### Crystals of time

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With title “crystals of time” I am jumping into the future. Indeed, yesterday I received email from one of my colleagues in Russia. He brought to my attention the recent news about “time crystals“: Scientists unveil new form of matter: Time crystals. He has pointed to me that it is something similar to what has been described by Kiwi Bird, and it may have something to do with Dzhanibekov effect that I ma currently in love with. I did not know about the Kiwi Bird, so I checked. Indeed – there is a lot to learn from this bird, and I will have to start studying its songs.

Силуэт птицы киви, соответствующий[1] автору.

Бёрд Ки́ви[2] (также, Киви Берд[3] и kiwibyrd[4]) (от англ. kiwi bird — птица киви) — псевдоним неизвестного автора (группы авторов), который вёл колонку в журнале «Компьютерра», ряде других печатных и онлайновых изданий ИД «Компьютерра» и публикует статьи в журнале «Популярная механика». Основные темы статей — криптография, конспирология и теория заговора.

In short: secret science, conspiracy theories, general scientific weirdness…. A soup made of information mixed with disinformation. Sometimes tastes good.

I looked into the book “Книга о странном”, (Book about Strange) by this strange bird, and of course I have found there the same picture mandala as in my blog post Dzhanibekov effect – Part 2.

The funny thing is that the book starts with “fractals” – “Глава 1. Фракталы истории” as it is with my post with the picture of the mandala. Strange indeed.

But, as I said, for me all this is in the future. For now I have to continue with Dzhanibekov effect. I am done, more or less, with the mathematics of elliptic functions (I did not even know what elliptic function is a year ago!). We have to return back to physics. In particular I am returning to the content of Dzhanibekov effect – Part 4: The equations of motion.

First recollections (Recollection definition, the act or power of recollecting, or recalling to mind; remembrance).

We consider rigid body. There are no rigid bodies in Nature. That is not a problem for us. Take a stone. It is rigid enough for us, unless someone is crashing it with a hammer. A rigid body has its center of mass. Center of mass of a body does not have to be in the body. Center of mass of an empty cup is somewhere in the air inside the cup. It is a point in space, not necessarily in the body. But somehow the body seems to know where its center of mass is. If there are no external forces or torques acting on the body, as it is, for instance, with a winged nut in Dzhanibekov experiment, then we can well assume that the center of mass is at rest with respect to the inertial frame attached to our laboratory. Classical mechanics tells us then that the angular momentum vector is of constant length and its direction is fixed in space. That is called the law of conservation of the angular momentum.

The fact that physicists call something “a law” does not mean that it is “a true law”. Take for instance a boomerang. It behaves strangely. Physicists explain that boomerang is not really free. There is an air. But what if space is always filled with some kind of “air” or “aether” or “vacuum energy”, and that each body can behave, under certain conditions, like the boomerang, or even stranger, by making use of this “vacuum air”? What if? An I think that it is not just “if”, but that it is really so. There is a lot that waits for being discovered. What then?

Interesting question, but we do not have to deal with this question now. We write it down, in order not to forget, an we simply accept the fact that angular momentum in our situation is conserved with a sufficient approximation to use this “law” in our mathematical idealization of reality. A bird in the hand is worth two in the bush.

What is this angular momentum? It can take a whole book to explain it in all details. But I will take a shortcut.

To a rigid body (a stone, a nut, spinning top) we can attach three mutually orthogonal “principal axes“, a “moving frame“, in such a way that the “inertia tensor” of the body is diagonal, it has three components $I_1,I_2,I_3.$ Here I am recalling the content of Dzhanibekov effect – Part 4: The equations of motion. The relation between components of vectors $\matbf{v}$ in the body frame, and components $\mathbf{V}$ of the same vectors in laboratory frame is given by the “attitude matrix” $Q.$ When the body rotates $Q$ in general depends on time.
…………
That being said I have to pause, and I will continue in the next post. I have to finish reading the book by Olga Kharitidi. I am done with 80% of this book. My impression is that she is mostly inventing her story. It does not sound like a true story. Somewhat similar to Castaneda. Perhaps some kind of “channeling” as well.

After I am done with Kharitidi, I have waiting for me “Quest: Evolution of a scientist“. This is autobiographical little book by Polish theoretical physicist Leopold Infeld. Infeld was a collaborator of Einstein and his book, published in 1942, has a lot of gossip. To the extent that when another famous Polish physicist, Mathisson, in a discussion with Infeld mentioned that he is reading “Quest”, Infeld asked: “where did you get it?”. Apparently later on Infeld did not want this book to be read. And then, perhaps, I will start reading the Kiwi Bird and time crystals.

### Elliptic m-deformed relativity

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According to Wikipedia special relativity theory was originally proposed in 1905 by Albert Einstein. But Wikipedia is not always the best source of information. For instance Wikipedia has a section about “Causality and prohibition of motion faster than light“. Quite often we can read sentences like that one:

” Since the moving clouds travel slightly slower than the speed of light, they do not actually violate Einstein’s theory of relativity which sets light as the speed limit.”

