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An “analytical-metaphysical” take on Special Relativity!


Doctordick

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After looking at this some I suspect that I may be considering a couple of things incorrectly. Firstly what exactly the transformation of the fundamental equation that moves it to a new reference frame is. I have been considering this to be any terms that can be added to the fundamental equation so that it is no longer valid. This clearly is not what the transformations that we are interested in are, but rather it seems that what transformations it is that we are interested in is the addition of mass or momentum operators to the fundamental equation.

 

There is one other issue that I have been having that is, how do we know that the Schrödinger equation tells us anything about the fundamental equation. That is, how do we know that the V(x) term wouldn’t be so complex or in a particular form so that we can no longer use Newtonian mechanics as an approximation to the fundamental equation.

 

Actually I think that this issue has already been brought up although I didn’t fully realize it or how you where solving it. Simply put it is that since the equation is scale invariant we can look at it on any scale and due to all interactions taking place due to a Dirac delta function, which has only a single point that it is not zero on, this allows us to look at the Schrödinger equation on a scale in which the V(x) function vanishes. Clearly on this scale the approximations, except perhaps the last approximation used to derive the Schrödinger equation, can be justified. From what I can tell the last approximation can be justified if we consider the rest frame of the object of interest.

 

In this way we can derive the Schrödinger equation and in so doing show that Newtonian mechanics is an approximation to the fundamental equation when looked at on some scale.

 

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

 

This would be correct it is also a topic that I am trying to start going into.

 

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

 

Then the question is vary much given an arbitrary set of elements (the ontological basis), arbitrary in that any particular element is identical so that any differences between elements is part of the explanation and not a property of the elements themselves. How is it that someone can explain such a set of points, that is, how can someone obtain expectations about how the elements will change given only the points.

 

What we are in fact doing is mapping such a set of points onto a Euclidean space and using a evolutionary parameter t to ask how might we explain how the system changes with t. But this change is itself part of the explanation and a consequence of new information becoming available and in order for us to conclude what the change is we must correspond elements at one value of the parameter t with elements at another value of t (that is they are considered the same element) and in so doing we conclude that the same elements existed at every point corresponding to a value of t.

 

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

 

So whether or not a flaw free explanation satisfies the fundamental equation is not the issue but rather the issue is that there exists a isomorphism between the explanation and a solution to the fundamental equation. That is, there is a mapping that preserves the result satisfying the fundamental equation.

 

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

 

The real question is how such a world view is constructed as there is nothing but arbitrary information to base it on even the way that we map the information before forming a world view must be part of the world view.

 

So what we must do is first realize that it is arbitrary information that we are dealing with then find a arbitrary method of mapping the information in such a way that we can analyze are world view and realize that it can be interpreted in such a way that it satisfies the fundamental equation?

 

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

 

That is in order to use their explanations we must be able to interpret the explanation in such a way that it obeys the constraints put down by the fundamental equation. What we must consider is that that explanation is in its rest frame because it is simply not a valid explanation anywhere else and we are not in its rest frame.

 

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

 

Certainly acceleration can exist I suspect that more of the problem here is why acceleration moves from one frame to a another how I am understanding this is that after an object has accelerated to a new reference frame in order for use to still use an explanation of the object in its rest frame we must consider that the momentum operators are no longer the same for the object in both explanations, that is our rest frame and its rest frame are no longer the same.

 

Now my question is what kind of number is the momentum operator that is added as it appears to be just a imaginary number that is needed for the object that is no longer in our rest frame. But wouldn’t an arbitrary complex number also be considered a new reference frame? Would acceleration still move it from one reference frame to the other or will such a reference frame even exist?

 

It may be that I am just trying to consider modifications that don’t or cannot exist in the fundamental equation.

 

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

 

In a sense the question here is, are we explanting the universe or is the universe the explanation. In the former we have no defense for saying if we have a valid explanation. Furthermore, we have no way to know if there is any kind of mapping between our explanation and the universe that we are explaining as we will have elements in our explanation to make it a valid explanation that need not be part of the universe and there is no way to know what the rest frame is as we don‘t know everything about the universe. In the latter case clearly the explanation has a rest frame even if it will move as new information becomes available to us either from realizing that the explanation is not valid or from new information that the explanation is based on becoming available to us. Furthermore any element that is part of a flaw free explanation that we are using is clearly part of the universe in that the universe would be inconsistent without it, which is not possible.

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After looking at this some I suspect that I may be considering a couple of things incorrectly. Firstly what exactly the transformation of the fundamental equation that moves it to a new reference frame is. I have been considering this to be any terms that can be added to the fundamental equation so that it is no longer valid. This clearly is not what the transformations that we are interested in are, but rather it seems that what transformations it is that we are interested in is the addition of mass or momentum operators to the fundamental equation.
I am not sure of exactly what you mean by “we are interested in”. We are interested in any transformation which can be defined and exactly how the defined transformation should be performed and what the meaning of the transformed solution is (that depends very much on exactly what kind of transformation one is talking about).
There is one other issue that I have been having that is, how do we know that the Schrödinger equation tells us anything about the fundamental equation. That is, how do we know that the V(x) term wouldn’t be so complex or in a particular form so that we can no longer use Newtonian mechanics as an approximation to the fundamental equation.
If you examine the deduction of my fundamental equation, you will discover that a very important step in that deduction is the proof that, no matter what patterns the valid elements may have, there always exists a set of hypothesized elements such that the rule [imath]\sum \delta (\vec{x}_i -\vec{x}_j)=0[/imath] will constrain the valid elements to exactly those required patterns. When one makes the approximations required to deduce Schrödinger's equation (doing all the required integration), that self same rule ends up being a function of “x”. Thus the meaning of the original proof is simply that, under the approximations made, there always exists some function of x which will require the behavior of any specific element to obey Schrödinger's equation where V(x) is that function. Is it possible that V(x) could be so complex that we can no longer use Newtonian mechanics? That depends on what you mean by “use Newtonian mechanics”. If you mean, “so complex that we can not solve the problem”; sure, there are a lot of Newtonian problems so complex that solution has eluded the scientific community for centuries. But can that be taken as evidence that Newtonian mechanics is false (false in the sense that even if the approximations used are valid, Newtonian mechanics is still false)? I think not.
Actually I think that this issue has already been brought up although I didn’t fully realize it or how you where solving it. Simply put it is that since the equation is scale invariant we can look at it on any scale and due to all interactions taking place due to a Dirac delta function, which has only a single point that it is not zero on, this allows us to look at the Schrödinger equation on a scale in which the V(x) function vanishes. Clearly on this scale the approximations, except perhaps the last approximation used to derive the Schrödinger equation, can be justified. From what I can tell the last approximation can be justified if we consider the rest frame of the object of interest.
It seems to me that you are confusing two very different issues here. Deriving the Schrödinger equation and showing that Newtonian mechanics is an approximation to the fundamental equation has nothing to do with any scale issues. Objects are defined to be collections of elements which remain in a coherent structure over a sufficiently long time to be thought of as individual entities. You should understand that one of the major approximations necessary to be made is that the energy of the entities must be approximately given by [imath]E=mc^2[/imath]. That means that the kinetic energy (the energy of motion) must be small compared to [imath]mc^2[/imath]: i.e., the elements going to make up that object cannot have net relativistic velocities with respect to the rest frame of the object. This is a fundamental constraint on Schrödinger's equation. That further means that V(x) cannot be so large to generate relativistic Newtonian velocities. The net result is that the collection of elements going to make up objects (such as my mirror assembly) cannot have relativistic velocities relative to one another. It follows that the directions of the individual elements making up the objects are all on essentially parallel paths, moving at v?. Any deviation from those parallel paths must be small.
Then the question is vary much given an arbitrary set of elements (the ontological basis), arbitrary in that any particular element is identical so that any differences between elements is part of the explanation and not a property of the elements themselves. How is it that someone can explain such a set of points, that is, how can someone obtain expectations about how the elements will change given only the points.
I think I will go over to Anssi's position on that. We are talking about an epistemological construct to explain a collection of elements which we know nothing about. We have a collection of elements associated with the index ti. It is the moment we presume persistence (that "element i" in a collection at tk is the same element as "element i" in a second collection at tq) that we need to attach a new numerical label xi in order to assure that these are still individual labels (in order to keep the fact that persistence is a presumption) which can be associated in any manner. This step starts us down that deduction of my fundamental equation.
What we are in fact doing is mapping such a set of points onto a Euclidean space and using a evolutionary parameter t to ask how might we explain how the system changes with t. But this change is itself part of the explanation and a consequence of new information becoming available and in order for us to conclude what the change is we must correspond elements at one value of the parameter t with elements at another value of t (that is they are considered the same element) and in so doing we conclude that the same elements existed at every point corresponding to a value of t.
I get the impression that you are trying to confuse yourself. There are three steps involved here and you need to understand each of those steps before proceeding to the next. First there is the deduction of my equation (which has to do with explaining an arbitrary past; that is why my first step is to define time). Second, is the proof that Schrödinger's equation is an approximation to that equation. And, third, is the demonstration that the picture requires the same relativistic transformations as does Maxwell's equation. Mixing and mushing with these three different issues is a procedure just crying to confuse you.

