Science Forums 2

[Doctordick]

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.

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 by and (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).

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.

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_?.

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.

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 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 using these equations and solving for the necessary transformation we arrive at the Lorenz transformations as the only possible transformation.

Now that is true.

Originally Posted by Bombadil

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

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.

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.

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.

Originally Posted by Bombadil

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

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