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An “analytical-metaphysical” take on Special Relativity!


Doctordick

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Sorry about the delays, been quite busy, but should have some more time at my hands for a little while now...

 

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]

 

Right, I see... I figure then that "second derivative" means "derivative of a derivative", and that that is exactly what you get by squaring the differential operator... Hmm, yeah, that bit seems to make perfect sense to me.

 

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).

 

Hmm... okay.

 

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]

 

Yup.

 

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.

 

Right, so they are waves plotted against the change in t...

 

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).

 

"...identical to change in t divided by..." I suppose you meant to say.

 

One thing that puzzled me there for a moment. I understood that the argument of the sine/cosine function being a constant will satisfy the equations you laid down, but I thought, wouldn't it also be allowed that the partial derivative of x was equal or negative to "the partial derivative of t times a"... But it dawned on me that I guess that could only satisfy either one or the other of those first order differential equations, in which case the only possibility is if indeed the argument itself remains constant. I'm just laying my thoughts here just so you can make sure I'm understanding this correctly.

 

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.

 

True.

 

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 ***.

 

Heh, I especially like your official "reason for editing: pure senility..." :D

 

And yeah, I know it's common human behaviour that we start making mistakes when we are good enough with something, as we become careless and start to overlook tiny mistakes without even realizing there ever was a mistake. I just view at it as a result of us interpreting the meaning of everything we see against our worldview, which includes our idea of the context of what we are seeing... I.e. we don't really look carefully but rather fill in most of the information with our expectations.

 

Fortunately, I'm so unfamiliar with this math stuff that I don't know what to expect, and thus I am unable to overlook the mistakes and blunt shortcuts; I just stumble over immediately when something doesn't make sense.

 

I'm sure that that tendency to fill the gaps with our expectations has caused you a lot of grief in trying to explain your analysis to people who interpret it in terms of their idea of what "physics" is (as in seeing it as describing "real objects" as oppose to an excercise of defining what "object" is).

 

Anyhow;

 

...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”.

 

Right, that seems to correlate with what you just explained in the previous post...

 

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.

 

As does that.

 

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”.

 

Heh, exactly, and I guess you could call it that :D

 

There is this old little trick which you've probably seen.

 

You are supposed to count the o's in:

"THE MISITRY OF UNAUTHORISED

CAR-OWNMENTSHIP OF MINORS

IN THE STATE OF WEST-VERGINIA"

 

If you are fluent with english and just a tiny bit careless, you may count less than there actually is, as you tend to skip some of the "of" words as irrelevant. Anyone not familiar with english would have to look much more carefully and wouldn't "know" how to see that without the "of"'s.

 

I wouldn't call it all just senility, it's just that you are "skipping a lot of of's" and other bits and pieces that are irrelevant when a person already knows what you are trying to say.

 

Back to the OP:

 

...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 first thing I need to point out is that the position of the point, x'=0, being the origin of the primed coordinate system must be at x=vt in the unprimed coordinate system as that is the definition of the primed frame's movement in the unprimed coordinate system. That implies that [imath]\alpha(vt)-\beta t = 0[/imath]: i.e., that is exactly the transformation which yields the origin of the primed coordinate system which is, by definition x'=0.

 

Yup.

 

From that we can immediately deduce that [imath]vt=\frac{\beta}{\alpha}t[/imath] or, dividing by t, that [imath]v=\frac{\beta}{\alpha}[/imath]. This is first of those four equations we are looking for.

 

Yup.

 

uuup, gotta go, I'll continue from here soon...

 

-Anssi

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

 

Just to be sure, "v" (speed of primed coordinate system) is different from "v?" (the constant speed of the elements of the explanation)?

 

I.e. we are NOT considering the coordinate system to move at the exact speed of the surface of the expanding sphere. (I guess if we did, we would run into infinities at the final transformation process)

 

We now need to lay out three additional valid independent equations involving the unknown coefficients. We know that both coordinate systems must yield a spherical surface originally defined by [imath]x^2+y^2+z^2+\tau^2=v_?^2t^2[/imath]: i.e., that surface must transform exactly into the surface [imath]x'^2+y'^2+z'^2+\tau'^2=v_?^2t'^2[/imath] in the primed coordinate system.

 

So, in other words, we are looking for a transformation process from "unprimed view" to the "primed view", which only modifies the "x" and the "t" parameters in such a way as to maintain a constant velocity of the elements; i.e. in such a way that the "primed explanation" also yields a perfect sphere, while referring to the same elements as the unprimed explanation.

 

(I.e. we are examining the logical roots of Lorentz transformation)

 

Simply performing the transformation defined above must yield exactly that result. When we use the proposed transformations perform the transformation (substitute the explicit forms for each primed coordinate) we get the following relationship: [imath](\alpha x -\beta t)^2+y^2+z^2+\tau^2=v_?^2(\gamma t -\delta x)^2[/imath] which expands algebraically directly into

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

 

Check.

 

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

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

 

Actually struggled with that part a bit, perhaps you can show some of the algebraic steps that gets us there. I can see though that the point of this step was to move all the terms having to do with position (x') onto the left side, and all the bits having to do with t' on the right side.

 

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

 

So we now have four equations in four unknowns:

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

 

Right, substituting 1, 1 and 0 accordingly into the above equation would immediately yield [imath]x^2+y^2+z^2+\tau^2=v_?^2t^2[/imath], so we know they must amount to 1, 1, and 0... Pretty clever.

 

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

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

 

Check.

And I guess that middle one also answers my first question also; "v" is indeed different from "v?" (I did almost mix them up in my mind couple of times)

 

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

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

 

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

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

 

Hmm, I don't understand where that middle term comes from, I would have thought that at this point we'd have:

 

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

 

Either way, I'm struggling with trying to understand the next step:

 

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

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

 

I understand how to get the first term, but not the second one...

I think I'll pause here.

 

-Anssi

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Even though I'm not through the first section, I can just take few things on faith and already start with the definitions and construction of a clock...

 

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

 

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

 

...

 

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

 

...

 

 

...

 

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

 

Check.

 

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

 

Yup.

 

Since every event in the system described has non-negligible momentum only in the [imath](y,\tau)[/imath] plane, we can display all important dynamic phenomena while considering only a cross section in that plane. Thus let us examine our standard clock as it appears in that cross section, paying particular attention to the associated velocity vectors.

...

 

[

 

It is interesting to note that T, the period of our standard rest clock, is identical to 1/v? times the distance the mirror moves in the [imath]\tau[/imath] direction during one clock cycle.

 

So it is...

 

Although actual position in the [imath]\tau[/imath] direction is a meaningless concept (as the entire object is infinite and uniform in that direction), our standard clock appears to be measuring the implied displacement of the mirror over time in that direction: i.e., we can infer that the mirror has moved a distance 2L0 in the [imath]\tau[/imath] direction during one complete cycle.

 

Yup.

 

This will turn out to be a very significant fact since the scale of the [imath]\tau[/imath] dimension is set by the form of the fundamental equation (setting the scale of any dimension sets the scale of all the others).

 

Hmmm... I'm struggling with this bit. I can take it on faith, but perhaps you could expand on it little bit.

 

Now consider an identical standard clock in a moving reference frame: i.e., identical to the clock just described except for the fact that I will allow the momentum of the mirror assembly to be non negligible in the y direction.

...

 

 

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

 

All clear up to here.

 

 

Note that the length of the moving clock is shown to be L'. This has been done because we know that the symmetry discussed in the previous section must require the Lorentz contraction to be a valid on any macroscopic solution if interactions with the rest of the universe may be neglected (up to this point the model was scale invariant): i.e., when we solve the problem in the moving clocks system we want the length of the clock as seen by the observer in that moving frame to be L0.

 

So in this context, the length contraction is a consequence of any explanation having to accommodate for the fact that the speed of the probability waves must be isotropic across moving coordinate systems.

 

(And it is analogous to how relativistic length contraction is a direct consequence of having defined relativistic simultaneity, which is a direct consequence of having defined isotropic speed to light)

 

Btw, looking at your drawing, it's kind of interesting how simple it is to visualize L' through L0 and the angle [imath]\theta[/imath]... I guess someone used to math can see the same thing as easily from [imath]L'=L_0\sqrt{1-sin^2(\theta)}[/imath] :P

 

We use the scale freedom in our model to set that length (as seen from the rest system) to be L'; then and only then can we seriously call the clocks identical. This will require [imath]L'=L_0\sqrt{1-sin^2(\theta)}[/imath] (the inverse of the relativistic transformation deduced earlier:

 

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

 

i.e., in order to get the length of the moving clock in the primed coordinate system we have to multibly by [imath]\alpha[/imath]). Note that [imath]sin(\theta)[/imath] is exactly the apparent velocity of the moving clock divided by the velocity of the elemental entities, v?, which actually has nothing to do with time.

 

Hmmm, so it is...

[imath] v = sin(\theta)v_? [/imath]

[imath] sin(\theta) = \frac{v}{v_?} [/imath]

 

So in other words: [imath]L'=L_0\sqrt{1-sin^2(\theta)}[/imath], just like you said.

 

Still haven't walked through the first section of the OP, but have no troubles taking some bits on faith as it appears to be quite analogous to standard relativity.

 

Since all velocities are v?, it follows directly that d1 + d2 = S. Please note that everything so far is being graphed as seen in the frame of the rest clock: i.e., S=v?Tm, where Tm is the period of the moving clock as seen from the rest frame.

