How can complex analysis be applied to Einstein's theory of relativity?

[Tom Piper]

As you know complex analysis has provided many useful tools for
harmonic analysis. However I think its application to Einstein's theory
of relativity is relatively limited. So I tried to modify complex
analysis in order to apply it to the theory of relativity more easily
in the following site.

http://www.geocities.com/tontokohirorin/mathematics/quadratic/complex2.htm

 

[Hurkyl]

I have some trouble with your article, since some of the symbols you use show up differently in different fonts.

For example, the symbol I think you're using for a right arrow shows up as a little R with a circle around it.


My first comment is on your choice of norm -- the Field Norm has somewhat nicer properties for this context. It is simply the product of the conjugates of the number:

N(a + bc) = a² - b² c²

It's nicer because:
(1) It takes values in your base field.
(2) It's a polynomial in the components of your number.

And it has the same order properties as your norm (N(p) < N(q) iff ||p|| < ||q||), so I don't think it changes anything you've done.


Also, it might be useful to treat Q(c) as a vector space over Q. (Again, borrowing from what can be done in complex analysis, or more importantly, what is done in Lorentzian geometry)

This treatment, I think, would clean up some of your definitions, and generalize them too. (you don't have to worry about things with zero or negative norm)


For example, if we write z = x + y c, then we have the differential of a function f is:

$$df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$$

And we have the two standard differentials: $dz = dx + c dy$ and $d\bar{z} = dx - c dy$.

Since these form a basis, we can actually rewrite a generic differential as:

$$df = \frac{\partial f}{\partial z} dz + \frac{\partial f}{\partial \bar{z}} d\bar{z}$$

And then decree a function to be "differentiable" with respect to z if and only if $\frac{\partial f}{\partial \bar{z}} = 0$.


Notice that ordinary complex differentiation can be treated in exactly this way.


The nifty thing about this treatment is that it's still a reasonable thing to do in more abstract settings. For example, there is still a reasonable way to define a differential of an algebraic function on, say, a finite field of characteristic 7, and all of the above carries over without any problem.



Another useful thing to look at are rings of formal power series over R. For an arbitrary ring R, one defines R[[x]] to be:

$$R[[x]] := \left\{ \sum_{i = 0}^{\infty} a_i x^i | a_i \in R \right\}$$

whether they converge or not. (Though, in the current context, I suspect there's a reasonable way to define a subring of "convergent" formal power series)

Derivatives on such a thing are defined termwise.

The trick with dealing with these sort of things is to avoid doing anything that asks what the "value" of the sum is.

There is also, of course, the field of formal laurent series over R. I think it's denoted R((x)). There is a slight deficiency, though -- they cannot be "left infinite" sums. Convergence really is needed to deal with doubly infinite sums.


I suspect you might enjoy learning algebraic geometry!

 

[Tom Piper]

Thank you for your detailed response.

"For example, the symbol I think you're using for a right arrow shows up as a little R with a circle around it."

Oh, really? I'm sorry for that mess. Your guess is exactly right.

"My first comment is on your choice of norm... (2) It's a polynomial in the components of your number."

I agree with you. By the way isn't it necessary that generally ordinary norm satisfies triangle inequality? For example, on that Field Norm, N(5)=N(4)+N(3).

"Also, it might be useful to treat Q(c) as a vector space over Q... Notice that ordinary complex differentiation can be treated in exactly this way."

It's a nice formulation, isn't it? Thank you for letting know me that.

"Another useful thing to look at are rings of formal power series over R...Convergence really is needed to deal with doubly infinite sums."

Indeed the convergence of Laurent expansion is significantly important and as written in my webpage, I've not found any effective way for convergence test on that expansion...

On physical side, as you may guess, the constant c means the light speed. Namely I've supposed (you may think ridiculous, though) that the light speed is irrational number but its square is rational number, and there is no irrational number in (classical) physical world except c.

 

[Hurkyl]

There are two number systems that you might also find interesting... they are both fairly natural analogs of Minowski space.


