Nothing here is particularly new, but as far as I know, the combination has not been proposed (or patented, I hope).

Instead of using public and private keys, we use public and private GUID’s together with a secure iterated hash function, **hash**(*N, message*) with 0 < **hash**() < ** H**, and a secure symmetric key function

Suppose Alice wants to sign messages. She chooses a random private GUID *pvt*, stores it safely, and computes her public GUID *pub* = **hash**(*N, pvt*), where *N* limits the number of signatures Alice can make using this GUID.

To sign her *K*-th message, Alice computes *msghash* = **hash**(1, *message*), *sigkey* = **hash**(*N* – *K*, *pvt*), and *sig* = **crypt**(*msghash*, *sigkey*), and submits {*pub*, *K*, *sig*} to the network together with a transaction fee in bitcoins. If the fee is large enough, and if the block chain does not already contain a signature with public GUID *pub* and a larger serial number *K*, the transaction will soon be included in a block. After the transaction is verified, Alice discloses *sigkey*. Anyone can compute *msghash* = **crypt**(*sig*, *sigkey*), verify that **hash**(*K*, *sigkey*) == *pub*, that **crypt**(*msghash*, *sigkey*) == *sig*, and given message, that **hash**(1, *message*) == *msghash*, but noone can use the newly disclosed *sigkey* to sign a different message unless they can succeed in subverting the blockchain. Unlike a public key signature, this signature requires reference to the blockchain for verification, but when quantum computers arrive, the public key free lunch is over.

It should be possible to use this approach to enhance the digital signature used in the bitcoin protocol without increasing the verification latency time by including {*pub*, *K*, *sig*} in the bitcoin transaction as well as the public key signature, and requiring publication of *sigkey *(but not necessarily inclusion of *sigkey* in the block chain) before the bitcoin transaction is accepted as verified.

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

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.

]]>If you want, look at Adobe’s instructions (click on the “yum” option), but they don’t tell you much about what gets installed where, and they don’t give the md5sum, sha1sum, or the gpg key fingerprint.

Archimerged knows it’s immoral to do this (sorry Richard, we all have our faults), and maybe he’ll get annoyed enough (with the advertisements and restrictions Adobe forces on him) to switch to gnash and never use reader, but seeing as he is neither virtuous nor annoyed enough to switch, he downloaded adobe-release-1.0-0.noarch.rpm into a directory by itself. Being naturally suspicious, he ran

**rpm -qlp *.rpm**

`/etc/pki/rpm-gpg/RPM-GPG-KEY-adobe-linux`

`/etc/yum.repos.d/adobe-linux.repo`

to see what installing it would do. He also recorded the checksums so you can see if you got the same file Ark did. (Adobe ought to publish these).

**md5sum *.rpm**

`e7fc18fa7e15e75be00814d2d4394634 [...]rpm`

**sha1sum *.rpm**

`26801152f27f01038fe1fd9f91df5faa44f2393b [...]rpm`

If you get a different result, please reply here. Next:

**rpm -Uvh *.rpm**

After the GPG key is unpacked, import it to see the fingerprint. Adobe ought to sign this key but last time Ark looked even Red Hat didn’t sign the keys in /etc/pki/rpm-gpg/.

**gpg --import /etc/pki/rpm-gpg/*adobe***

`gpg: key F6777C67: public key "Adobe Systems [...]" imported`

**gpg --fingerprint ff677c67**

pub 1024D/F6777C67 2007-02-28

Key fingerprint = 78A8 75E9 7F09 06BD 6355 73FA 3A69 BD24 F677 7C67

uid Adobe Systems Incorporated (Linux RPM Signing Key)

sub 2048g/7EB7D08B 2007-02-28

Then,

**yum -y install flash-plugin**

and it is mostly done. To check what got installed, Ark thinks there might be a yum command, but he knows the rpm command, which is

` rpm -ql flash-plugin`.

The results are

Finally, Ark looked at ` homecleanup` and ran it with

Adobe doesn’t provide any authentication on the adobe-release RPM, but actually Ark seriously doubts that his download was compromised. If a few other people report the same checksums and key fingerprints, it will be certain.

