Topology of Electromagnetism
(c) Robert Neil Boyd

The topic is, in general, topological physics. In particular, we are involved with the topological aspects of E/M. Topology involves dimensionality, in many cases, more than three physical dimensions. Please read the following quote from Tony Smith's website. I provided the links earlier. It helps one to understand, when they look at the links I provide. "The 4 GraviPhoton Special Conformal transformations are like the Moebius linear fractional transformations, that do deform Minkowski spacetime but take hyperboloids into hyperboloids and are the symmetries of superluminal solutions of the Maxwell equations. They are incompressible/linear from the point of view of a 6-dimensional SpaceTime, with 4 spatial dimensions and 2 time dimensions, because the conformal group over Minkowski spacetime is just SU(2,2) = Spin(2,4), the covering group of SO(2,4), and therefore the Lie algebra generators look like those of rotations in a 6-dim vector space of signature (2,4). This is the 4-dim space with 2-dim time suggested by Robert Neil Boyd, in which things look linear (even though from our conventional 3-dim spatial or 4-dim Minkowski point of view they might appear, due to our limited conventional perspective, to be nonlinear). If you regard Physical SpaceTime as the 6-dimensional vector space of Spin(2,4), and Internal Symmetry Space as 4-dimensional CP2, then the total space is 6+4=10-dimensional. With respect to the D4-D5-E6-E7 model, that 10-dim space corresponds:

to the 10-dim vector space of the D5 Lie Algebra Spin(2,8); and

to the 10-dim element of the decomposition of the 27-dim representation of the E6 Lie Algebra into 10 + 16 + 1 under its D5 subalgebra (see, for example, Lie Algebras in Particle Physics, 2nd edition, by Howard Georgi, Perseus Books (1999), page 308).

The Euclidean version of GraviPhotons, generators of Spin(6) = SU(4), correspond to Quaternion 2x2 Matrix Linear Fractional Moebius Transformations.

The corresponding Complex Linear Fractional Moebius Transformations can be used to visualize the usefulness of GraviPhoton SpaceTime Shape-Changing. For example, the Elliptic Complex Linear Fractional Moebius Transformations can be used to make a Complex version of a type of Alcubierre Warp Drive that moves through space by the Alcubierre mechanism (Classic and Quantum Gravity 11 (1994) L73) in which you "... create a local distortion of spacetime that will produce an expansion behind the spaceship, and an opposite contraction ahead of it. In this way, the spaceship will be pushed away from the earth and pulled towards a distant star by spacetime itself....". Since the spacetime around the spaceship is not distorted, the spaceship and its contents feel no G-forces from acceleration.

To try to visualize the full 4-dimensional Quaternionic SpaceTime version of the GraviPhoton Moebius Alcubierre Warp Drive, look at the RP1 x S3 SpaceTime of the D4-D5-E6-E7 physics model in terms of RP1 Time and S3 Space, and then look at the S3 Space part in terms of its Hopf Fibration.

The Hopf Fibration is described by William Thurston in his book Three-Dimensional Geometry and Topology, volume 1, Princeton University Press 1997, pages 103-108, where he says:

"... Identifying R4 with C2 ... If the coordinates in C2 are z1 and z2, the equation of a unit sphere becomes | z1^2 + z2^2 | = 1. Each complex line ... in C2 intersects S3 in a great circle called a Hopf circle. Since exactly one Hopf circle passes through each point of S3, the family of Hopf circles fills up S3, and the circles are in one-to-one correspondence with the complex lines of C2, that is, with the Riemann sphere CP1.

... Informally, the three-sphere is a two-sphere's worth of circles. Formally, we get a fiber bundle p : S3 -> S2, with fiber S1. ... We call this structure the Hopf fibration ...Figure 2.31 shows what the Hopf fibration looks like under stereographic projection. ... the vertical axis is the intersection of S3 with the complex line z1 = 0, and the horizontal circle is the intersection with z2 = 0. The locus { | z2 | < a }, for any 0 < a < 1, is a solid torus neighborhood of the z2 = 0 circle. Its boundary { | z2 | = a }, a torus of revolution, is filled up by Hopf circles, each winding once around the z1-circle and once around the z2-circle. ... any torus of revolution has curves winding around in both directions that are geometric circles ... [called] ... Villarceau circles. ...

