Archimerged subscribes to an ensemble of systems view, also called a multiverse of universes. This speculative background does not provide any reason to believe in Ark’s brainstorm, and there may be other equally valid interpretations of the same geometric mechanisms. It is in the details of building particles out of principal null curves that some excitement seems justified.
But to continue with the philosophy, each universe is a fixed photological space, specified by means of two principal null congruences (families of null space-filling curves). A given photological space is essentially a 4D spacetime with no additional structure or fields. Ark expects that the location and behavior of the matter and energy will be completely specified by the two principal null congruences and the four relations of the photology (before/after, here, there, neither-here-nor-there).
Ark doesn’t yet know if it is necessary to add a spacetime metric to the photological space, but if necessary, the principal null congruences specify the field equations which the Weyl tensor must satisfy. It might be that one can find rulers and clocks in the structure of a photological space. In that case, a metric tensor is unnecessary. But alternatively, solving for the Weyl tensor might invalidate certain photological spaces (no solution) or increase the probability of them (several solutions).
Anyway, back to the geometry.
Ark’s electron is a spinning sphere, the surface of which is infinitely far away (but you can walk around it). Actually, there are two surfaces, one in the infinite past from which the outgoing null curves arise, and one in the infinite future, toward which the ingoing null curves move. There really isn’t any sphere surface in the here-and-now. (Ark is very aware of another interpretation, in which it takes a finite time to reach the surface and the curves continue inside the sphere to minus infinity, but he will worry about that — and what happens on the equator inside the sphere — only if forced to. He has spent years sidetracked on those problems and has decided they are non-problems.)
The null curves of an electron twist around and around the electron axis, always staying at the same latitude. Eventually they straighten out, so that they are moving outward faster than they spiral around.
The image shows several null curves as they straighten out. An electron (in a photological space by itself) consists of three dimensions of these one-dimensional curves (the diagram shows curves at different longitudes): one curve for every longitude angle around the sphere, one curve for every latitude from north pole to south pole (the pole curves are straight lines), and a full set of these for every instant of time. Thus, the electron completely fills its 4D space. The space will be compactified by adding boundary points, and inserted into a 4D volume of a larger photological space. Ark’s animation of a single electron will show a single point moving along selected curves.
That is an electron by itself. In order to make up a photological space containing two electrons and a photon (not worrying about their origin or eventual fate), Ark starts with the flat photological space derived from flat 4-space, and decides where to put the two electrons. The photon is to connect them. So he removes a spaceline line segment with infinite duration — part of a plane with one time dimension and one space dimension. The removed points form a closed set, and can be replaced by another closed set in a standard way. (This needs to be proved in detail, but Ark has good reason to believe it works). The replacement set is a whole photological space with 4 dimensions, namely the photological space consisting only of two electrons and one photon. An number of these may be added. Later Ark will work out how the several electrons can exchange photons of finite duration, how two photons can collide to produce an electron and a positron, etc. First we want to precisely define the photological space consisting of two electrons and a photon.
In the previous post, Ark described the principal null curves of a photon. Each curve is trapped in a two-dimensional surface which is completely spacelike except for the null curves, which are lightlike (null). For points p and q on the same null curve, the photology will always have pNq (q is neither here nor there with respect to p), but for any point r not on that curve, the photology has pTr and qTr (with respect to r, p and q are both elsewhere). Also, importantly, for points p and q on the same trapped surface, the photology never has pHq (p and q can never be regarded as being in the same place at different times). In a coordinate basis in which two coordinates span that surface, the tangent vector has only two nonzero components: the other two coordinates are constant along the null curve. It is convenient to define this 2D surface as an infinitely long cylinder with the null curves twisting around and around as they move from the infinite past to the infinite future. It will be attached to both electrons at the same latitude.
Note that as usual in relativity, different reference frames result in different spacelike surfaces: there is no common definition of simultaneous for observers in different states of motion.
The third dimension of the photon corresponds to latitude on the electron. Each different latitude corresponds to a concentric cylinder at a different radius from the cylinder axis. The north pole of the electron connects to the cylinder axis, while the south pole connects to the cylinder surface. The topology of the cylinder is defined so that the outer surface has only one dimension, which is lightlike (null), just as the axis has only one dimension. Again, depending on the reference frame, different points on different null curves will be regarded as being simultaneous (and hence on a given 3D hypersurface of the photon), but for any of these 3D surfaces, no two points p and q on the surface can ever be regarded as being in the same place at different times.
That covers three dimensions. The fourth dimension is timelike. For a given observer, at every instant of time, the photon consists of a 3D hypersurface made of concentric cylinders, where each cylinder attaches to points at a single latitude of both electrons. All curves on the cylinders are spacelike except for the null curves. Absolutely all curves connecting the cylinders are spacelike. For a different reference frame, the concentric cylinders will comprise a different set of points, but all of the points on the 3D hypersurface are elsewhere with respect to each other.
Note that the photon has only one set of principal null curves specified, while the electron has two principal null curves passing through every point. Thus, really an electron is constantly emitting a photon (really a series of photons) along its outgoing principal null curves, and is constantly absorbing a (series of) photons along its ingoing principal null curves.
So as it turns out, every point in the two-electron space is mapped to a point of the first electron, a point of the photon, and a point of the second electron. (Later Ark will worry about photons which stay connected to an electron for only a finite time, and what to do with the second set of null curves).