User:Dc.samizdat/Euclidean spherical relativity

" Missing sections are noted: These notes are created by using the CITE button or "

"Spherical relativity and Euclidean relativity redirect here. For Euclidean spacetime models that are not four-dimensionally symmetric, see special relativity (alternative formulations)."

Euclidean spherical relativity is an alternative formulation of the theory of relativity based on a different geometric model of space and time. It is intentionally compatible with Einstein's theories of relativity (which have survived experimental verification to become a standard of physics), in that it attempts to reach equivalent physical laws and make essentially the same experimental predictions. It seeks to provide a simpler and more experientially satisfying explanation for the counter-intuitive and apparently paradoxical behavior of nature predicted by relativity theory.

Euclidean relativity formulations present an equivalent alternative to the non-Euclidean four dimensional Minkowski spacetime that Einstein adopted as the geometric frame for his theory of special relativity. This article is a survey of the several such alternative formulations based on a Euclidean metric which have been proposed. The term spherical relativity, and this article, apply only to formulations which are four-dimensionally symmetric.

Differences from the standard theory
Spherical relativity adopts a different reference frame than Einstein's relativity, as Einstein adopted a different reference frame than Galilean relativity. Ultimately the differences among these three reference frames are entirely due to differences in (a) their geometric shape (how many dimensions they have, and what those correspond to in our physical world), and (b) the speed that things move (relative to each other) in them.

Geometry
Minkowski spacetime features three spatial dimensions plus one of time. It defines Lorentz transformations of the four coordinates between moving reference frames (counter-intuitive distortions of the length of space and the rate of time) in order to preserve the experimentally verified invariance of the speed of light. The Minkowski metric is four-coordinate, but not symmetrically: one dimension is different than the other three, not just because time is (obviously) not the same thing as space, but because the time coordinate occurs in the Lorentz formulas with the opposite sign than the three space coordinates. The three dimensions of space by themselves (as used in Galilean relativity and Newtonian mechanics) form a regular Cartesian space (the kind of space we are most familiar with). But the four coordinates of Minkowski space do not form a regular four-dimensional Cartesian space, in which the four axes are orthogonal (perpendicular) to each other; they form a hyperbolic space.

Spherical relativity, on the other hand, does feature a symmetric Cartesian space of four (not three) orthogonal space dimensions, in addition to a time dimension. In standard relativity a reference frame (the space in which an observer moves over time) has three orthogonal axes (corresponding directly to the three dimensions of our ordinary day-to-day experience), and time is treated (in some respects) as a fourth dimension. But in spherical relativity, a reference frame has four orthogonal axes, and time, if treated in some respects as a dimension, is a fifth dimension.

The fourth spatial dimension is aligned with the observer's motion at any point in time, but it is not time, it is a real spatial dimension exactly like the other three. It is a distinguished direction only to the observer (because it is the direction in which he is currently moving). Spherical relativity is a theory that the space around us has four perpendicular spatial dimensions (though which we move over time), even though our universal day to day experience is that we only have access to three.

Our experience tells us that time and space are different kinds of dimensions (asymmetric axes of motion). There are only three orthogonal spatial dimensions in which we can see and move. Time, considered as an axis, isn't like a spatial dimension. We can't move backward in time, we can't see into the future (or really, into the past either), and so the asymmetrical Minkowski spacetime seems to be the only sense in which our reality can be said to have four dimensions. Spherical relativity challenges this common sense view. It says that despite appearances, if we look hard we can find four orthogonal spatial dimensions: eight perpendicular directions in space, not just the six we can see into and move freely in. The two we can't see into are where we came from, and where we're going to.

Cosmological model
Everything we can see (the earth, the solar system and the entire observable universe) seems to be arrayed around us in three dimensional space, but not in the sense that we are at the center (on the contrary, the cosmos appears to be isotropic, without any center). So the new theory demands a new view of our cosmological model as well. From the viewpoint of spherical relativity, everything we can see must be just a thin three-dimensional surface embedded in a four-dimensional space we can only imagine with difficulty. For example, all the atoms in the three-dimensional cosmos that we experience might be arrayed as a kind of cosmic soap bubble, in most places only one atom thick but some 14 billion light years in radius, and still expanding from its origin at the big bang (at the real center of the universe). Undoubtedly such a reconception would give as great a wrench to universally held ideas of our place in the universe as did Copernicus's discovery that the earth revolves around a central sun, or that even earlier revolution in mental space when mankind first realized that the earth is not flat, but round like a ball.

