value or is it used with tau in place of t in any reference frame?

 

I suspect that I may be misunderstanding something or perhaps assuming something here, if we take the distance that an object has moved to be [imath] ct=\sqrt{x^2+y^2+z^2+\tau^2} [/imath] I am under the impression that the functions [imath]x'=\alpha x-\beta t[/imath] and [imath]t'=\gamma t -\delta x[/imath] have some effect on the momentum, mass and energy of an object if in no other way then in the scaling of the t and x axis of the object of interest. If so, what is the connection between them. Is it simply a scaling of the equation? If so just where is the scaling taking place is it visible in the fundamental equation or is it only visible if we have defined a frame of reference in a flaw free explanation? In any case it seems that by changing the scale of either the mass ,momentum or energy we will change at least one of the remaining two and we can’t scale two of them without changing all three values.

[Doctordick]

So if any constraint for an explanation that we derive is not obeyed by an explanation then it must be due to an incorrect analysis of the known information?

Let us say that such an explanation contains an error somewhere!

So is it possible to make an explanation for any set of data or is any possible list of known information constrained in some way so that it must obey an explanation?

It is possible to “make an explanation” (do you “make” explanations) ... “constrained in some way”... “so that it must obey an explanation”? This doesn't sound like clear thinking.

I suspect that any set of data can form an explanation, if in no other way then by adding invalid objects until such an explanation exists.

I would have put this in another way; I would have said, “any set of data can be explained”. That is a true fact. Now, it is always possible that any given explanation can be invalidated by future data; however, the fact that it (and a future explanation which does explain the new data) can both be interpreted in such a way that they obey my equation will not be invalidated. You should note that this does not mean that the explainer interprets it in a way which obeys my equation (a proof that something can be done is not a proof that I, or anyone else, can do it). There are complex subtle things going on here.

So to rephrase, “any element that behaves in such a way that it makes the tau axis a visible part of the explanation would be an invalid element“.

You should instead say that, “any explanation which yields a way of measuring a unique value for the tau of a specific event” (as I have defined tau) has got problems. I would comment that, though Einsteinian perspective allows measurement of "change in tau" from one event to another (along the path of some entity) it does not provide any method of obtaining a unique value of tau, this is critical to the standard explanation of what is often referred to as the apparent "twin paradox".

I am thinking of an object as any set of points that if we take them to be the entire set of elements in the fundamental equation all of the derivatives have the same value.

I don't know where you picked this up. Nowhere have I said that “all of the derivatives have the same value”. I suspect you are interpreting momentum as mv and not as the partial with respect to x. The “mv” interpretation requires the concept of mass as a separate quality of the element. In my picture (that quantum gas) all elements are “massless” and, being massless, their velocity has nothing to do with their momentum. The fact that everything moves through the universe (that four dimensional frame including tau) at a fixed velocity is no more than a constraint imposed by the form of the fundamental equation.

 

The only requirement for the existence of an object is that the “direction” of the momentum (by the way momentum, in this picture, is a vector) of all the elements making up that object must be in the same direction (in the four dimensional space) so that there apparent velocities (in the observable three dimensional space) are the same. If that is not true, the elements making up the object won't stay together.

I have looked at that link unfortunately it is like many of the books in the google book search only a preview so many of the pages (in fact most of the book) are missing including the entire quantum safari.

No, the entire quantum safari is not missing; the first six pages are there. Go to the “preview this book” button under the picture of the book and then page down to page 120. There are enough errors in that picture to convince anyone who understands quantum mechanics that the picture is not consistent with quantum mechanics. In particular, the structure of the molecules which go to make up the antelopes, lions and the bushes are bound by quantum relationships and could never exist such that they would obey the scattering described there.

So is the way that you are using [math]\hbar[/math] equivalent to how it is used in quantum mechanics or are you using it in a slightly different way?

I am using it purely as a number being multiplied through my equation. Using it that way yields exactly the standard Schrödinger equation as presented in any standard introduction. The central point here is that time, as I have defined it, is not a measurable variable. When the experimenter sets time (in his frame) to be given by a clock (at rest in his frame) he essentially sets the relationship between the space derivative (momentum) and the time derivative (energy) to both be proportional the [math]\hbar[/math].

