Spacetime and anti-space time theory

 As promised in the last post, here is the beginning of the theory behind the images from the previous post:

The three sets correspond respectively to the yellow, blue and the separating areas on the figures in the last post.

Exercises:

1) Prove that [2u]=[u].

2) For any p  define -p, so that it is well defined.

3) To which of the three sets -p belongs? 

4) Show that 

-M_p⁺ = M_p^-, 

-M_p^- = M_p⁺ , 

-∞_p = ∞_p.

5) Prove that M_p⁺ and M_p^-  are non-empty.

P.S.1. Friday January 27, 9:30 AM: replaced yesterday's version of the file, changing a little bit the notation.

P.S.2. Saturday, January 28, 15:44. Added the last section - dealing with the construction work. It is being constructed right now. Pieces added and removed, until the foundation is firm so that it can survive even earthquakes.

Actually it ends with a "Hypothesis". I will try to check now if it is true or not? If it will prove to be correct, it will be a plus for my intuitions. If it will prove to be wrong - it will mean that my intuitions are not yet good, and that there is is an extra work needed in this particular area. And that is important: to state specific, clear and bold hypotheses, and to work hard to prove or disprove them!

P.S.3. Time "flows" backward in anti-space M_p^-. Whatever that means. That's evident to me. It must be so. Otherwise the Universe would be unbalanced - a dreadful thought.

P.S.4. 9:45 AM. If "our" universe is S^1xS^3, then there should be also the next level universe, which is S^1xS3xS^7. 

P.S.5. I will post the new version of the pdf file on Monday afternoon. The "Hypothesis" will then become a "Proposition".

P.S.6. Today I had the following conversation on my old Polish blog:

30.01.2023 X: I do not fit in your blog, so let me ask here.

"Logically, the sentence: if 1=2 then 3=33 is true. "Did the three squeeze into you, or am I missing something ?
30.01.2023 Arkadiusz Jadczyk
don't see the question....

X:30.01.2023
From the falsehood is false - "3=33", and the blog is that we get a true sentence.

30.01.2023 Arkadiusz Jadczyk

The whole sentence "If 1=2 then 3=33 " is logically true.
Whereas without this "if", the " 3=33" itself is false.
Context is important.

X: 30.01.2023

Mother of one, such a compromise of my favorite logic. I hope this remains a secret. ;)

The intuitive absurdity of the implication triumphed over memory, or maybe it was the stupefaction of discussing the integral from dx.

Thank you for your reply.

30.01.2023 Arkadiusz Jadczyk

Success with dx integrals. Thank you for your question. He who asks does not wander.

Source: https://www.ruspeach.com/en/handbook/proverbs/1878/
© www.ruspeach.com - Russian for foreigners

This causes me to think: such situations are not just restricted to abstract logic. They happen in real life. Suppose you tell your friend: If you do this, then I will do that. Time goes on, and, after a while, your friend goes around complaining about you: "He had promised me that he will do that, and he has failed to keep his promise!" The "If" part, that he has failed to fulfil,  is conveniently forgotten. His friends will never learn about the "if" part, and even if some of them do learn, it will be neglected as unimportant. Such is our tendency to look blameless, smart, and innocent in our own eyes. While such a tendency is probably hardwired in our brains - it only shows that our actions do obey the good old Aristotelian logic that we supposedly know so well.  

(Warning: no one should take the above as a "hidden message". Nothing is hidden there, and I did not have in mind any particular person. I am writing about what I consider a general useful wisdom.)

P.S.7. Monday 15:50: I am still not happy with the actual version of the pdf. It still needs verification and corrections. I am currently working on it.

P.S.8 Monday 18:50:  I am not finished yet, as I did not yet describe how the images has been created. This will come tomorrow.