The first (and most popular) is the quaternions. They're like the complexes, except there are three "imaginary" axes, and they're not commutative. (They're a division ring) One can envision the "real" axis as being the time axis, and the three "imaginary" axes as being the three spatial axes of 4-D Minowski space.

Their algebra is given in terms of the three basis elements (other than 1):

i^2 = j^2 = k^2 = -1
i * j = k = -j * i
j * k = i = -k * j
k * i = j = -i * k


The other is the "hyperbolic numbers", which are formed as R[h] / (h^2 - 1). In other words, they're defined like the complexes, except that we have that h is a square root of 1, rather than of -1. (And h is different from 1 and -1) This isn't a field, but it's still an interesting ring of numbers.

I don't think anything you've done had taken advantage of the fact that your Q(c) was a field -- I think everything carries directly over to the hyperbolic numbers. And, of course, they're a two dimensional vector space. And more importantly, (like the quaternions), they're a vector space over the reals so that convergence makes more sense!

 

[Tom Piper]

"The first (and most popular) is the quaternions...and the three "imaginary" axes as being the three spatial axes of 4-D Minowski space."

"The other is the "hyperbolic numbers"...This isn't a field, but it's still an interesting ring of numbers."

Thank you for letting know me those interesting number system. Actually I didn't use quarternion in my webpage because for some reason I think the space-time is composed of the direct product of three Q(c)s, namely there are three "times" in the world! For details, please visit;

http://www.geocities.com/tontokohirorin/physics/relativity3/relativity3.htm

"I don't think anything you've done had taken advantage of the fact that your Q(c) was a field...they're a vector space over the reals so that convergence makes more sense!"

That's true. Obviously Q(c) is not complete, and I haven't written any practical method to approximate real number to the number in Q(c). Actually I have no idea! However as you know, even if there doesn't exist the limit value in Q(c), we can check whether that limit converges or not. As there is no perfect circle in our real world, I think every (classical) physical value must be rational except light speed. Apart from philosophy, how do you think the "hyperbolic" Cauchy integral theorem mentioned in my webpage? It is satisfied by any function described with 2 dimensional wave equation, so I think it might be useful for engineering purposes.

 

[Tom Piper]

Sorry, for the definition of differentiability in my website was incorrect. I corrected it and updated the website.

 

[Tom Piper]

"..."I don't think anything you've done had taken advantage of the fact that your Q(c) was a field...they're a vector space over the reals so that convergence makes more sense!"

That's true. Obviously Q(c) is not complete, and I haven't written any practical method to approximate real number to the number in Q(c). Actually I have no idea! ...

However when Q(c) is UFD, the approximation can be done with a simple way. I added the webpage explaining how to approximate the value in R by that in Q(c) in the following website.

http://www.geocities.com/tontokohirorin/mathematics/quadratic/approx.htm

 

[Hurkyl]

It seems you had anticipated something I had planned on mentioning.

However, approximation using < is problematic, because it does not respect your norm.

For example, if I pick w^-1 = &radic;2 - 1, (in Q(&radic;2)) then the following series converges with respect to <:

w^-1 + w^-2 + w^-3 + ...

however, it appears wildly divergent with respect to your norm. The first few partial sums are, followed by their norm squared: (assuming I didn't make an algebraic mistake)

-1 + &radic;2 --> 1
4 - 2 &radic;2 --> 8
-7 + 6 &radic;2 --> 23
20 - 13 &radic;2 --> 62
...


The other problem is that Q(c) is a two-dimensional vector space over Q: you'd like its completion to look like a plane, not a line.


An interesting point I was going to make is this: once you decide upon a way to approximate a real number then you're going to want to declare two different approximations of the same real number to be equivalent... and in the end, you've simply reconstructed the real numbers!

(Compare to the model of the real numbers formed from equivalence classes of Cauchy sequences of rational numbers...)

 

[Tom Piper]

Oh, indeed! So when w^-1 is expressed as a+bc in Q(c), we should put both |a| and |bc| less than 1 in order to avoid such exponential explosion, shouldn't we? So, in your example, w^-1 should be substituted with, say, (1/2)(√2 - 1). Thank you for indicating my fault. I'll modify my webpage based on your indication.