It continues to be alleged that superluminal inﬂuences of any sort would be inconsistent with special relativity for the following three reasons: (i) they would imply the existence of a ‘distinguished’ frame; (ii) they would allow the detection of absolute motion; and (iii) they would violate the relativity of simultaneity. This paper shows that the ﬁrst two objections rest upon very elementary misunderstandings of Minkowski geometry and lingering Newtonian intuitions about instantaneity. The third objection has a basis, but rather than invalidating the notion of faster-than-light inﬂuences it points the way to more general conceptions of simultaneity that could allow for quantum nonlocality in a natural way.

The point is that very often physicists do not think. They repeat what someone told them, or what they read, without much thinking. To quote from “Superluminal motions?A bird-eye view of the experimental situation“, Found.Phys.31:1119-1135,2001, by Erasmo Recami

… Still in pre-relativistic times, one meets various related works, from those by J.J.Thomson to the papers by the great A.Sommerfeld. With Special Relativity, however, since 1905 the conviction spread over that the speed c of light in vacuum was the upper limit of any possible speed. For instance, R.C.Tolman in 1917 believed to have shown by his “paradox” that the existence of particles endowed with speeds larger than c would have allowed sending information into the past. Such a conviction blocked for more than half a century (aside from an isolated paper (1922) by the Italian mathematician G.Somigliana) any research about Superluminal speeds.

Science is not free from “religious wars”. But that is not the subject of my post today. My post is about a certain curious observation that gave me some idea, and I do not know whether this idea is new, or it already occurred to someone else before. And I do not care, because the idea may be not crazy enough to be worth of discussing. Nevertheless it fits the subject of discussion in my recent series, so I will tell it to you now, and, perhaps, ask some questions.

In Special relativity we have a strange formula for addition of velocities (here we will discuss only velocities in one space dimension):

Q & A: Relativistic velocity addition

To simplify the notation I will assume that $c=1,$ or, if you wish, I will understand my velocity $\beta$ as the quotient $u/c$ etc.
The relativistic addition of velocities is sometimes denoted as $u\oplus v$
$$\beta\oplus \beta’=\frac{\beta+\beta’}{1+\beta\beta’}.\label{eq:op}$$
John Baez, whom we know from my previous posts, has a web page on How Do You Add Velocities in Special Relativity? There he notices the well know fact that the relativistic addition of velocities is essentially the same as for hyperbolic tangent, where we have
$$\tanh (x+y)=\frac{\tanh x +\tanh y}{1+\tanh x\tanh y}.\label{eq:th}$$

One of the consequences of the above addition formula is that if, say $\beta=0.9$ and $\beta’=0.9$ then $\beta\oplus \beta’=0.994475.$
Your spaceship moves with respect to the Sun with velocity that of 90% of the speed of light, and you send from it, in the direction of its flight, a missile traveling with respect to the spaceship with another 90% speed of light, and yet, with respect to the Sun the missile has the speed of 99% of the speed of light, rather than 180% as we would expect from naive addition.

Now, in the recent series of posts we were discussing elliptic functions, and in particular Jacobi sinus function $\sn(u,m).$ We know that for the parameter $m=1$ we have $\sn(u,m)=\tanh u.$ We also have addition formula for $\sn(u,m)$. It is thus natural to ask how would special relativity look like when the formula (\ref{eq:op}) is replaced by one derived from the addition formula for $\sn(u,m)$ given in the post Elliptic addition theorem:

\mathrm{sn} (u+v,m)=\frac{\mathrm{sn}(u,m)\mathrm{cn}(v,m)\mathrm{dn}(v,m)+\mathrm{sn}(v,m)\mathrm{cn}(u,m)\mathrm{dn}(u,m)}{1-m\,\mathrm{sn}^2(u,m)\,\mathrm{sn}^2(v,m)}.
We can set $\beta=\sn(u,m),\, \beta’=\sn(v,m),$ then $\cn(u,m)=\sqrt{1-\beta^2},\, \dn(u,m)=\sqrt{1-m\beta^2},$ $\cn(v,m)=\sqrt{1-\beta’^2},\, \dn(v,m)=\sqrt{1-m\beta’^2},$ and the new, proposed addition formula, involving parameter $m$ not ncessrily equal to 1, reads:
\beta\oplus_m\beta’=\frac{\beta\sqrt{1-\beta’^2}\sqrt{1-m\beta’^2}+\beta’\sqrt{1-\beta^2}\sqrt{1-m\beta^2}}{1-m\beta^2\beta’^2}.
That is my candidate for the m-deformed relativity. How it compares with the non-deformed (that is “standard”) relativity? It looks weird.
Assume our space-ship travels with the speed 90% of the speed of light. Assume $m=0.9$, and assume we shoot a missile from our ship, in the direction of its motion. What will be the speed of the missile? Here are the plots:

The blue curve is the special relativity. The $0.9\oplus \beta’$ speed always increases, though slower and slower as $\beta’$ approaches 1. But the m-deformed relativity, represented by the red curve is even crazier. If the missile is shot with a speed over a certain value, it starts to move slower with respect to the Sun.