 

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

 

Once again, since the triangles A and B are identical as are the triangles a and b, we discover that one clock cycle, rather surprisingly, measures exactly the length of time it takes the mirror to move the distance 2L0 in the [imath]\tau[/imath] direction.

 

Now that took me a moment to dissect, and while I did find out that everything appears to be valid and without error, I walked through it in a reaaally topsy turvy way, and I really don't know in which order the logical steps should be carried out to end up with that graph.

 

Here's what I did to replicate your drawing:

 

- I laid down the mirrors in the (y,tau)-plane, setting [imath]L_0[/imath] between them.

- I drew velocity vectors for them (essentially [imath]v_?[/imath]) so that they were moving partially in y-direction (just like in your diagram)

- I set the distance between mirrors to L' (Got that by rotating the vector [imath]L_0[/imath] so that it was orthogonal to those velocity vectors)

 

That allowed me to see the triangle A, and I could draw the triangle B, and see the position you have marked as "reflection must occur at this point", without really understanding how you originally found it out.

 

But I did set up an animation with specific constant velocities, and found out that that is indeed exactly the position where the reflection occurs.

 

Also I could from that point on just draw those [imath]L_0[/imath] vectors along tau-axis, and also see the triangles a and b, and once again by animating the whole thing I found out that the second reflection indeed does occur exactly where you are saying it would, and everything overall lines up just like you drew them...

 

...but still I have no idea how did you originally get the magnitudes of d1 and d2 (I mean, how did you find out the the first reflection point etc.)... And how did you arrive to [imath]d_1 + d_2 = S[/imath]... I'm sure there is a better way to understand that than what I have in my mind right now.

 

btw, Arkain asked earlier if I could do some sorts of visualizations or animations that would be helpful in understanding this stuff, and I'm thinking this part might be something that should be relatively easy to visualize through animation... We'll see...

 

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

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

 

Which happens to be exactly the result expected from the standard Lorentz relation: i.e., the moving clock appears to run slow by exactly the factor [imath]\sqrt{1-sin^2(\theta)}[/imath].

 

Yes, so it appears...

 

I'll continue from here soon...

 

-Anssi

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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,
Somehow I get the impression that you are still missing the central issue. What we are talking about is a situation where we have an explanation of the universe as we see it. That means our explanation is not bothered by the fact that our reference frame (the coordinate system we use to represent our experiments) is moving with respect to the rest of the universe or not; the explanation includes the issues brought up by that circumstance. This has to do with absolutely any experiment we choose to examine. We choose to examine a few special circumstances because they point out, in detail, the problems with the old fashion Euclidean transformations used by Newton and, in fact, give us exactly what those transformations have to look like. The required transformations are a fact of life; your explanation must obey them and what kind of machinations you have to go through to achieve that is essentially beside the point.
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?
The “if we can construct objects then” does not need to be there. We don’t need the existence of “objects” (collections of elements which can be considered as universes unto themselves) in order to create a V(x,t) such that a single element will appear to approximately obey Newtonian mechanics. Given any collection of elements, it is always possible to imagine (or create fictional elements) such that we can see the impact of the rest of the universe (including those fictional elements) as causing an interaction of the form V(x,t). If we cannot create objects, then we cannot create rulers; this is another problem.
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.
I essentially agree with what you are saying except of the fact that I don’t feel that the existence of objects can be considered an axiom. We could certainly conceive of a “universe” without objects and even talk about some of the constraints on explaining that universe; it could even be a subset of our universe which had negligible impact upon that portion of the universe we deal with on a day to day routine. Say the inside of stars (or perhaps the interior of nuclei) or the structure of those great voids between the clusters of stars. We just couldn’t hypothesize rulers or clocks in such a realm.
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.
There is a big difference between “having to ignore” and “ignoring”. If a decent approximation can be achieved by “ignoring” then, to the same extent that the approximation is decent, the Lorenz transformations must be decent. And an object (which by definition presumes there is no impact upon its fundamental structure from the rest of the universe) requires the Lorenz transformation to be a central characteristic of any flaw free explanation. And, in order for that explanation to be flaw free, it cannot be changed by moving into one of those regions where objects can not exist. The explanation is a flaw-free explanation of “the universe” not just part of the universe.
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.
We cannot solve the general equation anyway so what difference does this make? What I am saying is that the elements of your explanation must obey that equation. I think what you are missing is the fact that most all common explanations are what is called “static”: i.e., none of the fundamental elements change in any way so obedience to the fundamental equation is a trivial issue. Just as the structure of my house obeys Newtonian mechanics: it just stands there without moving.

 

“Dynamic” explanations are a much more complex issue than are “static” explanations. In fact “physics” is perhaps the only field which makes much use of dynamic explanations of any kind at all. All the other fields are just far too unknown to even consider the existence of dynamic explanations. Chemistry and biology are now only beginning to think about dynamic explanation and those areas are often referred to as “physical chemistry” and “physical biology”. Physics sort of has ownership of dynamic explanations.

So that we must find a new way to define such things before we can answer such questions.
Well life is really not that complex; we have to think about what we want to know and how we might obtain it. Many many years ago, I was working on analyzing my fundamental equation (at a University) when an economist asked me what it had to do with economics. So I laid out the essentials of what economic knowledge was of interest and, using the variables as arguments of my fundamental equation, derived most of the common economic relationships taught in freshman economics. I gave it to him and failed to keep a copy for myself. I wish I had because some of it was rather interesting and might very well apply to situations where objects don't exist.
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?
I think it is the issue of the existence of “rulers” without which we couldn’t talk about “coordinate frames of reference”. Without “frames of reference” transformation between frames of reference is a pretty meaningless concept.

 

Have fun -- Dick

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"...identical to change in t divided by..." I suppose you meant to say.
Yes, you are absolutely correct.
And yeah, I know it's common human behavior that we start making mistakes when we are good enough with something, as we become careless and start to overlook tiny mistakes without even realizing there ever was a mistake.

 

...

 

I wouldn't call it all just senility, it's just that you are "skipping a lot of of's" and other bits and pieces that are irrelevant when a person already knows what you are trying to say.

On reading this whole section, I am moved to tell you about a book I bought in Golden, Colorado. I saw it in a “used book” store near where my wife was visiting a quilt store. (She is very much into quilting and, wherever we go, if there is a quilt store around, she wants to see it. New patterns and such.)

 

At any rate, the book I bought was “Consciousness Explained” by Daniel Dennett. I read the whole book in about three days. It was extremely interesting as he expressed a great number of very rational insights which had not occurred to me; but I don’t think he really “explained” consciousness for reasons I don’t have time to go into. I would very much recommend the book.

 

I googled Dennett and discovered two sites you might find interesting. They are videos which talk about some of the stuff covered in the book but are very limited as compared to the book itself.

 

I think he would find a lot of my work very applicable to some of the issues he brings up in his book.

(I.e. we are examining the logical roots of Lorentz transformation)
Absolutely correct!
I can see though that the point of this step was to move all the terms having to do with position (x') onto the left side, and all the bits having to do with t' on the right side.
Not quite accurate. The point was (now that we have eliminated the primed coordinates) to rearrange the terms such that we reproduced the correct form of the equation in the unprimed coordinates. Thus we must collect all the terms proportional to x2 on the left and all the terms proportional to v?t2 to the right. Where we go with the remaining terms (those proportional to 2xt; the only other terms) is actually immaterial as terms like 2xt do not exist in the equation in the unprimed coordinates. In detail, the algebra looks like this after one multiplies out the parenthesis on the right

[math]\alpha^2x^2-2\alpha x\beta t+\beta^2t^2+y^2+z^2+\tau^2=v_?^2\gamma^2t^2-v_?^22\gamma t\delta x+v_?^2\delta^2x^2[/math]

 

then rearranging the coefficient of each term, we have

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

 

Moving the x2 term on the right to the left and the t2 term on the left to the right (which changes the sign of both) we have

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

 

Moving the 2xt term on the left to the right and factoring out the appropriate terms this may be written

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

 

All that is left to do is divide the coefficient of the first term on the right by v? squared in order to factor that same term and one has

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

 

The rest you seem to have understood.

And I guess that middle one also answers my first question also; "v" is indeed different from "v?" (I did almost mix them up in my mind couple of times)
Context is often as important in mathematics as it is in any language. We tend to lack a sufficient number of symbols to make everything different (at least we don’t like to use that many).
Hmm, I don't understand where that middle term comes from, I would have thought that at this point we'd have:

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

 

You are absolutely right once again. That middle term just shouldn’t be there. When typing in the LaTex, I apparently retyped the coefficient. Anything to generate a little confusion right??? Sorry about that. I have edited out the error. Thank you again.
Either way, I'm struggling with trying to understand the next step:
Forgetting that added middle term is the most important thing here. Working from your (correct) expression, performing the indicated multiplication one has

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

 

In the first term, the denominator gets rid of the square of the [imath](1+v_?^2\delta^2)[/imath] term and the numerator just cancels out. In the second term, the [imath](1+v_?^2\delta^2)[/imath] term cancels out and, finally the right hand side consists of exactly what we are multiplying by.

 

It appears that, once again, your only confusion arises from my errors. Sorry about that. It is quite clear to me that you are the only one reading this stuff at all carefully.