 

I really think that what you are trying to do is to achieve epiphany; trying to get your brain (that source of squirrel thought) to give you solution you don't have to think about.

 

Most of what you say seems to be so “off the wall” that I don't have the interest to go down those paths. I have discovered a very good reason why our view of the world obeys the laws of physics; I have not discovered a way of creating a world view. As I have said many times, actually solving my equation is beyond possibility (it is that proverbial many body problem). Logical thought is insufficient to the job; only “squirrel thought” (which is beyond logical examination and thus can not be proved to be without flaw) is actually capable of providing even a rough solution. But we can talk about rules that the explanation must obey! And that is the very essence of science.

 

Actually finding solutions is another problem entirely and I will eventually talk about such things (as I believe there is a simple solution to AI); however, I will not approach that issue until I have communicated my deductions in their entirety.

 

Have fun -- Dick

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It is quite late when writing this, but I just wanted to give a quick sign of life here. I have read through the OP and my first reaction was that it seems to make perfect sense, and appears to reach - at least in my mind - quite expected conclusions. Many times, I would have expected that the relationships between isotropic C <-> simultaneity <-> geometry should be clear to everyone at least to the extent of being able to see how changing one definition (or call them postulates) affects the others, but I guess I've been proven wrong on that one many times :I

 

I also skimmed the responses and indeed seems like most of the reactions are based on little bit poor understanding of where the fundamental equation came from... That's a bit unfortunate but I guess it suffices that people then take that result on faith at first... :I

 

Anyway, as was the case with Schrödinger, I should still walk through the math in detail, to get an exactly proper view. And once again I will need help with that, so I'll try to get around to ask some questions...

 

But for now have to hit the sack,

-Anssi

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It seems that for the time being we shouldn’t consider the possibility of there existing any particular reference frames in which the fundamental equation is valid in and gives different predictions then in any other reference frame and just consider it just as valid in any reference frame as at the time being. I think that it may just be confusing the issue of when the fundamental equation is valid and where the Lorenz transformation comes from. I think that we have been doing this anyhow.

 

I am not sure of exactly what you mean by “we are interested in”. We are interested in any transformation which can be defined and exactly how the defined transformation should be performed and what the meaning of the transformed solution is (that depends very much on exactly what kind of transformation one is talking about).

 

Unless I am missing something I think that the only transformations that you have defined are the those resulting from multiplying [imath]\vec{\Psi}[/imath] by [imath]e^{ikx},e^{ik\tau}[/imath] and [imath]e^{ikt}[/imath] (where i is the square root of -1 and k is a real number) ( also I am only considering the one dimensional case here) performing these multiplications results in the addition of momentum mass or energy to the explanation.

 

The problem is that if a subset of the universe exists in which the fundamental equation can be considered valid in then the fundamental equation must be valid in both frames but the energy momentum and mass of the elements won’t agree in both frames.

 

If you examine the deduction of my fundamental equation, you will discover that a very important step in that deduction is the proof that, no matter what patterns the valid elements may have, there always exists a set of hypothesized elements such that the rule [imath]\sum \delta (\vec{x}_i -\vec{x}_j)=0[/imath] will constrain the valid elements to exactly those required patterns. When one makes the approximations required to deduce Schrödinger's equation (doing all the required integration), that self same rule ends up being a function of “x”. Thus the meaning of the original proof is simply that, under the approximations made, there always exists some function of x which will require the behavior of any specific element to obey Schrödinger's equation where V(x) is that function. Is it possible that V(x) could be so complex that we can no longer use Newtonian mechanics? That depends on what you mean by “use Newtonian mechanics”. If you mean, “so complex that we can not solve the problem”; sure, there are a lot of Newtonian problems so complex that solution has eluded the scientific community for centuries. But can that be taken as evidence that Newtonian mechanics is false (false in the sense that even if the approximations used are valid, Newtonian mechanics is still false)? I think not.

 

The sense that I keep thinking of, that I wonder if Newtonian mechanics would be useful in, is if the approximations made are not good approximations (that is if we can’t really ignore the influence of the rest of the universe). Seeing as we can’t solve the problem directly and the only explanation that I know of to compare it to is my experience I see no way to know one way or the other. Although it seems that if we look at the fundamental equation over small enough changes in the axis’s then this can be considered a good approximation or is this even an issue?

 

How I am understanding this is that if we look at the equation over a sufficiently small change in t then Newtonian mechanics will approximate the fundamental equation. And over those changes in t there will exist objects that can be considered universes onto themselves. That is, they have a rest frame in which the explanation of the object in its rest frame is just as valid as the explanation in the rest frame of the universe.

 

The problem is that when the Schrödinger equation is considered the change from the rest frame of the universe to the rest frame of just the object will result in changing the energy and momentum of the explanation so that the fundamental equation is no longer valid.

 

Now we want a transformation that transforms the measurements in one frame to any other frame. This would allow us to take the measurements taken in another frame and transform them to the measurements that they would be if we made them in our reference frame. The fundamental equation without the Dirac delta function is a wave that is expanding at a constant rate. This is given by the equation [imath]r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2}[/imath] now this equation must be valid in any reference frame that the fundamental equation is valid in. Considering the scale invariance of the fundamental equation and that any transformation must be invariant under any shift in the origin the only possible transformation is of the form [imath]\alpha(vt)-\beta t = 0[/imath] using these equations and solving for the necessary transformation we arrive at the Lorenz transformations as the only possible transformation.

 

In order for this to work it requires that the value of [imath]v_?[/imath] is the same in both frames but the actual value of [imath]v_?[/imath] is actually defined by the measures of length and time. Your clock will allow us to define time by counting the oscillations of the oscillator but this still requires that we define distance. In order for us to define distance to be the same in both frames we would have to use a property that is the same in both frames. The problem is that there is no property that is the same in both frames. Any property is a property of what we are explaining not a property of what the explanation must obey. So that all that we can do is use the same procedure to construct a unit of measure (The Schrödinger equation lets us do this?). If both explanations are valid then the procedure must be a scale of the procedure that the other observer is using. That scale is the very thing that we found to be the Lorenz transform.

 

It seems to me that you are confusing two very different issues here. Deriving the Schrödinger equation and showing that Newtonian mechanics is an approximation to the fundamental equation has nothing to do with any scale issues. Objects are defined to be collections of elements which remain in a coherent structure over a sufficiently long time to be thought of as individual entities. You should understand that one of the major approximations necessary to be made is that the energy of the entities must be approximately given by [imath]E=mc^2[/imath]. That means that the kinetic energy (the energy of motion) must be small compared to [imath]mc^2[/imath]: i.e., the elements going to make up that object cannot have net relativistic velocities with respect to the rest frame of the object. This is a fundamental constraint on Schrödinger's equation. That further means that V(x) cannot be so large to generate relativistic Newtonian velocities. The net result is that the collection of elements going to make up objects (such as my mirror assembly) cannot have relativistic velocities relative to one another. It follows that the directions of the individual elements making up the objects are all on essentially parallel paths, moving at v?. Any deviation from those parallel paths must be small.

 

But won’t the oscillator always be considered to move at [imath]v_?[/imath] as it has zero momentum in the [imath]\tau[/imath] direction or is it not considered part of the mirror assembly and relativistic effects wont effect it as it has a fixed speed after distance and time have been defined. This seems to be the case.

 

I get the impression that you are trying to confuse yourself. There are three steps involved here and you need to understand each of those steps before proceeding to the next. First there is the deduction of my equation (which has to do with explaining an arbitrary past; that is why my first step is to define time). Second, is the proof that Schrödinger's equation is an approximation to that equation. And, third, is the demonstration that the picture requires the same relativistic transformations as does Maxwell's equation. Mixing and mushing with these three different issues is a procedure just crying to confuse you.

 

I’m not sure that I follow exactly what the purpose in the first part of your previous post is. It looks like you were beginning to lay out some of the considerations for the first step in your deduction although I’m not sure I understand why, unless there is something about it that has some kind of influence on the current topic that you are trying to point out or you are suggesting that I take and go back and look at the original deduction. If that is the case, I think that it is best if I use your latest topic “What I believe an explanation is!” Thread for any questions as they come up in the thread as I have read all of the “what can we know of reality” thread as well as parts of the “Is time just an illusion?” thread, although it was quite some time ago and I can’t say how well I understand it while it at least made sense when I read it, and the “what can we know of reality” thread is beginning to get slightly confused in the topic originally meant to be disused as well as quite long. If this is the case just say so.

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Okay, time to start dissecting the this thread!

 

...I took the collection of ontological elements standing behind any explanation to be “unknowns” and then attempted to set down the relationships those unknowns had to obey: the result was the derivation of my fundamental equation. The presentation of that proof may be found here; where the following relationship is both defined and derived

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

 

I have already given a specific proof that Schrödinger's equation is an approximate solution to my fundamental equation...

