Thursday, 10 June 2010
Guest post: Carl Brannen, “Four Magnificent Papers by Authors Who Think I’m a Complete Idiot”
Carl Brannen is an electronics engineer with a penchant for theoretical physics, and speculations on alternative theories to mainstream physics describing what our world is made of. He has a master’s degree in Math and in Physics, and is quite skilled with programming. He also owns a blog where he discusses topics of his liking, especially physics. Let us see what Carl has to tell…
It’s quite an honor to be allowed to provide a guest post for Tommaso Dorigo. My working title for this post was “Four magnificent Papers by Authors Who Think I’m a Complete Idiot”. This was in preparation for my book, “A Complete Idiot’s Guide to Elementary Particle Physics.” The first was David Hestenes, of Geometric Algebra fame. The second was Lubos Motl, and his magnificent paper is the one that contains “tripled Pauli statistics”. The third was the somewhat obscure Mark Hadley, who has published a series of papers on a GR based theory of elementary particles. The fourth, David Bohm of Bohmian Mechanics, died too early to be provide an opinion on me, but one of his students says I’ve misunderstood Bohm’s easily understood opinions on relativity. I should link in an opinion of Lubos Motl, and a somewhat erroneous comment on Koide’s coincidences from Mark Hadley.
As if I were in a liferaft adrift on the ocean, I find myself wafting in the direction of the most recent wind. Louise Riofrio has kindly assembled together a series of posts, beginning with this one, discussing the evidence for a slow cosmological change in the speed of light, and promising a post for October, that is the direction I find that this post has written itself in.
The foundations of physics aren’t taught in grad school so much as picked up along the way, as one learns the techniques of calculation. Without classes devoted to the subject, it is easy to find that one has absorbed a certainty about the foundations which those who concentrate on the subject do not possess. This is a universal sociological fact pointed out by the author of Gravity’s Shadow. In bringing light to that peculiar form of blindness which is accompanied by belief that one already knows all that one can about the subject, one finds that the recipient has very little time, and less brain power available to analyze your effort. Accordingly, subjects that require time and effort to understand are off limits. Otherwise I’d be inclined to discuss Baylis’s paper on the problems that arise when one uses geometric algebra to geometrize spinors, and why I prefer density matrices.
So instead I am going to write a polemic against what I see as tendency of modern physics to misuse symmetry. To me, symmetry is a method that one uses to solve a set of equations. Symmetry cannot be an underlying principle in itself. Nor, as we show here, does a symmetry in observations necessarily imply the complete symmetry in the underlying physics.
While physics has been quite successful in abusing symmetry by worshipping it, that in itself is not evidence that symmetry is all there is to it. Nor is self-consistency and beauty proof against disproof. One remembers the elegant theory that the world is a flat plate, and rests on the back of a turtle. “And what is undernearth the turtle?” Another turtle of course. “And underneath that?” Yet another turtle. “And underneath the third turtle?” From there it’s turtles all the way down.
The most successful symmetry theory of physics is the special theory of relativity, from which we know that there can be no preferred reference frame. Newton out, Einstein in. Accordingly, let us derive the special theory of relativity from the very pracitical Newtonian engineering theory of Wave Motion in Elastic Solids, pp 274-281, by Karl F. Graff and kindly printed by Dover at a bargain price of $21.95.
We will assume that space-time is an isotropic elastic solid in 3 classical Newtonian dimensions. Such a media has a definite preferred reference frame, the media itself, and is the last thing one might suppose might lead to Lorentz symmetry, especially given the extreme efforts used to obtain Poincare invariance in the recent literature.
Let u(x,y,z;t) be the strain, that is, the deformation of the point (x,y,z) at time t. A strain in a material sets up a stress, that is, a force that reacts to undo the strain. For an elastic isotropic infinite solid media undergoing small linear deformations, there are two degrees of freedom available to characterize the media. We will use the Lamé parameters, lambda and mu. We will also assume a constant density, rho. Practical engineers need to apply external forces to media, but for our purposes we will leave these off and look only at waves propagating in the media itself. Then, from equation (5.1.3) of the above reference, we have the elastic equations of motion:
A very useful method of simplifying equations is to rewrite them in terms of a potential. For an arbitrary vector field like u, one requires a combination of scalar and vector potentials, the Helmholtz resolution. We write:
The scalar(vector) potential function is arbitrary in that one can add a constant (constant vector) to it without changing u. Of the two, the vector potential is even more arbitrary. Speaking in the physics language, we can make various gauge assumptions about it. The text assumes that the divergence of the vector potential is zero.
