An “analytical-metaphysical” take on Special Relativity!

Bombadil · 05-28-2009

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.

Quote:

Originally Posted by Doctordick

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 $\vec{\Psi}$ by $e^{ikx},e^{ik\tau}$ and $e^{ikt}$ (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.

Quote:

Originally Posted by Doctordick

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 $\sum \delta (\vec{x}_i -\vec{x}_j)=0$ 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 $r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2}$ 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 $\alpha(vt)-\beta t = 0$ 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 $v_?$ is the same in both frames but the actual value of $v_?$ 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.

Quote:

Originally Posted by Doctordick

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 $E=mc^2$ . That means that the kinetic energy (the energy of motion) must be small compared to $mc^2$ : 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 $v_?$ as it has zero momentum in the $\tau$ 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.

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Originally Posted by Doctordick

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.

AnssiH · 05-30-2009

Okay, time to start dissecting the this thread!

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Originally Posted by Doctordick

...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

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

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".

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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 $\vec{\Psi}$ by the simple function

$\prod_{j=1}^n e^{i\frac{Px_j}{n\hbar}}$ .

This change in $\vec{\Psi}$ 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 $\vec{\Psi}$ . 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 $\hbar$ at the denominator, so that's now fixed in the OP. The following quote is from a PM:

Quote:

Originally Posted by Doctordick

...just as the function $e^{\frac{-iqt}{K\sqrt{2}}}$ shifted the energy of by a constant factor $\frac{-iq}{K\sqrt{2}}$ (as a direct consequence of the differential with respect to “t”), the function $e^{i\frac{Px_j}{n\hbar}}$ shifts the momentum in the x direction of the jth entity by the constant factor $i\frac{P}{n\hbar}$ (as a direct consequence of the differential with respect to x_j). Both of these effects are a consequence of the product rule of differentiation. The product indicates addition of $i\frac{P}{n\hbar}$ for every value of j (all n entities) so the sum of all n terms (times $i\hbar$ will be “P”. Thus it follows that the change in $\Psi$ (changed to $\Phi$ times the product) produces an equation where the product can be divided out and thus creates a new function $\Phi$ 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:

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

The right hand side can be written:

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

$=$

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

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

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

I hope that's all valid, and judging from your comment "the function $e^{i\frac{Px_j}{n\hbar}}$ shifts the momentum in the x direction of the jth entity by the constant factor $i\frac{P}{n\hbar}$ (as a direct consequence of the differential with respect to x_j)" 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

Doctordick · 05-30-2009

Quote:

Originally Posted by Bombadil

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.

Quote:

Originally Posted by Bombadil

Unless I am missing something I think that the only transformations that you have defined are the those resulting from multiplying $\vec{\Psi}$ by $e^{ikx},e^{ik\tau}$ and $e^{ikt}$ (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).

Quote:

Originally Posted by Bombadil

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.

Quote:

Originally Posted by Bombadil

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_?.

Quote:

Originally Posted by Bombadil

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.

Quote:

Originally Posted by Bombadil

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 $r=v_?t=\sqrt{x'^2+y'^2+z'^2+\tau'^2}$ 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 $\alpha(vt)-\beta t = 0$ using these equations and solving for the necessary transformation we arrive at the Lorenz transformations as the only possible transformation.

Now that is true.

Quote:

Originally Posted by Bombadil

In order for this to work it requires that the value of $v_?$ is the same in both frames but the actual value of $v_?$ 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.

Quote:

Originally Posted by Bombadil

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.

Quote:

Originally Posted by Bombadil

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.

Quote:

Originally Posted by Bombadil

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.

Quote:

Originally Posted by Bombadil

But won’t the oscillator always be considered to move at $v_?$ as it has zero momentum in the $\tau$ 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”.

Quote:

Originally Posted by Bombadil

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.

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.

And more will be developed here.

