The Gospel of Q is a hypothetical source document that some scholars believe was used by the authors of the Gospels of Matthew and Luke, containing sayings and teachings of Jesus. It is thought to predate the canonical Gospels and focuses primarily on Jesus' wisdom teachings without a narrative framework.
The Lie quadric Q that we have introduced in Lie Sphere Geometry Part 6: The Gospel of Q is a "canonical Q". Like the canonical Gospel, it is a mixed bag. There is some lack of coherency there. While the spheres (m,t) are considered to be oriented, the sphere S3, on which they live, is considered to be unoriented. But S3 itself can be oriented and can have inside surface and outside surface, and they can be considered as different. The spheres (m,t) can slide either on the outside or on the inside of S3. Why should we be blind to the orientation of S3 itself?
Remark. Of course once we started on our path of seeking a logical consistency, we can follow the lead even farther along this path. Why should we put S3 in R4 instead of on S4? And then S4 on S5, S5 on S6 and S6 on S7? We can then stop with S7, because S7 is "special". S3 is related to quaternions, S7 to octonions, and this is the end of division algebras. Or we can continue to infinity of spheres of higher and higher dimensions, until we reach the unit ball in the infinite-dimensional Hilbert space, where the orientation does not make sense any more, and all infinite-dimensional separable Hilbert spaces are isomorphic - the natural end of our long journey.
In Part 3
we could see the animation of oriented circles gliding on the surface
of the sphere. There we could see the reason for the identification (m,t) = (-m,t+π mod 2π). But now there is no such identification. Before
(m,3π/2) and (-m, π/2) represented the same oriented circle. Now they are different. (m,3π/2) is already on the inner side of the sphere, while (-m,π/2) is on the outer side. Thus we do not need any more to take the quotient by Z2.
Let's go to the math. We modify the definition of Part 6:
Q+ = {[u]∈P+(V): (u,u) = 0},
where P+(V)=(V∖{0})/∼, with x∼y if and only if there exists λ>0 such that y=λx. Topologically P+(V) is S5. (Can you see it?).
Exercise 1. Show the isomorphism of Q+ and S3⨉S1.
Here is one possible speculation: if our universe indeed has two sides, the matter located on the other side would not be "visible", but still can have a gravitational influence on the matter on our side. It would modify the gravity, much like what we call "dark matter" does.