 

Hmmm... I'm struggling with this bit. I can take it on faith, but perhaps you could expand on it little bit.
The requirement that the probability of seeing an element at a future time (when it started at a specific point and interactions with the rest of the universe can be ignored) must be an expanding sphere requires all scale of all dimentions to be established if one is established. That is all there is to it.
Now that took me a moment to dissect, and while I did find out that everything appears to be valid and without error, I walked through it in a reaaally topsy turvy way, and I really don't know in which order the logical steps should be carried out to end up with that graph.
Self consistency is self consistency. I am not bothered at all by the way you stepped through the problem. Maybe there were a few steps which presumed the correctness of the diagram but they could have been reached by other methods. The fact that the total is internally self consistent is the single most important issue.
...but still I have no idea how did you originally get the magnitudes of d1 and d2 (I mean, how did you find out the first reflection point etc.)... And how did you arrive to [imath]d_1 + d_2 = S[/imath]... I'm sure there is a better way to understand that than what I have in my mind right now.
Well, first of all, I am somewhat proficient at plane geometry: i.e., I pretty well know where these lines will intersect. The fact that d1+d2 has to equal S comes directly from the idea that time (as I have defined it) is an interaction parameter (time being the same is equivalent to “interactions can take place”). The pulse of photons (that would be the oscillator) interacts twice with the left hand mirror. Once when the experiment begins and again at the end of the experiment. Once again, presuming all other interactions can be ignored (and presuming that interaction with the right hand mirror face does nothing except to reverse the travel direction of the oscillator), the position of the mirror and the oscillator must change by exactly v?t where t is the specified interaction time. Since the elapsed interaction time for both elements must be the same and v? must be the same, d1+d2 has to equal S.
...btw, Arkain asked earlier if I could do some sorts of visualizations or animations that would be helpful in understanding this stuff, and I'm thinking this part might be something that should be relatively easy to visualize through animation... We'll see...
that would be quite nice.

 

You seem to be picking up on everything quite well. Thanks for your attention.

 

Have fun -- Dick

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Not quite accurate. The point was (now that we have eliminated the primed coordinates) to rearrange the terms such that we reproduced the correct form of the equation in the unprimed coordinates. Thus we must collect all the terms proportional to x2 on the left and all the terms proportional to v?t2 to the right. Where we go with the remaining terms (those proportional to 2xt; the only other terms) is actually immaterial as terms like 2xt do not exist in the equation in the unprimed coordinates. In detail, the algebra looks like this after one multiplies out the parenthesis on the right

[math]\alpha^2x^2-2\alpha x\beta t+\beta^2t^2+y^2+z^2+\tau^2=v_?^2\gamma^2t^2-v_?^22\gamma t\delta x+v_?\delta^2x^2[/math]

 

....

 

Okay, I was able to follow that whole thing now. (And spotted a small typo, the last [math]v_?[/math] isn't squared there, but it was correct in the subsequent steps)

 

Btw, I found a very handy LaTex editor, should make life easier (and less error-prone :)

LaTeX Equation Editor for the Internet

It's exactly what I was dreaming of earlier, you can just type in some LaTex code and it is attempting to render it all the time. Or, you can just click on the buttons above to create whatever you need to, and paste the code into your post.

 

It seems to have some clever features in that you can just paint parts of the code and press parenthesis or fraction button, and it will put the painted parts inside the added code.

 

You are absolutely right once again. That middle term just shouldn’t be there...

 

...Working from your (correct) expression, performing the indicated multiplication one has

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

 

In the first term, the denominator gets rid of the square of the [imath](1+v_?^2\delta^2)[/imath] term and the numerator just cancels out. In the second term, the [imath](1+v_?^2\delta^2)[/imath] term cancels out and, finally the right hand side consists of exactly what we are multiplying by.

 

Right, okay, that seems pretty trivial now.

 

It is quite clear to me that you are the only one reading this stuff at all carefully.

 

I was just thinking about the same... Hmmm... :I

 

The requirement that the probability of seeing an element at a future time (when it started at a specific point and interactions with the rest of the universe can be ignored) must be an expanding sphere requires all scale of all dimentions to be established if one is established. That is all there is to it.

 

Oh, right, of course.

 

Self consistency is self consistency. I am not bothered at all by the way you stepped through the problem. Maybe there were a few steps which presumed the correctness of the diagram but they could have been reached by other methods. The fact that the total is internally self consistent is the single most important issue.

 

Yeah, I was thinking about the same thing, and certainly I was presuming the correctness, otherwise I would not have known which steps to take. I'm quite confident it's valid, and part of the reason I asked about that was to get a better ideas for how to represent the issue as clearly as possible in an animation. Anyway, I think I know what I'll do now...

 

But first, back to OP, the Lorentz transformation part:

 

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

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

 

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

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

 

So that's all clear now.

 

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

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

 

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

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

 

That step was throwing me off a bit, but I finally realized that, if I just focused on the equation on the right, I could just take the square root of it:

[math]

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

[/math]

 

[math]

\sqrt{1-\left(\frac{v}{v_?}\right)^2}v_?\delta=\left(\frac{v}{v_?}\right)

[/math]

 

And move everything but the [imath]\delta[/imath] to the right side, to arrive at:

 

[math]

\delta= \left(\frac{v}{v_?}\right)\frac{1}{v_?\sqrt{1-\left(\frac{v}{v_?}\right)^2}}

[/math]

 

So, I guess I just don't know what you meant by "divide through by the coefficient of [imath]\delta[/imath]...", as when I tried that (however I interpreted it), I wasn't finding my way to your result.

 

Since [imath]\alpha^2=1-v_?^2\delta^2[/imath],

 

Oop, looks like another typo there at the OP. I think that should be: [imath]\alpha^2=1+v_?^2\delta^2[/imath]

 

we know that

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

 

Hmmm... Okay, this once again involves algebraic steps that I'm not familiar with but have to guess, so let me know whether this is valid route. Just substituting [imath]\delta[/imath] first:

 

[math]

\alpha^2=1+v_?^2 \left(\left(\frac{v}{v_?}\right)\frac{1}{v_?\sqrt{1-\left(\frac{v}{v_?}\right)^2}}\right)^2

[/math]

 

[math]

\alpha^2=1+v_?^2 \left(\frac{v}{v_?}\right)^2

\frac{1}{v_?^2\left[{1-\left(\frac{v}{v_?}\right)^2}\right]}

[/math]

 

[math]

\alpha^2=1+ \left(\frac{v}{v_?}\right)^2

\frac{1}{{1-\left(\frac{v}{v_?}\right)^2}}

[/math]

 

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

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

 

Okay, once again new stuff to me, but after some head scratching I think it goes little bit something like this:

 

[math]

\alpha^2=1+ \left(\frac{v}{v_?}\right)^2

\frac{1}{{1-\left(\frac{v}{v_?}\right)^2}}

[/math]

 

[math]

\alpha^2=

\frac{1}{1}+ \frac{\left(\frac{v}{v_?}\right)^2}{{1-\left(\frac{v}{v_?}\right)^2}}

[/math]

 

[math]

\alpha^2=\frac{1-\left(\frac{v}{v_?}\right)^2}{1-\left(\frac{v}{v_?}\right)^2}+ \frac{\left(\frac{v}{v_?}\right)^2}{{1-\left(\frac{v}{v_?}\right)^2}}

=

\frac{1-\left(\frac{v}{v_?}\right)^2 + \left(\frac{v}{v_?}\right)^2}{1-\left(\frac{v}{v_?}\right)^2}

=

\frac{1}{1-\left(\frac{v}{v_?}\right)^2}

[/math]

 

So:

 

[math]

\alpha=

\sqrt{\frac{1}{1-\left(\frac{v}{v_?}\right)^2}}

[/math]

 

And I suppose that can then be written:

 

[math]

\alpha=

\frac{1}{\sqrt{1-\left(\frac{v}{v_?}\right)^2}}

[/math]

 

 

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

 

Hmmm, I remember [imath]\beta=\alpha v[/imath], but I don't remember (nor spot from the OP) where did we establish[imath]v_?^2\delta=\alpha v[/imath]... Also could not figure out how you got the [imath]\gamma=\alpha \frac{\beta}{v_?^2\delta}[/imath]

 

Perhaps you could expand on that a bit...?

 

Phew, the Lorentz-part almost done already!

 

I'd have a few things to say to lurkers, about what all this implies, but I'll save it for little bit later... I hope these explicit baby-steps make it easier for all the lurkers to follow the thing though, and perhaps think about the implications themselves.

 

-Anssi

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Okay, I was able to follow that whole thing now. (And spotted a small typo, the last [math]v_?[/math] isn't squared there, but it was correct in the subsequent steps)
Fixed it! Thank you.
Btw, I found a very handy LaTex editor
I have bookmarked it and will examine it later; thanks a lot I always liked WYSIWYG type editors.
So, I guess I just don't know what you meant by "divide through by the coefficient of [imath]\delta[/imath]...", as when I tried that (however I interpreted it), I wasn't finding my way to your result.
When you got to
[math]

\sqrt{1-\left(\frac{v}{v_?}\right)^2}v_?\delta=\left(\frac{v}{v_?}\right)

[/math]

 

And [moved] everything but the [imath]\delta[/imath] to the right side, to arrive at:

 

[math]

\delta= \left(\frac{v}{v_?}\right)\frac{1}{v_?\sqrt{1-\left(\frac{v}{v_?}\right)^2}}

[/math]

You were dividing through (through the whole equation) by the coefficient of [imath]\delta[/imath] : i.e., the factor [imath]\sqrt{1-\left(\frac{v}{v_?}\right)^2}v_?[/imath].

 

You just did the square root first instead of after. Perhaps I confused you by my comment, "divide through by the coefficient of [imath]\delta[/imath]...", as I should have said "divide through by the coefficient of [imath]\delta^2[/imath]...".

