Как складываются скорости в каждой из моделей при 0,6 c и 0,6 c?

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      "text": "Раздел 2. Три раздельных вывода.\nСопоставление даёт три РАЗНЫХ вывода, и смешивать их запрещено: различие ПРЕДПОСЫЛОК (одна модель принимает принцип, другая его отвергает); различие ПРЕДСКАЗАНИЙ (на одной постановке при одной наблюдаемой значения разные); согласие или несогласие С НАБЛЮДЕНИЕМ (предсказание против предъявленных данных опыта при объявленном допуске).\n\nНи один из трёх выводов не влечёт остальных. Модели с разными предпосылками могут дать одинаковое предсказание. Модели с одинаковым предсказанием остаются разными моделями. Согласие предсказания с наблюдением не делает модель верной, а несогласие называет неверным предсказание, а не всякое допущение, из которого оно получено."
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      "text": "Раздел 3. Восемь вопросов сопоставления.\nМетодика объявляет восемь вопросов, и полнота сопоставления определяется ответом на каждый из объявленных:\nвопрос 1 — картина мира: пространство, время, одновременность, статус среды и выделенной системы;\nвопрос 2 — одновременность и преобразование времени двух разнесённых событий;\nвопрос 3 — показания движущихся часов и длина движущегося стержня;\nвопрос 4 — сложение скоростей и скорость светового сигнала;\nвопрос 5 — интерферометр с двумя перпендикулярными плечами;\nвопрос 6 — совместимость с предъявленным наблюдением 1887 года;\nвопрос 7 — совпадение наблюдаемого при разных основаниях;\nвопрос 8 — расхождение моделей при малых скоростях и объявленном допуске."
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      "text": "Раздел 4. Полнота определяется покрытием вопросов.\nСопоставление полно, когда по каждому объявленному вопросу получен ответ и у каждого ответа названо основание. Полнота НЕ требует, чтобы модели разошлись: вопрос, на котором предсказания совпали, отвечен ровно так же, как вопрос, на котором они разошлись. Критерий, требующий различающего принципа, к физическому сопоставлению не применяется."
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      "text": "Раздел 8. Совпадение наблюдаемого при разных основаниях.\nНа постановке опыта 1887 года модель эфира С ГИПОТЕЗАМИ СОКРАЩЕНИЯ и специальная теория относительности дают один и тот же ответ о наблюдаемом сдвиге полос: сдвига нет. Основания при этом разные: у первой сдвиг исчезает потому, что плечо вдоль движения укорачивается ровно настолько, чтобы скомпенсировать разность времён; у второй его нет потому, что скорость света в лаборатории одна во всех направлениях и разности времён не возникает.\n\nИз этого совпадения НЕ следует эквивалентность моделей. Совпадение установлено для одной названной наблюдаемой на одной названной постановке; распространять его на другие постановки методика запрещает и проверяет этот запрет отдельной сценой."
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      "text": "§ 1.\nDefinition der Gleichzeitigkeit.\n\nEs liege ein Koordinatensystem vor, in welchem die Newtonschen mechanischen Gleichungen gelten. Wir nennen dies Koordinatensystem zur sprachlichen Unterscheidung von später einzuführenden Koordinatensystemen und zur Präzisierung der Vorstellung das „ruhende System“.\n\nRuht ein materieller Punkt relativ zu diesem Koordinatensystem, so kann seine Lage relativ zu letzterem durch starre Maßstäbe unter Benutzung der Methoden der euklidischen Geometrie bestimmt und in kartesischen Koordinaten ausgedrückt werden.\n\nWollen wir die Bewegung eines materiellen Punktes beschreiben, so geben wir die Werte seiner Koordinaten in Funktion der Zeit. Es ist nun wohl im Auge zu behalten, daß eine derartige mathematische Beschreibung erst dann einen physikalischen Sinn hat, wenn man sich vorher darüber klar geworden ist, was hier unter „Zeit“ verstanden wird.