 

[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]

 

Which, as you can see from the above expression of Schrödinger's equation, is essentially expressed there (note that, in classical Newtonian physics, kinetic energy is given by the square of the momentum divided by 2m) . The only problem then is the absence of [math]\hbar[/math] in the V(x); but in my derivation, V(x) is determined by an integral over the expectations of all the other entities in the universe. Since the energy of the universe is conserved, that is no more than the variation of the energy of the entire universe assigned to that element due to changes in x and nothing else (everything else has been integrated out). But, if energy of all elements (individually) is given by the time derivative of the wave function which yields their positional expectation, why would you not expect the total to be proportional to [math]\hbar[/math]. The standard physics doesn't include [math]\hbar[/math] in V(x) for the simple reason that potential energy is defined by different procedures.

 

The point being that physicists do not really worry about the full consequences of their definitions and essentially over define their variables which leads directly to what is commonly expressed as the “laws of physics”. As I said, [math]\hbar[/math] is circularly defined. The problem is, that circle is very big and involves most of physics and to actually trace the thing is really beyond me; nonetheless, it is eminently clear that the value is most certainly circularly defined.

I suspect that I may be misunderstanding something or perhaps assuming something here, if we take the distance that an object has moved to be [math] ct=\sqrt{x^2+y^2+z^2+\tau^2} [/math]

Yes, I think you are misinterpreting the expression. Remember, “t” is not a “measurable” variable. There exists no mechanism which actually measures it; however, the entire physics community presumes it is measurable and, in fact, claims its value is given by a clock (which is a macroscopic dynamic object designed by them), a rather complex device at rest in their reference frame. They are correct in that the dynamics of their universe as seen from such a “rest frame” do indeed evolve in a consistent way with that “clock”. They then extend that observation with the presumption that “clocks measure time”.

 

The problem there is the value of that constant “c”. In the four dimensional expression (my fundamental equation) that velocity is an unmeasurable thing (that "K" in my fundamental equation is a free variable which turns out to have no consequences) but in their three dimensional interpretation, c is their measurement of motion orthogonal to tau (apparent velocity of a massless object). Of issue is the fact that two very different definitions of time are used in modern physics: one is the definition associated with the idea that things physically interact when they exist at the same time and the other is that time is what clocks measure.

 

I use only the definition associated with the idea that things physically interact when they exist at the same time. So long as one is dealing with a fixed frame of reference, the two definitions only differ by the fact that one is measurable and the other is not; essentially the two definitions are proportional to one another. This fact is used to define that velocity “c”, essentially making the two definitions the same “so long as your frame of reference does not change”. Thus it is that two observers who are moving with respect to one another (and are not in the same frame) make different identifications for their measurements of “t”. Since measureing a distance involves knowing the positions of each end at a specific time (simultaneously) the difference in time definition leads to different x measurements (for exactly the same phenomena).

So is the change in x as given by the equation [math]x'=\alpha x-\beta t[/math] an un-measurable value or is it used with tau in place of t in any reference frame?

As that “t” is presumed to be measured, it is in fact the t displayed by the observers clock: each of those observers is using the idea “t is what is measured by clocks”. Both of these “t”s are proportional to the unmeasurable t used in my equation and thus there is a proportional relationship between them in special relativity since both observers are using fixed frames of reference (but this is not true when it comes to general relativity).

I am under the impression that the functions [math]x'=\alpha x-\beta t[/math] and [math]t'=\gamma t -\delta x[/math] have some effect on the momentum, mass and energy of an object if in no other way then in the scaling of the t and x axis of the object of interest.

I don't think you are looking at this issue from the proper perspective. Both observers are examining exactly the same phenomena and they are obtaining results which are consistent with the assumption they are at rest in the universe (see the Michelson-Morley experiment). The “FACT” is that they both obtain results which are consistent with the presumption they alone are at rest: i.e., both observers see the other's experiment as showing the other person is moving. The difference in their perceptions of the measurements resolves down to the fact that they do not agree with one another's measurements of x or t. The relationships, [math]x'=\alpha x-\beta t[/math] and [math]t'=\gamma t -\delta x[/math], specify exactly the difference they see in those measurements.

If so, what is the connection between them. Is it simply a scaling of the equation? If so just where is the scaling taking place is it visible in the fundamental equation or is it only visible if we have defined a frame of reference in a flaw free explanation?

