Archimerged was about to delete all his pages and disappear, planning to pretend he never wrote all of those manic posts about extracting energy from 30 Kelvin delta T, and plotting to publish his latest work with no reference to his manic and embarrassing alter-ego, who is presently submerged thanks to Lamictal. But there is no point in pretending to be what you aren’t, and Archimedes is an impressive namesake (provided you ignore Archimedes Plutonium of USENET infamy). So, here are the results of the last few years of occasional bouts of study, particularly of O’Neill’s chapter on the Petrov classification, and visualization of moving points in 3D.

PND: principal null direction. ** n**D TGSM:

*(2 or 3) dimensional totally geodesic sub-manifold of a Lorentz spacetime.*

**n**First and foremost, unrepeated principle null directions do * not* make null geodesics. (The hazards of skipping the hard-to-understand stuff and cutting to the chase).

Second, the Newman-Penrose formalism is really useful. Totally geodesic sub-manifolds (TGSM’s) are important. Lorentz TGSM’s in which all directions are spacelike but one repeated PND, similar to the trapped surfaces found in the Kinnersley case II metrics, are ** really** important. Study of Kinnersley’s type D metrics yields a lot of insight. But a multi-particle spacetime cannot be type D everywhere.

Third, by specifying repeated principal null directions on 2D and 3D sub-manifolds, we get initial conditions for solving the PDE’s for a metric (up to conformal factor) which specifies that those directions ** are** principal null. This is similar to my earlier plots regarding abandoning the stress-energy tensor as predecessor to the metric, quoting someone who paraphrases Einstein about the G tensor being of marble while the stress-energy tensor is of cheap plywood, and trying to paste Kinnersley’s metrics together while keeping all of spacetime type D. But this time, the supposed route to an actual metric seems to lead somewhere.

Fourth, some details:

Only the particles have repeated principal null directions, usually one double direction and two unrepeated directions. Where particles intersect, there is a one-dimensional surface which may have two double directions, one from each particle. The principal null geodesics always eventually form loops, see below.

The spacetime manifold is as described in Einstein and Rosen, *Physical Review*, 1935. That paper is rarely referenced because the details contained a significant error: the metric was not free of singularities as advertised, because it could be extended to a metric which has singularities (see Visser, 1996, *Lorentzian Wormholes*, p. 47). But I use the basic idea: two world-sheets connected by bridges.

The theory matches the program (Kuhn would call it a “new paradigm”) repeatedly pleaded by Einstein (see, *e.g.,* Schilpp, 1949, p. 675).

An accusation, …, ‘Rigid adherence to classical theory’. … demands … a defense. … Newton’s theory deserves the name of a classical theory,… abandoned [because] the idea of forces at a distance has to be relinquished…

Consequently, there is, strictly speaking, today no such thing as a classical field-theory; one can, therefore, also not rigidly adhere to it. Nevertheless, field-theory does exist as a program: “Continuous functions in the four dimensional continuum as basic concepts of the theory.” …

Archimerged’s latest guess (Feynman, “Guess [and] compute the consequences,” *Character of physical law*, 1994, p. 165) specifies initial condition surfaces for four continuous unit vector fields on the 4D continuum, the principal null directions. A metric is computed up to conformal factor from these four fields and the PDE’s which arise when Weyl tensor is contracted against each principal null field in turn. There are no additional fields. But the initial condition surfaces for the principal null fields comprise all of the particles, and at the same time avoid “infinitely high frequencies and energies” (Schilpp, 1949, p. 676) which arise if point particles are specified in the stress-energy tensor. The statistics inherent in quantum theory arise when an ensemble of spacetimes is considered. Quantum interference in the two-slit experiment arises from the requirement that the fields be continuous (see below).

The particles arise on closed 2D or 3D surfaces where two or more of the fields coincide (up to sign, which is irrelevant). The repeated principal null directions are tangent to null geodesics, which are trapped in those surfaces and which eventually form closed loops.