Oop, looks like another typo there at the OP. I think that should be: [imath]\alpha^2=1+v_?^2\delta^2[/imath]
And once again I bow to your clarity of observation. I didn't realize just how sloppy my presentation was.
Hmmm... Okay, this once again involves algebraic steps that I'm not familiar with

...

And I suppose that can then be written:

 

[math]

\alpha=

\frac{1}{\sqrt{1-\left(\frac{v}{v_?}\right)^2}}

[/math]

The process you performed was exactly the process I had in mind when I said, “use 'common denominators' to add the two terms above”.
Hmmm, I remember [imath]\beta=\alpha v[/imath], but I don't remember (nor spot from the OP) where did we establish [imath]v_?^2\delta=\alpha v[/imath]...
You just proved to yourself (by that common denominator move) that

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

 

which implies (multiplying both sides by [imath]v_?^2[/imath] that

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

 

but we have also just shown that

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

 

Those two equations taken together imply that [imath]v_?^2\delta=\alpha v[/imath]. You're just not as familiar with seeing algebraic relationships as most physicists would be. We would notice immediately that both alpha and delta had exactly the same factor in common and differed only by the factor [imath]\frac{v}{v_?^2}[/imath].

Also could not figure out how you got the [imath]\gamma=\alpha \frac{\beta}{v_?^2\delta}[/imath]
If you look back at the OP I think you will find that is one of the original four equations we started with.
Phew, the Lorentz-part almost done already!
Unless you have more questions, I think we are done. :) I appreciate the time you have put into this very much. I really didn't realize how bad my presentation was until you took the trouble to try and understand it. :eek_big: I hope I can do a better job of presenting the GR deductions. The math there is nowhere near as complex as Einstein's Rieman geometry but neither is it "easy". The calculus required includes use of the Euler–Lagrange equation which is rather advanced mathematics.

 

I hope I can drag you through it. I have googled it and found no presentation which shows the history of its development (which was the way it was introduced to me). I am sorry but it seems to me that some of the aspects of classical mechanics are just not taught any more. Everyone seems to take the position that the standard modern presentation is the only rational presentation and that makes the subject hard to comprehend (in my opinion :shrug: ). Sometimes I think scientists just want to confuse people.

I'd have a few things to say to lurkers, about what all this implies, but I'll save it for little bit later... I hope these explicit baby-steps make it easier for all the lurkers to follow the thing though, and perhaps think about the implications themselves.
I am looking forward to those comments and I would like to hear the “lurker's” comments also. I am always wondering what is going on in their heads.

 

Have fun -- Dick

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Just a quick reply...

Oh btw, I started to comment on Dennet and Consciousness Explained, but it went totally off-topic so quickly that I decided to not post it here. Anyhow, have not read the book, but been meaning to many times.

 

You were dividing through (through the whole equation) by the coefficient of [imath]\delta[/imath] : i.e., the factor [imath]\sqrt{1-\left(\frac{v}{v_?}\right)^2}v_?[/imath].

 

You just did the square root first instead of after. Perhaps I confused you by my comment, "divide through by the coefficient of [imath]\delta[/imath]...", as I should have said "divide through by the coefficient of [imath]\delta^2[/imath]...".

 

Hehe, actually it was just another silly ambiguity that got me there. [imath]\delta[/imath] was also a coefficient by itself so I interpreted "divide through by the coefficient of [imath]\delta[/imath]" as "divide through by the coefficient [imath]\delta[/imath]", i.e. I attempted to divide through with [imath]\delta[/imath] itself. That didn't get me anywhere! :D

 

You just proved to yourself (by that common denominator move) that

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

 

which implies (multiplying both sides by [imath]v_?^2[/imath] that

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

 

but we have also just shown that

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

 

Those two equations taken together imply that [imath]v_?^2\delta=\alpha v[/imath].

 

Ah, I see it now.

 

If you look back at the OP I think you will find that [[imath]\gamma=\alpha \frac{\beta}{v_?^2\delta}[/imath]] is one of the original four equations we started with.

 

Ah, spotted it now :)

 

Back to OP:

 

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

 

So now I understand that bit.

 

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

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

 

Right, so transformation for x' is:

[math]x'=\alpha x -\beta t[/math]

 

And

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

[math]\beta=\alpha v[/math]

 

So clearly

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

 

I was struggling with t', and after looking at it for a while, it seems like it was due to an error earlier in the OP;

 

...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]

 

I think that should be [imath]t'=\gamma t -\delta x[/imath] in there. That's how it's used in the subsequent steps in the OP, including the final result (me thinks).

 

So, assuming I'm right:

[math]t'=\gamma t -\delta x[/math]

 

And

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

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

 

Then

[math]

t'= \frac{1}{\sqrt{1-\left(\frac{v}{v_?}\right)^2}} t - \left(\frac{v}{v_?}\right)\frac{1}{v_?\sqrt{1-\left(\frac{v}{v_?}\right)^2}} x

[/math]

 

And to tidy it up:

[math]

t'= \frac{1}{\sqrt{1-\left(\frac{v}{v_?}\right)^2}} t - \frac{1}{\sqrt{1-\left(\frac{v}{v_?}\right)^2}} \left(\frac{v}{v_?}\right) \frac{x}{v_?}

[/math]

 

[math]

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

[/math]

 

Which is your result. Yup, looks valid.

 

Just for convenience, one can define [imath]sine\theta = v/v_?[/imath] as this makes the square root in the above equations equal to [imath]cos(\theta)[/imath] yielding a simpler representation. If that constant velocity v? were to be c, those would become exactly the standard relativistic transformations.

 

Yup.

 

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

 

Yup, so we've drawn out - with explicit logical steps - that the transformation required to maintain a specific constant velocity between moving coordinate systems, must (unsurprisingly) affect the way that all the data is laid down in a given coordinate system. And that the transformation is, unsurprisingly, Lorentz transformation.

 

And from that I also know that if the data is plotted in a coordinate system where "t" is also expressed as an axis of its own, the transformation can be intuitively understood as a straightforward scale procedure.

 

And in this context the transformation is simply a requirement of the explanation to remain self-coherent when it's referring to the same data expressed, only expressed in different (moving) coordinate systems.

 

The fact that my model requires them for internal consistency implies that my model actually requires any conceivable universe to satisfy the relations associated with special relativity.

 

Yup, certainly looks that way.

 

Unless you have more questions, I think we are done. :)

 

I think I'll still walk through the bits about definition of simultaneity carefully, albeit the result there seems to me to be a very obvious consequence of everything we've talked thus far. But at least I might spot some more typos :)

 

Also I'll do those clarifying animations for all the lurkers, and have a stab at explaining what all this means as it appears there is quite a bit of confusion there still, and I think I know where that confusion is (people try to view this result as if it says something about the ontological reality of nature, as otherwise they don't understand how could any result be meaningful)

 

I appreciate the time you have put into this very much. I really didn't realize how bad my presentation was until you took the trouble to try and understand it. :eek_big: I hope I can do a better job of presenting the GR deductions. The math there is nowhere near as complex as Einstein's Rieman geometry but neither is it "easy". The calculus required includes use of the Euler–Lagrange equation which is rather advanced mathematics.

 

Okay, as it is with SR, I'm not really familiar with the math of GR. I just look at it in a kind of "intuitive" way, imagining a spacetime transformation in my mind. But I'm sure if I'm careful enough and ask enough questions, I can walk through it as well.

 

I hope I can drag you through it.

 

Certainly. A new thread?

 

-Anssi

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Oh btw, I started to comment on Dennet and Consciousness Explained, but it went totally off-topic so quickly that I decided to not post it here. Anyhow, have not read the book, but been meaning to many times.
There is one question I would like to ask Dr. Dennet regarding “Consciousness Explained”. Since he claims a supposed explanation of Consciousness, he should be able to answer any question regarding the presence or absence of consciousness. The question I have in mind is, “is one conscious when one is dreaming?” I think the answer to that question could be interesting.
I was struggling with t', and after looking at it for a while, it seems like it was due to an error earlier in the OP;

 

I think that should be [imath]t'=\gamma t -\delta x[/imath] in there. That's how it's used in the subsequent steps in the OP, including the final result (me thinks).

And you are right on the money again. I have looked at my original notes (the work from which I essentially copied the OP) and all of the errors you found are actually errors in my effort to copy and are correctly expressed in the original. At least it is nice to know that my original is not that contaminated with such off the wall errors. Thanks again (and I edited the OP earlier this morning when I first looked at your post).
I think I'll still walk through the bits about definition of simultaneity carefully, albeit the result there seems to me to be a very obvious consequence of everything we've talked thus far. But at least I might spot some more typos :)
I can only hope you don't. It is quite clear to me at this point that, when it comes to the kind of errors you have found, I can't see the woods for the trees. I guess, when I proof read these things, all I do is just scan the stuff. I do find a lot of errors on my own you know. In fact, I find quite a few on first reading.
Okay, as it is with SR, I'm not really familiar with the math of GR. I just look at it in a kind of "intuitive" way, imagining a spacetime transformation in my mind. But I'm sure if I'm careful enough and ask enough questions, I can walk through it as well.
I have been thinking about the approach by which the Euler-Lagrange equation was presented to me and I don't know if I can remember the details. It was a long time ago and, as I remember it, not taken from a book; at least not from any book I have a copy of (my math professor was educated back before quantum and relativity were really serious physics subjects and I suspect his view was closer to the point of view more common in the late 1800's). So I may just have to do a little hand waving there. We will see what I can come up with when I get into it.
Certainly. A new thread?
Yes, but a tad down the line. Before we go into GR, I would like to show you my deduction of Dirac's equation. That deduction is quite straightforward and answers a few interesting questions mostly avoided by conventional modern physics. I will try to work up my first post this weekend.

