in Multiple Dimensions

Now I know that other old men have been very foolish in saying things like this, and, therefore, I would be very foolish to say this is nonsense. I am going to be very foolish, because I do feel strongly that this is nonsense! I can't help it, even though I know the danger in such a point of view. So perhaps I could entertain future historians by saying I think all this superstring stuff is crazy and is in the wrong direction....I don't like that they're not calculating anything. I don't like that they don't check their ideas. I don't like that for anything that disagrees with an experiment, they cook up an explanation -- a fix-up to say, "Well, it still might be true." For example, the theory requires ten dimensions. Well, maybe there's a way of wrapping up six of the dimensions. Yes, that's possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, there's no reason whatsoever in superstring theory that it isn't eight of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesn't produce anything; it has to be excused most of the time. It doesn't look right.

Richard Feynman, quoted in

Not Even Wrong, The Failure of String Theory and the Search for Unity in Physical Law, by Peter Woit [Basic Books, 2006, p. 174-175]

Reading popular accounts and textbooks on physics and astronomy, for years I tended to think of Einstein's notion of the curvature of spacetime as primarily a matter of space being curved. This was reinforced by things like diagrams or museum exhibits that showed the "gravity well" of the sun, for instance, as a funnel with a hole at the bottom, around which ball bearings could roll and then fall in. These models seemed to presuppose rather than explain the existence of motion induced by gravity. Indeed, the exhibits relied on it. Only as explained in the essay "The Ontology and Cosmology of Non-Euclidean Geometry" did it finally occur to me that space was not curved at all. As time passes, what it is that displaces an object in space is curvature along the *temporal* axis.

Hence the accompanying diagrams, which illustrate Einstein's Equivalence Principle in General Relativity -- where time is the vertical [y] axis and space has been reduced to one dimension in the horizonal [x] axis. Thus in the first two diagrams, we have "free fall," i.e. one does not *feel* weight or acceleration. At left, this is in the absence of an gravitational field, and there is in fact no acceleration -- a straight vertical line is rest in the frame of reference. At right, however, in the presence of gravity, an actual acceleration exists (a curved line -- *velocity* would be a straight line at a non-zero angle to the vertical) -- although it cannot be detected by internal measurements and will only be noticed by observing motion relative to external objects, like the surface of the planet towards which one is falling. For Einstein, these situations are not just analogous, but identical. In the illustration, they *are* identical because the (red) paths of the objects remain in the same relationship to the coordinate grids -- even in the diagram at right, the object is still "at rest" in relation to the frame of reference. It is just the grids that have been distorted. The distortion is not in the axis that represents space, it is in the axis that represents time. As we move up the temporal axis, the path is displaced laterally in space. This is the appearance of the "force" of gravity, whose literal existence Einstein has eliminated by way of geometry.

In the next two diagrams, we have weight. At left, weight is felt because a surface prevents free fall in the gravitational field. Motion actually is *not* induced, but the acceleration of gravity presses the path against the surface, which exerts a force that can be measured and felt. At right, in the absence of a gravitational field, the force must come from some mechanical expenditure of energy, whereby a surface presses against and accelerates the passenger, observer, or instrument. In each case, weight is felt as the path of motion cuts across the grid of temporal coordinates. Again, to Einstein, this makes the two cases identical, as are the two free fall cases above. And again, the identity consists in the relation of the path of motion to the coordinate grid.

Now, this produces a reasonably satisfying picture, but further considerations introduce puzzles. Einstein's theory is directed towards the explanation of gravity, but he realized himself that this was only the beginning of the project. There was another force of nature that Einstein knew about in the years he was working on Relativity, electromagnetism. Indeed, the fundamental principle of Special Relativity, that the velocity of light in a vacuum is a constant no matter how it is measured, is a consequence of its being a factor in Maxwell's equations for electromagnetism. Einstein figured that it wouldn't *be* a constant if one got different velocities from difference measurements. This brilliant simplicity was the axiom from which the rest of Special Relativity followed. Similarly, the brilliant axiom of General Relativity was that there was no force of gravity, simply a curvature in spacetime. A planet in orbit was moving in what was, for it, a straight line, i.e. a "geodesic," the shortest path, in its geometrical manifold.

