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.
P.S. 14-04-25 17:43 From the Preface to the book by Pavel Florensky, suggested by Anna:
"... And indeed, in a book published some time later. “Dielectrics and their technical applications” (Moscow, 1924) Florensky came to the conclusion that there are possible unconventional, previously unrealizable ways of obtaining energy at a certain combination of material and spatial medium, which later on in quantum theory called the physical vacuum [aether]."
"... This process is theoretically justified by a new representation of spacetime , according to which a spatio-temporal manifold has its own “SIDE”, that transforms the energy dissipated in the observed physical processes into low-entropic valuable types of energy."
P.S. 15=-4-25 9:51 Yesterday I have participated in a two and half hour long seminar
"A conceptual solution to the hidden mass problem"
Maxim Grigorievich Godarev-Lozovsky
Chairman of the St. Petersburg Philosophy Club of the Russian Philosophical Society, Head of the Scientific and Philosophical Seminar of the Russian Philosophical Society in St. Petersburg, in the House of Scientists in Lesnoye.
A conceptually new scientific and philosophical approach to the solution of the hidden mass problem by astronomers on the basis of the assumption of the logical possibility that gravitation is one-dimensional and electromagnetism is three-dimensional, but these interactions coincide on the two-dimensional fractal surface of the Metagalaxy is proposed.
Long and somewhat chaotic discussion followed the presentation. Some of the participants were sharing their "crazy" ideas. Others were critical about the scientific level of the presented ideas - no peer review publications. I have asked my question about mathematics of "one-dimensional gravitation" to learn that there is none. Just a philosophical hypothesis. Lot of talk about different approaches to "dark matter" and "dark energy". Several aether theories mentioned several times. All penetrating De Broglie's longitudinal photons (Tesla "scalar" waves) were accompanying almost every displayed slide.
hiherg ->
ReplyDeletehigher
where P=(V)=(V∖{0})/∼ ->
where P+(V)=(V∖{0})/∼
"Topologically it is a 3-sphere. (Can you see it?)."
ReplyDeleteNo, despite of what "it" means.
Corrections made. Thanks.
Deletewhere P+(V)=(V∖{0})/∼ ->
DeleteP bold
"Topologically it is S6." ->
DeleteWhat "it" is? P+ or Q+?
Anyway rather not S3 and not S6.
Fixed again. Thanks.
Delete"I would modify the gravity" ??
DeleteMaybe you meant "It would"...?
@Anonymous
DeleteIndeed. Thanks.
"Before (m,3π/2) and (-m, π/2) represented the same oriented circle. Now they are different"
ReplyDeleteYes, surely, they are different - one is inside the sphere and the other is outside it.
Similar ideas go back to Pavel Florensky https://en.wikipedia.org/wiki/Pavel_Florensky
who said that any planar triangle consists of two, one with positive area and the other with negative one. Area is negative when we look at the triangle "from the inside out" (change orientation). The side lengths of such triangle are thus purely imaginary values.
Thanks. Found this:
Deletehttps://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%B8%D0%BC%D0%BE%D1%81%D1%82%D0%B8_%D0%B2_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B8
Interesting! Will have to read "Мнимости в геометрии"
Very glad that you found it interesting! Last summer we went to Solovki to feel the atmosphere where Florensky worked in the end of life, greatly impressive!
DeleteOne more book by Florensky "The inverse perspective" is also in my list to be read
ReplyDeleteThanks. Downloaded for reading. Found this:
Deletehttps://paulbourke.net/miscellaneous/reverseperspective/
Rublev's Icons are also there.
"a new representation of spacetime, according to which a spatio-temporal manifold has its own “SIDE” that transforms the energy dissipated in the physical processes into low-entropic valuable types of energy"
ReplyDeleteThis is precisely what happens inside a living creature - despite of the 2nd thermodynamic law acting in the outside world, it manages to maintain low entropy in the small volume inside its body.