Substituting our potentials into the elastic equations of motion, we find that they are satisfied if:
The above equations are massless examples of the Klein Gordon equation, the relativistic generalization of Schroedinger’s equation. In a source-free region, the components of Maxwell’s equation satisfy the Klein Gordon equation, as do the components of the Dirac equation. A slight difference is that the wave speeds depend on the Lamé parameters and the density, rather than being the speed of light. Moreover, the wave speeds for the two wave types are different.
The upper, scalar potential, wave equation corresponds to longitudinal waves and has the faster wave speed. The lower, vector potential, wave equation has the slower wave speed and corresponds to transverse waves.
Suppose elastic creatures made of such a media wish to determine the preferred reference frame. They can find it by looking at the speed difference between the two types of elastic waves; the preferred reference frame is the only one where space appears isotropic. If, however, the creatures in the media are restricted to only measure one of the two types of waves, there will be only one wave speed, and, as with the situation with Maxwell’s equations, they will be unable to distinguish a preferred reference frame. An elastic creature might notice this, and become the elastic Einstein by promoting the idea that contrary to intuition, there can be no preferred reference frame.
The other day at the Crossroads Shopping Center chess club, a friend told me that he had great difficulty understanding how it could be that matter could have so much energy built into it that one could manipulate it into a nuclear explosion. I thought about it for a move or two. I told him that from the point of view of elementary particle physics it was not at all surprising. What was surprising is not that hydrogen bombs are so hot, but instead that the world as we see it, is so cold.
Our experiments in physics are restricted to particles with energies many orders of magnitude less than the Planck energy. The Planck energy is about the amount that a citizen of a developed country uses in electrical power in two weeks. In particle physics, it is difficult to explain why most particles don’t have this much energy. That’s right all that energy in just one electron. Maybe one explanation for the cold temperature of the world as we see it is that the energy per particle dropped due to inflation.
Among the particles that we CAN experiment with, it seems that Lorentz symmetry is exact. Is this because Lorentz symmetry is an exact principle of nature? Or is it because we do not have the resources to excite the higher velocity elastic deformations of space-time?
As far as a unified field theory goes, the elastic equations of motion discussed above are missing a few key details. The most obvious one is that elastic deformations are not quantized. They can come in any size and any energy. In quantizing them, it would be natural to find that the minimum excitation energy for a quanta is on the order of the Planck mass. In our very cold condition, we are interested only in excitations that have energies far far below the Planck mass. Among the elastic deformations, we can eliminate one of the two branches, say the faster one, by assuming that its quantum excitations are all of the order of the Planck energy, and hence are not observed in the cold universe that we see. The remaining excitations will all satisfy the same Klein Gordon equation, and so will satisfy Lorentz symmetry.
Of course there are several other defects in the elastic proposal. The number of deformations is far too few, so the known elementary particles would have to be composites. Accordingly, researchers pursuing these sorts of ideas work on preon theories. But that is another story. What we intend to point out by this post is that Lorentz symmetry is a very slippery rock to stand on, if one’s highest energy experimental measurements for its exactness are 10 orders of magnitude below the natural energy scale. One should not be too surprised if advances in physics are made by people who do not cripple themselves with a slavish devotion to symmetry principles “all the way down”.
Light is a vector, or transverse wave rather than a scalar, or longitudinal wave. In the Standard Model of elementary particles, only the Higgs is a scalar particle, but the Higgs has never been observed. All the observed elementary particles are, like light, vector particles. Well, technically the fermions, such as an electron or quark, are Dirac spinors. A spinor is more or less the square root of a vector. I reject Dirac spinors, preferring the density matrix form. The density matrix squares the spinors, again returning them, more or less, to vector form.
In addition to scalar particles being absent from the experimenter’s observations, research on the interaction between black holes and elementary particles suggest that scalar particles would be rather stranger than is currently expected. See the section on “tripled Pauli statistics” in the above linked paper by Lubos Motl.
The elastic equations of motion were defined under the assumption that the density of the media, rho, is constant. For the usual engineering problems, this is a good approximation for both longitudinal and transverse waves. But of the two types of waves, it is only the longitudinal (scalar) deformations that change the density of the media, the transverse waves preserve density to first order.
Let’s get back to the subject of the constancy of the speed of light. Since the big bang, the universe has considerably thinned out. If we were to naively model space-time as a classical isotropic media, this will result in a decrease in the density. But the spreading of an elastic media is also accompanied by changes to its elastic parameters. The speeds of the longitudinal and transverse (scalar and vector) wave speeds depend on these parameters as follows:
As the universe expands, presumably its lambda, mu, and rho change. And this returns us to Louise Riofrio’s equations for the changing speed of light.