User:Dc.samizdat/Dimensional relativity

" Missing sections are noted: These notes are created by using the CITE button or " The dimensional theory of relativity is an equivalent formulation of the standard theory of relativity based on a spherical Euclidean metric in four spatial dimensions. It is one of several Euclidean relativity theories with the same simple geometry, which differ in their precise formulation and interpretation.

This theory holds that the universe has four orthogonal spatial dimensions (in addition to time), one of which is hidden from each observer as a consequence of the fact that everything in nature is constantly in motion at the speed of light. Nothing material is at rest, or in motion at any speed other than lightspeed, in any proper frame of reference, even and especially each observer in his own frame. The theory implies that the large scale structure of the universe (cosmology) and its fine structure (quantum mechanics) can only be visualized as the motions of objects of four dimensional extent (4-polytopes), all at the same constant speed, in a space with four orthogonal axes (rather than as the motions of three dimensional objects, at various speeds, in a space with three orthogonal axes, which is an accurate visualization only at everyday scales).

Origin
"“The universe is a sphere whose center is wherever there is intelligence.”"This statement by the 19th century American essayist and natural philosopher Henry David Thoreau is the original precise articulation of the principle of dimensional relativity. Thoreau did not provide a theory of how it could be raised to the status of geometric physical law alongside Copernicus’ heliocentric principle and Newton’s laws of motion, but he wrote it more than half a century before Einstein’s theory of relativity.

Einstein himself was the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere (the first written articulation of the principle of Euclidean relativity). He did this as a gedankenexperiment in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe. But when in his 1921 Princeton lecture he invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", he was careful to note parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."

Informally, the dimensional theory of relativity may be given as a sort of dual inverse of this formulation of Einstein's: The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding the observed universe as a quasi-spherical manifold of three dimensions, embedded in the physical Euclidean continuum of four dimensions.

The dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the Minkowski spacetime geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, as it was first worked out by the Swiss schoolteacher and self-taught mathematician Ludwig Schläfli around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of Euclidean geometry to spaces of any number of dimensions. He coined the general term polyscheme to mean geometric forms of any number of dimensions, including two-dimensional polygons, three-dimensional polyhedra, four dimensional polychora, and so on, and in the process discovered all the regular polyschemes that are possible in every dimension, including, in particular, the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five Platonic solids in three dimensional space). Thus he was the first to explore the fourth dimension and discover its geometric properties. Because his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the same ground, and competing terms were coined; today polytope is the commonly used term for polyscheme.

Poetry
Millay knew that poetry is the invention of symmetries, and at its best the discovery of nature's symmetries. Poetry is metaphor, which is to say dimensional analogy, and the sonnet is a strict form of it, like the analogy between regular polytopes in three and four dimensions discovered by another woman poet, Alica Boole Stott. Poetry and mathematics have common origins and their greatest practitioners use the same method, which is simply to look, see, and find the symmetry. One of Millay's sonnets begins "Euclid alone has looked on beauty bare". When she went off to Paris for her Fatal Interview with him, perhaps she sensed in George Dillon the soul of an earlier Parisian youth who burned brightly, Évariste Galois who discovered the mathematics which underlies geometry, and invented symmetry group theory before his own fatal interview at 20. Millay's contemporary poet Emmy Noether, the greatest mathematician of a time which is remembered for the emergence of the great physicists, found that Galois's poetry underlies all physics, too. Noether's theorem, the deepest mathematical finding in physics, is her intricate sonnet that expresses how each great formula of physics expresses a conservation law, which in every instance is itself the expression of an exact symmetry group. These poets knew that great poetry is the discovery, or the rediscovery, of nature's mathematics.

Formulation
Like Einstein’s theories of special relativity and general relativity, the precise form of the dimensional theory arises in two developmental phases: a special form restricted to conditions of negligible gravity, and its generalization applicable to all kinds of acceleration. Also by analogy to Einstein’s theory, it is predicated on a complementary pair of postulates: a relativity principle, and an observation of constant velocity.

The special theory of dimensional relativity:


 * The laws of physics are the same in all inertial reference frames of a Euclidean space of four dimensions.
 * All objects with proper mass, including the observer, move with constant velocity c through four dimensional Euclidean space, in the inertial reference frames of all observers. Light signals propagate through four dimensional vacuum at the constant velocity 2c for all observers, regardless of the motion of the signal source.

The general theory of dimensional relativity:


 * The laws of physics are the same in all reference frames of a Euclidean space of any number of dimensions.
 * All tangible objects, regardless of mass, including light signals and the observer, move with constant velocity c through their proper Euclidean space, in the reference frames of all observers, regardless of the motion of their source.

Cosmology
These are the voyages of starship Earth, to boldly go where no man has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since.

Conclusion
Of course it is strange to consciously contemplate this world we inhabit, the planet, the solar system, the vast galaxy, as the merest film, a boundary no thicker than the diameter of an electron. But is not our unconscious traditional concept of the boundary of our world even stranger? We are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the polyschemes Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more interesting than those we can discover in three. The strange thing, when we think about it, is that there is a boundary between three and four dimensions. Why can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have three dimensions, and not four, or five, or the n dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three?

We know that in Euclidean space the boundary between three and four dimensions is itself a three dimensional space, so we should suspect that we are confined within the curved boundary itself. Again, our unconscious provincial concept is that there is nothing else: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else (all the regular objects that exist in higher dimensions), so this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of no thickness, a mere abstraction with no physical power to separate, be a more suitable explanation?

The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."

I believe our universe is properly considered as a Euclidean space of four orthogonal spatial dimensions. But other people will have to work out all the math, because I don't have any mathematics myself; quite unlike Coxeter, I am illiterate in that language.