If you go back to the derivation of my fundamental equation, you will discover that it is only valid in a frame of reference where the observer is at rest with respect to the universe. The issue being brought up here is the fact that we are not talking about reality but rather about one's expectations: i.e., a given observer may ignore the majority of the universe under the presumption that it has no bearing upon the particular experiment he is observing. If this is the case, the universe he is explaining is presumed to be at rest with respect to that observation.

 

Now, if two observers moving with respect to one another (both presuming the universe is at rest with respect to themselves) are observing exactly the same experiment, they can't both actually be at rest with respect to the “real” universe (absolutely everything) but my equation must, nonetheless, yield their individual expectations. On the surface this appears to be a contradiction; however, it is not as it can be explained away by presuming the given differences in their measurements of the coordinate in the direction of motion. I pointed out that the scale of those measurements are set by the structure of the universe they find themselves in. Once again we are led to a circular definition of something. In this case it is the circular definition of distance along that coordinate.

In any case it seems that by changing the scale of either the mass ,momentum or energy we will change at least one of the remaining two and we can’t scale two of them without changing all three values.

These things are defined and measured via complex physical experiments which involve representations of specific events via coordinates of x and t (and here I am referring to the x and t specification used by the experimenters). Again, I want to make it clear that the two observers described in this “special relativistic” experiment are describing exactly the same experiment but from a slightly different perspective.

 

Lastly I would like to point out the fact that the assumption that one is at rest with respect to the entire universe is certainly “WRONG”. The microwave background is certain evidence of the fact that the rest frame of the universe certainly is unique; another “FACT” that the physics community refuses to recognize. This could very well lead to consequences not recognized in their theories. And I will get to that issue when we get into general relativity.

 

Have fun -- Dick

Edited February 1, 2016 by Doctordick

[Kharakov]

Hi Dr.Dick,

 

I couldn't PM you back (to low of a post count), but thank you for the thoughtful reply. I will have to work on the semantic issues you pointed out in my next revision.

 

Bare in mind I searched you out after I came up with the whole thing (after doing an earlier very rough version of the idea on my old haunt) and am not familiar with the mathematics you are using (matrix algebra and the like) which is why I used simple trigonometric and algebraic equations in my paper (which is enough for the purposes of describing another dimension).

 

If you skip the first babble section and check out the math, ponder the implications of the de Broglie wavelength (confirmed physically) having an exact trigonometric relationship with a rest mass energy equivalent photon wavelength (of a particle), you might perceive what caught my mind. BTW... de Broglie and 3dspace wavelength are the same thing, it's just 2 ways of describing the same thing (although 3dspace is more... succinct? although de Broglie deserves the credit for the idea... sheesh).

 

Anyways, I have a version of a paper with all of the maths (and simple examples of equations that you can use to calculate values). If you check out the last 2 pages (titled: 5d physics notes...) you'll see them. Also, the biggest deal that I noticed was the "de Broglie relationship" between rest mass energy (wavelength equivalent photon) and the de Broglie wavelength, so I think if you could check out those equations as well it would be nice.

 

I haven't corrected this online version yet (semantically).... and will have to go through it thoroughly, but for now, the math is as it is. Replace <dot> with . (if you have the patience to look at the math) It may require the latest version of Acrobat Reader.

 

www<dot>personal<dot>psu<dot>edu/kfv100/boyfriends%20theory/The%204th%20dimension%20of%20space.pdf

[Bombadil]

It is possible to “make an explanation” (do you “make” explanations) ... “constrained in some way”... “so that it must obey an explanation”? This doesn't sound like clear thinking.

 

Perhaps a better way of saying it is that instead of making an explanation we solve or at least approximate an explanation ,it seems that an explanation is constrained in that it must satisfy the fundamental equation this says nothing though about what this implies. Now, it seems to me that I can pick any set of data that I wish to explain, so my question is can any set of date be explained with an number of body’s in the fundamental equation? Or to rephrase, could a two body solution have the same behavior for two of the body’s as a three or four body solution could?

 

I don't know where you picked this up. Nowhere have I said that “all of the derivatives have the same value”. I suspect you are interpreting momentum as mv and not as the partial with respect to x. The “mv” interpretation requires the concept of mass as a separate quality of the element. In my picture (that quantum gas) all elements are “massless” and, being massless, their velocity has nothing to do with their momentum. The fact that everything moves through the universe (that four dimensional frame including tau) at a fixed velocity is no more than a constraint imposed by the form of the fundamental equation.

