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AnssiH · 06-07-2009

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Originally Posted by Doctordick

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

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

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

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

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

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

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

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

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

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

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

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

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

is the differential equation of a traveling wave.

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

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

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

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

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

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

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

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

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

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

Ahha, true.

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

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

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

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

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Originally Posted by Doctordick

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

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

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

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

-Anssi

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