In those terms, one might think that another force of nature, which produces accelerations from electrical charge or magnetism, could also be handled by geometry. However, while gravity affects every object in the universe, electromagnetic forces only act on certain things, things with electrical charge or magnetic polarity. Somehow, electrically and magnetically neutral objects move unconcerned through space without "seeing" any curvature induced by electromagnetic fields. In those terms, electromagnetism could hardly be inducing a curvature in space or spacetime itself, since even uncharged objects could not fail to "see" that. Einstein's 4-D spacetime needs to be unaffected by electromagnetic forces. So, presumably, geometry is not going to work for electromagnetic, or any other, forces. If that is the case, Einstein's great insight and technique, only working for gravity, ends up limited, awkward, and incommensurable with however other forces of nature are going to be understood. This is inelegant and, truth be told, ugly and discouraging. The laws of nature, at least to Einstein and many other physicists, are expected to be beautiful and sublime. If one force is really geometry, then they all should be.

However, in 1919, soon after General Relativity itself was published in 1915, Theodor Kaluza and Oskar Klein noticed that adding an extra dimension to Einstein's equations effectively *incorporated* Maxwell's equations into them. Thus 4-D spacetime would not be curved by electromagnetic forces -- there is now just an *additional* dimension of spacetime, five in all, and only electrical charge and magnetic polarity "see" curvature in the fifth. Despite its elegance, Einstein was curiously uneasy about this theory, hesitated to recommend its publication, despite having no real objections, and then apparently forgot about it for the rest of his life, even as his publicly advertised project was a "Unified Field Theory" that would incorporate gravity and electromagnetism. He never made much progress on such a theory, and the rest of Physics, soon enamoured of Quantum Mechanics, figured that Einstein's approach was leading nowhere and that QM would turn out to have all the answers.

But as the decades went by, a functioning Quantum theory of gravity eluded formulation. Physics began to come back around to Einstein, and the Kaluza-Klein approach was actually rediscovered. Now there were other forces of nature to explain, the Strong and Weak Nuclear forces. As it happened, adding even more dimensions, up to *26*, and now down to ten or eleven, ended up accommodating everything. According to Roger Penrose [*The Road to Reality, A Complete Guide to the Laws of the Universe*, Jonathan Cape, London, 2004, Alfred A. Knopf, New York, 2005], however, there were some problems with the approach, and enthusiasm for it has ebbed and flowed. But the most recent avant-guarde physics today, **string theory**, follows this path. Penrose himself seems less than enthusiastic about strings, for mathematical reasons, empirical reasons, and because he is skeptical of introducing additional dimensions beyond 4-D spacetime -- seeing problems there that are both mathematical and, well, ontological.

There was, indeed, the little question of what Kaluza and Klein's extra dimension meant. If it was simply another dimension of space, it seems like we would notice it. While science fiction always enjoys adding extra dimensions to reality, and mathematics happlily contemplates Hilbert Space, with an infinite number of dimensions, there is so far no empirical evidence for more than three spatial dimensions. Indeed, in Quantum Mechanics itself, more than three dimensions would alter, for instance, the number of possible magnetic substates of atoms, which fill all possible orientations in space, and this would affect the chemical properties of atoms. With four spatial dimensions, the Periodic Table would look much different. So if Kaluza and Klein's fifth dimension existed, there would need to be something peculiar about it. The notion proposed was that it would be "curled up into a tiny loop" [Penrose, *op.cit.*, p.326]. Unlike the other dimensions of space, this one did not stretch out, was not *extended*. However, while this removes the fifth dimension from macroscopic space, objects affected by electrical and magnetic fields *do* "stretch out" as they move, just like all objects affected by gravity. These things certainly were not *moving* in a "tiny loop." If the idea of curved spacetime was to provide a *path*, geodesic, for moving objects, a curled dimension isn't exactly going to do that. But whether extra dimensions like this represent physical realities or are just tricks of mathematical formalism now seems a matter of dispute, and perhaps even disinterest, among the physicists who employ such devices. Perhaps we don't need to care about how the fifth dimension (or other extra ones) needs to exist -- the math works, and that's all that matters. Penrose, who isn't sure the math works all that well anyway, also is impatient, as I am, with positivist approaches that dismiss concern about the physical reality being described. If there are going to be extra dimensions, they should make some sense, both physically and metaphysically.