 

...and expanded that proof to show that a three dimensional representation of any explanation may be achieved via a three dimensional representation of that self same equation: i.e., it leads to a three dimensional form of Schrödinger equation.

 

...I also showed that the Dirac delta function could be used as a rational representation of any rules that universe might have. This step provided another subtle alteration in the original numbers which was somewhat unexpected. Notice that multiplication of each of the original numbers representing any observation by any specific constant also has utterly no consequences within this model: i.e., the model is scale invariant.

 

...if the data belonging to a given observation could be divided into two (or more) sets having negligible influence on one another, those sets could be examined independently of one another: i.e., these collections would end up being constrained by exactly the same relationship which constrained the original universe. This is to say that these subsets (or “objects”) could be analyzed as a universes unto themselves

 

...there is a subtle problem here: the fundamental equation was constrained (see appendix 3 of the original proof) to be valid only in the rest frame of the universe. The central issue here is that the two collections of elemental entities either have significant influence on one another or they do not. If they do not have any significant influence on one another, the constraint that the equation is only valid in the rest frame of “the universe” cannot be a valid constraint as either object may be considered to be a universe unto itself: i.e., the rest frame of one collection of elemental entities may not be the same as the rest frame of the other. The solution to this problem lies with the scaling of the geometry between the two systems: there must exist a consistent way of converting a solution in one system to a solution in the other independent of any influence between the two.

 

Up to this point I can only say "check".

 

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

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

 

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

 

Yeah, I had to scratch my head with that, simply because I can't immediately see what happens there, due to my unfamiliarity with these math tricks...

 

I asked about it in a PM, and DD noticed it was missing the [imath]\hbar[/imath] at the denominator, so that's now fixed in the OP. The following quote is from a PM:

 

...just as the function [imath]e^{\frac{-iqt}{K\sqrt{2}}}[/imath] shifted the energy of by a constant factor [imath]\frac{-iq}{K\sqrt{2}}[/imath] (as a direct consequence of the differential with respect to “t”), the function [imath]e^{i\frac{Px_j}{n\hbar}}[/imath] shifts the momentum in the x direction of the jth entity by the constant factor [imath]i\frac{P}{n\hbar}[/imath] (as a direct consequence of the differential with respect to xj). Both of these effects are a consequence of the product rule of differentiation. The product indicates addition of [imath]i\frac{P}{n\hbar}[/imath] for every value of j (all n entities) so the sum of all n terms (times [imath]i\hbar[/imath] will be “P”. Thus it follows that the change in [imath]\Psi[/imath] (changed to [imath]\Phi[/imath] times the product) produces an equation where the product can be divided out and thus creates a new function [imath]\Phi[/imath] which gives exactly the same probability distribution except for the fact that the momentum of the universe in the x direction is no longer zero but turns out to be “P”. In is no more than a common trick done in quantum mechanics.

 

I am getting old and careless. I will leave it to you to check the algebra.

 

Yup, so if I understand this part correctly, here goes... To make things simpler for myself, I just concentrated on a partial derivative of a single x. I.e:

 

[math]

\frac{\partial}{\partial x} \vec{\Psi} = \frac{\partial}{\partial x} \vec{\Phi} e^{i\frac{Px}{n\hbar}}

[/math]

 

The right hand side can be written:

 

[math]

\left\{ \frac{\partial}{\partial x} \vec{\Phi} \right\} e^{i\frac{Px}{n\hbar}} + \vec{\Phi} \frac{\partial}{\partial x} e^{i\frac{Px}{n\hbar}}

[/math]

 

[math]

=

[/math]

 

[math]

\left\{ \frac{\partial}{\partial x} \vec{\Phi} \right\} e^{i\frac{Px}{n\hbar}} + \vec{\Phi} i\frac{P}{n\hbar} e^{i\frac{Px}{n\hbar}}

[/math]

 

And at that point the e's could be factored out, leaving us with:

 

[math]

\left\{ \frac{\partial}{\partial x} + i\frac{P}{n\hbar} \right\} \vec{\Phi}

[/math]

 

I hope that's all valid, and judging from your comment "the function [imath]e^{i\frac{Px_j}{n\hbar}}[/imath] shifts the momentum in the x direction of the jth entity by the constant factor [imath]i\frac{P}{n\hbar}[/imath] (as a direct consequence of the differential with respect to xj)" I think I am on the right track with this. That operation applied to each element would seem to do exactly what you are saying it would.

 

I'll try and continue from here tomorrow...

 

-Anssi

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It seems that for the time being we shouldn’t consider the possibility of there existing any particular reference frames in which the fundamental equation is valid in and gives different predictions then in any other reference frame and just consider it just as valid in any reference frame as at the time being.
But we can't do that. The actual equation was proved to be valid only in the rest frame of the universe. If you are not in the rest frame of the universe, the equation simply is not valid.

 

However, since two scientists moving with respect to one another may indeed “presume” their personal frame is “the rest frame of the universe”(i.e., they ignore information which might settle the question) their physics must be valid in both frames (otherwise they will obtain different results, invalidating that presumption). It is that fact which requires the special relativistic transformations.

 

The whole thing is quite simple. The fundamental equation is essentially a wave equation with fixed velocity and as such requires exactly the same transformation properties required by Maxwell's equation. I just go through that derivation in detail with a detailed defense which I think you are having difficulty following.

Unless I am missing something I think that the only transformations that you have defined are the those resulting from multiplying [imath]\vec{\Psi}[/imath] by [imath]e^{ikx},e^{ik\tau}[/imath] and [imath]e^{ikt}[/imath] (where i is the square root of -1 and k is a real number) ( also I am only considering the one dimensional case here) performing these multiplications results in the addition of momentum mass or energy to the explanation.
That shift is completely analogous to the ordinary Galilean transformation of non-relativistic physics. It gives the phenomena being described as seen by the rest observer's construct of a moving inertial frame. It omits changes due to the moving observer's different definition of simultaneity but it still yields the correct results (just not in the perspective of the moving observer).
The problem is that if a subset of the universe exists in which the fundamental equation can be considered valid in then the fundamental equation must be valid in both frames but the energy momentum and mass of the elements won’t agree in both frames.
Again, I get the feeling you are confusing things here. It is the actual phenomena which must be the same from both reference frames; it will just be seen differently by the two observers.
Although it seems that if we look at the fundamental equation over small enough changes in the axis’s then this can be considered a good approximation or is this even an issue?
Small changes are not the issue; the issue is that the projected velocities (velocities perpendicular to the tau axis) are small compared to v?.
How I am understanding this is that if we look at the equation over a sufficiently small change in t then Newtonian mechanics will approximate the fundamental equation.
Again, small change in t is of little significance. Newton's equations are essentially two body equations whereas my equation is a many body equation. If you have the correct solution for the rest of all those bodies (which is, from Newtonian position, they can be ignored) then only those velocities are significant.
Now we want a transformation that transforms the measurements in one frame to any other frame. This would allow us to take the measurements taken in another frame and transform them to the measurements that they would be if we made them in our reference frame. The fundamental equation without the Dirac delta function is a wave that is expanding at a constant rate. This is given by the equation [imath]r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2}[/imath] now this equation must be valid in any reference frame that the fundamental equation is valid in. Considering the scale invariance of the fundamental equation and that any transformation must be invariant under any shift in the origin the only possible transformation is of the form [imath]\alpha(vt)-\beta t = 0[/imath] using these equations and solving for the necessary transformation we arrive at the Lorenz transformations as the only possible transformation.
Now that is true.
In order for this to work it requires that the value of [imath]v_?[/imath] is the same in both frames but the actual value of [imath]v_?[/imath] is actually defined by the measures of length and time.
Clocks do not measure time (if time is defined by interactions) but rather measure changes in tau. If both observers define time via clocks (at rest in their frames) then they are essentially using the same units for space and time (tau is being referred to as if it were time). This means that the Lorenz transformation of distance measure is all we need to make the two velocities identical.
Your clock will allow us to define time by counting the oscillations of the oscillator but this still requires that we define distance. In order for us to define distance to be the same in both frames we would have to use a property that is the same in both frames.
Yes, some stable solution to the fundamental equation: what we have defined to be “an object”. A ruler of some sort.
The problem is that there is no property that is the same in both frames.
If “objects” (collections of elements which are stable structures over reasonable times) can exist, then there certainly exist things which have the same properties in both frames.
Any property is a property of what we are explaining not a property of what the explanation must obey. So that all that we can do is use the same procedure to construct a unit of measure (The Schrödinger equation lets us do this?).
Newtonian mechanics leads us to quantum mechanics (I will show Dirac's equation is also an approximation to my equation) which leads to structures called atoms and from there to molecules which takes us to chemistry, which leads to biology. In fact, it seems that almost all of science can be traced to solutions of these relationships. If these things are to be independent of what frame we choose as our rest frame, then the Lorentz transformation must be valid.
But won’t the oscillator always be considered to move at [imath]v_?[/imath] as it has zero momentum in the [imath]\tau[/imath] direction or is it not considered part of the mirror assembly and relativistic effects wont effect it as it has a fixed speed after distance and time have been defined. This seems to be the case.
We aren't transforming to the “rest frame of the oscillator”; we are examining these phenomena in two specified frames of reference both of which consist, for the most part, of massive elements, thus it follows that their apparent velocity (the portion perpendicular to tau) will be less than v? except for the oscillator itself. The oscillator itself is also an object (a collection of elements remaining in a stable structure over substantial time) called “a pulse”; all the elements are traveling in the same direction and maintaining a structure over time. Since they have no momentum in the tau direction, they will appear to be moving at exactly v? but, since they constitute “a pulse” (they are essentially located in a specifiable though moving position) they cannot be momentum quantized in the x direction. The pulse fulfills the definition of “an object”.
I’m not sure that I follow exactly what the purpose in the first part of your previous post is. It looks like you were beginning to lay out some of the considerations for the first step in your deduction although I’m not sure I understand why, unless there is something about it that has some kind of influence on the current topic that you are trying to point out or you are suggesting that I take and go back and look at the original deduction.
What I was trying to point out is that there are a number of different proofs going on here and that one should not confuse one with another. Each one builds on the earlier one but takes nothing from the earlier proof except the conclusion of that proof; what was proved is thus taken as fact.