]]>“Link to this article in your own blogs, by all means, also, if you write about that article and quote a sentence or two from it. That way when that LinuxInsider article appears in searches on Google and other search engines, so will your correction. It all helps to protect my good name, which I obviously care about and should care about.” — PJ

Misleading story in LinuxInsider which quotes Laura DiDio’s inaccurate statements, and which inaccurately summarizes Ubuntu founder and Canonical CEO Mark Shuttleworth’s statement:

Shuttleworth himself places the emphasis of his announcement not on the open source nature of his company’s software, but on the fact that it’s distributed for free.

But what Shuttleworth actually said is not quite the same:

In the Ubuntu community, we believe that the freedom in free software is what’s powerful, not the openness of the code,

“Distributed for free” is not the same as “freedom in free software”.

]]>He knows a lot of Wald’s *General Relativity,* but never became fluent in the language of wedge products *x* ∧ *y* (also known as bivectors), Λ^{2}, Hodge star, etc. He did read Wald’s chapter on Spinor methods once, and realizes there is some connection between the multiplicity of roots of a certain fourth order polynomial and the principal null directions and Petrov classification, but it remains hazy. He used *Mathematica* on a 1-gig system to fully expand the Riemann and Weyl tensors for an arbitrary metric and figure some things out that way, never realizing that O’Neill (and others) have methods of finding the useful results with pencil and paper. He has lots of photocopies of papers, like Newman-Penrose, Goldberg-Sachs, but never became fluent in them either. When he tries to start at the beginning and work through, he has always gotten sidetracked on a tangent, for instance, reading topology books he got interested in whether you can or cannot exhibit a well-ordering of the reals and worked on that for a year. But right now, Ark needs to know exactly what the principal null directions are. (So why is he writing about section 1.4 of O’Neill? Well, probably it really is relevant…)

Ark is planning to make revisions to the following, which is partly wrong.

Some comments on the book. It has has *lots* of typos; perhaps Ark will publish a list *(after* he finishes that animation).

In Section 1.4, “Extending Manifolds,” subsection “Gluing topological spaces,” O’Neill considers two topological spaces *M* and *N* with subsets Μ ⊂ *M* and Ν ⊂ *N* that are to overlap in the final space *Q*. A matching map *μ*: Μ → Ν specifies how the subsets are to overlap and especially how the open sets of *Q* are to be determined. The topology of *Q* needs to be Hausdorff: for every pair of points *p* and *q* ∈ *Q*, there must be disjoint open sets *O*_{p} and *O*_{q} with *p* ∈ *O*_{p}, *q* ∈ *O*_{q}, and *O*_{p} ∩ *O*_{q} = ∅.

O’Neill’s method of establishing a Hausdorff result involves a constraint on the matching map *μ*: Μ → Ν. Consider any sequence of points {*p _{n}*} in Μ, together with {

{

p} →_{n}p∈M \Μ and {μ(p)} →_{n}q∈N \Ν.

In example 1.4.6, two planes *M* and *N* (each a copy of R^{2}) are to be glued together along the open half-planes Μ ⊂ *M* and Ν ⊂ *N* (each {(*r*, *t*): *r* r = 0) axes (each infinitely long) and right half-planes. O’Neill specifies matching map *μ*(*r,t*) = (*r, t*/*r*), neglecting to note that when *t* is held constant at zero, *t*/*r* fails to diverge as *r* approaches zero, so the constraint on *μ* is not met. (Ark was proud of discovering this). When this approach is actually used in Chapter 4, O’Neill does not make this mistake, using something like (*r, t* + 1/*r*), which does diverge for all *t* as *r* → 0.

Ark proposes an equivalent method where the topology of the result and its Hausdorff nature is much easier to see. In the example, before even thinking of gluing the spaces together, he maps each plane onto an open subset of another plane, such that the parts to be glued overlap, while the parts which are to be distinct are mapped to disjoint regions. After gluing the two new planes together using the identity map, the resulting space *Q* consists of a plane less the positive *r* axis, *Q* = R^{2} \ {(*r*, *t*): *r *≥ 0 & *t* = 0}. As a topological space, *Q* is the desired result, and as a manifold, it already contains the necessary coordinate systems. In particular, we have a set of (*r*, *t*) coordinates which cover the left half-plane and the top right quadrant, and another set which cover the left half-plane and the lower right quadrant, although they do not assign the same coordinates to a given point in the left half-plane.

