Lecture 12: Special theory of relativity derived from Siemens’ unipolar machine
2024/2
NIDEC Technical Adviser
Takashi Kenjo
Continuing from the last article, I would like to consider non-Lorentz forces and the theory of relativity. Electrodynamics and relativity theory are closely related.
Einstein was the first to realize this fundamental relation. But Maxwell had begun arguing that light is an electromagnetic wave around 1864, which prepared the path for Einstein’s breakthrough. In 1905, Einstein presented five epoch-making papers, two of which were on his ideas about special relativity. In the first paper On the Electrodynamics of Moving Bodies , the word “Unipolarmaschinen” appears only once in parenthesis, as seen in Figure 1 . This is the main theme of this article, and has also been my long-standing question.
技術的装置で考える哲学者
According to him, Einstein wrote in 1930 in a letter, “I never stopped working on technical things. This was also beneficial for scientific research.” The article further says: … a dynamo machine designed by Werner Siemens in 1881 – known as a unipolar machine – was the cause for him to think deeply about the principle of electromagnetism in a new way and mention it in his paper on the special theory of relativity.The photo in Figure 2 is a unipolar machine I set up to examine Einstein’s idea, and Figure 3 illustrates the setup to measure the generation of voltage.
今回はここで，ニデックのある研究員を紹介してみようと思います。大学では航空宇宙を含む機械工学を専攻し，修士課程を修了し，20 11 年入社の研究員です。T 君とします。2022年10 月に佐野茂氏と『一般相対性理論』[1]を技術評論社から出したときに早速読後感を書いてくださいました。Ｔ君の的確なコメントの一部を紹介します。その中に，第8 章で「特殊相対性理論があまりにあっさり導かれることに驚いた。」とあります。この部分はドイツ語による込み入った論理の解読は困難と悟って独自に導いたものです。T 君は第9 章での一般相対性理論の根底に関するドイツ語の読解が面白かったということです。
一般相対性理論は重力に関係する大宇宙の時空の法則です。これはまさに理論物理ですから技術的な装置とは無縁のように思えます。しかし，等価原理というものを発想させたエレベータが機械的な装置です。当時の欧州ではモ―タを使うエレベータは最先端技術の一つでした。
『一般相対性理論』 T君のご読後感から
大学の講義では相対性理論にはあまり触れませんでした。しかし一般相対性理論に興味があり，豊富でカラフルな図形に助けられ，数式を追って本書を読んでみたというよりも，読めました。また，アインシュタインだけでなく，様々な科学者・技術者が相対性理論の発展に寄与してきたのが物語として分かります。彼らが宇宙の謎を追い求める物語は気持ちをワクワクさせ，難しい理論の理解を助けてくれます。
第1～4章は・・・・（略）
第5章で時間の概念が導入され波動の話がでてきます。大学1年生の化学の講義でいきなり波動関数がでてきて面喰ったのですが，本章では「波動」の意味や取り扱い方が分かりやすく書かれています。
第6章は電磁気学です。相対性理論と電磁気学にどんな関係があるのだろうと訝ったのですが，第8章で相対性理論を思いついたアインシュタインの考察につながります。著者の見城氏はモータの実用化への理論では日本の第一人者であり，難解な概念的な電磁気学と実用的なモータの繋がりがわかり，このような観点で電磁気学を学べるのも良いと思います。
第7章にはアインシュタインが相対性理論を考案する前の子供のころや学生時代の科学史が展開され，相対性理論の発見はどのような歴史的背景で導かれていったのかがわかります。・・・・略
第8章で特殊相対性理論の解説になり，第7章の電磁気学を背景として特殊相対性理論が，あまりにあっさり導かれることに驚きました。・・・・（略）
第9章は一般性相対性理論を導くのですが，ここでも難しい式が前半の幾何の話と結び付けられ，図形的な意味と物理的な意味がつかみやすくなっています。また，アインシュタインのドイツ語を読み解くことで，その奥にある考察を読み解く試みが面白いです。時制の使い方の分析からアインシュタインの深い物理的な考え方に触れることができたと思います。
以下略
Electromagnetic motor and Galilean transformation - Reexamination of fundamental physical laws
Those who have attended a course on or taught themselves relativity may be quick to conclude that the theory of relativity deals with the Lorentz transformation, but the Galilean transformation is the starting point.
