Lie Sphere Geometry Part 4: oriented spheres

 Circles are one-dimensional. They live in a two-dimensional space. If you throw a stone on a water surface - circular waves appear and move away from the source of the disturbance.

Circular waves appear and move away from the source of the disturbance

Their radii measure time. But we live in a three-dimensional space. If, from outside of space,  we throw a stone on the ether of space, spherical waves appear and move away from the source point. Their radii measure time. So, let us move from circles to spheres. This is harder to imagine, but more realistic.

But our space is spherical itself. Thus we are going to discuss the space of spheres S² on the sphere S³.

To draw S³ we need four-dimensional space R⁴. The scalar product in R⁴ is

x·y = x¹y¹+x²y²+x³y³+x⁴y⁴.

and the norm

||x||² = x·x

There the equation of S³  is

S³ = {x∈R⁴: x·x = 1} .                        (1)

For each m∈ S³ we will create now the family spheres S_t(m):

S_t(m) = {x∈S³: x·m = cos(t)} .           (2)

Remark 1. In (2) both x and m are in S³. From the Cauchy inequality we have |x·m| ≤ ||x|| ||m|| = 1. Therefore -1 ≤  x·m  ≤ 1. Thus setting x·m = cos(t) for some t in [0,2π] makes sense. The set S_t(m) is invariant under all rotations R of R⁴ that fix the rotation axis m. Indeed, if Rm=m, from invariance of the scalar product under rotations Rx·Ry =x·y we get Rx·m =x·m. Thus  S_t(m) has a spherical symmetry. I will not go into why and in which sense it is a "sphere". It is clearly a "point" for t=0 and t=π. Can you see it?

But cos(t) = cos(2π-t). To remove the redundancy we introduce oriented spheres. Or, in other words, we introduce also sin(t) into the description. Every nonpoint sphere S_t(m) is a two-dimensional surface in a three-dimensional manifold S³. Given any point x∈S³, the tangent space to S_t(m) at x is two-dimensional. It is in the three-dimensional space T_x(S³) tangent to S³ at that point. Thus there is a one-dimensional subspace of T_x(S³) perpendicular to T_x(S_t(m)). Therefore there are two opposite unit vectors perpendicular (or "normal") to S_t(m) at x. Each of them define an orientation, it defines what is "inside" the sphere (the opposite is then considered as being "outside").

We choose the following formula to define the normal unit vector n(x) for each x in a nonpoint S_t(m):

n(x) = ( m-cos(t)x )/sin(t).        (3)

Making sense of Eq. (3). Making sense of Eq. (3) requires some work - a dissection of the formula. For that we need some tools, and to take appropriate safety measures.

Safety measures should come first.

Safety measures should come first. Are we safe? There is a dangerous denominator sin(t). What if it is zero? That would be a catastrophe! But we are discussing nonpoint circles, thus t is different from 0 and  π. In the interval [0,2π), and for t different from these two dangerous points, sin(t) is nonzero. We are safe!

Now comes the dissection work. The vector n(x) should be perpendicular to any vector tangent to S_t(m) at x. Is it? How can we see it? To answer this question we need an algebraic characterization of the tangent space T_x(S_t(m)). Let u be a vector tangent to S_t(m) at x. Let x(s)∈S_t(m), s∈R, be a smooth curve through x, with x(0)=x, (dx(s)/ds)|_s=0 = u.  Now x(s) is on the unit sphere S³, thus

x(s)·x(s) =1 for all s.

Differentiating with respect to s at s=0 we get

u·x = 0.                   (4)

On the other hand x(s) is on the same sphere S_t(m) for all s. Thus

x(s)·m = cos(t)   for all s.

Differentiating with respect to s at s=0 we get

u·m = 0.                   (5)

Equations (4) and (5) characterize vectors tangent to S_t(m) at x. Vector u has four components. These four components satisfy two equations. 4-2 =2, thus we have two-dimensional space of vectors tangent to the two-dimensional sphere. It looks good. Now we need to check that our vector n(x) given by (3) is orthogonal to all such u. Since n(x) is a linear combination of m and x, both being orthogonal to u, thus n(x) is also orthogonal to u. But n(x) should be tangent to S³. This condition gives us an additional requirement:

n(x)·x = 0.

This we need to check. So we check, suppressing the inessential denominator:

(m-cos(t)x)·x = m·x -cos(t)x·x = cos(t) - cos(t) = 0.

This can be also understood by writing m-cos(t)x as m - (m·x)x. We subtract from m its orthogonal projection on x, therefore we obtain vector perpendicular to x.

Exercise 1. Verify that n(x) given by Eq. (3) is of unit norm.

Exercise 2. Verify that (m,t) and (-m, t+π) define the same oriented sphere - the same sphere as a set, and the same normal vector field n(x).

Conclusion: The set of all oriented spheres on S3 is isomorphic to

(S³⨉S¹)/Z₂,

where Z₂ = (+1,-1) acts by

(-1)(m,t) = (-m,t+π mod 2π).

We are now ready to add dimensions and to move to the world of conformal transformations.

 The world of conformal transformations.

P.S. 09-04-25 14:44 From yesterday's seminar "What is photon?"

Nikolai Magnitskii

Federal Research Center “Computer Science and Control”, Moscow State University, Ltd “New Inflow”, Russian Federation
"Theory of compressible oscillating ether Results in Physics 12 (2019) 1436–1445"

The paper considers ether as a dense inviscid compressible oscillating medium in the Euclidean three-dimensional space, given at each instant of time by the velocity vector of propagation of the ether density perturbations and satisfying the continuity equation and the ether momentum conservation law. It is shown that a generalized nonlinear system of Maxwell-Lorentz equations that is invariant with respect to Galileo transformations, the linearization of which leads to the classical system of Maxwell-Lorentz equations; laws of Biot-Savart-Laplace, Ampere, Coulomb; representations for Planck’s and fine structure constants are obtained from the system of the two ether equations as well as formulas for electron, proton and neutron, for which the calculated by formulas values of their internal energies, masses and magnetic moments coincide with an accuracy to fractions of a percent with their experimental values which are anomalous from the point of view of modern science. A concept of an ethereal theory of atom and atomic nucleus is presented, which makes it possible to answer many questions about the structure of atom, on which modern science is unable to answer.

Keywords: Ether, Maxwell-Lorentz equations, Ampère and Coulomb laws, Planck’s and fine structure constants, Proton, Electron, Neutron, Atomic nucleus.

Slide from the presentation