However, the worry about "curled up" dimensions may begin from a false premise. If space as such was never going to be curved in the first place, and it was *time*, Einstein's fourth dimension, that distorted spacetime and induced motion through curvature, then Kaluza and Klein's fifth dimension need not be one of space either. We may simply be dealing with another dimension of time. Indeed, if electromagnetic fields induce motion, it must be along a temporal dimension. Since we only see one point in time, it is then obvious why a fifth dimension does not contribute another macroscopic direction to the world. And if uncharged objects do not "see" electromagnetism, then they simply follow the gravitational fourth dimension but not respond to or follow the electromagnetic fifth. Other forces will then involve the addition of more dimensions. It may seem strange that time would pass along all the temporal dimensions, and that different objects "feel" the curvature in at least one (gravitational) and possibly more of the dimensions; but this is surely no stranger than the "curled up" dimensions that physicists may not even believe in, and it does explain how the various forces accelerate objects in space.

However, it now should be clear why the "curled up" dimensions are appealing. They are the way in which **spatial dimensions effectively become temporal dimensions**. Thus, physicists can treat their extra dimensions as though they add to space, but their spatial extension has practically been reduced to a point, which means that they

Thus, at left we see one dimenion of space, on the x axis, and two of time, on the y and z axes, which are here folded out flat above and below the x axis. The electromagnetic field is here shown as neutral. It would not affect any object in it, electromagnetically neutral or not.

Now, at right we have the case where an electrical field might be acting counter to the gravitational one, tending to move a charged object in one direction rather than the other. So which way does the object move? Will it rise up off the ground, as the electrical field shows? Well, physically, we need to sum the forces. Geometrically we need to find the vector that sums the coordinates. It will not lie in either plane, but in the 3-D space between them, were we to have the y and z axes at the proper right angle to each other. Since the *position* of the object in the gravitational field does not show the gravitational acceleration on it, while the field lines do, what must be summed is where the object *would have been* in the gravitational field had it not be stopped by a surface. If the gravitational curvature matches the electromagnetic, then the object remains where it is. If the electromagnetic curvature is greater, then the object will feel a net force and rise above the surface -- something we can see happen with simple magnets or static electric charges.

Now, proposing that there is more than one dimension of time makes it sound like objects are progressing down different time lines into different futures. But objects would exist in the multiple temporal dimensions just as they exist in multiple spatial dimensions. Or perhaps we should not be thinking of them as *temporal* dimensions at all. Traditionally, time has been viewed as the real thing and motion as derivative from it. If time stopped, things would be fixed at rest. However, Relativity changes this picture at bit. It is for something going the velocity of light, not something at rest, that time stops. Time exists for things going less than the velocity of light -- the subluminary rather than sublunary (i.e. beneath the sphere of the moon) world. So perhaps it is motion more than time that is fundamental. Similarly, although we imagine time as existing along a continuous axis, and science pretty much assumes this picture, that is not the way time occurs in nature. Only the present exists, while the past and the future do not. There is no good evidence that time is a kind of clothesline across eternity. If it were, and the future somehow already existed, this would have the disturbing consequences, not just of (fatalistically) denying free will, but even of denying things essential to Quantum Mechanics: chance and indeterminacy. If the future already exists, then the collapse of the wave function is not really random -- it is already "written" what will be produced by the collapse.

So perhaps time is simply something *shared* by multiple temporal dimensions, and their own essential nature is about something else. What this might be comes from a different consideration. In Quantum Mechanics, matter and energy, before they are observed, exist as waves. The Copenhagen Interpretation of Quantum Mechanics disregards the physical existence of waves and takes as significant only the "probability clouds" that result from the square of the wave function. The only physical realities, then, are the particles that result when observation collapses the wave function. However, it seems to me, as discussed elsewhere, that the physical reality of the waves themselves cannot be denied. Now, if the waves are real, we need to think about what a wave is. A wave in a line, a one dimensional object, is only possible if the line is on a two dimensional surface, within which the line oscillates. A wave on a surface, like waves on the ocean, a two dimensional object, is only possible if surface is on a three dimensional object, the ocean. An ocean wave takes a three dimensional form. Now, the waves we get of particles in atoms are *three dimesional* waves. Extending the principle, these three dimensional waves are only possible by *oscillating into a fourth dimension*.

