Archive for the ‘Einstein’s big toe’ Category

Significant Progress on Einstein’s big TOE.

2011-05-16

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.  nD TGSM:  n (2 or 3) dimensional totally geodesic sub-manifold of a Lorentz spacetime.

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.

There *is* no royal road to geometry

2007-05-11

Archimerged has been studying. In particular, O’Neill’s 1995 book on The geometry of K___ B____ H___s, which title Ark is leaving out so that googling for that misleading keyword will not lead here. (This doesn’t mean he doesn’t believe in b____ h___s. He just feels that the metric has other applications). He bought it around 1996 and has read parts in great detail but skipped some things which he is now working through. He wants a very firm grasp of the Petrov classification (he skipped the beginning of chapter 4) and the precise meaning of principal null directions. Misled by the title into thinking that it dealt only with things he wasn’t interested in, he also skipped section 2.4, and so missed the observation that ∂t is spacelike in a small region just outside of the trapped surface at r = m. So now he knows that the metric is spinning even though outside of r = 2m, it is stationary, and the pretty picture in the previous post is misleading because the vertical dimension is not always timelike. He still hopes to find that there is a way to animate this metric so it looks like the word picture in his earlier musings.

He knows a lot of Wald’s General Relativity, but never became fluent in the language of wedge products xy (also known as bivectors), Λ2, 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 qQ, there must be disjoint open sets Op and Oq with pOp, qOq, and OpOq = ∅.

O’Neill’s method of establishing a Hausdorff result involves a constraint on the matching map μ: Μ → Ν. Consider any sequence of points {pn} in Μ, together with {μ(pn)}, the image in Ν of the first under the matching map μ. Then space Q will be Hausdorff provided that there is no sequence which satisfies both

{pn} → pM \ Μ and {μ(pn)} → qN \ Ν.

In example 1.4.6, two planes M and N (each a copy of R2) 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 = R2 \ {(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): r r, t): t > 0} and {(r, t): r r, t): t 2 \ {(r, t): r ≥ 0 & t ≤ 0} and R2 \ {(r, t): r ≥ 0 & t ≥ 0},

via

ir(t) = (rt − et) / (r − 1) for r ir(t) = (0t − et) / (0 − 1) = et for r = 0

ir(t) = et for r > 0

and

jr(t) = (rt + et) / (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.

Null curves in time

2007-05-05

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 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):

Outgoing null two space dimensions one time dimension

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

Matching a photon up to an electron

2007-05-04

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

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.

Null curves of electron by itself

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).

Why a photon carries only one null congruence

2007-05-03

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.

Spinning spin-half electron, wavy twisting photons, and other puzzles

2007-05-03

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

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).

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

2007-05-01

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.

Photology and Photological spaces

2007-05-01

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

Archimerged finds that it is a bad idea to use well-known terms when subtle changes in the rules invalidate a lot of what people know about the terms. Since he is changing a bunch of rules, he needs some new terms. The systems in an ensemble (i.e., the universes in a multiverse) will be sets of points with certain additional structure.

Ark’s approach starts with a topological space, i.e., a set of points and a topology, which is a set of subsets of the space known as the open sets. He includes a manifold structure so as to be able to label the points. Next, instead of adding a metric, he adds the most abstract form of time and causality. Ark has decided to call a topological space with this added structure a photological space. Photology is an obsolete word for optics, and it seems appropriate.

So, a photological space is a topological space together with a photology, an ordered set (B, H, T, N) of four relations. (A relation is a set of ordered pairs of points. We denote the fact that the pair (p, q) is in the relation R by writing pRq.)  The four relations of a  are

  • the before / after relation denoted bBa or aAb (when point b is before a and a is after b),
  • the here relation denoted pHq (when p and q can be considered to be in the same place at different times),
  • the there (or elsewhere) relation denoted pTq (when p and q can be considered to be in different places at the same time),
  • and the neither here nor there (or null) relation denoted pNq (when p and q can neither be in the same place at different times, nor in different places at the same time, but are on a path light could follow).

Relative to a given point p, all sufficiently nearby points q must satisfy exactly one of pHq, pTq, or pNq. Points which satisfy pHq or pNq must also satisfy either pAq and qBp, or pBq and qAp.

A photology (B, H, T, N) has a close connection to an equivalence class of conformally equivalent metrics, and Ark’s approach uses a manifold together with a photology instead a manifold together with a metric.

This machinery is not really necessary, but it serves to emphasize the difference between this approach and standard general relativity. Ark used to say “all conformally equivalent metrics are really the same thing.”  See Wald, General Relativity, p. 445: conformally equivalent spaces have identical causal structure, and their Weyl tensors are equal.

Back to foundations of physics for a while…

2007-05-01

It may be possible to blow Archimerged’s cover given the information herein. If you succeed, please do not publish your results. Ark likes to toil in anonymity, and believes the basic credit for the work described herein belongs to Einstein and others, and would rather not receive the blame if it fails completely.

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

When Archimerged was in college (the first time), he never knew what he wanted to do, and kept switching majors. Starting in engineering school, he would have studied four semesters of physics, with the last one quantum mechanics. But he switched to anthropology (physical anthro actually taught evolution in detail) and so always felt deficient in quantum mechanics. Within a year or so he dropped out and worked in software for around 10 years, but managed to finish his undergraduate work, transferring to a nearby university just before his freshman credits would have become non-transferable.

Anyway, over the years he has learned quite a lot about quantum physics and related subjects. Intertwined in those years, he went to grad school and got a PhD in chemistry (related to bio-molecules) but by the time he finished he was more interested in solving the big physics problem: unifying gravity with quantum physics. At the beginning of grad school, he had to take undergraduate physical chemistry, not having taken it before. The derivation of the Maxwell-Boltzmann distribution was very interesting, and he considered ensembles to be very fundamental to the nature of the universe.

His approach has always been ensemble-based. At first, he thought of considering every possible pattern of dots on a display, varying in time. Any one sequence of frames makes a movie of sorts, but essentially all of them are just “snow.” But if you add some rules for how the dots start out and how they can change from frame to frame, more meaningful movies might emerge. He felt (and still feels) that the world is like this: particles move about according to certain laws (including the law of gravity, or maybe only the law of gravity) in a given “system” (where an ensemble is a set of systems), and the “real” universe we see is just one of these systems. Not the best of all possible worlds, but the most likely.

When he read a popular account of the “Many Worlds” interpretation of quantum mechanics, he thought that’s right, but why this branching stuff? Eliminate the branching, just start with all possible distributions of particles and all ways for them to move.

So he said, particles just move around according to the law of gravity, nothing else, and that is enough. But, what is the law of gravity? So he studied general relativity. A book club happened to offer an introductory package including Wald’s General Relativity, and he studied the first chapters in great detail. He formed the opinion that exact solutions are the only thing of interest. He also formed the opinion that you start with the metric and calculate the curvature and the energy tensor, rather than starting with the energy tensor and trying to find a metric. While reading Gravitation (usually called MTW after the authors), page 901, he formed the opinion that the principal null geodesics of exact solutions are very important.

It’s now been almost 20 years from the start and 10 years since Ark first ran across the metrics, and Ark has finally figured out a way to use certain simple vacuum metrics as electrons and photons. It really works! (see also blog motto). So he will try to explain it in the following posts.