map the planes onto the open sets

{(

r,t):rr,t):t> 0} and {(r,t):rr,t):t2 \ {(r,t):r≥ 0 & t ≤ 0} and R^{2}\ {(r,t):r≥ 0 &t≥ 0},

via

i_{r}(t) = (rt− e^{t}) / (r− 1) forri_{r}(t) = (0t− e^{t}) / (0 − 1) = e^{t }for r = 0

i_{r}(t) = e^{t }for r > 0

and

j_{r}(t) = (rt+ e^{t}) / (r− 1).

…

]]>

Obviously the above is incomplete, and it turns out, partly wrong. But Figure 1.3 in O’Neill is very misleading, and Ark will put his version here. Meanwhile he wants to get this post out the door.

Archimerged has been playing with gnuplot. (He no longer uses *Mathematica*, being a supporter of free/libre software. Sage is a nice free computer algebra package, and it uses gnuplot for graphics.) He got reminded of gnuplot because some images on Wikipedia were made with it. It has a very clunky interface, like VMS, which it was modeled after. (VMS was about the fifth operating system Ark learned). But Ark spent time log ago learning gnuplot, so it’s easier for him than to learn a 3D modeling package like k3d, which he intends to learn, but its faster not to.

Anyway, here we have a nice colored plot of the outgoing null curves of the metric Ark models his electron after (Ark isn’t dropping the name of the metric or giving its usual interpretation):

The third dimension is time and each color is a different principal null curve. You can tell these curves are outgoers because they start at small diameter and as time increases (upward) they move outward from the circle at the center. All of these curves are at the same latitude of the sphere. Note how they are straightening out.

Thinking in 4D is not easy, and intuitions can be misleading. In the end, you have to prove things in algebra to be sure the geometry is right. Ark is presently trying to figure out how to smoothly deform those curves so they all go in the same direction. That’s fairly easy. The hard part is the fact that there is another set of them (the ingoers) to deal with. Maybe it can’t be done. Ark hopes not, but it wouldn’t be the first time. See also the blog motto.

For each pair of distinct points *p* and *q*, either *pNq*, or *pTq*. Relative to *p*, *q* is neither here nor there (lightlike, null), or else *q* is elsewhere (there, spacelike). Every curve on the surface is either the null curve, or it is spacelike. If *pNq*, then either *pBq*, or *pAq*. One is before the other, but no points off a given null curve are before or after any of the points on the null curve.

Another one of these surfaces fits just above (after) this one, but rotated about the axis a little. That is, Ark’s electron is always sending these null curves out somewhere. But another diagram is needed… Off to make it.

Here is the code which produced that plot. Ark used gimp to convert the postscript to png.

set size ratio -1 0.5, 0.5

unset zeroaxis

unset xtics

unset ytics

unset ztics

unset border

unset colorbox

r(a,b) = sqrt(a**2+b**2)

mod2pi(a) = a - 2*pi*floor(a/(2*pi))

phi(a,b) = (r(a,b) > 1 ? mod2pi(atan2(a,b) - 1/(r(a,b) - 1)) : 0)

T(a,b) = Tr(r(a,b))

Tr(r) = (r > 1 ? r - 2/(r - 1) + 2*log(r - 1) : -100)

set parametric

set terminal table

set output "3D.table"

set samples 600

set isosamples 600

splot [0:2*pi] [1/1.05:1/16.0] cos(u)/v, sin(u)/v, T(cos(u)/v,sin(u)/v)

set terminal postscript color enhanced solid linewidth 0.5 portrait

set output "3D.ps"

set cbrange [0:2*pi]

set palette model HSV functions gray,0.75,1

set view 60,0

set pm3d at s explicit corners2color c1

splot "3D.table" using 1:2:3:(phi($1,$2)) with pm3d notitle

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).

]]>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.

]]>Real soon now, Archimerged is going to make that animation. But he has some puzzles first. (Ark likes puzzles).

- The total energy of the electron ought to be proportional to its rate of spin.
- What (in this model) causes interference?
- How does one photon end and another begin?
- How does a photon carry energy?
- What is the meaning of the second (ingoing) principal null congruence?