"An interesting point I was going to make is this: once you decide upon a way to approximate a real number then you're going to want to declare two different approximations of the same real number to be equivalent... and in the end, you've simply reconstructed the real numbers!"

two different approximations of the same real number to be equivalent... That's true. But please see that my method is for the analysis on Q(c), not R. Practically every irrational number is expanded with infinite series. Therefore if the finite series in Q(c) instead of that infinite series is used as an approximation of that irrational number in Q(c), then I think analytical calculations such as limit or integral also give some approximate result, i.e. converge to the true value when that finite series converges to infinite series.

 

[Hurkyl]

For the series w^1 + w^2 + w^3 + ... to converge in your norm, I think what you want is for |w| < 1 and |w*| < 1. (w* is the conjugate). I'm not sure what happens if |w| = 1 or |w*| = 1.

I determined this by looking at the recursion:

(a + bc) --> (a' + b' c) = w^-1 (a + bc) + 1

or, in matrix form:

(a, b, 1)A = (a', b', 1)

which transforms a partial sum into the next partial sum. Its eigenvalues are w, w*, and 1.


Of course, the coefficients won't always be 1's for a general approximation... I still worry that, no matter what w is, there might be some real numbers for which the partial sums won't converge in your norm... but I haven't yet worked out a proof either way.


I guess the thing I was trying to suggest is this: it might pay to do analysis in the reals, and the study how the rational approximations behave in comparison.

I guess, in the end, the proof will be in the pudding. If you're given functions f, g:Q(c) --> Q(c), you'll need something analogous to solving a differential equation:

y'(t) + f(t) y(t) = g(t)


Or, maybe not! While I have little hope that differentiation will be well-behaved, it is quite possible that some form of integration will be well-behaved. If you could do all the physics in integral form, that might be all you need.


There's another problem that it looks like I forgot to bring up before: functions from Q --> Q like:

$$f(x) := \left\{ \begin{array}{ll} 0 \quad & x^3 < 2 \\ 1 & x^3 > 2 \end{array}$$

are infinitely differentiable... I don't know what other ugly things can happen.


Okay, let me collect my thoughts...

I guess my reservations center around the fact you haven't really worked out the appropriate analogue of real analysis for Q(c), though you have sketched out how you'd like things to behave.

A worked example would be great -- full details of working out, say, the trajectory of a free particle, or a harmonic oscillator. Things for which one would normally have the power of existence and uniqueness theorems for differential equations, et cetera.

 

[Tom Piper]

However as we know, since Q(c) is a field (under the supposition that Q(c) is UFD), the value of every finite polynomial in Q(c) is also in Q(c). My basic idea is that if the infinite series is approximated by finite series, the differential is approximated by difference, and the integral is approximated by summation, then every result of calculation is still in Q(c) (how primitive!) Of course, the problem is whether that approximation converges to the true value or not. As you've indicated in the last post, at the moment I must say the convergence check should be done for each case independently...

 

[Tom Piper]

Hi, Hurkyl,

Following your note in the post #10, I abandoned to approximate the real number by power series in Q(c) (it's a safer way, isn't it?) Instead, I clarified the difference between true limit and "approximate" limit in the following webpage.

http://www.geocities.com/tontokohir...atic/approx.htm

The limit appeared in the main website means always the latter.

 

[Tom Piper]

Sorry, for the above URL seems wrong. The correct URL is,

http://www.geocities.com/tontokohirorin/mathematics/quadratic/approx.htm

 

[Tom Piper]

I've found the former description of calculation for residue in my webpage was ineffective. So I modified the integral path in order to calculate the residue of f(x-yc) on the origin of Q(c) and updated the webpage;

http://www.geocities.com/tontokohirorin/mathematics/quadratic/complex2.htm

 

[Tom Piper]

I updated the site;

http://www.geocities.com/tontokohirorin/mathematics/quadratic/complex2.htm

with some examples of hyperbolic holomorphic functions, which are shown graphically. Also I added some graphical examples of hyperbolic Moebius transformation and the simulator for it written by Java applet. It is available to simulate hyperbolic Moebius transformation with your own parameters by yourself there.