 

Have fun -- Dick

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These animations should make it fairly easy to comprehend the diagrams (and they will get embedded to the OP too):

 

 

Click on the video to get to YouTube for high resolution version

YouTube - Presentation of a stationary clock http://www.youtube.com/watch?v=jbNqSnVgUZs

 

 

As is explained in the OP, this is simply a representation of a clock from its own rest frame. I.e. rest frame in terms of x,y,z-directions (tau corresponds to the definition of mass and gets integrated out).

 

 

 

 

 

 

 

 

Click on the video to get to YouTube for high resolution version

YouTube - Presentation of a moving clock http://www.youtube.com/watch?v=-UDrWCIgmTk

 

The same clock shown from a different frame, after exactly the kind of transformation that is required for the picture to remain self-coherent.

 

I will try and provide some more clarification on this issue soon.

 

-Anssi

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Somehow I get the impression that you are still missing the central issue. What we are talking about is a situation where we have an explanation of the universe as we see it. That means our explanation is not bothered by the fact that our reference frame (the coordinate system we use to represent our experiments) is moving with respect to the rest of the universe or not; the explanation includes the issues brought up by that circumstance. This has to do with absolutely any experiment we choose to examine. We choose to examine a few special circumstances because they point out, in detail, the problems with the old fashion Euclidean transformations used by Newton and, in fact, give us exactly what those transformations have to look like. The required transformations are a fact of life; your explanation must obey them and what kind of machinations you have to go through to achieve that is essentially beside the point.

 

So the requirement that the Lorenz transformation is part of our explanation is a consequence of being able to construct an object in our explanation where we have defined an object to be a collection of elements that maintain their orientation over changes in t. Would this be equivalent to saying that an observer at rest with an object will always use the same explanation even after they both have accelerated?

 

As a consequence of defining an object like this, if we take an object and then transform all of the elements that it is composed of so that it is in a moving frame (we accelerate the object for instance) it will appear to contract according to the Lorenz transformation. What seems more important to me is that if we suppose that an observer measures the object while at rest with it and then accelerates with it, since anything that might change the length of the object he is measuring will effect his ruler as well as anything else he might measure, he will still consider the object to have the same measurements.

 

The actual requirements, though, for a observer at rest with something not necessarily an object, to agree on the measurements before and after an acceleration, and for the object to be Lorenz contracted for a observer that remains at rest and doesn’t accelerate with it is not that it is an object but rather that the construct is unaffected by the rest of the universe. That is, we can still use the same explanation of the construct for an observer that remains at rest with it. This can be accomplished by being able to consider the construct separately of the universe that is of any influences resulting from the Dirac delta function, effecting the construct outside of it can be ignored. This is just saying though, that there is no preferred reference frame in which the fundamental equation is valid in, which is not necessarily a true statement about any explanation but for the time being is considered a useful approximation as it shows that the Lorenz transformation is a consequence of being able to explain things with the use of objects.

 

The “if we can construct objects then” does not need to be there. We don’t need the existence of “objects” (collections of elements which can be considered as universes unto themselves) in order to create a V(x,t) such that a single element will appear to approximately obey Newtonian mechanics. Given any collection of elements, it is always possible to imagine (or create fictional elements) such that we can see the impact of the rest of the universe (including those fictional elements) as causing an interaction of the form V(x,t). If we cannot create objects, then we cannot create rulers; this is another problem.

 

Then is it even possible to create a V(x,t) such that the elements won’t appear to obey Newtonian mechanics? Also, if you are saying that even if we cannot construct objects that Newtonian mechanics will be an approximation to how elements behave I can’t see how we could use Newtonian mechanics and so call it an approximation without the use of a clock which so far requires the use of objects to construct.

 

I essentially agree with what you are saying except of the fact that I don’t feel that the existence of objects can be considered an axiom. We could certainly conceive of a “universe” without objects and even talk about some of the constraints on explaining that universe; it could even be a subset of our universe which had negligible impact upon that portion of the universe we deal with on a day to day routine. Say the inside of stars (or perhaps the interior of nuclei) or the structure of those great voids between the clusters of stars. We just couldn’t hypothesize rulers or clocks in such a realm.

 

It seams clear that such systems will appear to be Lorenz contracted from an outside prospective like a ship going by at relativistic speeds. But in considering possible explanations of such objects, what will happen to the expanding sphere? I know that the fundamental equation is still the equation of an expanding wave, we just can‘t say that it is expanding at a constant speed because we can‘t define the distance that it has expanded or how long it has expanded for.

 

We cannot solve the general equation anyway so what difference does this make? What I am saying is that the elements of your explanation must obey that equation. I think what you are missing is the fact that most all common explanations are what is called “static”: i.e., none of the fundamental elements change in any way so obedience to the fundamental equation is a trivial issue. Just as the structure of my house obeys Newtonian mechanics: it just stands there without moving.

 

But isn’t the only thing that is important is that there is some way that we can map the behavior of the elements in however we are explaining them into a solution to the fundamental equation? How we are explaining them and what we are explaining is of no importance as long as such a mapping exists.

 

I think it is the issue of the existence of “rulers” without which we couldn’t talk about “coordinate frames of reference”. Without “frames of reference” transformation between frames of reference is a pretty meaningless concept.

 

Is there a way though, to define a coordinate system without the use of objects that we can use to define distance? Isn’t it possible that we may be able to use some sort of consistent repeating behavior of elements to set up a coordinate system even if it isn‘t what we would ordinarily consider to be a coordinate system?

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There is one question I would like to ask Dr. Dennet regarding “Consciousness Explained”. Since he claims a supposed explanation of Consciousness, he should be able to answer any question regarding the presence or absence of consciousness. The question I have in mind is, “is one conscious when one is dreaming?” I think the answer to that question could be interesting.

 

Why, what do you have in mind?

 

I personally wouldn't know what to answer, as it's kind of a matter of what one supposes "being conscious" means.

 

But, running with the idea that dreams are a product of the brain re-ordering information, and looking for new self-coherent connections between ideas (which would explain why in dreams there often exists rather strange combinations of ideas, why dreams often include things that have been in our mind recently, why we sometimes come up with a solution to a difficult problem in our dreams, and why good night sleep seems to aid learning), then I think it's to be expected that sometimes that ordering doesn't include the idea of "self" at all, and thus cannot really be remembered later (it's not "something that happened to me"), and can't really be considered a conscious experience.

 

On the other hand, maybe one doesn't want to call those instances "dreams" at all, in which case a dream is always something consciously experienced.

 

Of course I understand some people associated "being conscious" with the everyday idea of being awake...

 

Anyway, I read the simultaneity stuff from the OP and didn't spot any errors. Albeit I did read it little bit less carefully than the rest, because that relationship between the definition of simultaneity and geometry seems fairly obvious to me.

 

At this point, it is very important to examine the reverse case...

 

...

 

...Thus it is that he will hold forth that his clock is correct and the (so called) rest clock appears to be contracted by exactly the standard Lorentz contraction. The only reason I went through that, was because many people have difficulty comprehending how the moving observer (who' entire perspective appears to be Lorentz contracted) can see the rest observer as Lorentz contracted. (I strongly suspect that many trained physicists can't see it either; they are just throughly indoctrinated in conventional relativity. But that is only an opinion :lol:)

 

Well I have not heard any physicist talking about that relationship explicitly either, but then where would I...? The dumbed down presentations are so silly they make my head hurt, and I always end up thinking "someone needs to be fired for this..."

 

Incidentally, as I'm writing this there's a History Channel program "The Universe" on, and it's an episode about space travel. They are bringing up the speed limit of light speed as per relativity, and sure enough, they have once again dumbed it down so much as to make completely false assertions. Here's a direct 1:1 quote straight from Mr. Neil deGrasse Tyson (PhD in Astrophysics)

 

"Light's fast, but the universe is huge. So even if we could ride on a beam of light, if we wanted to cross the galaxy - the way they do it on the science fiction programs - it would take a hundred thousand years to do it".

 

And the narrator continues:

"And even at light speed, traveling to the nearest galaxy, would take several million years"

 

And then of course they had a computer animation where "a beam of light" is travelling "really fast through space"...

 

It is really really really beyond me, how any self-respecting scientist - who supposedly understand what relativistic speed limit means - can spew something like that from their mouth... Is it possible, that some of them don't really understand what it means? That they just heard somewhere that C is the speed limit and run that against their newtonian worldview without understanding what it means and where it comes from? Because if they understood it, surely they'd see it doesn't really play down the way they talk about it... Is that it? I'm quite puzzled. And to be honest, a bit annoyed.

 

How common is that misconception then? How many people reading this thinks what they are saying is valid, according to relativity? And why?

 

Well, then they go on to talk about space warps in terms of static spacetime, and about tachyons in a really stupid and incoherent manner (They seem to think tachyon simply refers to an "infinitely fast" object or something). I can't go on because I'm getting too depressed over that, so back to the topic :)

 

The only reason I find that relationship between simultaneity and geometry so obvious is because I understood it myself via visualizing Lorentz-transformation in my head. I have ran into many people who "know" relativity but did not understand it enough to see that relationship for some reason. I suspected it was just because they were somewhat casually interested of the subject and had never thought about it much.

 

At any rate, it should be quite obvious that if you set about to measure the length of a box, you are measuring "where the extends of the box are simultaneously". Change your idea of simultaneity, and you change your idea of the geometry of the box.