 

“all elements are massless” I suspect that this is due to there being no negative mass in this interpretation. If so why is there no negative mass? I suspect that this is due to the infinite uncertainty on the tau axis resulting in having no way to tell the difference between movement in the tay axis so that only a positive derivative is possible, and since they sum to zero all of them must be zero.

 

I thought that the fact that everything moves through the universe with a constant velocity was a consequence of how you constructed the ideal clock or am I still missing something about how the equation [imath] ct=\sqrt{x^2+y^2+z^2+\tau^2} [/imath] is derived?

 

The problem there is the value of that constant “c”. In the four dimensional expression (my fundamental equation) that velocity is an unmeasurable thing (that "K" in my fundamental equation is a free variable which turns out to have no consequences) but in their three dimensional interpretation, c is their measurement of motion orthogonal to tau (apparent velocity of a massless object). Of issue is the fact that two very different definitions of time are used in modern physics: one is the definition associated with the idea that things physically interact when they exist at the same time and the other is that time is what clocks measure.

 

While in the interpretation that you are suggesting tau is what clocks measure and things interact at the same t, this offers no problems though because t is just used as an evolution variable, while no constraints for the value of tau was placed in the derivation of the fundamental equation?

 

As that “t” is presumed to be measured, it is in fact the t displayed by the observers clock: each of those observers is using the idea “t is what is measured by clocks”. Both of these “t”s are proportional to the unmeasurable t used in my equation and thus there is a proportional relationship between them in special relativity since both observers are using fixed frames of reference (but this is not true when it comes to general relativity).

 

So for the time being tau and t should be considered to be proportional to each other. Is this the reason that t was scaled in the derivation of the Lorenz transformation and not tau?

 

I don't think you are looking at this issue from the proper perspective. Both observers are examining exactly the same phenomena and they are obtaining results which are consistent with the assumption they are at rest in the universe (see the Michelson-Morley experiment). The “FACT” is that they both obtain results which are consistent with the presumption they alone are at rest: i.e., both observers see the other's experiment as showing the other person is moving. The difference in their perceptions of the measurements resolves down to the fact that they do not agree with one another's measurements of x or t. The relationships, [imath]x'=\alpha x-\beta t[/imath] and [imath]t'=\gamma t -\delta x[/imath], specify exactly the difference they see in those measurements.

 

But if we where to work directly from the fundamental equation for each frame (something that we cannot actually do because we don’t have the solutions for it) is it the same fundamental equation that they both use, or is it a transformed fundamental equation, or do we use just one fundamental equation that may not be what either of them sees and transform what it says the measurements are from one frame to the next?

 

I’m thinking that it is the last one and that such a frame would be at rest (actually this just says that all the sums in the fundamental equation vanish in this frame) with the universe or at least with what we know of the universe and that in such a frame the measurement of tau is equivalent to that of t and that this is the reason behind using t as one of the elements that is being scaled in the derivation of the Lorenz transformation.

 

 

P.S I’m going to be a little slower then usual to reply as I’m not going to be near a computer for the next week and a half or so.

[Doctordick]

... it seems that an explanation is constrained in that it must satisfy the fundamental equation ...

No, that is not true. What I am saying is that “there always exists an interpretation of any explanation which requires the ontological elements underlying that explanation to obey my fundamental equation. I don't think you understand that statement. I suggest you go back to the opening post of this thread and make an attempt to understand the apparent behavior of the hypothetical balls I define in the thought experiment I put forth there.

I have a thought experiment you really need to perform. Suppose, for the fun of it, that I am an individual from a technologically advanced society and I meet with you to show you a couple of devices we have invented. I can't show you why it works the way it does because I, personally, don't know the science behind it; but I do know exactly what it does. The first device looks exactly like what you would see as an old fashion analog pocket watch. It has a dial with three hands which show hours, minutes and seconds, and has a knurled stem at the top which would appear to be for setting and/or winding the watch.

 

But I tell you it is not a watch; it is a one way time machine. When the stem is turned it will move the holder (and the holder only) into the future. When the stem is not turned, the reading on the time machine will read exactly the correct time (we won't worry about relativistic effects here, just assume that, for practical purposes, we live in a Newtonian universe). When the stem is turned, the reading on the face can be advanced. When the reading is advanced, the holder will be moved to exactly the time indicated on the face. The reverse is not possible. It is my understanding that one can not move to the past because doing so would cause paradoxes, but moving to the future will cause no such problems.