Constant velocity
Spherical relativity and the standard theory characterize the velocity of moving objects differently, with respect to the constant velocity of light.

Descriptive adequacy
A theory that claims to be equivalent to standard relativity must show the math.

Descriptive utility
''How does this equivalent theory provide a simpler, more symmetric or more elegant description of physical reality? How does it make it easier to think about physics, or to visualize objects and their motions?''

Explanatory adequacy
''A theory must contain adequate explanations for its consequences, particularly any novel ones that challenge common experience. (The theory that the earth is round like a ball is only adequately explained if it includes a theory of gravity, something which makes down always toward the center of the ball wherever you are, rather than the same direction everywhere. Otherwise in some places on the earth you would fall off.)''

Hidden fourth dimension
To be successful, a theory that the universe has four spatial dimensions (eight perpendicular directions instead of six) must explain why, as a natural consequence of geometry and the laws of motion, one spatial dimension is nearly invisible and inaccessible to us. If there are really four equivalent spatial dimensions, why should one of them appear different than the other three?

Explanatory utility
''Is this theory merely equivalent, or does it tell us anything new? Does it provide a more satisfying why to stand behind the what of any facts we already know? Can it resolve any outstanding mysteries or paradoxes? Does it give us answers to any unsolved problems in physics? Does it point us toward a solution of any unsolved problems, or provide a new method or toolset enabling further inquiry, for example by providing a new way to test existing unproved theories? Does it imply any such tests of its own validity (is it falsifiable independently of the original theory it is supposedly equivalent to)?''

Relativistic effects
Although spherical relativity predicts the appearance of the same counter-intuitive relativistic effects as special relativity, it resolves their paradoxes by explaining them as mere appearance: perspective distortions due to a physical rotation between the observer's frame of reference and the observed frame. The effects are exactly the distortions of optical perspective, only due to a rotation in four spatial dimensions instead of three.

History
Although Einstein presented his special theory of relativity in a non-Euclidean Minkowski spacetime with three spatial dimensions, he was aware of its equivalence to a Euclidean space of four spatial dimensions. In fact, Einstein was the inventor of Euclidean relativity (confirming it as another instance of Stigler's law, though we already knew Euclid didn't discover Euclidean relativity).

In 1921, Einstein gave four lectures at Princeton describing all of relativity: the Galilean, special and general theories. (It took two of the lectures to cover his theory of general relativity.) He explained that measurements in the three dimensions of space and one of time are distorted in the vicinity of gravitational masses, such that spacetime is curved, and moreover the degree of curvature varies from place to place (depending on the observer's proximity to gravitational masses). Anywhere there is any gravitational effect, three-dimensional space is non-Euclidean (curved), because the radius of a sphere around a gravitational body is not proportional to its surface area in the expected Euclidean way: instead of $r = \surd(A/4\pi)$ we get a slightly longer $r$, exactly analogous to the way the radius of a circle drawn on the surface of the earth is slightly longer than $$r = C/2\pi$$ (because the radius is really a curved line, and the center from which it is drawn is lifted above the plane of the circle). Einstein then proceeded to ask what is the overall shape of the universe? What does all this local curvature add up to? Does it cancel out, resulting in an approximately flat universe, or does it produce an overall curvature?


 * Is the universe infinite (a three dimensional space that goes on forever in every direction), but overall and at large enough scale a Euclidean (uncurved) three-dimensional space?
 * Or is it finite (a three dimensional manifold that curves back on itself), and so overall and at large enough scale a non-Euclidean (curved) three-dimensional space?

He noted that because stars are apparently evenly distributed everywhere, and the relative velocity of the stars is small compared to the velocity of light, it follows that the curvature of the universe must be roughly the same overall. The fact that the curvature is greater in some places (near gravitational masses) is due only to the uneven distribution of mass (its tendency to clump together into planets and stars and galaxies). So in one of his famous gedankenexperiments he invites us "to ignore these more local non-uniformities of the density of matter and of the [gravitational] field in order to learn something of the geometrical properties of the universe as a whole".

Criticism
Critics of the viability of the theory of spherical relativity question whether it really resolves the apparent paradoxes of relativity theory, or just pretends to do so by invoking a hidden dimension. Critics of the usefulness of the theory find it more difficult to accept the existence of an unseen fourth dimension, and to visualize four-dimensional objects and their motions, than they do continuing to live with the long-standing apparent paradoxes of the standard theory.