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Doctordick · 05-31-2009

Quote:

Originally Posted by arkain101

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

What I was trying to do is explain to Bombadil exactly what this is all about. The central issue being that my fundamental equation is maybe a pretty thing but is essentially a useless construct since we can't solve many body problems. He keeps wanting to find something that equation says about reality and the correct answer is “absolutely nothing”. It is indeed the subsequent relationships which give us something to think about. Exactly what does Newtonian mechanics and relativity tell us about our universe? Now that is a serious philosophical question.

Quote:

Originally Posted by AnssiH

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

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

Quote:

Originally Posted by AnssiH

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

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

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

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

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

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

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

Quote:

Originally Posted by AnssiH

...but I understand almost nothing of the above

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

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

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

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

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

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

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

Quote:

Originally Posted by Doctordick

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

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

Thus my conclusion is,

Quote:

Originally Posted by Doctordick

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

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

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

Have fun -- Dick

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