Is that crazy enough to have a chance to be useful?

Can these elliptically deformed addition formulas be included in some geometrical setting? Will it follow from some algebra involving a generalization of the Lorentz group? I do not know.

### Some day are diamonds (living spiraling force) some days are stones (elliptic)

Yes, elliptically speaking: some days are stones. They whirl on a string and when released hit no target. I was writing about slinging stones with a string, like here

Even if you are good at it, not every shot results in a hit. As John Denver puts it:

Some days are diamonds some days are stones
Sometimes the hard times won’t leave me alone

That happens with me as well. Today is one of these stones.

I’d like to say I’ve been fine and I do
But we both know the truth is hard to come by

My attempts in understanding mathematical derivation of the periodicity of Jacobi elliptic functions with imaginary argument failed. I looked into ten or so different derivations by different authors, and each time there was a hole in the argument that I was not able to fill in. That’s depressing.

But I will continue. Perhaps I will find a good argument, perhaps I will invent one, perhaps someone will help me. It would be not fair to give up!

In the meantime, late in the evening, I am reading the book “Entering the Circle: Ancient Secrets of Siberian Wisdom Discovered by a Russian Psychiatrist“, by Olga Kharitidi

Olga Kharitidi’s debut book is a remarkable account of her spiritual adventure in snowbound Siberia. Joining an ailing friend on a spontaneous trip to the Atai Mountains, Dr. Kharitidi is taken into apprenticeship by a native Shaman who guides her through bizarre, magical, and often terrifying experiences that open her eyes to a wellspring of deeper learning. On the road to Belovedia, a fabled civilization of highly evolved beings, she encounters revolutionary mystical teachings while discovering ancient secrets of magic and healing. At once a modern odyssey and a timeless dreamscape, Entering the Circle is an inspiring story of personal growth and an insightful work about the limitless potential of human spirit.

I am reading this book because I have found this piece on Internet:

I first heard about Kozyrev in Olga Kharitidi’s book, Entering the Circle. This non-fiction book is about the author who is a Russian psychiatrist and her journey into the fascinating world of Siberian shamanism. She recounts her experience of traveling to the Altai Mountains and interacting with shamans. They lead her on a series of trance journeys in which she experiences other Times and places. The Shamans tell her Time is not what she thinks; Time is spirals, two of which are currently intersecting, meaning an era of great change on the planet.

Upon return to her “normal” life, she meets a prominent physicist who invites her to his lab. When she goes there, he shows her a sort of Time machine made out of tubular mirrors. He explains that its design is based on the work of a Nikolai Kozyrev. Olga goes into the tube and experiences similar Time journeys to the ones she had with the shamans in the Altai Mountains.

Nikolai Kozyrev was an important astrophysicist sent at the height of his career to the Gulag by the KGB. While imprisoned he had the opportunity to meet Siberian Shamans also jailed there. They taught him about the nature of Time. Upon being released from prison Kozyrev dedicated all his scientific research to the subject of Time. He concluded Time is not an abstract concept or a static reality but rather an “Alive Energy.”

Kozyrev theorized that all of life is drawing energy off an unseen, spiraling force. Time is nothing more than pure spiraling movement. Time is created by spiraling energy patterns. He called these spiraling energy patterns made by time “torsion fields.”

Torsion fields or torsion waves are words used to describe the spiraling flow of “time energy” that Kozyrev discovered. The word torsion means spinning or twisting. These are often also referred to as scalar waves. Time is three-dimensional spinning vortices of energy existing in all sizes. Kozyrev proved that torsion fields can move at speeds exceeding the speed of light. These vortices of time can interact or intersect.

Rotating objects or sources that release energy like the sun, the planet, the center of the galaxy create torsion fields. Torsion fields are electrically neutral, meaning they cannot be detected with current scientific instruments. Kozyrev’s methods can be used and explored and are still being investigated at the Moscow Institute of Studies in Temporology.

The sun, because of its size, is the main generator of torsion fields in our solar system. The torsion field or time spiral coming from the center of our galaxy is another spinning source that has a major effect on life in our solar system. We are intersecting this torsion field and have been since 1998. This intersection ends in 2012.

Now you can understand why I am interested – “Time is three-dimensional spinning vortices of energy ” Very inspiring (in fact I am reading the book in Russian: Харитиди Ольга — «Вхождение в круг» (скачать)).

Perhaps tomorrow. Will tell you about my progress.