7年かけて作った「人工ブラックホール」でホーキング放射を初観測。ブラックホールが完全にブラックではない証拠になるか
BY MUNENORI TANIGUCHI 2016年08月17日 16時00分 0
イスラエルの科学者ジェフ・スタインハウアーが人工的なブラックホールを製作し、その振る舞いからスティーブン・ホーキング博士が1974年に発表した理論「ホーキング放射」に似た現象を観測したと発表しました。
スタインハウアーが作った人工的なブラックホールは本物のブラックホールのように光を含めて何でも吸い込むというものではなく試験用のチューブ内に流体を流し、ある地点でそれを音速以上に加速させることで音響的な事象の地平面を生み出すというもの。本物のブラックホールでは光が逃げられなくなる位置で事象の地平面が発生しますが、この人工ブラックホールでは音が逃げられなくなる位置を事象の地平面とします。
ブラックホールは一般に、すべての物質を引き寄せ、光であっても逃さないほどの強力な重力を有するため、宇宙空間にポッカリと真っ黒く何も見えない穴が開いたように見えると思われています。しかし、ホーキング博士は量子効果を考慮するとそこからは熱的な放射がなければならず、事象の地平面(想像図に描かれるブラックホールの縁の部分)から電磁波のようなエネルギーが発せられるため「ブラックホールは完全なブラックではない(何も放出しないわけではない)」としました。
もう少し詳しく説明すれば、真空中には何も存在しない部分から常に一対の粒子が発生しており、それらはすぐに結合してまた消滅するという動作を繰り返していると考えられます。事象の地平面よりブラックホールの中心に近いところでは、粒子が発生してもブラックホールに引き寄せられるだけで終わってしまいます。ところが、事象の地平面では片方の粒子のみがブラックホールに落ち、もう片方が外へ逃げる可能性が考えられます。するとこの逃げた粒子が、外界からはブラックホールが放出したエネルギー、つまりホーキング放射として観測できることになります。
一方、ブラックホールに落ちた方の粒子は、相棒を失ってエネルギー的にマイナスとなっているため、ブラックホール全体からエネルギーを吸収します。ホーキング博士は、この現象が長期間繰り返されることで最終的にブラックホールは蒸発するように消えるだろうとしました。
スタインハウアーは実験のために、「人工ブラックホール」を7年かけて製作しました。人工ブラックホールではガスを絶対零度近くまで冷却して得られる、ボース・アインシュタイン凝縮と呼ばれる状態のガスをチューブ内に流し、ある位置でそれをレーザーを使って音速以上に加速させることで音響的な事象の地平面を生み出します。
そしてこの音響的な事象の地平面に一対のフォノン(音子、音響量子、音量子)を発生させ、フォノンの片方がブラックホールから放射されるように移動し、一方のフォノンがブラックホールにとらわれるホーキング輻射と同様の現象が発生するのを確認したとのこと。スタインハウアーは特に高エネルギーのフォノンのペアが量子的なもつれの状態を示すこともわかったとしており、これはホーキング放射の重要な証拠となる特徴だとしています。
この実験結果を聞いた学者の一部には、フォノンのもつれ状態が低エネルギー状態で観測していないことを理由に不完全な結果だとする向きもあるものの、他方ではホーキング放射を擬似的に再現したのはこれが初のことであり、実験結果に欠陥がないと確かめられれば、ヒッグス粒子の発見と同じぐらい大変な出来事だとする声もあります。
もしかするとこの実験を発端として、ホーキング博士がノーベル賞を受賞するというストーリーもありえるかもしれません。
論文はNature Physics 「Observation of quantum Hawking radiation and its entanglement in an analogue black hole : Jeff Steinhauer」http://japanese.engadget.com/2016/08/17/7/
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\title{\bf Announcement 185 : The importance of the division by zero $z/0=0$}
\author{{\it Institute of Reproducing Kernels}\\
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall state the importance of the division by zero $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
{\bf Introduction}
\bigskip
%\label{sect1}
By {\bf a natural extension of the fractions}
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers. The result is a very special case for general fractional functions in \cite{cs}.
