Put in spin-half, and transverse waves and polarization fall out.

See also the other posts in this category. Reading them in the order posted might be a good idea.

In the last of seven lectures on “The Character of Physical Law” which Richard Feynman gave at Cornell University in 1964, he said,

One of the most important things in this “guess — compute consequences — compare with experiment” business is to know when you are right. It is possible to know when you are right way ahead of checking all the consequences. You can recognize truth by its beauty and simplicity. It is always easy when you have made a guess, and done two or three little calculations to make sure that it is not obviously wrong, to know that it is right. When you get it right, it is obvious that it is right — at least if you have any experience — because what happens it that more comes out than goes in. Your guess is, in fact, that something is very simple. If you cannot see immediately that it is wrong, and it is simpler than it was before, then it is right.

Archimerged has been improving his “guess” for over 10 years now, and once again, it looks good. This time it looks a lot better than it ever did before. So, once again, he tries to explain it.

Ark still holds the opinion that the key to everything is the principal null directions (eigenvectors) of the Weyl tensor.  These were discovered after Einstein’s death, and Ark also holds the opinion that Einstein would have seen their significance.  Note that specifying the principal null directions puts strong constraints on the metric, since the Weyl tensor is calculated from the metric, and the principal null directions specify a set of equations which the Weyl tensor must satisfy.

In general relativity, you start by specifying the position and momentum of the matter and energy (using the metric in this specification). The “field equations” simply specify that the Einstein tensor calculated from the metric must equal a constant times the stress-energy-momentum tensor, also calculated from the metric. They are horribly complicated because the metric occurs on both sides, and you can’t solve for it.

Ark notes that Einstein said that the left-hand-side of his equation (the Einstein tensor) was like marble, while the right-hand-side was just an ad-hoc addition, like a cheap plywood annex to a marble building. So he feels no particular loyalty to the field equations of general relativity. Instead, he proposes a different set of field equations. Instead of specifying the stress-energy tensor in advance and trying to find a matching metric, he specifies the principal null vector fields and tries to find matching metrics. Then, given a metric, he thinks he can calculate the stress-energy-momentum tensor to find the mass and momentum distribution.

Each principal null vector field is tangent to a space-filling family (a congruence) of curves, so to specify one of these vector fields is to divide spacetime up into disjoint one-dimensional light-like curves. As a guess to be confirmed, Ark assumes that there are only two distinct fields, in particular, the four fields are a pair of identical twins. A metric which yields these fields is called Petrov type D.

So, what do the principal null curves have to do with particles? Well, if you specify a reference frame (three space axes and one time axis), then at a given time, each curve corresponds to a single point in three dimensional space. So light up selected points in a 3D display and watch as they move along the null curves.

Ark knows how to lay out these curves for a pair of spin-half fermions exchanging spin-one bosons, and he plans to make an animation and post it on YouTube with a link to this blog. The amazing thing is that once you satisfy the spin-half condition (go around twice to get to the starting point), the lighted points moving in space from one fermion to the other take the form of a transverse wave. Things like polarization also fall out. Ark says more came out than he put in.

A fermion is like a sphere rotating in 3D space, with threads attached to every point. The spin-half condition specifies that the sphere must rotate continuously, yet the threads are bundled into a rope with the other end fixed to a point, and they never get twisted or tangled.

Ark actually tried it with four ribbons attached at one end to a tin can and the other to a table. Two ribbons were taped to opposite sides of the top, and two to the bottom. He raised the can above all ribbons and rotated it one turn, and the ribbons got twisted. Then he lowered the can below all the ribbons and continued to rotate it a second turn in the same direction. All the ribbons came untwisted. This is literal spin-half. Ark is pretty sure a picture like this is in MTW Gravitation, but he didn’t quite grasp it the first (or second) time around.

If instead of moving the can up and down, you move the bundle of ribbons so the bundle starts below the can and after one rotation of the can, the bundle has revolved half way and is at the top, and after two rotations, the bundle is back to the bottom, the bundle is describing a cylinder. You can do this with one set of ribbons and two cans. Both cans rotate. The bundle of ribbons revolves about the axis connecting the two cans (which is perpendicular to each can’s axis), but none of the ribbons twist, and the bundle as a whole does not twist or rotate about its own axis.

Now for the finale. Everything described so far is in 3D space. But the ribbons are supposed to define the path of light. They do. In the animation Ark is going to figure out how to make real soon now, a point of light starts at the sphere surface and moves along the ribbon until it comes to the other sphere, where it vanishes. The ribbons are always straight (after they enter the bundle), and the points of light move along the path which the ribbon described at the time the point of light first appeared. So these points move straight from one end of the bundle to the other. But at the next instant, after the first point has moved away from the sphere, the next point starts a little higher. When the spheres have rotated one full turn, the bundle reaches the top of both spheres and starts back down. If the bundle is outlining a cylinder, then the points of light form a helix wrapping around the cylinder. Viewed from the side, it looks like a sine wave. This is circular polarization. If the bundle is moving almost up and down, it is vertical polarization.

For an encore, Ark will exhibit anti-particles. Obviously, if we mark the north pole of a rotating sphere, and say particles rotate clockwise as viewed from the north pole, while anti-particles rotate counterclockwise, we can tell an upside-down particle from an anti-particle. But Ark’s particles don’t seem to have any difference between the north and south poles. But actually, since the arrangement of ribbons in the bundle is fixed, we can put the north pole’s ribbon on the outside of the bundle, and the south pole’s ribbon at the center.

Next time, Ark hopes to explain why a pair of particles move apart, while a particle and an anti-particle move together. And why high frequency is high energy, etc., etc.


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