From SU(1,1) to the Lorentz group

From the two-dimensional disk we are moving to three-dimensional space-time. We will meet Einstein-Poincare-Minkowski special relativity, though in a baby version, with x and y, but without z in space. It is not too bad, because the famous Lorentz transformations, with length contraction and time dilation happen already in two-dimensional space-time, with x and t alone. We will discover Lorentz transformations today. First in disguise, but then we will unmask them.
[latexpage]
First we recall, from The disk and the hyperbolic model, the relation between the coordinates $(x,y)$ on the Poincare disk $x^2+y^2<1$, and $(X,Y,T)$ on the unit hyperboloid $T^2-X^2-Y^2=1.$ [caption id="attachment_3054" align="aligncenter" width="640"] Space-time hyperboloid and the Poincare disk models[/caption]

\begin{eqnarray}
X&=&\frac{2x}{1-x^2-y^2},\\
Y&=&\frac{2y}{1-x^2-y^2},\\
T&=&\frac{1+x^2+y^2}{1-x^2-y^2}.
\label{eq:XYT1}\end{eqnarray}

\begin{eqnarray}
x&=&\frac{X}{1+T},\\
y&=&\frac{Y}{1+T}\label{eq:Xx1}.
\end{eqnarray}

We have the group SU(1,1) acting on the disk with fractional linear transformations. With $z=x+iy$ and $A$ in SU(1,1)
$$A=\begin{bmatrix}\lambda&\mu\\ \nu&\rho\end{bmatrix},$$
the fractional linear action is
$$A:z\mapsto z_1=\frac{\rho z+\nu}{\mu z+\lambda}.$$
By the way, we know from previous notes that $A$ is in SU(1,1) if and only if
$$\nu=\bar{\mu},\,\rho=\bar{\lambda},\quad|\lambda|^2-|\mu|^2=1.$$

Having the new point on the disk, with coordinates $(x_1,y_1)$ we can use Eq. (\ref{eq:XYT1}) to calculate the new space-time point coordinates $(X_1,Y_1,T_1).$ This is what we will do now. We will see that even if $z_1$ depends on $z$ in a nonlinear way, the space-time coordinates transform linearly. We will calculate the transformation matrix $L(A),$ and express it in terms of $\lambda$ and $\mu.$ We will also check that this is a matrix in the group SO(1,2).

The program above involves algebraic calculations. Doing them by hand is not a good idea. Let me recall a quote from Gottfried Leibniz, who, according to Wikipedia

He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal’s calculator, he was the first to describe a pinwheel calculator in 1685[13] and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is the foundation of virtually all digital computers.

I used Mathematica as my machine. The same calculations can be certainly done with Maple, or with free software like Reduce or Maxima. For those interested, the code that I used, and the results can be reviewed as a separate HTML document: From SU(1,1) to Lorentz.

Here I will provide only the results. It is important to notice that while the matrix $A$ has complex entries, the matrix $L(A)$ is real. The entries of $L(A)$ depend on real and imaginary parts of $\lambda$ and $\mu$

$$\lambda=\lambda_r+i\lambda_i,\, \mu=\mu_r+i\mu_i.$$

Here is the calculated result for $L(A)$:

L(A)=\begin{bmatrix}
-\lambda_i^2+\lambda_r^2-\mu_i^2+\mu_r^2 & 2 \lambda_i \lambda_r-2 \mu_i \mu_r & 2 \lambda_r \mu_r-2 \lambda_i \mu_i \\
-2 \lambda_i \lambda_r-2 \mu_i \mu_r & -\lambda_i^2+\lambda_r^2+\mu_i^2-\mu_r^2 & -2 \lambda_r \mu_i-2 \lambda_i \mu_r \\
2 \lambda_i \mu_i+2 \lambda_r \mu_r & 2 \lambda_i \mu_r-2 \lambda_r \mu_i & \lambda_i^2+\lambda_r^2+\mu_i^2+\mu_r^2
\label{eq:a2l}\end{bmatrix}

In From SU(1,1) to Lorentz it is first verified that the matrix $L(A)$ is of determinant 1. Then it is verified that it preserves the Minkowski space-time metric. With $G$ defined as
$$G=\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&1\end{bmatrix}$$
we have
$$L(A)GL(A)^T=G.$$
Since $L(A)_{3,3}= \lambda_i^2+\lambda_r^2+\mu_i^2+\mu_r^2 =1+2|\mu|^2\geq 1>0,$ the transformation $L(A)$ preserves the time direction. Thus $L(A)$ is an element of the proper Lorentz group $\mathrm{SO}^{+}(1,2)$.