... the maps gt : S3 -> S3 given by multiplication by exp( i t ), for t in R, are isometries and ... leave the Hopf fibration invariant. Thus S3 has isometries that don't have an axis: the motion near any point is like the motion near any other point. This is one way in which S3 seems rounder than S2. The one-parameter family { gt } is called the Hopf flow. ...

... The descriptions of S3 via quaternions and via complex numbers can be combined. If we look at [the quaternions] as a complex vector space, multiplication on the left by the quaternion i is the same as multiplication by the complex number i, so the vector field Xi(p) = i p , for p in S3, induces the Hopf flow on S3. ... because i plays no special role among the quaternions ... any pure quaternion of unit norm can be used in lieu of i ... Thus there are many Hopf flows and many Hopf foliations on S3. In particular, we can take three mutually orthogonal vector fields Xi, Xj, and Xk to get three mutually orthogonal families of Hopf circles. ...

... If g in S3 is not +/- 1, the transformation x -> g x fits into the unique Hopf flow generated by the Hopf field x -> p x , where p is the unit quaternion in the direction of the g purely imaginary component of g . Similarly when h in S3 is not +/- 1, the transformation x -> x h fits into a unique Hopf flow, generated by a vector field x -> x p . The two kinds of Hopf flows are distinguished as left-handed or right-handed: the circles near a given circle wind around it in a left-handed sense or in a right-handed sense ... any right-handed Hopf flow commutes with any left-handed Hopf flow. ...". In a given Hopf Fibration, any two Hopf circles are linked.

In terms of the Space S3, the GraviPhoton Moebius Alcubierre Warp Drive looks like a vortex in which the Destination Space is pulled along the Hopf Flow Circles in toward the ship and, after the ship passes through it, is expelled behind the ship. Physically, the 4 covariant polarizations (t,x,y,z) of a GraviPhoton moving in the z-direction correspond to:

t-polarization = Flow in Time circle RP1 = Timelike movement;

z-polarization = Hopf Flow in Longitudinal Hopf circles (with plane of Hopf circle parallel to z-axis) of Hopf Fibration = Spacelike movement in the z direction; and

x and y polarizations = transverse polarization of conventional Far Field GraviPhoton.

The structure is that of a Penrose Twistor, as described in the book Spinors and Spacetime, volume 2, by Penrose and Rindler (Cambridge 1986), pages 61-62: "... These curves are ... circles ... They twist around one another (hence the term twistor!) in such a way that every pair of circles is linked ...

... They lie on the set of coaxial tori ... [that] ... are the rotations about the z-axis of a system of coaxial circles in the (x,z)-plane. From the point of view of the compactified space-time ... we should regard the hyperplane t = tau as being compactified conformally) by a point at infinity. It then becomes topologically a three-dimensional sphere S3 (of which the hyperplane t = tau may be regarded as the stereographic projection). The vector field on S3 is everywhere nonsingular and nowhere vanishing. The circles constitute what is known as a Hopf fibring of S3. With a suitable scaling they become Clifford parallels on S3. ... all the circles in the hyperplane thread through the particular (smallest) circle ... the radius of this smallest circle ... is given, generally, ... by the spin divided by the energy ...".

A sequence of Hopf tori, each made up of Hopf-Clifford circles, is shown in a .mpg movie on a UBC web page.

If these limited descriptions are insufficient to contribute to your understanding, please, if it is possible for you to do so, examine the materials on Tony Smith's website at

Also, please examine the materials on R. M. Kiehn's website at

I am especially fond of his CONFORMAL THEORIES, CURVED PHASE SPACES, RELATIVISTIC WAVELETS AND THE GEOMETRY OF COMPLEX DOMAINS to be found at:

Another rather nice one is ALGEBRAS, SYMMETRIES, SPACES.

I am particularly fond of his description of Shipov boundaries here. Also, see "Infinite Dimensional Geometry, Non Commutative Geometry, Operator Algebras, Fundamental Interactions":

in which Ark has a major contribution.

Perhaps you might find some time to discover the various algebras and their relationships to manifolds and dimensionality.