 

Any explanation of anything can be interpreted in a manner which makes it a solution to my fundamental equation. :sherlock: A fact of little real use!

 

Schrödinger's equation (and thus Newtonian mechanics) constitutes an approximate solution to that equation. :sherlock: Perhaps this is of some use; it sure justifies Newtonian mechanics. :read:

 

The fact that the fundamental equation is essentially a wave equation with fixed velocity demands SR transformations. :sherlock: That is interesting; it implies there cannot be an explanation which violates SR. :thumbs_up That is worth knowing.

 

And more will be developed here. :candle:

 

Have fun -- Dick

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

 

There was a tricky sentence in there, and just to be absolutely sure I interpreted it correctly, did you mean to say:

 

What we have is the fact that our mental model of reality must include the fundamental symmetry that all solutions, in the absence of outside influence, must transform to valid solutions to the "fundamental equation" in the center of mass of a system of any collection of data.

 

At least that would make sense to me.

 

It turns out that we are quite lucky in that the consequences of the above symmetry have already been completely worked out long ago by others. Notice that, if one ignores the Dirac delta function (as it has no spacial extension) my fundamental equation is a simple linear wave equation in four dimensions with wave solutions of fixed velocity.

 

Once again due to my lacking math knowledge, I'm unable to see that clearly. That it's "a linear wave equation with wave solutions of fixed velocity".

 

I googled "linear wave equation" and came up with a lot of stuff that looks partially familiar but thought maybe it's best if you just point me out to the correct direction.

 

At any rate, I have no problems with taking that on faith for now, as I figure the important bit is that the elements are expected to travel at fixed velocity.

 

The constraint spoken of above is exactly the same constraint placed on the conventional Euclidean mental model of the universe by the fixed speed of light in Maxwell's equations. As we all know, if we constrain ourselves to linear scale changes, it turns out that there exists one very simple (and unique) relativistic transformation which maintains a given fixed velocity for all reference frames moving with constant velocity with respect to one another.

 

Indeed, Lorentz transformation.

 

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

 

Yup.

 

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

 

We need to have a formula for translating coordinate points in the first frame, [imath](x,y,z,\tau)[/imath], into the identical points represented in the second frame, which have to be [imath](x',y',z',\tau')[/imath] in a way which continues the validity of the fundamental equation. In order to do that, I will use the fact that the fundamental equation is (sans interactions) a wave equation where the wave velocity, v? is constant; thus, we can use an opening circumstance where (at t=0), [imath]\Psi[/imath], the wave function of an object consisting of a single element (i.e., all interactions with the rest of the universe are being ignored), consists of a spike at the origin in both frames and is zero elsewhere (that means we are starting with the origins of both frames of reference exactly aligned origins).

 

Yup.

 

Anyone familiar with wave equations understands that the solution here is quite simple, [imath]\Psi(t)[/imath] is thereafter a spike at r=tv? (where r is the radius of a four dimensional sphere centered on the origin) and zero elsewhere from then on. (Think of a flashbulb going off at the moment the origins of the two coordinate systems are exactly in the same point and then picture the sphere of light expanding at the speed of light.) The fact that our case is a four dimensional sphere is only of passing significance here, as we are still speaking of uniform radial expansion: i.e., the radius to that pulse of probability must be given by [imath]r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2}[/imath]. Please notice that this means that once a scale is set for one coordinate, it is likewise set for all the others (otherwise we wouldn't have a sphere).

 

Not really familiar with wave equations but still that all seems to be trivially true.

 

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

 

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

 

Yup, it all seems clear up to this point...

 

Nevertheless, it is fairly easy to show that the transformation from one coordinate system to the other can be no more complex than [imath]x'=\alpha x -\beta t[/imath] and [imath]t'=\gamma x -\delta t[/imath].

 

For those who don't believe that, consider the terms in a power series expansion of some supposed arbitrary function. The constant terms of that power series can be dropped as they move the primed origin at all times even t=0 where we have already defined it to be in exactly the same position as the unprimed coordinate system.

 

...but I understand almost nothing of the above :(

I suspect alphas and betas refer to something different than they do inside the fundamental equation, and that [imath]\gamma[/imath] is the Lorentz factor...? Also don't know what to make of the [imath]\delta t[/imath]. I have no idea what's a "power series". Needless to say, I am quite lost once again :)

 

I think I should stop here until I understand that step properly.

 

-Anssi

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I think I should stop here until I understand that step properly.

 

Me too. I was doin okay untill that. (Following along :):) )

 

I was also quite impressed with the subsequent(right word?) proofs

 

Any explanation of anything can be interpreted in a manner which makes it a solution to my fundamental equation. A fact of little real use!

 

Schrödinger's equation (and thus Newtonian mechanics) constitutes an approximate solution to that equation. Perhaps this is of some use; it sure justifies Newtonian mechanics.

 

The fact that the fundamental equation is essentially a wave equation with fixed velocity demands SR transformations. That is interesting; it implies there cannot be an explanation which violates SR. That is worth knowing.

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I was also quite impressed with the subsequent(right word?) proofs
What I was trying to do is explain to Bombadil exactly what this is all about. The central issue being that my fundamental equation is maybe a pretty thing but is essentially a useless construct since we can't solve many body problems. He keeps wanting to find something that equation says about reality and the correct answer is “absolutely nothing”. It is indeed the subsequent relationships which give us something to think about. Exactly what does Newtonian mechanics and relativity tell us about our universe? Now that is a serious philosophical question.
What we have is the fact that our mental model of reality must include the fundamental symmetry that all solutions, in the absence of outside influence, must transform to valid solutions to the "fundamental equation" in the center of mass of a system of any collection of data.
Other than the fact that I would have said “so long as outside influence can be ignored”. If that is what is meant by “in the absence of outside influence”, then we agree.
Once again due to my lacking math knowledge, I'm unable to see that clearly. That it's "a linear wave equation with wave solutions of fixed velocity".
Actually, it is quite simple; anyone competent in modern physics is totally familiar with both the trigonometric functions and exponential functions and the relationships between them. If you check out the wikipedia entry for “Exponential_function”, about two thirds of the way down the page, you will find the expression,

[math]e^{a+bi}=e^a[cos(b)+i\;sin(b)][/math].

 

Since [imath]e^{a+bi}=e^a e^{bi}[/imath] that implies [imath]e^{bi}=cos(b)+i\;sin(b)[/imath]. That means that waves (described by sine and cosine functions) are describable with exponential functions. Put this together with the fact that the differential of the sine function is the cosine function (and vice versa) and one has the fact that

[math] \frac{\partial^2}{\partial x^2}\Phi (x \pm vt)=-\frac{1}{v^2}\frac{\partial^2}{\partial t^2}\Phi(x \pm vt)[/math]

 

is the differential equation of a traveling wave. The shape of Phi can be a sine or cosine wave where a specific value is maintained at any point where [imath]x=x_0 \mp vt[/imath] (in other words, [imath]x \pm vt = x_0[/imath]: i.e., the shape of the wave is unaltered and only moved to a greater or lesser value as t increases. The solution has nothing to do with the wave length of the wave and thus a pulse can be created by summing a whole set of different wave lengths. That is what is displayed on the wikipedia entry for “Wave_equation”. Notice further that the squared relationship can be factored into a product of two first order equations with solutions moving in opposite directions. A lot of people think of the first order equations as more fundamental than the squared expression.

 

The important fact is that, anytime one sees a differential equation of such a form, one is working with wave phenomena.

