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The New Flatland
SilenceKit writes "The New York Times has a cool story today on a sequel to Flatland, the classic geometry/social satire which "it may be no exaggeration to say has been read by every self-respecting physicist, mathematician and science fiction writer." The new one, by Ian Stewart, is called "Flatterland" and is a tour of a century of strange geometry -- from fractals to "Minkowski space," whatever that is. The story (free registration required) is at the Times" I was loaned Flatland: A Romance Of Many Dimensions by one of my college profs - it's a great book, and this come from someone who really hated geometry (What bad high school teachers can do). It's still available on fatbrain - pretty good for a 19th century text about geometry, to still be in print.
Rudy Rucker, author of the Software, Wetware, Freeware, Realware series. He later wrote a somewhat more accessible introduction to thinking about dimension called The Fourth Dimension and How to Get There which I recommend highly. Rudy Rucker is lots of fun and most things he gets involved with are cool - check out his list of works.
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Xenu loves you!
Here is a copy of flatland online from project gutenberg. Great fun to read.
Jerry Thorwayn is an author of a book called "Proffessor Ovilstar's Infinate Paradox", and it deals with all kinds of math theorums, as well as these flat land principals. You can see it here: www.ovilstar.net. He has written a whole book about this type of thing and inter-weaves romance and excitement into it. Kind of a sci-fi educational book :)
Dimensions are only orthogonal if the metric is trivial (i.e., diagonal).
The "metric" as I stated above is a 4-dimensional tensor (matrix, essentially) which (basically) provides the coefficients in the distance measure.
i.e., in 3-space, we have a distance measure of
ds^2 = dx^2 + dy^2 +dz^2
Here, the metric has a 1 in the dx-dx term, a 1 in the dy-dy term, and a 1 in the dz-dz term: in terms of a matrix, this would be just the three-dimensional unit matrix.
General relativity says that the metric is affected by the presence of matter and energy, so therefore, depending on the spacetime configuration, the dimensions may not be orthogonal. Near a black hole, this is actually distinctly not true - there is significant radial/temporal mixing in certain coordinate systems. Granted, you can always define a coordinate system with orthogonal coordinates.
In any case, you were mainly talking about special, rather than general, relativity here. The 4-dimensional metric being talked about is a Minkowski space - time is distinctly separate from any others, because in the Minkowski metric, we have
ds^2 = dx^2 +dy^2 +dz^2 - c^2 dt^2
note that time has a changed sign. We call this 3+1 spacetime.
All special relativity does is define exactly how you transform from one coordinate system to the next - in this case, via Lorentz boosts. Lorentz boosts in one direction have both a spatial boost and a temporal dilation, which is what you are talking about. It doesn't mean the coordinates aren't orthogonal - it just means that when you boost from one frame to a faster-moving frame, both the boost direction and the temporal direction are affected.
Short answer: you answered your question yourself - you said "rate of movement", but above you said "movement" - there's no reason to expect that "moving" is the same as "steady motion" - or, that a simple translation is the same as a velocity shift. In truth, it isn't.
Think about it this way. It's true, time and space are orthogonal. If I do a time translation (i.e., in my coordinate system I redefine t = t + t0) it doesn't affect the x, y, or z coordinates. Likewise for spatial coordinates.
But there's no a priori reason to expect that when you start talking about another frame of reference that is in constant motion with another that it shouldn't affect the other coordinates. After all, you're already mixing things: you have to talk about z = z0 + vt, for instance. Here, instead of the boost simply being z = z0 + vt, there's a Lorentz factor as well.
The first is Sphereland, written in 1960 by Dionys Burger. The English translation I have has a forward by Isaac Asimov [cool little useless tidbit].
The second is Geometry, Relativity, and the Fourth Dimension, written by Rudolf v. B. Rucker in 1977. Aside from the <sarcasm>eloquent title</sarcasm> it follows in the same format as the previous two.
The following is part of the blurb from Sphereland:
And Geomerty, Relativity, and the Fourth Dimension:---
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"Of course, that's just my opinion. I could be wrong." --Dennis Miller
Never read Bearden's paper. From the OP, it's total crap.
(a) I can write Maxwell's equation in whatever form I like : quaternions, vectors, component by component (go read Gell-Mann's book to see how cumbersome that is) covariant-indice form, and (most beautifully) in differential forms. It expresses a bunch of coupled differential equations that's all.