I don't think this ever comes up for much discussion because usually so little attention to paid to the physical reality of particle waves. On the other hand, we do have popular objects in contemporary physics, *strings*, which are regarded as vibrating, as waves indeed, into the multiple dimensions provided by the Kaluza-Klein approach. Now, every object with mass and/or momentum has a "de Broglie wavelength," where **= h/p** (the wavelength equal Planck's Constant divided by momentum). Anything with mass or momentum creates a gravitational field, so a three dimensional de Broglie wave, if it oscillates into a fourth dimension, enters the dimension relevant to gravity. Now, following the considerations above, this dimension would be the first temporal dimension, Einstein's fourth dimension.

What would this mean? A mass in 3-D space oscillating into a fourth dimension would create a gravitational field and would be responsible for the curvature in spacetime introduced by that dimension. If the extension of that dimension doesn't even exist, except in the imagination (vanishing into the past and future), then the mass itself, the wave itself, is the temporal existence. Mass, gravity, and time would thus all be phenomena of the same underling reality, a standing wave in four dimensions. Objects without (rest) mass, like photons, which spontaneously exist at the velocity of light, do not themselves undergo the passage of time.

In these terms, a multiplicity of temporal dimensions is simply a multiplicity of forces. Perhaps seven of these [7 + 3 = 10] are necessary for the mathematics, but just in terms of the forces that induce motion, there might only be three, one for gravity, one for electromagnetism, and one for the strong nuclear force. The weak nuclear force mediatates interactions but doesn't seem to involve attraction or repulsion. On the other hand, electromagnetism does operate as separate electrical and magnetic forces, so perhaps that gets us back to four extra dimensions. Three dimensions of time and three of space would have been a nice touch, but there is no counting on it.

Still, we must ask, What does this mean? Why would a fourth or fifth dimension contribute force, motion, and time to mere space? I can only give such an answer in pure metaphysics. Outside of space, if not of time, what we may have is a dimension of transcendence. In my view, what this contributes is possibility, the very stuff of the Copenhagen wave function (squared). This is the equivalent of **matter** in Aristotle's metaphysics, which is the power, potential, and possibility of all existence. This is transcendence because it is more, and it underlies, phenomenal objects, which are immanent in the world. Possiblities may, indeed, seem to belong to the world, but they are undoubtedly hidden from us, often to our complete bewilderment as they unfold. Just what *is* possible in the long term is a good question.

The diagram at above right provides one model for the metaphysics of value and of necessity. It is based on the manner in which the magnetic substates in an atom involve vectors of angular momentum that correspond to interger multiples of the "reduced" Plank's Constant () in the conventional z axis. Here, it is not value or necessity that are in question, except for the necessity of the laws of nature ("conditioned" necessity in the diagram) in so far as this is manifest in the forces of nature. Where the z axis can represent three dimensional space, a vector into the transcendent then represents oscillation into one of the dimensions of time, or of force, as defined by the laws of nature. Indeed, what everyone from Einstein onward has wanted is a single ultimate force, the "unified field," or the Grand Unified Theory of Everything (GUTE). So in those terms, there may be just one ultimate temporal dimension, after all, but it splits up with the "symmetry breaking" that differentiates the forces. We may get just as many dimensions as the symmetry breaking generates.

Again, we must worry about what kind of sense it makes to speak of the wave function as existing in the transcendent, which usually means a different reality or universe. Of course, in science fiction, all kinds of things exist just across the line in a different dimension, other universes, other time lines of alternate realities, "subspace," etc. Here, all we need is Kantian Quantum Mechanics, i.e. transcendence which is not really separate or different, but a bit like the plumbing, the backstage, of the phenomenal world. Under the surface, the future is being prepared, but the future is limited by the laws of nature and the possibilities that they allow. But even understanding the laws of nature cannot uncover all the possibilities, all the potentialities of existence. The imagination can do that sometimes, but it is rare indeed.