...but I understand almost nothing of the above :(

I suspect alphas and betas refer to something different than they do inside the fundamental equation, and that [imath]\gamma[/imath] is the Lorentz factor...? Also don't know what to make of the [imath]\delta t[/imath]. I have no idea what's a "power series". Needless to say, I am quite lost once again :)

Let me begin with “a power series”. As they say in that page, power series are very useful when it comes to analysis. In general, most well behaved functions can be “expanded” into a power series such as,

[math]f(x)=\sum_{n=0}^\infty a_n x^n = a_0+a_1x+a_2x^2+\cdots+a_n x^n+\cdots[/math]

 

This expansion is useful to analyze the behavior of that function f(x) and that is what I am doing here. I am starting with the idea that x' is some arbitrary function of x, y, z, tau and t (where we are looking in the original coordinate system). My first step is to eliminate y, z and tau. The transformation can not depend upon y, z or tau because markers designating all points for any specific value of these arguments will end up being on the same line in both coordinate systems so a direct comparison is available (both observers will use the same value). Either party has the ability to move his origin by any specific distanced along these axes and the other party can do likewise; thus the change in x can not depend upon these values.

 

So I am down to the fact that the function I am looking for can, at worst, depend upon x and t. Now, if I make a power series expansion of that function, I can look at the impact of the various terms. My first conclusion is that a0 must vanish because, in either coordinate system, adding a constant to any x measurement is totally equivalent to moving the origin and the observers must be free to do so independent of the transformation (I have already taken advantage of that capability by setting their origins to be in the same place when t=0).

 

The second observation is a little more complex. Let us suppose that an is non zero for some n not equal to one and then look at an event which starts (at t = 0) at some point which is not the origin of their coordinate systems. As I have already said, both observers are free to move their origins to this new point. When they do that, the actual transformation changes by that factor an(-)xn back at the original origin so they now get a different transformation at the origin. That simply can not be correct.

 

The net effect of the above is that the worst case scenario is that only the linear, a1 term can have any impact. Whatever the change is to be, it must be the same everywhere (or they can't change their origins). Exactly the same arguments go for the dependence of x' on time and also apply directly to the form of the function which is to yield t'. This is essentially exactly what I said in the original post:

For those who don't believe that, consider the terms in a power series expansion of some supposed arbitrary function. The constant terms of that power series can be dropped as they move the primed origin at all times even t=0 where we have already defined it to be in exactly the same position as the unprimed coordinate system.

 

Furthermore, all terms not linear in x or t will generate changes which will create different answers when we simply transform the origin (something both coordinate systems must allow).

Thus my conclusion is,
... the transformation from one coordinate system to the other can be no more complex than [imath]x'=\alpha x -\beta t[/imath] and [imath]t'=\gamma x -\delta t[/imath].
The alpha, beta, gamma and delta are nothing more than numbers (those linear factors in that expansion I just discussed). As you say, these alphas and betas have utterly nothing to do with the operators appearing in the fundamental equation.

 

I hope that clears things up a bit. If you have any more questions let me know.

 

Have fun -- Dick

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However, since two scientists moving with respect to one another may indeed “presume” their personal frame is “the rest frame of the universe”(i.e., they ignore information which might settle the question) their physics must be valid in both frames (otherwise they will obtain different results, invalidating that presumption). It is that fact which requires the special relativistic transformations.

 

Isn’t it also possible though that they could just use explanations that require that elements exist that would place them at the rest frame of the universe? That is, choose invalid elements so that they are at the rest frame of the universe. I can see no reason why this couldn’t be done as long as the resulting explanation is flaw free. It just seems that as of yet there has been no property that such a reference frame would have other then it being “the rest frame of the universe” that would suggest that it is the rest frame. And so there is no way to tell if it is the rest frame or not.

 

That shift is completely analogous to the ordinary Galilean transformation of non-relativistic physics. It gives the phenomena being described as seen by the rest observer's construct of a moving inertial frame. It omits changes due to the moving observer's different definition of simultaneity but it still yields the correct results (just not in the perspective of the moving observer).

 

So are these the quantum mechanical transformations and so still not the relativistic transformations, so that if we considered a Newtonian universe that is one in which the Lorenz transformation would not be needed (which is not a possibility considering that all explanations must obey the Lorenz transformation) then these transforms would correspond to the corresponding acceleration, but in using these transformation there will be an error that will only be noticeable at relativistic speeds. In which case is it now possible to correct the transformations for relativistic speeds?

 

Again, I get the feeling you are confusing things here. It is the actual phenomena which must be the same from both reference frames; it will just be seen differently by the two observers

 

But won’t they only be seen differently by the two observers because they won’t agree on the mass, momentum and energy of the objects? If they agreed on these then they would agree on the measurements of the objects that they are explaining. And so, will agree on what they see.

 

Again, small change in t is of little significance. Newton's equations are essentially two body equations whereas my equation is a many body equation. If you have the correct solution for the rest of all those bodies (which is, from Newtonian position, they can be ignored) then only those velocities are significant.

 

But still don’t we have to know what condition is necessary for us to use that two body solution as a solution and that is that we must be able to ignore influences from the rest of the universe which happens when the Dirac delta function has no effect on the equation for the elements we are ignoring? Maybe I’m just wondering to much about how big of an effect those elements that we are ignoring are going to have on the problem, but it seems that they will have some kind of effect, it is just a question of how big of an effect.

 

Clocks do not measure time (if time is defined by interactions) but rather measure changes in tau. If both observers define time via clocks (at rest in their frames) then they are essentially using the same units for space and time (tau is being referred to as if it were time). This means that the Lorenz transformation of distance measure is all we need to make the two velocities identical.

 

But don’t we still need to either define a measure of t (which makes little séance as it can’t be measured) or [imath]\tau[/imath] so that we can define the value of [imath]v_0[/imath]. Or maybe we are just using [imath]v_0[/imath] as a measure of distance and so measuring it as length rather then velocity. In which case all that we need to do is define the value of [imath]v_0[/imath].

 

If “objects” (collections of elements which are stable structures over reasonable times) can exist, then there certainly exist things which have the same properties in both frames.

 

But won’t they appear to have different properties when observed from a different frame? That is, they won’t appear to be the same in their rest frame as in any other frame? For instance, if a object is defined to be a unit rod in its rest frame and measured in a moving frame then observers in both frames won’t agree on the length of the rod.

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Bombadil, I just don't know how to reach you. I get the feeling you either didn't read post #23 or you didn't understand what I meant.

Any explanation of anything can be interpreted in a manner which makes it a solution to my fundamental equation. :hihi: A fact of little real use!

 

Schrödinger's equation (and thus Newtonian mechanics) constitutes an approximate solution to that equation. ;) Perhaps this is of some use; it sure justifies Newtonian mechanics. :read:

 

The fact that the fundamental equation is essentially a wave equation with fixed velocity demands SR transformations. :sherlock: That is interesting; it implies there cannot be an explanation which violates SR. :thumbs_up That is worth knowing.

 

And more will be developed here. :candle:

You keep trying to use the fundamental equation to deduce something about the explanation. That is absolutely impossible because any explanation of anything can be interpreted in a manner which makes it a solution to my fundamental equation. That fact contains no information of any kind!
Isn’t it also possible though that they could just use explanations that require that elements exist that would place them at the rest frame of the universe?
No! They are trained scientists who are well aware of the supposed explanation of whatever phenomena they are investigating: i.e., they are both using exactly the same explanation. If they are not, then there is no reason to even dream there is any association between their experiments. If they are using exactly the same explanation then both their valid elements and their hypothesized elements are the same. I repeat, there is “NO” information in the fundamental equation; all information is in their explanation!
So are these the quantum mechanical transformations and so still not the relativistic transformations, so that if we considered a Newtonian universe that is one in which the Lorenz transformation would not be needed (which is not a possibility considering that all explanations must obey the Lorenz transformation) then these transforms would correspond to the corresponding acceleration, but in using these transformation there will be an error that will only be noticeable at relativistic speeds. In which case is it now possible to correct the transformations for relativistic speeds?
You don't seem to understand what relativity is all about. The central issue of relativity is that physics (the laws, equation and such) apply independent of your frame of reference. If you have the physics correctly specified and do all your calculations in one specified inertial frame then the issue of relativity does not even come up! You can use whatever frame you wish. In fact, that is the very central issue of relativity.
But won’t they only be seen differently by the two observers because they won’t agree on the mass, momentum and energy of the objects? If they agreed on these then they would agree on the measurements of the objects that they are explaining. And so, will agree on what they see.
You are talking about the consequences of the necessity of the relativistic transformations, not the basis of the relativistic transformations. You are confused about the issues under examination. The central issue is, “will they agree on the physics calculations!” You are taking the results of that conundrum and seeing them as reasons for the problem. You have it dead backwards.
... and that is that we must be able to ignore influences from the rest of the universe which happens when the Dirac delta function has no effect on the equation for the elements we are ignoring?
We are not ignoring the Dirac delta function. If you followed the proof that Schrödinger's equation is an approximation to my fundamental equation you would be well aware of that fact. One cannot obtain Schrödinger's equation if you omit the impact of the Dirac delta function.
... but it seems that they will have some kind of effect, it is just a question of how big of an effect.
The effect is exactly as important as the probability that xi=xj. If that is not true, the impact of the Dirac delta function vanishes exactly. In our explanation of reality, our world view (which is the explanation we are working with), the probability that xi=xj for most of the elements making up our universe is so insignificant as to be non existent! So that two body relationship (Schrödinger's equation) is a very reasonable approximation. We are talking about that specific explanation and not the general implications of my equations (you should be well aware that there are none associated with my equation).
But don’t we still need to either define a measure of t ...
The t is an interaction parameter; as such it is not directly measurable (no device exists which will provide a specific answer to the question “what time is it”); however, it none the less describes evolution of mechanical devices. Everyone uses clocks as the standard for physics evolution. So, saying “clocks measure time” does nothing except define the velocity to be used in the fundamental equation.
But won’t they appear to have different properties when observed from a different frame? ... For instance, if a object is defined to be a unit rod in its rest frame and measured in a moving frame then observers in both frames won’t agree on the length of the rod.
Again, you are confusing the fundamental equation with your explanation of reality. It is the physics (your explanation of reality) which must agree with the measures of both observers. Now, what does your world view say about a ruler you have in your office compared to that same ruler when you take it with you on a drive in your car. You want to get relativistic? If you get on a star cruiser and head for Alpha Centauri at 99% the speed of light and pull that same ruler out of your pocket. Does your world view suggest that you will find that ruler has changed its length? Or will it weigh down your pocket? Gee, if it did, you could use that fact to tell how fast you were moving (but that's a violation of relativity, the physics would be different). The observer on earth (who is using his Galilean inertial frame for his measurements) will look through the telescope and deduce the weight and length of the ruler. What will he say? My god, look how short that ruler has gotten and gee, it must weigh twenty pounds. Think about these things a little.