Probably, the double covering has a direct relation to orientability. The manifold has two sides, one with positive area (volume, n-shape generally), the other (from the inside) with negative area. To go from one surface to another, you must literally PASS THROUGH INFINITY (we do not poke holes in the canvas of space).
ReplyDeleteAnd further. The lengths on one side are purely real, on the other - purely imaginary. And the complex numbers (spinor coordinates) keep control of both sides at once! Spinors connect them and, in a sense, "overcome the infinity".
Both S1xS3 and (S1xS3)/Z2 are orientable. S1xS3 is orientable because S1 and S3 are both orientable. (S1xS3)/Z2 is orientable because it carries a Lie group structure. Spinors, for me, are still a mystery.
Delete"Spinors, for me, are still a mystery"
DeleteSpinors are then a transcendental structure that steps over infinity and are always incomprehensible mystery for our mind. They are like hinges on the door into infinity.
Topologically P+(V) is S5. Can you see it?
ReplyDeleteTrying to show that projective space is topologically a sphere.
Projectivity means that [u]=[u'] if u=λu'
So we can write
u^2=(u')^2=(λu)^2=λ^2 u^2
Then, (1-λ^2)u^2=0
Taking, e.g., λ^2= 1-1/(u5)^2, we get
(u0)^2+(u1)^2+(u2)^2+(u3)^2-(u4)^2=1
which is the equation of sphere.
По-моему, всё намного проще. Стереографическая проекция сферы S^5 использует лучи (а не прямые) пространства R^6, то есть половинку проективного пространства P+(V).
DeleteКстати, отсюда следует, что проективное пространство топологически эквивалентно двум соприкасающимся сферам S^5+S^5, то есть двойной оболочке сферы S^5, причём, это относится к любой размерности.
DeleteIt would be more transparent if the proof would be split as follows:
DeleteThe map x -> [x] is from S^5 to P^+(R^6)
a) injective
b) surjective
Unfortunately, I have no practice at all in proving either injectivity or surjectivity. This skill will certainly be needed in doing "Exercise 1. Show the isomorphism of Q+ and S3⨉S1".
DeleteSo i will eagely wait for someone else to present a solution.
Можно попробовать доказать геометрически. Нарисовать на плоскости две соприкасающиеся окружности и проведите через точку касания прямую, которая разбивается на два луча. Затем можно исследовать взаимную однозначность отображения S^1+S^1 --> P(V)=P+(V)+P-(V). Далее надо как-то распространить это доказательство на большую размерность.
DeleteActually, proof of the isomorphism of Q+ and S3⨉S1 repeats that of the isomorphism of Q and S3⨉S1/Z2 in Part 6, and the mapping formula (1) is the same.
DeleteExcept that now (m, t) and (-m, t+π mod 2π) define different spheres, and though x(-m,t+π mod 2π) = -x(m,t), these two points are not equivalent any more since λ cannot be -1. So, two different spheres of S3⨉S1 correpond to two different points of Q+.
@Anna "Actually, proof of the isomorphism of Q+ and S3⨉S1 repeats that of the isomorphism of Q and S3⨉S1/Z2 in Part 6,"
DeleteThat's right!
Уши спинора S1+S1 = 360+360
Delete@Anna
Delete" I have no practice at all in proving either injectivity or surjectivity."
The map x -> [x] is from S^5 to P^+(R^6)
a) injective
b) surjective
Injectivity: We prove that if x,y are in S^5, x≠y, then [x]≠[y].
Indeed, suppose, on the contrary, that [x]=[y]. That means y=λx, λ>0. But, by assumption, x·x = y·y=1. It follows that λ^2=1. But then λ>0 implies λ=1, thus x=y.
Surjectivity: Given any [x] we ant to show that that there exists y in S^5 such that [x]=[y]. Indeed define y=x / (x·x)^(1/2). Then y is on S^5 and [y]=[x].