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

Archimerged has a collection of well-known metrics he is using to model elementary particles. (The discoverer originally said they were particles). Ark doesn’t feel constrained to follow them exactly, which is a good thing because they don’t exactly work. They serve simply as examples of what is possible. Anyway, these metrics have only two null congruences, and have two Killing (pronounced keyling) vectors (along which the metric is constant). The one he is using for photons is particularly simple. In the usual coordinate basis, one principal null vector is constant with only one nonzero component. The other principal null vector has three nonzero components, but there is a special three-dimensional hypersurface where that vector has only two nonzero components. This makes the surface a *trapped* surface because principal null curves tangent to the surface must stay tangent. Further, the 3D trapped hypersurface naturally divides into 2D trapped surfaces. This is convenient, since the null curves of the metric Ark is using for electrons always stay in a constant “latitude” surface.

Only one of the congruences of the photon metric has a trapped surface. The other congruence is exactly perpendicular to this surface. So, Ark figures a photon *is* this three-dimensional hypersurface. And only one congruence fits into this hypersurface (answering the title question). Now, the surface is kind of funny. All curves in the surface are spacelike except for the principal null curves, which are lightlike. As a photological space, all pairs of points (*p*, *q*) in the trapped hypersurface belong to either the “there” relation (so *pTq* is true) or they belong to the “neither here nor there (null)” relation (so *pNq* and either* pBq* or *pAq* is true).

In the usual coordinate basis, the metric depends on only two coordinate vectors, and the other two coordinate vectors are Killing vectors. Well, on the 3D hypersurface, the metric depends on only *one* coordinate, which corresponds to the latitude of the spinning sphere metric. For the other two, Ark switches to another system in which one coordinate specifies which null curve of the constant latitude surface the point is on, and the other specifies how far along the curve the point appears.

This specifies how to match up all of the ingoing (or outgoing) null curves of the first electron to the null curves of the photon, and at the other end, to outgoing (or ingoing) curves of the second electron.

The four-dimensionally acute reader will notice that so far, Ark’s photon stays attached to the sphere for zero time. In order to work, the photon must occupy a 4D volume of spacetime. But Ark’s photon metric has only a 3D trapped hypersurface. Ark has been stuck on problems like this for years, and he has finally realized that they are non-problems. By abstracting the metric into a photology (before/after, here, there, neither-here-nor-there relations), Ark can take the 3D surface, add a time coordinate, and have a continuous time-series of these 3D surfaces which correspond to the continuous time-series of 3D spheres.

Still to be worked out: a lot, including where one photon (as a light quanta) ends and another begins, and how to attach two photons to the same electron.

2009-10-30 at 09:54 |

Nice try, but it is because photos don’t have pockets that they can only handle one congruence at a time :P