The above questions are at least partly answered today. These are not:

- Besides its literal meaning (go around twice to return to the original state), spin-half also means the angular momentum is quantized in units of
*half**h*-bar. - What
*is*Planck’s constant anyway?

In the process of making the animation, and solving the puzzles, Ark is doing some concrete thinking in four dimensions. The spinning metric isn’t really spinning so long as the principal null curves end up moving straight to infinity. Rather, the outgoing null curves twist round and round the spherical surface (initially, they move an infinitesimal distance outward for each turn around the sphere), until at a large distance, they began moving outward faster than around. This original metric (Ark is withholding its name because it has irrelevant connotations and also because he converts it to a photological space) is a stationary sphere with null curves wrapped around it which eventually end up proceeding in all directions out to infinity.

When many spinning particles are incorporated into a target photological space, the principal null curves of the particles (both ingoing and outgoing curves, separately) all begin and end wrapped around the spherical surface of particles. None of them ever make it to (or come from) “outer” infinity. The null curves of the target space which reach infinite distance also come from infinite distance. When a group of null curves of a pair of particles are bundled into a photon, all of them run from the source particle to the destination.

Obviously, a null curve which starts as the outgoing null of one particle becomes the trapped null curve of the photon (in a preferred frame, the principal null vector has two zero components), and when it reaches the second particle, it becomes the ingoing null curve. What about the other congruence?

Because principal null curves form a congruence (they fill four-space and never cross), at any given time and place, there is exactly one null curve (from a given principal null vector field) at that event. Ark’s animation is going to show points moving along some of these null curves in three-space. Two of these points will never end up colliding.

Since the original spinning metric isn’t really spinning, Ark is making the sphere spin on its axis so that at infinite distance (in a photological space containing nothing but the sphere), the null curves go around at a specified rate, which is the total energy of the particle.

Now Ark knows that photons of high frequency have high energy. And electrons absorb and emit photons, gaining and losing energy in the process. He is right now investigating exactly how his particles do this (if they do it). The principal null curves of a spin-half electron are arbitrarily straightened and bundled into a rope (the photon) perpendicular to the spin axis, with the north pole curve on the outside and the south pole curve at the center of the rope. The rope describes a cylinder or an elliptical cylinder as it moves above the north pole and below the south pole to permit the sphere to spin without making the rope twist. At the other end of the cylinder, there is another spinning particle where the principal null curves can unbundle and dive back into the maelstrom.

The rope must trace out the cylinder (the straight rope revolving about the cylinder axis) once for every two turns of the sphere. Obviously, the angular frequency of the photon is the angular frequency of this revolving rope. (Ark expects that actually, a photon consists on just one turn around the cylinder, after which a new photon starts, pointing usually in a different direction. If the rope goes around twice, that is two photons one after the other.)

Ark believes that his model fits very closely to Cramer’s Transactional interpretation of quantum mechanics (TIQM), which is also related to Feynman and Wheeler’s advanced and retarded absorber theory. [ref to be added]. One nice thing about this is that Cramer has already established (at least it got published in *Reviews of Modern Physics*) the equivalence of his interpretation with ordinary quantum mechanics, and to some extent with quantum field theory. Of course, Ark would rather use an ensemble of systems (or multiverse of universes, with each universe a fixed 4D spacetime) view in place of Cramer’s “offer wave,” but he isn’t that far into working it out.

One striking thing about TIQM is the advanced and retarded waves, and the negative and positive energy they carry. In thinking about this puzzle, Ark has decided that photons carry *twist* as well as having a frequency: an outgoing photon can cause an increase or decrease in spin, and the simultaneously arriving ingoing photon must exactly match this increase or decrease. Because of the spin-half pattern (the rope revolves around the cylinder once, going above the north pole and below the south pole, as the sphere rotates twice), the twist carried by the rope is precisely the *change* in rotational frequency of the sphere. There are always two of these ropes (which are completely oblivious to each other — they correspond to two separate null congruences), and if they didn’t both twist by exactly the same amount, a great tangle would ensue. Thus, the sphere does not go around precisely twice for one revolution of the rope, but the difference corresponds to the amount of energy delivered or removed by the advanced and retarded photons (ingoing and outgoing congruences).