 

The question is, if I operate my time machine, what do you see? If you think about it a little, you should realize that, as I turn the stem, I move to whatever time is indicated on the face: i.e., I don't disappear and then reappear at the new time, I instead move through each and every time indicated on the dial. If you look at the face of the device while I am turning the stem, you will simply see the correct time as, whatever time you are at, I am there too (the second hand will appear to advance just as it did when I wasn't touching the stem). You will see me standing very still with my hand on the stem. If I advance the dial one hour while I take one breath (during the breath I turn the minute hand entirely around the face), you will see that breath as taking the entire hour. If my pulse were sixty beats a minute, you could perhaps detect my heart as beating once or twice during that hour (depending of course on how fast I personally am turning the stem). We won't worry about other effects; you could push me but I don't think either of us would like the results.

 

My second device is a toy we make for our children. It appears to be a standard baseball but it is not. It contains exactly the same time machine which I just demonstrated for you; however, it has no stem. Within the ball is a second device which, via internal dynamic effects enables it to know exactly where it is (remember, for practical effects of this thought experiment, we live in a Newtonian universe). When it is moved, the time machine (it and the ball within which it is contained) is advanced in time proportional to the distance it is moved. It is advanced exactly one second for every foot it is moved. Such a ball will display some rather interesting dynamic effects. (For the sake of simplicity, we won't consider rotation; the system is not designed to be rotated as rotation brings up some very complex effects.)

 

Consider two children ten feet apart playing catch with such a ball. From the children's perspective, how long does the ball take to cross the room? Suppose you replace the children with professional baseball pitchers? Then try firing it out of a canon. If you cannot figure out the logical consequences let me know and I will explain (and justify) the results. Another thing you might look at is tying a string to the ball and swinging it is a circle. I think you will find the consequences are quite interesting.

What you need to do is very carefully work out exactly what the apparent behavior of that ball is. I think that solving that intellectual problem should clear up a lot of the difficulties you are having.

 

Have fun -- Dick

[Bombadil]

No, that is not true. What I am saying is that “there always exists an interpretation of any explanation which requires the ontological elements underlying that explanation to obey my fundamental equation. I don't think you understand that statement. I suggest you go back to the opening post of this thread and make an attempt to understand the apparent behavior of the hypothetical balls I define in the thought experiment I put forth there.

 

How I am understanding your thought experiment involving a ball that advances one second in timer for every foot that it moves in space is as follows. When we throw the ball the time that it takes the ball to travel is the time that it would take had we thrown a regular ball plus one second for every foot that it traveled. So if we throw it at what would normally be ten feet a second and we throw it ten feet it will take ten seconds for having gone ten feet plus one second for how fast we throw it so that it will take eleven seconds. In fact as you have posted previously if we consider the maximum possible limiting speed it would be ten seconds to travel ten feet. Such a speed could not be reached by the ball though.

 

There are a couple of things that do seem to be of interest here. Firstly that it seems that the same force is required to throw the ball, that is it takes as much force to through the ball at about .9 feet a second as it would take to through a regular ball of the same size and mass ten feet a second. I am not considering the possibility of additional mass increase and I don‘t know if any would occur. Also if we made the ball so that a clock could be put inside of it the clock would say that it only took one second to go the ten feet. If we where in a room in which all objects in the room where so effected that is they travel one second for every foot they travel then I think that we would have to conclude that the ball only traveled at most one foot. This seems to lead to the distance having been scaled as well. I will note that this seems to be the exact effect that we see only with what is considered to be c or 1/k in place of the maximum possible speed.

 

I will now go through how I understand the requirements to derive the Lorenz transformation as I suspect some misunderstanding. To start with how I am understanding how you have defined an object, you have defined it such that the internal relationship of the elements of interest is unchanged. Now you also say that

 

The only requirement for the existence of an object is that the “direction” of the momentum (by the way momentum, in this picture, is a vector) of all the elements making up that object must be in the same direction (in the four dimensional space) so that there apparent velocities (in the observable three dimensional space) are the same. If that is not true, the elements making up the object won't stay together.

 

Now I’m not sure of just how you are defining velocity but I suspect that you are defining it according to the equation [imath]v=\frac{x}{\tau}[/imath] in which x is the arc length in the direction of movement in the x,y,z coordinate system although I suspect that you may have a different way of defining it.