Remark: Of course we could have chosen $G$ with $(+1,+1,-1)$ on the diagonal. We would have the group SO(2,1), and we would write the hyperboloid as $X^2+Y^2-T^2=-1.$ It is a question of convention.

In SU(1,1) straight lines on the disk we considered three one-parameter subgroups of SU(1,1):

\begin{eqnarray}
X_1&=&\begin{bmatrix}0&i\\-i&0\end{bmatrix},\\
X_2&=&\begin{bmatrix}0&1\\1&0\end{bmatrix},\\
X_3&=&\begin{bmatrix}i&0\\0&-i\end{bmatrix}.
\end{eqnarray}
\begin{eqnarray}
A_1(t)&=&
\exp(tX_1)=\begin{bmatrix}\cosh(t)&i\sinh(t)\\-i\sinh(t)&\cosh(t)\end{bmatrix},\\
A_2(t)&=&
\exp(tX_2)=\begin{bmatrix}\cosh(t)&\sinh(t)\\ \sinh(t)&\cosh(t)\end{bmatrix},\\
A_3(t)&=&
\exp(tX_3)=\begin{bmatrix}e^{it}&0\\ 0&e^{-it}\end{bmatrix}.
\end{eqnarray}

We can now use Eq. (\ref{eq:a2l}) in order to see which space-time transformations they implement. Again I calculated it obeying Leibniz and using a machine (see From SU(1,1) to Lorentz).

Here are the results of the machine work:

L_1(t)=L(A_1(t))=\begin{bmatrix}
1 & 0 & 0 \\
0 & \cosh (2 t) & -\sinh (2 t) \\
0 & -\sinh (2 t) & \cosh (2 t)\end{bmatrix},

L_2(t)=L(A_2(t))=\begin{bmatrix}
\cosh (2 t) & 0 & \sinh (2 t) \\
0 & 1 & 0 \\
\sinh (2 t) & 0 & \cosh (2 t)
\end{bmatrix},
L_3(\phi)=L(A_3(\phi))=\begin{bmatrix}
\cos (2 \phi ) & \sin (2 \phi ) & 0 \\
-\sin (2 \phi ) & \cos (2 \phi ) & 0 \\
0 & 0 & 1 \\
\end{bmatrix},
The third family is a simple Euclidean rotation in the $(X,Y)$ plane. That is why I denoted the parameter with the letter $\phi.$ In order to “decode” the first two one-parameter subgroups it is convenient to introduce new variable $v$ and set $2t=\mathrm{arctanh}(v).$ The group property $L(t_1)L(t_2)=L(t_1+t_2)$ is then lost, but the matrices become evidently those of special Lorentz transformations, $L_1(v)$ transforming $Y$ and $T$, leaving $X$ unchanged, and $L_2(v)$ transforming $(X,T)$ and leaving $Y$ unchanged (though with a different sign of $v$). Taking into account the identities
\begin{eqnarray}
\cosh(\mathrm{arctanh} (v))&=&\frac{1}{\sqrt{1-v^2}},\\
\sinh(\mathrm{arctanh} (v))&=&\frac{v}{\sqrt{1-v^2}}
\end{eqnarray}
we get
L_1(v)=\begin{bmatrix}
1 & 0 & 0 \\
0 & \frac{1}{\sqrt{1-v^2}} & -\frac{v}{\sqrt{1-v^2}} \\
0 & -\frac{v}{\sqrt{1-v^2}} & \frac{1}{\sqrt{1-v^2}}\end{bmatrix},

L_2(v)=\begin{bmatrix}
\frac{1}{\sqrt{1-v^2}} & 0 & \frac{v}{\sqrt{1-v^2}} \\
0 & 1 & 0 \\
\frac{v}{\sqrt{1-v^2}} & 0 & \frac{1}{\sqrt{1-v^2}}
\end{bmatrix},

In the following posts we will use the relativistic Minkowski space distance on the hyperboloid for finding the distance formula on the Poincare disk.

When China Rules the World

The title of the post can be found on Wikipedia with additional information. But this post is more personal. Indeed China rulez! Few weeks ago I received the following email:

Quantum World-2017

Time: 16th-18th October, 2017

Place: Changsha, Hunan Province, China

On behalf of the organizing committee of CQW-2017, we sent you a letter a few days ago invited you to join us and give a speech at first Annual Conference of Quantum World (CQW-2017), which will be held on 16th-18th October, in Changsha, Hunan Province, China. It seems you have not received that letter yet, so I am writing again to extend our sincere invitation. As we have learnt your valuable contribution to Asymptotic Formula for Quantum Harmonic Oscillator Tunneling Probabilities…, we believe your inspirational speech and participation will highlight this congress a lot!