 

Have fun -- Dick

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What we have is the fact that our mental model of reality must include the fundamental symmetry that all solutions, in the absence of outside influence, must transform to valid solutions to the "fundamental equation" in the center of mass of a system of any collection of data.

Other than the fact that I would have said “so long as outside influence can be ignored”. If that is what is meant by “in the absence of outside influence”, then we agree.

 

Actually apart from the bolded up parts, it was a quoted from the OP, and yes that's how I interpreted it myself. The bolded up parts were replacing the stuff I found confusing in the OP, and actually, now I think you may have intented to write "in the center of mass of any collection of data". Ehh, at any rate, might be worthwhile to tidy it up in the OP, just in case :)

 

Actually, it is quite simple; anyone competent in modern physics is totally familiar with both the trigonometric functions and exponential functions and the relationships between them. If you check out the wikipedia entry for “Exponential_function”, about two thirds of the way down the page, you will find the expression,

[math]e^{a+bi}=e^a[cos(b)+i\;sin(b)][/math].

 

Since [imath]e^{a+bi}=e^a e^{bi}[/imath] that implies [imath]e^{bi}=cos(b)+i\;sin(b)[/imath]. That means that waves (described by sine and cosine functions) are describable with exponential functions.

 

Hmmm, okay, after a lot of head scratching, given that [imath]e^{bi}=cos(b)+i\;sin(b)[/imath], I can understand how [imath]e^{bi}[/imath] can be seen as a unit vector on a complex plane, and I can see how that can be plotted as a wave against change in "b"... Only, of course you can easily plot 2 different waves; one for the real part and one for the imaginary part of that unit vector... I mean;

 

plot cos(b) + i sin(b) from b=0 to b=2pi - Wolfram|Alpha

 

Is there just a convention that they always use just the other part or something? (really just guessing here :)

 

And toying around with Wolfram Alpha more, looks like the [math]e^a[/math] part affects the magnitude of the result... If it's set to zero, the magnitude is 1 etc, following the properties of e.

 

So with that I can understand how [imath]e^{a+bi}[/imath] could be used as a way for encoding a wave; the real part gives the amplitude and the imaginary part gives the phase through some convention. Is that the idea? That people build functions that exploit [imath]e^{a+bi}[/imath] within to come up with a wave.

 

Put this together with the fact that the differential of the sine function is the cosine function (and vice versa) and one has the fact that

[math] \frac{\partial^2}{\partial x^2}\Phi (x-\omega t)=-\frac{1}{\omega^2}\frac{\partial^2}{\partial t^2}\Phi(x-\omega t)[/math]

 

is the differential equation of a traveling wave.

 

Well after some head scratching, I could not understand that stuff above. Nor your further commentary about the issue. I'm guessing [imath]\omega[/imath] means angular frequency here, and I suppose that is essentially the rotation rate of the unit vector (the phase) or something like that. Also I can see it looks similar to the fundamental equation.

 

I would like to understand how waves equations work so if you can provide more help with that, it would be good. Still in the meantime, I can proceed forwards with the OP as I can take it on faith that indeed your equation is a wave equation with waves traveling at fixed velocity.

 

Let me begin with “a power series”. As they say in that page, power series are very useful when it comes to analysis. In general, most well behaved functions can be “expanded” into a power series such as,

[math]f(x)=\sum_{n=0}^\infty a_n x^n = a_0+a_1x+a_2x^2+\cdots+a_n x^n+\cdots[/math]

 

Once again after toying with Wolfram alpha and reading the wikipedia explanation, I think I understand that little bit. Seems like it is basically a handy general way to represent (an approximation of) any sort of curve that any "well behaved function" might plot, i.e. to represent that function itself. The coefficients control the shape and the position of the curve in a completely general fashion; [imath]a_0[/imath] moves the whole curve along y-axis and [imath]c[/imath] moves it along x-axis. [imath]a_1[/imath] controls the linear component and the rest control the shape of the curve, yeah I think I got it.

 

 

This expansion is useful to analyze the behavior of that function f(x) and that is what I am doing here. I am starting with the idea that x' is some arbitrary function of x, y, z, tau and t (where we are looking in the original coordinate system). My first step is to eliminate y, z and tau. The transformation can not depend upon y, z or tau because markers designating all points for any specific value of these arguments will end up being on the same line in both coordinate systems so a direct comparison is available (both observers will use the same value). Either party has the ability to move his origin by any specific distanced along these axes and the other party can do likewise; thus the change in x can not depend upon these values.

 

So I am down to the fact that the function I am looking for can, at worst, depend upon x and t.

 

Yup, quite reasonable as we are looking at the impact of "speed" along the x-axis between different coordinate systems.

 

So, just to re-summarize, essentially we are talking about a function that, upon the input of "the X-axis position of a specific event in the unprimed coordinate system", would give us the X-axis position of that same event in the primed ("moving") coordinate system...?

 

Now, if I make a power series expansion of that function, I can look at the impact of the various terms. My first conclusion is that a0 must vanish because, in either coordinate system, adding a constant to any x measurement is totally equivalent to moving the origin and the observers must be free to do so independent of the transformation (I have already taken advantage of that capability by setting their origins to be in the same place when t=0).

 

Ahha, true.

 

The second observation is a little more complex. Let us suppose that an is non zero for some n not equal to one and then look at an event which starts (at t = 0) at some point which is not the origin of their coordinate systems. As I have already said, both observers are free to move their origins to this new point. When they do that, the actual transformation changes by that factor an(-)xn back at the original origin so they now get a different transformation at the origin. That simply can not be correct.

 

Okay, yeah, thinking of this in terms of "function that upon the input of the X-position of an event in first coordinate system gives us the X-position of the same event in the second coordinate system", then yes non-linear answer would give completely different results when just moving the origin. So, "ahha, true".

 

The net effect of the above is that the worst case scenario is that only the linear, a1 term can have any impact. Whatever the change is to be, it must be the same everywhere (or they can't change their origins). Exactly the same arguments go for the dependence of x' on time and also apply directly to the form of the function which is to yield t'.

 

Yup, definitely sounds like I understood the "power series analysis" correctly.

 

...the transformation from one coordinate system to the other can be no more complex than [imath]x'=\alpha x -\beta t[/imath] and [imath]t'=\gamma x -\delta t[/imath].

The alpha, beta, gamma and delta are nothing more than numbers (those linear factors in that expansion I just discussed). As you say, these alphas and betas have utterly nothing to do with the operators appearing in the fundamental equation.

 

So yeah now I think I understand that bit...

 

Sorry I was slow, I wrote this reply over the course of many days, taking a hour from here and hour from there teaching myself the relevant wave function and power series stuff... I'll try to get around to continue from here soon...

 

-Anssi

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Ehh, at any rate, might be worthwhile to tidy it up in the OP, just in case :)
I have edited the OP. Not exactly what you suggested but I think it is better now.
Is there just a convention that they always use just the other part or something? (really just guessing here :)
Not really; they often use the exponential representation because of the convenience of representing some waves via complex amplitude particularly in electronics because of the ninety degree phase differences in voltage to current functions for different kinds of components (in a linear circuit, voltages in resistors and capacitors are ninety degrees out of phase when plotted as a function of the current). It provides a major simplification in the mathematics. A very similar thing occurs in quantum mechanics. A wave represented by [imath]e^{bi}[/imath] can be seen as having a constant amplitude everywhere (it is just a vector rotating in a complex space). I wouldn't worry about it as there is a lot of serious physics behind the representation. The only reason I bring that up is that your analysis is perfectly correct except that when you say, “that people build functions that exploit [imath]e^{a+bi}[/imath] within to come up with a wave” you sort of have things backwards.
I would like to understand how waves equations work so if you can provide more help with that, it would be good.
What you are missing is knowledge of differentiation of trigonometric functions. The derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x). That being the case, the second derivative of either is a simple change of sign.