The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
for all
$$
a, b, c, d \in {\bf C }
$$
and
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $
$$
F (b, 0) = 0.
$$}
\medskip
Furthermore, note that Hiroshi Michiwaki gave the important interpretation of the division by zero $z/0=0$ by the intuitive meaning of the division, independently of the concept of the product (see \cite{ann}).
We shall state the importance of the division by zero $z/0=0$.
\bigskip
\section{}
On AD 628, the zero was appeared in India, and the zero division $z/0=0$ was discovered on Feburary 2, 2014, definitely with the clear definition and motievation. The uniquess and the natural interpretation were given in \cite{taka, ttk,kmsy} and \cite{ann}, respectively. Several physical interpretations of the division by zero were given in \cite{kmsy}.
\bigskip
\section{}
By the introduction of the division by zero $z/0=0$, four arithmetic operations; that is,
addition, subtraction, multiplication, and division are always possible; note that for division, we were not able to divide by zero. There was one exceptional case for the division by zero.
\section{}
For the Euclidean (B.C. 3 Centuary ) geometry, two non-Euclidean geometries were appered about 2 hundred years ago, and in particular, in the elliptic type non-Euclidean geometry, the point at infinity was introduced by the stereoprojection of the Euclidean plane to the sphere and the concept is a standard one in complex analysis around over one hundered years. And then we have considered as $1/0= \infty$ (\cite{ahlfors}). However, surprisingly enough, the division by zero means that $1/0=0$.
\section{}
We shall recall the fundamental law by Newton:
\begin{equation}
F = G\frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, we obtain the important interpretation:
\begin{equation}
F = 0 = G \frac{m_1 m_2}{0}.
\end{equation}
Of course, here, we can consider the above interpretation for the mathematical formula (4.1) as the new interpretation (4.3). We can find many physical formulas with the division by zero.
\section{}
In complex analysis, linear fractional functions
$$
W = \frac{az + b}{cz + d}, \quad ad -bc \ne 0,
$$
map the extended complex plane onto the extended complex plane containing the point at infinity, one to one, conformally, beautifully. This beautiful property is changed as the beautiful formula that linear fractional functions map the whole complex plane onto the whole complex plane, one to one, however, at one point of the singular point, the linear fractional functions have strong discontinuity.
The division by zero excludes the infinity from the numbers.
\section{}
We did, essentially, not consider the division by zero, and so the property of the division by zero; that is, at the isolated singular points of analytic functions, to consider the analytic functions are new mathematics and new research topics, essentially.
\section{}
The impact to complex analysis is unclear, we, however, obtain a typical new theorem:
\medskip
{\bf Theorem :} {\it Any analytic function takes a definite value at an isolated singular point }{\bf with a natural meaning.} The definite value is given by the first coefficient of the regular part in the Laurent expansion around the isolated singular point.
\medskip
This will be the fundamental theorem on the division by zero in Complex Analysis and we have many applications for the Sato hyperfunction theory, generating functions theory and singular integral theory (\cite{mst}).
\section{}
In particular, the divison by zero gives new interpretations on the finite part of Hadamard
for singular integrals and the Cauchy's principal values. The division by zero will represent discontinuity properties on the universe.
\section{}
Even for middle high shool students, the division by zero may be accepted as the beautiful result with great pleasures:
For the elementary function
$$
y = f(x) = \frac{1}{x},
$$
we have $f(0) = 0$; that is, $1/0=0$.
\section{}
We can introduce the division by zero $100/0=0,0/0=0$ with the simple and natural definition for the division by the Hiroshi Michiwachi method (\cite{ann}) in the elementary school. The division by zero will request the change of all the related books and scientific books.
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$
The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
\section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\medskip
{\bf Announcement 179} (2014.10.22): Division by zero is clear as z/0=0 and it is fundamental in mathematics
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{mst}
H. Michiwaki, S. Saitoh, and M. Takagi,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields, Tokyo Journal of Mathematics (in press).
\bibitem{ann}
Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics,
Institute of Reproducing Kernels, 2014.10.22.
\end{thebibliography}
\end{document}
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