Under the theme “From E=MC2 to Quantum Industry”, the first Annual Conference of Quantum World (CQW-2017) aims at 200+ oral presentations in Quantum Physics and Mechanics, Quantum Information Science, Quantum Chemistry, Quantum Optics, Quantum Materials, and Quantum System, Quantum Engineering and Application, which cover hot topics with both theoretical and experimental contributions.

The conference venue Changsha, as the capital of Hu’nan province, is a beautiful, creative, historical and cultural city with comfortable climate, unique scenery and convenient transportation. It will give you a special experience on the colligation and integration of the Huxiang Culture with the modern civilization.

Worth mentioning that partial scientific program with speakers’ profile and excellent speech titles has been updated on website, kindly click here to view and give us your valuable advice.

Sincerely yours,

CQW-2017 Organizing Committee

Of course I was surprised, because I do not expect anybody but few experts in the whole world care about my paper. Looks to me like a huge conference industry.

In my blog post Lorentz transformation from an elementary point of view – from blogging to science publishing I wrote about a paper that came as the result of blogging. This is the second paper that I wrote together with prof. Jerzy Szulga, a continuation of our previous paper, A Comment on “On the Rotation Matrix in Minkowski Space-time” by Ozdemir and Erdogdu, http://arxiv.org/abs/1412.5581, Reports on Mathematical Physics, 74(1), 2014, 39-44,
DOI: 10.1016/S0034-4877(14)60056-2. Today I received from my coauthor a message with a copy of another invitation from China:

Dear Dr. ….,

I’m writing to follow-up my last invitation as below, would you please give me a tentative reply? Thank you very much. I apologize for the inconvenience if the letter disturbed you more than once.

It is our great pleasure and privilege to welcome you to join the 8th World Gene Convention-2017, which will take place in Macao, China during November 13-15, 2017. We would like to welcome you to be the chair/speaker in Theme 902: Agriculture, Food and Plant Biotechnology while presenting about A Comment on “On the Rotation Matrix in Minkowski Space-Time” by Ozdemir and Erdogdu…….

If the suggested thematic session is not your current focused core, you may look through the whole sessions and transfer another one that fit your interest (more info about the program is available athttp://www.bitcongress.com/wgc2017/ProgramLayout.asp

Under our SAB members’ contributions and endeavor, BIT’s 7th World Gene Convention-2016 (WGC-2016), successfully held in Shanghai, China during November 13-15, 2016. Totally, there were nearly 300 world-renowned experts, professors, laboratory principals, project leaders and representatives of well-known enterprises attended the WGC-2016. Participants from the international enterprises, academic and research institutions enjoyed the three days scientific program. Depending on the warmly support and good suggestions from all of the participants, we are confident in organizing WGC-2017 which would be better and more successful than WGC-2016.

WGC-2017 features a very strong technical program, mainly focused on: breakthroughs in gene, advances genomics & genetics, new research of DNA and RNA, focus on basic research, the frontier research of life sciences, new biotherapy discovery, emerging areas for medicine applications, robust technology development, and cutting-edge Biotechnology. It aims to provide a platform for all experts from academia, industry and national labs to discuss latest hot researches and achievements. Attendees will hear world-class speakers discussing the challenges and opportunities facing the gene, biotechnology and life sciences field. The business & academic experts who are from home and abroad will give excellent speeches.

In addition to the dynamic scientific program, you will benefit from the wonderful experience in Macao, China. Macao is an international free port. It’s famous for light industry, tourism, and hotel. Macao is also one of the most developed and richest regions in the world, this is a city of amazing and fascinating cultural wealth. The unique blend of European and Oriental cultures existing here creates a pleasurable and laid back atmosphere in a truly beautiful city. We hope you will enjoy your stay in this beautiful city with all its feature, beauty, architecture and hospitality!

We expect your precious comments or suggestions; also your reference to other speakers will be highly appreciated. We look forward to receiving your replies on the following questions:

1. What is the title of your speech?

2. Do you have any suggestions about our program?

We look forward to see you in Macao in 2017 for this influential event.

Sincerely yours,

Ms. Teresa Xiao

Organizing Committee of WGC-2017

It is only Chinese people can have such a broad and brave imagination and vision of the future. To connect Lorentz and spin groups of matrices with genes and agriculture – it is a real feat.

Elliptic m-deformed relativity

[latexpage]
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.