[math]\frac{d^2}{dx^2}sin(x)=-sin(x)[/math]

and

[math]\frac{d^2}{dx^2}cos(x)=-cos(x)[/math]

 

It follows that anytime you see a differential equation of the form

[math]\frac{\partial^2}{\partial x^2}\Phi(x,t)=a^2\frac{\partial^2}{\partial t^2}[/math]

 

you immediately know that the solution is a wave function (a sine or cosine) with a simple argument of the form ax+t or ax-t (the sign of t is immaterial as when you take the second derivative the factor ends up squared). Furthermore, the second order differential equation can be written

[math]\left\{\frac{\partial^2}{\partial x^2}-a^2\frac{\partial^2}{\partial t^2}\right\}\Phi(x,t)=0[/math]

 

which factors into

[math]\left(\frac{\partial}{\partial x}-a\frac{\partial}{\partial t}\right)\left(\frac{\partial}{\partial x}+a\frac{\partial}{\partial t}\right)\Phi(x,t)=0[/math]

 

yielding two first order differential equations which will satisfy that equation.

[math]\left(\frac{\partial}{\partial x}-a\frac{\partial}{\partial t}\right)\Phi(x,t)=0[/math]

and

[math]\left(\frac{\partial}{\partial x}+a\frac{\partial}{\partial t}\right)\Phi(x,t)=0[/math]

 

The first is solved by a sine or cosine function with the argument (ax+t) and the second by a sine or cosine function with the argument (ax-t). These two functions are simple waves moving in opposite directions. If we call the argument of the sine and/or cosine function “z” then z=ax+t or z=ax-t. If z (the argument of the sine or cosine function) is a constant, we are talking about a specific point on that function. The question you need to ask yourself is, if t changes, how must x change in order for z (the argument) to remain constant? Obviously, if t increases by some amount, ax must either decrease by the same amount (for z=ax+t) or increase by the same amount (for z=ax-t). This can only be true if the change in x is identical to t divided by minus a or plus a (for the two cases). It should be clear to you that the change in x has to be given by x=x0+vt so the velocity of the wave must be one over a.

Well after some head scratching, I could not understand that stuff above. Nor your further commentary about the issue. I'm guessing [imath]\omega[/imath] means angular frequency here, and I suppose that is essentially the rotation rate of the unit vector (the phase) or something like that.
Sorry about that. The [imath]\omega[/imath] just stands for a constant. What it means depends upon your specific definition of angles in the sine function and I was being very careless there as x is clearly in radians. Normally angles are expressed in degrees but in physics (for differential simplicity) angles are almost always expressed in radians. 360 degrees equal [imath]2\pi[/imath] radians. I am going to edit that post and replace [imath]\omega[/imath] with v. I really should have done that originally but you know I am getting senile and sometimes what pops into my head is just wrong. Getting old is a pain in the ***.

 

I hope I have made it a little clearer; I sincerely apologize for making this difficult for you.

Sorry I was slow, I wrote this reply over the course of many days, taking a hour from here and hour from there teaching myself the relevant wave function and power series stuff... I'll try to get around to continue from here soon...
You owe me no apology! It is I who owe you an apology as I was the one who made it difficult for you to understand. As I said, the only rational for my errors was pure senility.

 

It reminds me of a joke on work performance I heard a long time ago.

 

When you are young and you know nothing, you have to think everything out. As you spend time learning your job you discover things here and there that you don't have to think about; it's just routine. If you spend enough time at a specific job you will eventually reach the point where it is all routine and you don't have to think at all -- and that is called “senility”.

 

Have fun -- Dick

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Any explanation of anything can be interpreted in a manner which makes it a solution to my fundamental equation. :sherlock: A fact of little real use!

 

Schrödinger's equation (and thus Newtonian mechanics) constitutes an approximate solution to that equation. :sherlock: Perhaps this is of some use; it sure justifies Newtonian mechanics. :read:

 

The fact that the fundamental equation is essentially a wave equation with fixed velocity demands SR transformations. :sherlock: That is interesting; it implies there cannot be an explanation which violates SR. :thumbs_up That is worth knowing.

 

And more will be developed here. :candle:

 

Firstly the fact that anything can be interpreted in a way that makes it a solution to the fundamental equation is of little real use because the fundamental equation is a n-body equation and as such it can’t be solved directly.

 

Now since you have derived the Schrödinger equation all that this says is that Newtonian mechanics can be used as an approximation to any explanation. We can construct objects, bounce things off of each other, things like that. Also it tells us that if we know the form of V(x,t) then we can explain the behavior of one element.

 

Finally that “The fact that the fundamental equation is essentially a wave equation with fixed velocity demands SR transformations.” means that if two observers are in different reference frames and can use the same explanation then their explanations must obey SR transformations.

 

No! They are trained scientists who are well aware of the supposed explanation of whatever phenomena they are investigating: i.e., they are both using exactly the same explanation. If they are not, then there is no reason to even dream there is any association between their experiments. If they are using exactly the same explanation then both their valid elements and their hypothesized elements are the same. I repeat, there is “NO” information in the fundamental equation; all information is in their explanation!

 

But even if there was a way of telling which frame was moving wouldn’t they still be using the same explanation in the example you put forward in that any explanation that would be considered useful must explain whatever property it was that separated the reference frames. Both being trained scientists trained in the same explanation must use an explanation that would explain this or the explanation would be so flawed that they would be forced to fined a new one. That or they will have to agree to ignore whatever information it is that doesn’t agree with their explanation.

 

Is this the point though, that no matter what explanation that they decide to use if it is a flaw-free explanation then it has an interpretation that satisfies the fundamental equation and in that interpretation all elements must have a constant speed as seen from any frame. Which leads us to the Lorenz transformation.

 

So my question is, is this one of the defenses for the two reference frames using the same explanation without any way of telling which one is moving? From what I can tell the only other defense lies in their observations (that is what they are explaining) and not in the fundamental equation.

 

There is no information about any particular explanation in the fundamental equation all that we know is that any flaw-free explanation can be interpreted in a way that makes it a solution to the fundamental equation.

 

You keep trying to use the fundamental equation to deduce something about the explanation. That is absolutely impossible because any explanation of anything can be interpreted in a manner which makes it a solution to my fundamental equation. That fact contains no information of any kind!

 

But didn’t you use the fundamental equation to deduce that there exists an interpretation of an explanation which must obey SR or is this considered a subtly different topic and not something that we deduced about the explanation?

 

You don't seem to understand what relativity is all about. The central issue of relativity is that physics (the laws, equation and such) apply independent of your frame of reference. If you have the physics correctly specified and do all your calculations in one specified inertial frame then the issue of relativity does not even come up! You can use whatever frame you wish. In fact, that is the very central issue of relativity.

 

I suspect that the key words here are “If you have the physics correctly specified” and suspect that Newtonian mechanics is not the correct physics to use quite simply because using Newtonian mechanics it would be possible to construct an object that could accelerate to any speed and as far as I can tell if we obey the Lorenz transformation it is impossible to move an object faster then [imath]V_0[/imath] (the speed of light).

 

The effect is exactly as important as the probability that xi=xj. If that is not true, the impact of the Dirac delta function vanishes exactly. In our explanation of reality, our world view (which is the explanation we are working with), the probability that xi=xj for most of the elements making up our universe is so insignificant as to be non existent! So that two body relationship (Schrödinger's equation) is a very reasonable approximation. We are talking about that specific explanation and not the general implications of my equations (you should be well aware that there are none associated with my equation).

 

So we are only interested in explanations that will be equivalent to what we experience? That is when that two body equation (Schrödinger’s equation) is a close approximation in that no differences will be noticeable. Or the probability that [imath] x_i=x_j[/imath] is sufficiently small that it can be ignored.

 

Also aren’t there only no general implications because there is no general solutions and in order to get even a close approximation we must simplify it into something that we can solve (obtain implication of the explanation).

 

Again, you are confusing the fundamental equation with your explanation of reality. It is the physics (your explanation of reality) which must agree with the measures of both observers. Now, what does your world view say about a ruler you have in your office compared to that same ruler when you take it with you on a drive in your car. You want to get relativistic? If you get on a star cruiser and head for Alpha Centauri at 99% the speed of light and pull that same ruler out of your pocket. Does your world view suggest that you will find that ruler has changed its length? Or will it weigh down your pocket? Gee, if it did, you could use that fact to tell how fast you were moving (but that's a violation of relativity, the physics would be different). The observer on earth (who is using his Galilean inertial frame for his measurements) will look through the telescope and deduce the weight and length of the ruler. What will he say? My god, look how short that ruler has gotten and gee, it must weigh twenty pounds. Think about these things a little.