A boson seems to be a closed 2D surface, which can be visualized as two 1D loops both starting at a fermion bridge, and moving along two different paths, one on each world-sheet, until the two loops reach a second fermion bridge where they complete the closed loop geodesics. The loop comprises points (events) on parallel PN geodesics. Exactly which points belong to the loop at a specific instant in time depends on the observer. To an observer moving with respect of the fermion, the loop starts as a single point on the fermion surface, splits into two points on the fermion joined by a curve looping out of the fermion, and forms a closed loop with one point on the fermion when boson emission is complete.

The boson is obviously a highly curved surface which will distort spacetime significantly. Hence, a distant observer will observe acceleration of the fermion in the direction of the boson when it absorbs or emits a boson. Two fermions of opposite electric charge will emit and absorb the boson on adjacent sides. But when fermions of the same charge exchange a boson, the boson will have to approach one or the other from the distant side in order to achieve repulsion. The details of the topology of bosons and fermions must be dictated by this and other experimental observations.

A pair of fermions seems to be a 3D surface which bridges the two world sheets. One possibility can be visualized as a torus which immediately splits into two tori. Each torus can be visualized as a 2D surface in space, moving in time. Each point on the 2D surface is part of a different principal null geodesic. They all are separated by spacelike intervals. The points along each geodesic are (obviously) separated by null intervals. Thus, no point of the 3D surface is in the past or future of any other point. They are all either on the light cone, or “elsewhere”. If a few of the points on a 2D fixed-time slice of the surface are visualized, they are seen to move around the circumference of the torus and to describe a helix wrapped through the hole.

The “inside” of the torus “contains” all of the second world sheet. If you visualize approaching the surface, so that it flattens out, and pass through it, you see a surface behind you, and if you keep moving in the same direction, you discover that it is a torus with a helix of opposite handedness, which “contains” all of the first world sheet.

A pair of fermions is created when a pair of bosons of sufficiently high curvature meet (details to be determined), allowing two oppositely charged fermions to diverge, with the principal null geodesics of one fermion of the pair forming a closed loop with the geodesics of the other. They remain separate until eventually they collide and annilhilate. So Wheeler’s idea that there is only one electron which moves back and forth in time might work out. (Gleick, *Genius*, 1992, p. 122). In that case, the closed loop of principal null geodesics of the electron would traverse all of the electrons and positrons in the universe before reaching its starting point.

From the requirement that the positron be “an electron moving backward in time,” I expect to find rotation which reverses when the spin axis is reversed. From the observation of spin splitting in electron magnetic resonance, I expect to find a rotation which is unchanged when the spin axis is reversed. But if the topology is of a sphere, there is only one kind of rotation. This suggests the torus. If a given principal null geodesic wraps once through the hole and once around the circumference, when you turn the torus over in space, the rotation about the circumference reverses, but the handedness of the helix created when the curve loops through the hole does not change.

From the existence of the muon and tau, I expect to find a minimum mass, and variations with higher mass. Once around and once through is the minimum curvature which creates both types of rotation. If the geodesic wraps twice through the hole, the curvature would be higher, corresponding to a muon. Three times through would be the tau. More times through the hole would yield such high curvature that the W boson is not heavy enough to create such a fermion, or that the accessible photon energies are not high enough to create a pair of such fermions.

The two slit experiment requires that motion of a particle from A to B always involves two paths, and the probability of reaching B must depend on the phase of the total path length, either the sum for bosons, or the difference for fermions (IIUC).

The phase of a fermion involves the path of one geodesic around the circumference of the torus, relative to a fixed point on the torus as viewed by a distant observer. Each point of the 3D surface is shared by each world-sheet, but the paths may vary. Also, the phase can vary because a given “point” (which is never on the same null geodesic from moment to moment) as seen by a distant observer in the first world sheet would be fixed, while the “same point” would be seen to rotate around the 2D torus by the distant observer in the second world sheet, permitting each path to cover a different distance involving a different number of full turns. It remains to be proved that the difference in the phases of the paths controls (as cos^2 of phase difference) the number of ways a given pair of paths can exist.

The phase of a boson involves the path of the geodesic “around” a spacelike loop as seen by a distant observer in one world sheet compared to the motion as seen by a different observer in the other world sheet. When the two bosons meet at the destination fermion, each geodesic must close so that it loops exactly once between the source and destination fermions. It remains to be proved that the sum of the two phases controls (as cos^2 of phase sum) the number of ways a given path can exist.