Here, we will review some basic facts about the motion of objects and electromagnetic forces, as shown in Figure 4 . Fleming explained the complex relationship in an easy-to-understand manner; Figure 4(a) . Fleming explained the complex relationship in an easy-to-understand manner;
Relative motion
The relationship between the stator and rotor is one of relative motion. Let us consider a linear motor, as shown in Figure 5 , instead of a rotary motor. In this figure two coordinate systems K and K’ are set: the K coordinate frame is fixed on the stationary windings,
while K’ is fixed with the linear train which is moving in the●-direction with speed●. Their relative movement is depicted in Figure 6 . In connection to this figure I’m going to put forth a question about a certain asymmetry that was whispered among the engineers at the factory run by Albert’s uncle Jakob Einstein when Albert was a little boy. You will see that what seems obvious is not so obvious.
Between the two systems in Figure 6 there is the following relationship:
Now, let's take a closer look at Fleming's right-hand rule on Figure 4(c) . When the magnetic field● is cut by a moving conductor, an electric field●is generated, where Figure (a) the fingers of the right hand show the directions of each vector. This phenomenon is expressed by:
Next, let's consider a symmetry related to relative motion.●n Figure 6 is a moving system, but for the observer who is moving with it, this is a stationary system, and●is a coordinate system that moves with●. If we rearrange the left and right terms in the two equations above, we get the following equations.
Great! The symmetrical relationship of relative motion appears to hold for electromagnetic systems as well.
However, this was true up until Faraday. With the advent of Maxwell, an asymmetry arose. Einstein solved this problem and proposed a new symmetry, which is the special theory of relativity.
Before we discuss this, we shall take a closer look at an important observation made by Faraday. Figure 7 shows his experiment with a generator, which he allegedly conducted in 1831. In this experiment a current was detected when a resistor was placed across the two terminals. Today it is called a unipolar generator, but I don't think that term existed at that time. Also, we may imagine that it was an experiment in relation to Arago's disk. If we look only at the closed circuit shown here, the magnetic flux linkage is constant whether the disk rotates or not, so the law of electromagnetic induction
cannot by itself explain the generation of electromotive force. When there is motion in the elements that make up the closed circuit, an electromotive force is generated by cutting of the magnetic field. In Faraday's unipolar machine, however, it is not simple to theoretically treat it using mathematical equations because the velocity●of the conductor in question varies with the distance from the axis. Meanwhile, the engineers in Jacob Einstein's factory were discussing Siemens’ unipolar generator, and “Unipolarmaschine” must have been used as a technical term.
construction was similar to the one shown in Figure 3 , and the motion velocity●isverywhere. Now, I would like to make a few additions using Figure 8 and Figure 8 . Figure 8 shows how to derive one of Maxwell's equations:
Here, principle of electromagnetic induction, equation (5), is applied to a tiny rectangular coil with a single turn, where the relationship between the electric field●along the coil and the magnetic flux surrounding the coil is as follows:
From this, we obtain
By generalization we obtain above (6). This was the case with a coil at rest.
Next, let’s apply (5) to the case of the H-type conductor in Figure 9 , where the sliding conductor A’B’ is in motion. Here, both the magnetic field and the electric field are uniform. Since the area surrounded by the red integral path changes with time, the magnetic flux linkage also changes with time. And (2) is obtained, which is regarded as Fleming's right-hand rule.
Asymmetry appears
Asymmetry appears●●can be modified to●●.
Readers may know that it was James Maxwell, born in Scotland, who took Faraday's discoveries and insights seriously. He assumed that a change in the electric field can generate a magnetic field, similar to how changes in the magnetic field generate an electric field. Based on this assumption, in 1864 he rewrote Ampere’s integral formula, using differential expression, as follows:
(Note, however, that the vector analysis notations as we know them today was later developed by Heaviside.)