 

Anyway, let me walk through the final part of the OP still, maybe we'll uncover some typos...

 

Finally, in order for the moving observer to measure the period of the rest clock, he must receive two signals...

 

...

 

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

 

Yup

 

or, solving for S,

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

 

While I was able to understand that that expressions is true, I do not know what the algebraic steps are in between there. It's just once again my unfamiliarity with math, I'm sure it's something really simple :P

 

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

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

 

Yup.

 

I'll continue from here soon...

 

-Anssi

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So the requirement that the Lorenz transformation is part of our explanation is a consequence of being able to construct an object in our explanation where we have defined an object to be a collection of elements that maintain their orientation over changes in t. Would this be equivalent to saying that an observer at rest with an object will always use the same explanation even after they both have accelerated?
Your use of the word “orientation” here bothers me! An object was defined to be a collection of elements which could be considered an entity unto itself for sufficient time to examine or use that fact. The only orientation of interest is its momentum direction in the four dimensional space being used here. No “orientation” is being maintained; the only thing being maintained is the fact that the elements making up that object have to remain in the vicinity of one another. Since they are all moving at the same speed (v?) that fact has important logical consequences.
... he will still consider the object to have the same measurements.
Well, of course, that is exactly the phenomena under examination.
... but for the time being is considered a useful approximation as it shows that the Lorenz transformation is a consequence of being able to explain things with the use of objects.
Ok.
But in considering possible explanations of such objects, what will happen to the expanding sphere?
Somehow I don't think you are following the deduction at all. If you followed what I was doing, you wouldn't ask such a question.
I know that the fundamental equation is still the equation of an expanding wave, we just can‘t say that it is expanding at a constant speed because we can‘t define the distance that it has expanded or how long it has expanded for.
Why don't we just drop these issues for the time being as they have nothing to do with the problem presented in this thread.
But isn’t the only thing that is important is that there is some way that we can map the behavior of the elements in however we are explaining them into a solution to the fundamental equation? How we are explaining them and what we are explaining is of no importance as long as such a mapping exists.
Ok.
Is there a way though, to define a coordinate system without the use of objects that we can use to define distance?
Again, I don't think this has anything to do with what this thread is talking about. You are trying to discuss things far beyond the issues presently being argued here.
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Since I've carefully followed the earlier steps of the analysis up to this point, I can see quite clearly what this thread is about, and I thought I should try and clarify this thread for all the interested lurkers out there.

 

First of all, this is all part of an epistemological analysis, meaning it is not supposed to uncover any sort of ontological reality. Instead, it is shown how relativistic time relationships arise from circumstances that are necessary for any world model that conceives "unknown data patterns" in terms of "persistent objects", as long as the model preserves self-coherence very carefully every step of the way.

 

Issues discussed earlier in different threads:

 

Any world model needs to define objects, based on some unknown data patterns (it needs to be known "what constitutes an object"). In the earlier thread (Deriving Schrödinger Equation), it was shown that while many valid sets (of defined objects) would always exist for any finite amount of data, any self-coherent set of defined objects will end up obeying quantum mechanical relationships. Or another way to put it, it was shown that quantum mechanical relationships are already embedded in the original symmetry requirements, and will be a feature of a self-coherent worldview regardless of the content of the raw data patterns (It is simply a matter of ordering the information in a specific way).

 

(Note; the idea that some "ontological identity" exists behind the raw data patterns is always an open question, and probably easiest to just suppose there aren't any ontological identities out there. The idea of "persistent identity" is clearly something required by a mental conception of reality, but I would not expect the actual reality to require it)

 

Once you have a world model, that defines objects, that obey quantum mechanical relationships (which newtonian mechanics are an approximation of), you already hold a perspective where there are "objects" that are persistent through time, that are moving around in a coordinate system, and that are interacting upon touch.

 

If you have no patience to follow the analysis up to that point, you can still follow this thread if you just let yourself take the above on faith for time being.

 

This thread is explaining exactly how, and exactly why, those defined objects must obey relativistic time relationships, as a result of carefully preserving self-coherence every step of the way. I.e. how relativity is not a feature of ontological reality, but instead spring from the way we classify and order data through our world models.

 

About the presentation form:

 

Note, that [imath]x,y,z,\tau[/imath] parameters, and all the other definitions related to the fundamental equation, are there for the sole purpose of being able to investigate relationships between things that we have defined in our head.

 

I.e. that the analysis is using a tau dimension to communicate this issue, is of little significance here. It does NOT mean that there exists an ontologically real tau dimension.

 

It does not even mean, that there literally exists a tau dimension as such in our minds. The presentation form (of [imath]x,y,z,\tau[/imath]) is not important, what is important is the exposed relationships between defined things; While we do not consciously think about the world in terms of a tau dimension, the timewise relationships that are expressed in the [imath]x,y,z,\tau[/imath] form, are necessary for any self-coherent world model; i.e. these same relationships are always to be found in one form or another (as long as that self-coherence is carefully preserved).

 

Onto the relativistic relationships

 

First note that the relationships that were exposed before this thread, were expressed in terms of a coordinate system that is at rest with the entire set of defined entities. I.e. the probability function of that form would behave exactly correctly ONLY in that one coordinate system.

 

Then it was shown what sort of transformation between coordinate systems (that are moving within each others) must be performed for the world model to remain self-coherent. It was shown why the transformation must be Lorentz-transformation (to [imath]x,y,z,\tau[/imath] space), as otherwise the new perspective would not map properly over the same raw data anymore. Don't take this as consequential to the chosen presentation form, as it is more properly a consequence of the original symmetry arguments, and the same relationship could certainly be expressed in many different forms. (Note; this is not exactly how Lorentz transformation is used in relativity, I can talk about the difference separately if it's not already obvious)

 

Next, note that the relationships exposed before this thread, were shown to map perfectly onto the relationships given by modern physics (often taken as features of ontological reality as oppose to features of our world model), which yields us a way to see exactly how does a defined macroscopic object such as "a clock" map into this picture (of [imath]x,y,z,\tau[/imath] space).

 

When that clock assembly is expressed in terms of the [imath]x,y,z,\tau[/imath] space, we can show the logical consequences of that self-coherent transformation between moving coordinate systems. It is shown, that the dynamic behaviour of the clock absolutely must be plotted in a way where the supposed geometry and the dynamics (incl. the observable cycle count) of the construction must exhibit relativistic behaviour as a function of the chosen coordinate system.

 

In other words, the reasons for that relativistic behaviour are entirely epistemological, as oppose to being a feature of the ontological reality itself.

 

About simultaneity and standard perspective of relativity

 

I think the real significance of this result shows up when we consider the concept of "simultaneity" and how it's historically been treated.

 

Let me start from the beginning of this whole conundrum, just in case someone doesn't know.

 

One-way speed of information

 

First, there is no way for you to tell "when" some distant event occurred, unless you know how fast the information about that event reached yourself.

 

Second, there is no way to measure the speed of information from one location to another, unless you have placed synchronized clocks in those locations.

 

Third, there's no way to tell if two spatially separated clocks are synchronized, unless you know how fast the information reached you from both of them. -> You are back to square one.

 

The idea of synchronizing the clocks in one place and then moving them to their respective locations, is invalidated once you realize that the dynamic behaviour of those clocks is governed by the same phenomena that we are set to measure. I.e. we will not know how the clocks are affected by us moving them until we know the one-way speed of information.

 

It follows from those facts that there is no way to measure one-way speed of information. You can only measure two-way speed of information (information starting its propagation from the same clock where it ends up).

 

And it follows that, to consider two spatially separated events as simultaneous, is to make an assumption about how fast the information about those events reached you.

 

Now, the epistemological analysis displays in very explicit manner this exact same problem. One-way speed of information is completely hidden from the view, in the sense that two observers can completely disagree with the one-way speed, and still map the same raw data in their respective coordinate systems. It displays exactly how and why we are free to make any assumption about the one-way speed (as long as the two-way speed remains unchanged) without generating any observable predictions.

 

In other words, each observer is entirely free to plot the data in their personal coordinate system in a manner where the speed of the information is considered equal in all directions against themselves. Of course when they do that, they also define which events they consider as simultaneous.

 

Of course, in standard relativity, that is exactly the assumption that is made by the postulate that the speed of light is isotropic across inertial frames, and people tend to assign varying degrees of ontology to this postulate.

 

Notice how, it is one thing to say "speed of light is the same for all observers", and another thing completely to say "all macroscopic objects measure the speed of light as the same". The latter assertion has to do exactly to the fact that "all clocks are defined macroscopic objects governed by exactly the same dynamics we are set to measure" (which is a very important issue, as here - under a careful analysis - it yields an explanation to time dilation measurements)

 

If you instead take the first assertion as literally true, then when being asked the question "what is happening in Alpha Centauri right now?", your answer of course depends entirely on the coordinate system you happen to choose for your answer.

 

Imagine that mankind had scattered all across the universe, and everyone were supposed to throw a party whenever it is the New Year's Eve on earth. If each space colony were to throw the party whenever they figure is appropriate according to their coordinate system, then hardly no one would be throwing the party at the same time (in terms of ANY coordinate system).

 

You would be having a party onboard of your ship, while another ship passing you would figure it's still 2 weeks until the party should be thrown. Of course, if they are planning to change their direction drastically within the next 2 weeks, they might be facing quite a difficult decision...