(b) Maxwell's theory is a classical field theory. You can attach words like " unified ", "successful", and blah to it, and it's still a classical field theory. I don't understand what the heck the OP is rambling about "quartenions" blah. There is no point to be made. The use of "big words" and impressive looking references only serve to give the false impression that there is actually "something" there.
(FYI, we now have quantum field theory, where we promote variables to operators and then impose canonical quantization on it. Which, btw, is totally incompatible with general relativity).
I don't blame the OP for being ignorant about the finer points of physics (after all, it's a career choice which pays very little!). But the sad truth is that nowadays, there are so many "wannabes" like Bearden who writes crap and mislead the general public. The people should be careful about what they read.
Mode (3) smart-aleck mode. Press * to return to main menu.
In 1873, Maxwell succeeded in uniting a couple hundred years of electrical and magnetic scientific observations into a comprehensive, overarching electromagnetic theory of light vibrations ... carried across space by this "incompressible and highly stressed universal aetheric fluid ..." Maxwell's mathematical basis for his triumphant unification of these two great mystery forces of 19th Century physics were "quaternions" -- a term invented (adopted would be a more precise description) in the 1840s by mathematician Sir William Rowan Hamilton, for "an ordered pair of complex numbers" (quaternion = four).
Complex numbers themselves, according to Hamilton's clarifications of long-mysterious terms such as "imaginary" and "real" numbers utilized in earlier definitions, were nothing more than "pairs of real numbers which are added or multiplied according to certain formal rules." In 1897, A.S. Hathaway formally extended Hamilton's ideas regarding quaternions as "sets of four real numbers" to the idea of four spatial dimensions, in a paper entitled "Quaternions as numbers of four-dimensional space," published in the Bulletin of the American Mathematical Society [4 (1887), 54-7]. It is obvious from Maxwell's own writings that, even before Hathaway's formalization, his choice of quaternions as mathematical operators for his electromagnetic theory was based on his belief that three-dimensional physical phenomena are dependent upon higher dimensional realities.
In a tragedy for science, after Maxwell's death, two other 19th Century "mathematical physicists" -- Oliver Heaviside and William Gibbs -- "streamlined" Maxwell's original equations down to four simple (if woefully incomplete!) expressions. Because Heaviside openly felt the quaternions were "an abomination" -- never fully understanding the linkage between the critical scalar and vector components in Maxwell's use of them to describe the potentials of empty space ("apples and oranges," he termed them) -- he eliminated over 200 quaternions from Maxwell's original theory in his attempted "simplification."
This means, of course, that the four surviving "classic" Maxwell's Equations -- which appear in every electrical and physics text the world over, as the underpinnings of all 20th Century electrical and electromagnetic engineering, from radio to radar, from television to computer science, if not inclusive of every "hard" science from physics to chemistry to astrophysics that deals with electromagnetic radiative processes -- never appeared in any original Maxwell' paper or treatise! They are, in fact--
"Heaviside's equations!"
You can check this out by read a highly revealing paper on the subject by another renowned British mathematical physicist of this century, Sir Edmund Whittaker, titled simply "Oliver Heaviside" (Bulletin of the Calcutta Mathematical Society, Vol. 20, 1928-29, p.202); or, another overview of Heaviside by Paul J. Nahin, "Oliver Heaviside: Sage in Solitude" (IEEE Press, New York, 1988, p.9, note 3.).
The end result was that physics lost its promising theoretical beginnings to becoming truly "hyperdimensional" physics ... over a century ago ... and all that that implies.
Lt. Col Thomas E. Bearden, retired army officer and physicist, has been perhaps the most vocal recent proponent for restoring integrity to the scientific and historical record regarding James Clerk Maxwell -- by widely promulgating his original equations; in a series of meticulously documented papers on the subject, going back at least 20 years, Bearden has carried on a relentless one-man research effort regarding what Maxwell really claimed. His painstaking, literally thousands of man-hours of original source documentation has led directly to the following, startling conclusion:
Maxwell's original theory is, in fact, the true, so-called "Holy Grail" of physics ... the first successful unified field theory in the history of Science ... a fact apparently completely unknown to the current proponents of "Kaluza-Klein," "Supergravity," and "Superstring" ideas ....
To investigate this further you should take a look at --
"Possible Whittaker Unification of Electromagnetics, General Relativity, and Quantum Mechanics,"
by T.E. Bearden
(Association of Distinguished American Scientists
2311 Big Cove Road, Huntsville, Alabama, 35801)
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