Here,  is the actual current density. Actual current is the current produced by electrons moving in a vacuum or in a conductor. Meanwhile, the second term on the right-hand side is called the displacement current. To examine the essence of the matter,
we consider the case where
Since this has a similar form to equation (6), the following algebraic form is obtained from association:
●Since this has a similar form to equation (6), the following algebraic form is obtained from association:ベクトルの方向成分を考慮すると(11) 式が正しいです。
By putting● , equation (4) can be rewritten as
Since (11) and (12) hold simultaneously, we obtain:
Thus the symmetry like that in (3) and (4) has been lost and an asymmetry comes out). In other words, a contradiction appears in plain expected relativity. As a practical matter,●, so●is negligible compared to 1, and symmetry holds as an approximation. But theoretically, it doesn't make sense.
Elimination of asymmetry – resulting in Lorentz transformation of the electromagnetic field!
Einstein says in his 1905 paper that he resolved this question. Instead of the rambling explanation of this paper, let's use a simple linear algebraic approach here to solve this essential matter. In place of (13) and (14), using coefficient● , we put
Let's rewrite this using a matrix with a first-order transformation expression:
Here the subscripts are omitted for simplicity.
Let's rewrite this using a matrix with a first-order transformation expression:
where the determinant of the transformation matrix is
By replacing●with ●in (17), symmetrical relationship must appear:
From comparison with (20),●must be:
What does●● mean?
In order for●in equation (22) obtained above to be a real number,●is required. What exactly does this mean? The rotor we are currently considering in the unipolar machine is composed of a metal such as copper or aluminum, and its●magnetic permeability●and permittivity●are almost the same as in a vacuum. Then,●is the speed of light●
We thus have●which is the key factor that Lorenz presented!
Lorentz transformation of spacetime
Now, will Maxwell's electromagnetic field equation itself be changed to include●? No, it is not necessary when the following spacetime relationship described in terms of the Lorentz transformation is taken into account.
Note that if●he position and time of the moving system become imaginary numbers. And if●they become ●, which is unrealistic. Therefore, the velocity of relative motion●must satisfy●In other words, even if we design a high-speed spacecraft, it will be unable to exceed the speed of light ●
Principle of constant speed of light
The speed of light is●●The speed of light is●●and permittivity ●●. Assuming that we cannot distinguish between a stationary vacuum and a moving vacuum, the speed of light is independent of the speed of the light source.  is the speed limit and cannot be exceeded. “Let's use this as an axiom to unify electromagnetic theory and mechanics.” I think this was the young Einstein's idea of electrodynamics. Will readers easily accept Einstein's theory? I think it's natural to not be easily convinced. Andrew Warwick's Masters of Theory [2] describes the difficulty that scholars at Cambridge University, which has a long tradition in theoretical physics, experienced before they finally accepted the theory of relativity around 1919.
あとがきに代えて： ―必ずやる，できるまでやるーる
「モータには（回生）発電作用が宿されていることは天からの恩恵ともいえます」と書きました。この意味は深いことを実感します。2023 年の初夏のことですが，私は思いがけない連絡を北尾副社長から受けたのです。太陽誘電(株)様がモータが宿す回生発電を有効活用して１充電当たり1000km走行の電動アシスト自転車の開発に成功し，近い将来の実用化に目途をつけたというのです。同社によるこの技術的挑戦は，2009 年に品川区大崎でのME/L(日本電産モータ基礎研究所)にお越しくださった保坂康夫様を激励した筆者の言葉によって，社内の反対の声が大きいなかで，執行役員高木満男様が説得なさって研究開発をし続けたということでした。ここに論じたようにモータと発電機に内在する物理法則は宇宙を貫く相対性原理です。 これはあくまで物理学の領域の事柄です。さらに電池を含む電気化学と生理学として人の運動能力と健康増進の機能，さらには，違和感なく快適に 回生発電させる仕組みの研究によってsustainable society を目指す事業への視野が大きくなりそうです。（次は充電不要な電動アシスト自転車が目標とのことです。）
References
１．Stefan Siemer : Einstein in München, Kultur & Technik, 02/2005
２．Andrew Warwick：Masters of Theory, University of Chicago Press, 2003