 

Firstly I can say that the laws of physics appear to be the same in every reference frame in so far as the laws that I can measure agree with each other. Further more there has not been one found that is not the same in any reference frame however this says nothing about why they are the same or what seems more important what are the requirements for this to be the case. The Lorenz transformation seems to be a consequence of this not a requirement.

 

This does however lead to what seems to be an interesting question which is, if the ruler was now shorter would we notice it. If it is shorter due to being in a moving reference frame won’t everything else now be shorter by the same amount so that, once again, we can’t tell if we are moving. If it is heaver, would it be possible to design a way to measure it so that we would be able to tell. Again I don’t think that we could, due to how things must be defined. What I’m saying is that even if the laws of physics do change if all that we are interested in is what is on the ship then I’m not sure that we can tell the difference, at least not if there exist a interpretation which obeys the fundamental equation. Again even with the speed of light (the oscillator) if it did change speed, could we tell. If we are truly using it to define simultaneity then there would be no way to time how long it takes to go one direction over the other (as we are timing it with its self) and so we would come to the same conclusions about how fast it moves.

 

Basically it seems that we have circularly defined the laws of physics in such a way that if one changes they all will change in such a way that it will not be noticeable, at least if there exist a interpretation which obeys the fundamental equation. That is, we can’t perform an experiment to find out which law has changed because it means that they all have changed.

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Firstly the fact that anything can be interpreted in a way that makes it a solution to the fundamental equation is of little real use because the fundamental equation is a n-body equation and as such it can’t be solved directly.
Actually that is not quite true. There are some special circumstances where it can be but even then the amount of work required is beyond us.
Now since you have derived the Schrödinger equation all that this says is that Newtonian mechanics can be used as an approximation to any explanation.
What it actually says is that the elemental components upon which that explanation is based will approximately obey Newtonian mechanics. What you need to remember is that Newtonian mechanics includes the behavior of static structures. The issue there is that most explanations are like rocks and statues; their basic elements are presumed to be static.

 

The correct statement here is, if “we can construct objects, bounce things off of each other, things like that” there are conceivable entities whose hypothetical behavior will generate a V(x,t) such that Newtonian mechanics will approximately “explain the behavior of one element”. You have to keep your mind open to all the possibilities, not just presume the first one that pops into your head.

Finally that “The fact that the fundamental equation is essentially a wave equation with fixed velocity demands SR transformations.” means that if two observers are in different reference frames and can use the same explanation then their explanations must obey SR transformations.
Einstein proposed the axiom that the speed of light is the same in all frames and from that axiom deduced SR. My position is that his axiom could very well be “wrong” (and, down the road, I will demonstrate the power of dropping that axiom). I show that it is the fact that objects can be defined which requires SR. Objects are collections of elemental entities which can be seen as stable structures unto themselves: i.e., the rest of the universe can be ignored. If SR is not valid, you cannot define a structure such as clocks and rulers. Thus it is that any explanation which depends upon the existence of objects can not be valid. This is a much more powerful conclusion than Einstein’s SR.
But even if there was a way of telling which frame was moving ...
Try physically defining a frame of reference in a universe where objects can not exist.
That or they will have to agree to ignore whatever information it is that doesn’t agree with their explanation.
Ignoring information is one of the favorite moves of any unthinking person.
Is this the point though, that no matter what explanation that they decide to use if it is a flaw-free explanation then it has an interpretation that satisfies the fundamental equation and in that interpretation all elements must have a constant speed as seen from any frame. Which leads us to the Lorenz transformation.
You seem to overlook the fact that the fundamental equation is valid only in the rest frame of the universe. That fact was explicitly used in the derivation.
There is no information about any particular explanation in the fundamental equation all that we know is that any flaw-free explanation can be interpreted in a way that makes it a solution to the fundamental equation.
What you say is true but I wonder what is going on in your mind. Exactly what do you think such a statement means?
But didn’t you use the fundamental equation to deduce that there exists an interpretation of an explanation which must obey SR ...
I would have said that SR is embedded in the fundamental equation via that fixed velocity. Without it the universe cannot contain any objects and sans objects all we have is a manybody equation which can not be solved in general. How would you yourself exist in a universe without objects.
I suspect that the key words here are “If you have the physics correctly specified” and suspect that Newtonian mechanics is not the correct physics to use quite simply because using Newtonian mechanics it would be possible to construct an object that could accelerate to any speed and as far as I can tell if we obey the Lorenz transformation it is impossible to move an object faster then [imath]V_0[/imath] (the speed of light).
So you have pointed out a flaw in Newtonian mechanics. Isn’t it a much more powerful statement to say that if the elemental entities of your explanation do not obey SR, the explanation is flawed and just leave it there?
Basically it seems that we have circularly defined the laws of physics ...
That is indeed the apparent consequence of my deductions isn’t it. :shrug:

 

Have fun -- Dick

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What it actually says is that the elemental components upon which that explanation is based will approximately obey Newtonian mechanics. What you need to remember is that Newtonian mechanics includes the behavior of static structures. The issue there is that most explanations are like rocks and statues; their basic elements are presumed to be static.

 

Then even if an object is moving in our explanation we still use the same explanation of the object that we would if it was at rest, we just have to remember that it is now moving and transform the explanation of the object so that it still has a constant speed but is now moving partly along some x axis in addition to the [imath]\tau[/imath] axis. Which simply requires the Lorenz transformation?

 

The correct statement here is, if “we can construct objects, bounce things off of each other, things like that” there are conceivable entities whose hypothetical behavior will generate a V(x,t) such that Newtonian mechanics will approximately “explain the behavior of one element”. You have to keep your mind open to all the possibilities, not just presume the first one that pops into your head.

 

Are you saying that if we can construct objects then it is possible to choose elements in such a way as to create a V(x,t) that will explain the behavior of the elements as though they where obeying Newtonian mechanics?

 

Einstein proposed the axiom that the speed of light is the same in all frames and from that axiom deduced SR. My position is that his axiom could very well be “wrong” (and, down the road, I will demonstrate the power of dropping that axiom). I show that it is the fact that objects can be defined which requires SR. Objects are collections of elemental entities which can be seen as stable structures unto themselves: i.e., the rest of the universe can be ignored. If SR is not valid, you cannot define a structure such as clocks and rulers. Thus it is that any explanation which depends upon the existence of objects can not be valid. This is a much more powerful conclusion than Einstein’s SR.

 

So you have actually shown that rather then using the axiom, that the speed of light is the same in every reference frame to derive the Lorenz transformation. We can use the axiom that an object can exist in our explanation, where an object is simply a collection of elements that maintain their orientation and so can be explained separately of the rest of the universe.

 

Ignoring information is one of the favorite moves of any unthinking person.

 

But haven’t you already said that we have to ignore any information that will settle the question of what is the rest frame so that we can use the same explanation of an object that we would if the objects where at rest with the universe.

 

What you say is true but I wonder what is going on in your mind. Exactly what do you think such a statement means?

 

If we are using a flaw free explanation, meaning that whatever explanation it is that we are using, it is consistent with all of our observations, we certainly don’t have to be explaining things with a solution to the fundamental equation. There are in fact many ways to explain such information, however there must be a mapping between however we are explaining our observations and possible solutions to the fundamental equation. Whatever this maps our explanation into is an interpretation of our explanation. Or perhaps the mapping is the interpretation. (I’m not sure which one would be considered the interpretation) Either way what I am saying is that there is such a mapping that maps our explanation into a solution to the fundamental equation and whatever mappings satisfy this must depend on the observations, not on the possible solutions, to the fundamental equation.

 

I would have said that SR is embedded in the fundamental equation via that fixed velocity. Without it the universe cannot contain any objects and sans objects all we have is a manybody equation which can not be solved in general. How would you yourself exist in a universe without objects.

 

This question seems somewhat unanswerable as all of the explanations that I know any thing about require the use of objects, so I don’t even know how I would explain a universe where an object couldn’t exist., so I certainly cant say how I would exist in such a universe. More importantly (and perhaps more to the point you are trying to make) is that the word I or you would seem to imply that we are referring to an object which we are considering to not exist in the first place, So that we must find a new way to define such things before we can answer such questions.

 

Such explanations are perhaps for the time being of little interest as we can’t form objects, although it seems that it would be interesting to know what can be said about such explanations. While I may be wrong about this it seems that it may be outside of our current interests. It still seems that the Schrödinger equation could be used to explain a single element if we can choose the proper V(x,t) although perhaps the approximations that were made particularly setting [imath]E=mc^2[/imath] are too limiting to make this of much use in such a case.

 

So you have pointed out a flaw in Newtonian mechanics. Isn’t it a much more powerful statement to say that if the elemental entities of your explanation do not obey SR, the explanation is flawed and just leave it there?

 

Wouldn’t we be best off, though, to say that any explanation containing objects must obey SR or the explanation is flawed? or can any element be considered an object if objects can exist in our explanation?

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