Have fun -- Dick

Doctordick · 05-30-2009

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Originally Posted by AnssiH

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

I am looking forward to it.

Have fun -- Dick

AnssiH · 05-31-2009

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Originally Posted by Doctordick

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 $\vec{\Psi}$ 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 $\vec{\Psi}$ . 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.

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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.

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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.

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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 $c=\frac{1}{K\sqrt{2}}$ .) For the moment (since K is actually a totally open parameter) I will set this constant velocity to v_?.

Yup.

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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., $\tau'=\tau$ .

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

Yup.

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

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

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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 $\Psi(t')$ 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 $\tau$ 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 $\tau'=\tau$ (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...

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

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 $\gamma$ is the Lorentz factor...? Also don't know what to make of the $\delta t$ . 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

arkain101 · 05-31-2009

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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

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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.

Doctordick · 05-31-2009

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Originally Posted by arkain101

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.

Quote:

Originally Posted by AnssiH

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.

Quote:

Originally Posted by AnssiH

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,

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

Since $e^{a+bi}=e^a e^{bi}$ that implies $e^{bi}=cos(b)+i\;sin(b)$ . 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

$\frac{\partial^2}{\partial x^2}\Phi (x \pm vt)=-\frac{1}{v^2}\frac{\partial^2}{\partial t^2}\Phi(x \pm vt)$

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 $x=x_0 \mp vt$ (in other words, $x \pm vt = x_0$ : 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.

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Originally Posted by AnssiH

...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 $\gamma$ is the Lorentz factor...? Also don't know what to make of the $\delta t$ . 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,

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

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 a_n 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 a_n(-)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:

Quote:

Originally Posted by Doctordick

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,

Quote:

Originally Posted by Doctordick

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

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

Bombadil · 06-03-2009

Quote:

Originally Posted by Doctordick

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.

Quote:

Originally Posted by Doctordick

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?

Quote:

Originally Posted by Doctordick

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.

Quote:

Originally Posted by Doctordick

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.

Quote:

Originally Posted by Doctordick

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 $\tau$ so that we can define the value of $v_0$ . Or maybe we are just using $v_0$ 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 $v_0$ .

Quote:

Originally Posted by Doctordick

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.

Doctordick · 06-04-2009

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.

Quote:

Originally Posted by Doctordick

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.

And more will be developed here.

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!

Quote:

Originally Posted by Bombadil

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!

Quote:

Originally Posted by Bombadil

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.

Quote:

Originally Posted by Bombadil

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.

Quote:

Originally Posted by Bombadil

... 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.

Quote:

Originally Posted by Bombadil

... 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 x_i=x_j. 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 x_i=x_j 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).

Quote:

Originally Posted by Bombadil

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.

Quote:

Originally Posted by Bombadil

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

AnssiH · 06-07-2009

Quote:

Originally Posted by Doctordick

Quote:

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

Quote:

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,

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

Since $e^{a+bi}=e^a e^{bi}$ that implies $e^{bi}=cos(b)+i\;sin(b)$ . 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 $e^{bi}=cos(b)+i\;sin(b)$ , I can understand how $e^{bi}$ 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 $e^a$ 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 $e^{a+bi}$ 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 $e^{a+bi}$ within to come up with a wave.

Quote:

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

$\frac{\partial^2}{\partial x^2}\Phi (x-\omega t)=-\frac{1}{\omega^2}\frac{\partial^2}{\partial t^2}\Phi(x-\omega t)$

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 $\omega$ 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.

Quote:

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,

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

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; $a_0$ moves the whole curve along y-axis and $c$ moves it along x-axis. $a_1$ controls the linear component and the rest control the shape of the curve, yeah I think I got it.

Quote:

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...?

Quote:

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.

Quote:

The second observation is a little more complex. Let us suppose that a_n 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 a_n(-)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".

Quote:

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.

Quote:

Originally Posted by Doctordick

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

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

An “analytical-metaphysical” take on Special Relativity!

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