 

So, is the speed of light ontologically isotropic? I.e. does the state of reality actually change as a function of which coordinate system you happened to choose to describe it? I have some doubts :)

 

I expect everything above to be abundantly obvious to anyone who understands relativity, and I suppose that is why you often hear physicists say that relativity is just a handy convention, without dwelling more on the subject of ontology. Certainly you can build many sorts of ontological interpretations (like static spacetime), and I guess some people choose to view it simply as an expression of realistic dynamics, intentionally leaving the question of underlying ontology unanswered.

 

Yet, I'm sure everyone wonders to some extent, what does the validity of relativity actually imply about reality? Everybody does assign some level of ontology to some aspects of relativity, and certainly until now, it has not been trivial to answer the question "where does observable time dilation come from if not from isotropic speed of light?".

 

Notice that the epistemological analysis explains that issue, without saying anything about ontological simultaneity. It is entirely possible that "real simultaneity" is absolute, relativistic, ontologically meaningless, bubbly, or anything at all. What is significant is that relativistic time dilation is found to be a logically unavoidable consequence of "self-coherent object definition", "macroscopic clocks", and "the need to express the same reality in terms of different coordinate systems". It occurs in your head, without ANY ties to what sort of simultaneity exists ontologically.

 

The title of this thread could perhaps more properly be "Relativity demystified", as it, in my opinion, explains completely why relativistic world conception is valid, without adding any mysterious implications to reality.

 

And it should be clear at this point, that supposing an ontological reality of spacetime is also a case of taking the relativistic concepts way too literally. Likewise, you will often notice that physicists are using the idea that the one-way speed of light literally is C in all inertial frames, in their further musings. For instance, a violation of locality - meaning superluminal speed of information - is considered to destroy causality. But that is true IF and only if you actually assume each observer has got ontologically different simultaneity -> only if you suppose reality really does care about your chosen coordinate system in some sense (via static spacetime or something else).

 

Whereas the epistemological analysis shows explicitly that the supposed simultaneity of each observer is simply an idea in their head, which hinges completely on their assumption about the one-way speed of light. That opens up quite a few doors that are normally thought to violate relativity.

 

I do think it would be a valuable thing for anyone interested in relativity to understand this analysis. It might take time, but it is quite illuminating, I can assure you. Think about it.

 

-Anssi

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I personally wouldn't know what to answer, as it's kind of a matter of what one supposes "being conscious" means.
My point exactly! Primitive outlooks have almost always included dreams as being as real as what we call reality. Certainly, in my dreams, I feel as if I am conscious; how is it that I remember what happened if that were not the case? And Dennet's definition of consciousness seems to me to include dreaming. The issue is, what philosophical position is appropriate here.
Of course I understand some people associated "being conscious" with the everyday idea of being awake...
The reason I brought the question up.
Incidentally, as I'm writing this there's a History Channel program "The Universe" on, and it's an episode about space travel. They are bringing up the speed limit of light speed as per relativity, and sure enough, they have once again dumbed it down so much as to make completely false assertions. Here's a direct 1:1 quote straight from Mr. Neil deGrasse Tyson (PhD in Astrophysics)

 

"Light's fast, but the universe is huge. So even if we could ride on a beam of light, if we wanted to cross the galaxy - the way they do it on the science fiction programs - it would take a hundred thousand years to do it".

It's all in your perspective. If you could offset the effects of acceleration (some kind of internal force field local to the ships cabin which would accelerate every element in your local environment at exactly the same rate), the ship could attain astounding accelerations. Now of course that is a science fiction idea but it isn't one which violates any law of physics except perhaps the energy problem which is not trivial. Let us suppose we have some way of using the free matter in space to obtain this energy (if we are moving fast enough, the amount of material we intercept could be quite large).

 

So let us say any desired acceleration is possible. If that is the case, we can go anyplace in the universe in as little time as we desire (insofar as ship time is concerned). Using a “earth rest frame of reference map of the universe” together with a ship time clock for time measurements, we can achieve any velocity right up to infinity and, strange as it might seem, we can use Newtonian physics to plot our course through the universe. That is an interesting consequence of using that mixed coordinate system. I suspect, if humans ever do achieve interstellar travel, they will use my geometry for plotting their course; it is much simpler than using Einstein's picture. For any physicists reading this note that one “g” acceleration (in the mixed coordinate system) is not one “g” acceleration in the ships frame so the standard relativistic comparisons are not valid. (I just pointed that out because I know you will do the relativistic calculations and point out my answer is different).

 

But back to the reason I mentioned this case. It is much easier to see rapid travel as “forcing” one into the future instead of in terms of a limiting velocity. For example, Alpha-Centauri is roughly four light years away. It is somewhat surprising that one “g” (32 feet per second per second) is almost exactly equal to one light year per year per year. So if we accelerated off towards Alpha-Centauri at one “g” (as measured on our mixed coordinate system) until we got half way there (two light years) [imath]t=\sqrt{\frac{2d}{g}}[/imath] it would take two years (ship time). We could then de-accelerate for another two years and arrive at rest at Alpha-Centauri. We could spend what time we needed there and then return to earth in another four years (ship time). So we would say the round trip was eight years long. How much time would pass on earth?

 

The answer is quite simple: our actual path in my space would be four segments, each two light years in the x direction plus two light years in the tau direction, c being one light year per year. Since these directions are orthogonal to each other, our actual total distance of travel would be [imath]4\sqrt{2^2+2^2}=8\sqrt{2}=11.31[/imath] light years. So the earth observers would say the trip took roughly eleven years and four months. That is, we could see ourselves as being forced into the future a distance of roughly three years and four months.

 

Suppose we wanted to go to the other side of the galaxy, some 200,000 light years away. Half way would be 100,000 light years; [imath]t=\sqrt{200,000}[/imath]=447 years so, at one “g” it would take roughly 1,800 years ship time for the round trip. At four g's (something the crew could probably get used to) we could do the round trip in roughly a little over 900 years (ship time). But how long would we be gone, earth time? [imath]t=4\sqrt{100,000^2+225^2}[/imath] which is roughly 410,000 years. Just a tad short (by a mere 10,000 years) of the apparent speed of light. In this case we have been pushed about 409,100 years into the future. But what will the crew say their velocity was? They went some four hundred thousand light years in a little over 900 years. That would be almost 450 light years per year.

 

It's all how you look at these things. And I also get annoyed with professional physicists.

Well, then they go on to talk about space warps in terms of static spacetime, and about tachyons in a really stupid and incoherent manner (They seem to think tachyon simply refers to an "infinitely fast" object or something). I can't go on because I'm getting too depressed over that, so back to the topic :)
You should notice that “tachyons” don't exist in my representation. “Apparent” velocities in excess of light are just plain -???- “not possible” -???- (see the above analysis). That is one reason I do not like Einstein's picture: it suggests the existence of phenomena which violate the logic of his own construct. A very poor characteristic for any theory to contain.
While I was able to understand that that expressions is true, I do not know what the algebraic steps are in between there. It's just once again my unfamiliarity with math, I'm sure it's something really simple :P
We started with the expression

[math]S=2L_0+Ssin(\theta)[/math]

 

subtracting [imath]Ssin(\theta)[/imath] from both sides, we obtain

[math]S -Ssin(\theta)=S(1-sin(\theta))=2L_0[/math]

 

and then dividing by [imath]1-sin(\theta)[/imath] one obtains

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

 

And that should take you to the end of the OP. I just read your latest post and agree with most all of it; however, I don't think it would be possible to chart a path which would allow you to party all the time by accelerating your ship in a manner where it would be the New Year's Eve on earth all the time! The general relativistic transformation has to yield a stopped clock on the earth as seen from your ship! That, I think might require an infinite acceleration. But maybe you could get far enough away from the earth such that there was enough mass between you and the earth to yield a black hole solution. :shrug:

 

Have fun -- Dick

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It's all in your perspective. If you could offset the effects of acceleration (some kind of internal force field local to the ships cabin which would accelerate every element in your local environment at exactly the same rate), the ship could attain astounding accelerations. Now of course that is a science fiction idea but it isn't one which violates any law of physics except perhaps the energy problem which is not trivial. Let us suppose we have some way of using the free matter in space to obtain this energy (if we are moving fast enough, the amount of material we intercept could be quite large).

 

So let us say any desired acceleration is possible. If that is the case, we can go anyplace in the universe in as little time as we desire (insofar as ship time is concerned).

 

Yeah, and what bothers me is that they talk about relativity and the relativistic speed limit as if it was some sort of speed limit against some metaphysical space, and back that idea up with a computer animation which gives everyone exactly the idea of racing through space really fast alongside a beam of light (i.e. they give the idea that at light speed, light is at rest with you). Certainly someone might argue that they were talking about the travel taking this and this long in earth's frame, just to simplify things or whatever, but I really don't think that sort of simplification helps in communicating what relativistic speed limit means. It only serves to confuse people.

 

I don't mind them talking about special relativistic view only, i.e. ignoring gravitational effects or energy questions for simplicity, as you can just imagine that you shift the perspective between different space ships as they pass each others (they can certainly communicate when they pass, so this just serves as a means to discuss our ideas of relativistic ontologies).

 

So, for all casual lurkers who have fell prey to exactly these sorts of assertions about relativistic speed limit, let it be said that relativity certainly allows for a spaceship to get from Earth to Alpha Centauri in any arbitrarily short period of time, even in 1 second (we can allow them to gain speed before passing the start line on earth) according to its own perspective (its own clocks and its own idea about the distance of the travel)

 

What the relativistic speed limit more properly means, is that no given object can move faster than C in terms of any inertial frame. That is a consequence of Lorentz transformation being valid (which is a scale-like procedure to the apparent speeds, as oppose to a velocity addition procedure). I.e. in terms of Earth's frame, the clocks on space ship would be said to be at almost standstill. In ship's frame, the distance between Earth and Alpha Centauri would be said to be shrunk into some tiny distance)

 

The point is, it is not a speed limit against any "space", it is a speed limit against any chosen coordinate system. For anyone who has followed this analysis, that should ring some bells...