P.S.9. Tuesday 7:50: Today is this "tomorrow". I decided to leave the creation of the images as an exercise for the Reader. All the theory that is needed is already described in details. More important is now to connect the SO(4,2) description with the SU(2,2) description. In their ingenious paper (see the reference in the pdf file) Wojciech Kopczyński and Lech Stanisław Woronowicz provided such a connection. 

W. Kopczyński and L.S. Woronowicz, A geometrical approach to the twistor formalism, Rep. Math. Phys. Vol 2, pp. 35-51 (1971). It is a paper of an indescribable beauty!!!  A true Spring Sonata. Therefore in the next posts I will write about my own little variations on this subject. My main addition, my little twist,  will be: adding the orientation. In the SU(2,2) description, till now, there is no place for "anti-space". Anti-space, anti-time, anti-matter, anti-gravity - that's what I love! There would be no life, no stability without these "antics".  

P.S.10. Tuesday 16:15: Change of mind. I still have to define the conformal structure (i.e. metric up to a variable scale factor). And then to study null geodesics (light rays). This can be done without using SU(2,2). Though SU(2,2) will add an extra perspective. But that's for later. Practical question: can light travel between space and anti-space? We need to know!

P.S.11. Wednesday 9:11: By "chance", searching the net for "graph of a unitary operator" I have found this amazing video demonstrating how linear algebra can enhance and lighten our lives:

 It also shows that in fact there exists a left-handed anti-universe.

Imagine Infinity - A Challenge

 At the end of the Universe there is Infinity. What shape does it have? No abstract mathematics is needed in order to answer this question. You need just Light. No algebraic topology, no Hopf algebras, no category theory. It is as simple as it can be - and yet imagining it can be challenging. And so here I am presenting you with this challenge. And I am accepting this challenge myself, since at present I do not have a clear answer.

So here it is.

We are in 4D, coordinates X,Y,Z,W. The infinity is given by the intersection of these two cylinders:

1) X² + Y² + Z² = 1

2) Z² + W² = 1

We notice that necessarily X,Y,Z,W must be in the interval [0,1]. So our infinity is bounded, it is contained within the unit cube of the four-dimensional space.

Next thing we notice is that there are two equations for four variables. Therefore infinity is a two-dimensional surface (4-2=2). So it should be possible to visualize it in a 3D space in which we live. Of course there will be infinitely many ways of visualizing it, depending on our creativity. Some will be more appealing than others.

I will be working on this problem myself with my poor imagination. You can use yours.

The theory behind this challenging simple problem I will present in one of the next posts.

The fact is: This is the double cover of the conformal infinity of the Minkowski space as discussed by Roger Penrose - the mathematician who in 2020, was awarded one half of the Nobel Prize in Physics for the discovery that black hole formation is a robust prediction of the general theory of relativity.  In the past I have already discussed it here.  But today I doubt  if my proposed solution is a good one. I think it may be either wrong or incomplete.

This photo is not presenting a solution of our challenge 

 but perhaps the true solution will be in some sense similar? I am not sure...

P.S.1. The challenge problem  addresses a "toy infinity", good for (2+1)-dimensional spacetime, thus stripped of one space dimensions. 

P.S.2. Is it a "twisted torus"? Whatever it means.....

P.S.3. Seeking for the solution

The Seekers - Walk With Me

The Carnival Is Over

by The Seekers

P.S.4. We need to know ALL about this surface at infinity. It is infinitely important. Light can travel there in no time, and come back bringing us information. It is our past and our future. It is a homogeneous space for the Poincare group. Therefore a specie of elementary particles is living there. What are these particles? What do they do? What kind of a geometry this infinity carries? All these questions MUST be answered.

I have found a book "Surfaces in 4D space" by three authors Scott Carter,  Seiichi Kamada and Masahico Saito. But it is too difficult for me. I do not know the techniques. And my surface is probably extremely simple: it has no knots etc. But I want to know all the topological invariants - if there are any non-trivial.

P.S.5. Laura is telling me that I should write a book about this Conformal Infinity. Perhaps. But first I need the results. A lot of them. And good quality results.