 

At this point I suspect that I have been mistakenly understanding that the differentials are effected by the frame of reference. I am now suspecting that this is in fact not the case and that the differentials in the fundamental equation are not effected by the reference frame being used. How I understand it at this point is, we have to consider that every element considered in the fundamental equation must satisfy the equation [imath] ct=\sqrt{x^2+y^2+z^2+\tau^2} [/imath] now you have said that.

 

That falls directly out of my fundamental equation and is essentially a differential description of that expanding sphere I just mentioned above.

 

Now I don’t have a problem in believing this but as of yet I can’t seem to find anywhere where you have proven it to be the case. I don’t know if this is due to my not understanding the derivation of it or that you have not yet done it because either you consider it trivial or because it is sufficiently complex that it will take us too far off topic to be worth going into at present.

 

Now combining this with the fact that we have not defined a scale and such a thing is in fact defined by the explanation we can conclude that the fundamental equation is scale invariant (meaning that any axis can be scaled and the fundamental equation is still valid) combined with the fact that we are using a Euclidean geometry we can use this to derive the Lorenz transformation.

[Doctordick]

Hi Bombadil, my problem here is that I don't think you have put much thought into the problem. You just seem to re-express the behavior I pointed out to you.

When we throw the ball the time that it takes the ball to travel is the time that it would take had we thrown a regular ball plus one second for every foot that it traveled.

With the first five words, you have simply brushed aside the question being asked. All your comments are simple re-expressions of the specific facts I have given you, no real description of how this toy appears to behave differently from other items you might handle.

So if we throw it at what would normally be ten feet a second...

And how do you know that you have thrown it “at what would normally be ten feet a second”?

I am not considering the possibility of additional mass increase and I don‘t know if any would occur.

Why not? What happens when you try to throw something? Throw it softly and its velocity will be small; throw it hard and its velocity will be higher. When you throw it, the force you use yields the acceleration (F=ma) that is how you judge its mass isn't it? Suppose you are throwing it relatively hard and you want to throw it faster; you use more force right. Now how do you interpret the fact that no matter how hard you push this thing, you can't make it move any faster? Isn't that exactly what you mean by apparent mass increase?

My sole purpose in presenting this problem was to point out that the simple presumption that movement through space causes movement through time yields exactly the kind of phenomena relativity was designed to explain: a finite maximum velocity and apparent mass increases as apparent velocity increases. The only conundrum which arises in this picture is, why does one move through time when one is standing still? The obvious answer is, perhaps we only think we are standing still. Could it be that we are moving through space in a direction which is being projected out of our perception? That is, could we be living in a four dimensional universe where the fourth dimension is simply projected out by some overlooked phenomena?

It turns out that, if physical objects are nothing more than coherent collections of massless elements which just happen to be momentum quantized in some specific direction, the resulting appearance is, not just the kind of phenomena relativity was designed to explain, but turns out to be in exact agreement with special relativity in every detail. Conceptually, it is far simpler picture then Einstein's space-time continuum. You need to understand the intrinsic qualities of such a universe before you can comprehend how the ordinary relativistic relationships arise in such a picture.

How I understand it at this point is, we have to consider that every element considered in the fundamental equation must satisfy the equation [math] ct=\sqrt{x^2+y^2+z^2+\tau^2} [/math] now you have said that....

Now I don’t have a problem in believing this but as of yet I can’t seem to find anywhere where you have proven it to be the case.

It is little more than the mathematical statement “that movement through space causes movement through time”. Distances have clearly not been defined (thats the scale issue I brought up). In order to establish distances, you have to define some procedure and any procedure requires specification of much more than the simple path of the entity under examination (it is dependent upon structures which display your explanation of the rest of the universe).

It is probably best to drop the subject at the moment an concentrate on the derivation of Schroedinger's equation. Once you have a mental picture of that connection to reality, we can discuss what your reasonable expectations can be for the rest of the universe.

Have fun – Dick

 

Edited February 1, 2016 by Doctordick

[Doctordick]

Well, I have just read this topic and corrected all the math functions expressed ( I hope) by replacement of "[imath] and [/imath] (which originally stood for "inline" expression. (I can't replace others usage as I don't have the capability to edit their posts.)

 

I also notice that we never got to General Relativity and that is a sad failure.  I have reread all my posts and am rather proud of the rationality of the presentation.

 

Have fun -- Dick