 

The macroscopic dynamics that we are expressing in any given coordinate system, must transform in a very specific manner (for self-coherence reasons) when we choose to express them from a different coordinate system. Quote from my previous post: "it is shown, that the dynamic behaviour of the clock absolutely must be plotted in a way where the supposed geometry and the dynamics of the construction must exhibit relativistic behaviour as a function of the chosen coordinate system."

 

You could say that it is simply consequential to how persistent objects have been defined, that they exhibit such relationships. It doesn't matter what coordinate system you choose to describe those objects, but as long as they appear in speeds less than C in one coordinate system, they must appear in speeds less than C after transformation to any coordinate system.

 

Using a “earth rest frame of reference map of the universe” together with a ship time clock for time measurements, we can achieve any velocity right up to infinity and, strange as it might seem, we can use Newtonian physics to plot our course through the universe. That is an interesting consequence of using that mixed coordinate system.

 

Hmmm, yeah that is interesting idea.

 

I suspect, if humans ever do achieve interstellar travel, they will use my geometry for plotting their course; it is much simpler than using Einstein's picture. For any physicists reading this note that one “g” acceleration (in the mixed coordinate system) is not one “g” acceleration in the ships frame so the standard relativistic comparisons are not valid. (I just pointed that out because I know you will do the relativistic calculations and point out my answer is different).

 

My brain went all twisty and it's late so I can't figure this out; would the ships acceleration in the mixed coordinate system show up as constant if it shows up as constant in the ships own measurement?

 

But one interesting thought that popped to my mind while thinking about this was that all the massive things in the universe appear to be quite close to being rest against each others... I mean, shouldn't we expect massive things to form quite uniformly across "all the inertial frames" so to speak. And if they were spread across all inertial frames in somewhat uniform manner, then from our perspective, most galaxies etc, should appear to be moving at very near the speed of light, so close that they would be almost indistinguishable as galaxies... Hmmmmm, my brain went all twisty again.

 

But back to the reason I mentioned this case. It is much easier to see rapid travel as “forcing” one into the future instead of in terms of a limiting velocity. For example, Alpha-Centauri is roughly four light years away. It is somewhat surprising that one “g” (32 feet per second per second) is almost exactly equal to one light year per year per year. So if we accelerated off towards Alpha-Centauri at one “g” (as measured on our mixed coordinate system) until we got half way there (two light years) [imath]t=\sqrt{\frac{2d}{g}}[/imath] it would take two years (ship time). We could then de-accelerate for another two years and arrive at rest at Alpha-Centauri. We could spend what time we needed there and then return to earth in another four years (ship time). So we would say the round trip was eight years long. How much time would pass on earth?

 

The answer is quite simple: our actual path in my space would be four segments, each two light years in the x direction plus two light years in the tau direction, c being one light year per year. Since these directions are orthogonal to each other, our actual total distance of travel would be [imath]4\sqrt{2^2+2^2}=8\sqrt{2}=11.31[/imath] light years. So the earth observers would say the trip took roughly eleven years and four months. That is, we could see ourselves as being forced into the future a distance of roughly three years and four months.

 

Suppose we wanted to go to the other side of the galaxy, some 200,000 light years away. Half way would be 100,000 light years; [imath]t=\sqrt{200,000}[/imath]=447 years so, at one “g” it would take roughly 1,800 years ship time for the round trip. At four g's (something the crew could probably get used to) we could do the round trip in roughly a little over 900 years (ship time). But how long would we be gone, earth time? [imath]t=4\sqrt{100,000^2+225^2}[/imath] which is roughly 410,000 years. Just a tad short (by a mere 10,000 years) of the apparent speed of light. In this case we have been pushed about 409,100 years into the future. But what will the crew say their velocity was? They went some four hundred thousand light years in a little over 900 years. That would be almost 450 light years per year.

 

Seems like a fairly simple way to figure out these things.

 

You should notice that “tachyons” don't exist in my representation. “Apparent” velocities in excess of light are just plain -???- “not possible” -???- (see the above analysis). That is one reason I do not like Einstein's picture: it suggests the existence of phenomena which violate the logic of his own construct. A very poor characteristic for any theory to contain.

We started with the expression

 

Yeah it's a bit mixed up concept because people usually keep talking about tachyon's "motion" and "speed", but really the only coherent way to incorporate the idea of tachyons into the picture is just to take something in the past as being affected by something that "happened in the future", i.e. think of it in terms of static spacetime... No matter which way you discuss that concept, it makes zero sense to talk about their motion. I think the idea they have in their head is literally a spacetime changing as a function of another time over and beyond spacetime. I would advise a person to take a deep breath and start over at that point :I

 

[math]S=2L_0+Ssin(\theta)[/math]

 

subtracting [imath]Ssin(\theta)[/imath] from both sides, we obtain

[math]S -Ssin(\theta)=S(1-sin(\theta))=2L_0[/math]

 

and then dividing by [imath]1-sin(\theta)[/imath] one obtains

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

 

And that should take you to the end of the OP.

 

Okay, that's pretty clear now, it was just a procedure I had never seen before and didn't pop into my mind. I'll try and walk through the rest of the OP soon...

 

I just read your latest post and agree with most all of it; however, I don't think it would be possible to chart a path which would allow you to party all the time by accelerating your ship in a manner where it would be the New Year's Eve on earth all the time! The general relativistic transformation has to yield a stopped clock on the earth as seen from your ship! That, I think might require an infinite acceleration. But maybe you could get far enough away from the earth such that there was enough mass between you and the earth to yield a black hole solution. :shrug:

 

Heh, "slightly" out of my depth to work that out, and certainly things get complicated when you start taking all things into consideration... I was intentionally just thinking about it in terms of special relativity and with the idea of changing continuously from one inertial frame to another when you are accelerating, where you'd just assume the simultaneity of that frame (and if you are a party-loving hippie, that's probably the interpretation you want to use :)

 

But yeah, even then you couldn't keep it up absolutely forever no matter how far from earth you started :I Well, pretty long party anyway!

 

At any rate, one more comment to all the lurkers. I'd think anyone can understand, that it makes very little sense to suppose that reality is affected by whichever coordinate system you choose to describe it. Reality probably is whatever it is no matter how you plot it in your spacetime diagram, unless you want to go with some sort of idealistic philosophy.

 

Yet, the relativistic transformation between coordinate systems is valid. So, shouldn't one be interested to understand why?

 

At least I've been interested to find an answer to that question, therefore it is pretty hard for me to understand the reluctance to look at work which explains exactly that conundrum... :shrug:

 

Maybe part of the problem is that people are drawn to mysterious views of the ontological reality... Look at almost any mainstream presentation of relativity / QM, it's always bunch of way too enthusiastic physicists talking about modern physics in ways that suggest very mysterious ontology to nature. They ALWAYS imply that this is the ontology that scientists have somehow practically proven to be true.

 

I guess, the more mysterious the presentation makes things sound, the more it sparks the interest of general public, and therefore those descriptions gain the most attention. Everybody loves the possibility of "Many-Worlds" and relativistic time and length contraction and 11 dimensions without understanding where those concepts are coming from...

 

-Anssi

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Your use of the word “orientation” here bothers me! An object was defined to be a collection of elements which could be considered an entity unto itself for sufficient time to examine or use that fact. The only orientation of interest is its momentum direction in the four dimensional space being used here. No “orientation” is being maintained; the only thing being maintained is the fact that the elements making up that object have to remain in the vicinity of one another. Since they are all moving at the same speed (v?) that fact has important logical consequences.

 

Perhaps my use of the word orientation was improperly chosen and I have been having a slight misunderstanding about just what an object in general is considered, and, when I was referring to a construct here:

 

This can be accomplished by being able to consider the construct separately of the universe that is of any influences resulting from the Dirac delta function, effecting the construct outside of it can be ignored.

 

I should have used the word object instead. If so this is likely just a minor misunderstanding on my part of understanding just what an object is. I have been under the impression that an object is a collection of elements such that all of the elements that it is composed of will maintain the same order over changes in t so that it would be static. While this is certainly a subset of possible elements it may not be the entire set of objects.

 

Well, of course, that is exactly the phenomena under examination.

 

Yes but it is part of assuming the existence of objects and need not be considered an assumption of its own which it could very easily be taken to be.

 

Somehow I don't think you are following the deduction at all. If you followed what I was doing, you wouldn't ask such a question.

 

Maybe I’m just not fully understanding how the sphere was arrived at but it seems that it may be related to the fact that the solution to the wave equation in odd dimensions is determined entirely by the boundary values of the initial conditions while in even dimensions the entire sphere has an effect on the equation. The point being if we don’t assume the existence of objects might not an expanding solid ball be more of the proper description of the equation then the expanding sphere that you used.

 

Maybe this would be more closely related to why the universe appears three dimensional. Either way I agree that the topic should probably be dropped as it seems to be outside of the topic being discussed. I just wanted to give some idea of where my question was coming from.

 

From the remainder of your post it seems that either I’m not asking the right questions or you don’t have any problem with how I‘m understanding the topic at hand. With that in mind I’m going to return to the original post and see what there is that I have a problem following and when I come up with some new questions I will post them unless you bring my attention to something that you think I should pay particular attention to first.

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