\nWir haben zu berücksichtigen, daß alle unsere Urteile, in welchen die Zeit eine Rolle spielt, immer Urteile über gleichzeitige Ereignisse sind. Wenn ich z. B. sage: „Jener Zug kommt hier um 7 Uhr an,“ so heißt dies etwa: „Das Zeigen des kleinen Zeigers meiner Uhr auf 7 und das Ankommen des Zuges sind gleichzeitige Ereignisse.“ [Anm: Die Ungenauigkeit, welche in dem Begriffe der Gleichzeitigkeit zweier Ereignisse an (annähernd) demselben Orte steckt und gleichfalls durch eine Abstraktion überbrückt werden muß, soll hier nicht erörtert werden.]\n\nEs könnte scheinen, daß alle die Definition der „Zeit“ betreffenden Schwierigkeiten dadurch überwunden werden könnten, daß ich an Stelle der „Zeit“ die „Stellung des kleinen Zeigers meiner Uhr“ setze. Eine solche Definition genügt in der Tat, wenn es sich darum handelt, eine Zeit zu definieren ausschließlich für den Ort, an welchem sich die Uhr eben befindet; die Definition genügt aber nicht mehr, sobald es sich darum handelt, an verschiedenen Orten stattfindende Ereignisreihen miteinander zeitlich zu verknüpfen, oder — was auf dasselbe hinausläuft — Ereignisse zeitlich zu werten, welche in von der Uhr entfernten Orten stattfinden.\n\nWir könnten uns allerdings damit begnügen, die Ereignisse dadurch zeitlich zu werten, daß ein samt der Uhr im Koordinatenursprung befindlicher Beobachter jedem von einem zu wertenden Ereignis Zeugnis gebenden, durch den leeren Raum zu ihm gelangenden Lichtzeichen die entsprechende Uhrzeigerstellung zuordnet. Eine solche Zuordnung bringt aber den Übelstand mit sich, daß sie vom Standpunkte des mit der Uhr versehenen Beobachters nicht unabhängig ist, wie wir durch die Erfahrung wissen. Zu einer weit praktischeren Festsetzung gelangen wir durch folgende Betrachtung.\n\nBefindet sich im Punkte $A$ des Raumes eine Uhr, so kann ein in $A$ befindlicher Beobachter die Ereignisse in der unmittelbaren Umgebung von $A$ zeitlich werten durch Aufsuchen der mit diesen Ereignissen gleichzeitigen Uhrzeigerstellungen. Befindet sich auch im Punkte $B$ des Raumes eine Uhr — wir wollen hinzufügen, „eine Uhr von genau derselben Beschaffenheit wie die in $A$  befindliche“ — so ist auch eine zeitliche Wertung der Ereignisse in der unmittelbaren Umgebung von\n$B$ durch einen in $B$ befindlichen Beobachter möglich. Es ist aber ohne weitere Festsetzung nicht möglich, ein Ereignis in $A$  mit einem Ereignis in $B$ zeitlich zu vergleichen; wir haben bisher nur eine „$A$-Zeit“ und eine „$B$-Zeit“, aber keine für $A$ und $B$ gemeinsame „Zeit“ definiert. Die letztere Zeit kann nun definiert werden, indem man durch Definition festsetzt, daß die „Zeit“, welche das Licht braucht, um von $A$  nach $B$ zu gelangen, gleich ist der „Zeit“, welche es braucht, um von $B$ nach $A$ zu gelangen. Es gehe nämlich ein Lichtstrahl zur „$A$-Zeit“ $t_{A}$ von $A$  nach $B$ ab, werde zur „$B$-Zeit“ $t_{B}$  in $B$ gegen $A$ zu reflektiert und gelange zur „$A$-Zeit“ $t'_{A}$ nach $A$ zurück. Die beiden Uhren laufen definitionsgemäß synchron, wenn\n\n$t_{B}-t_{A}=t'_{A}-t_{B}$\n\nWir nehmen an, daß diese Definition des Synchronismus in widerspruchsfreier Weise möglich sei, und zwar für beliebig viele Punkte, daß also allgemein die Beziehungen gelten:\n\n1. Wenn die Uhr in $B$ synchron mit der Uhr in $A$ läuft, so läuft die Uhr in $A$ synchron mit der Uhr in $B$.\n\n2. Wenn die Uhr in $A$ sowohl mit der Uhr in $B$ als auch mit der Uhr in $C$ synchron läuft, so laufen auch die Uhren in $B$ und $C$ synchron relativ zueinander.\n\nWir haben so unter Zuhilfenahme gewisser (gedachter) physikalischer Erfahrungen festgelegt, was unter synchron laufenden, an verschiedenen Orten befindlichen, ruhenden Uhren zu verstehen ist und damit offenbar eine Definition von „gleichzeitig“ und „Zeit“ gewonnen. Die „Zeit“ eines Ereignisses ist die mit dem Ereignis gleichzeitige Angabe einer am Orte des Ereignisses befindlichen, ruhenden Uhr, welche mit einer bestimmten, ruhenden Uhr, und zwar für alle Zeitbestimmungen mit der nämlichen Uhr, synchron läuft.\n\nWir setzen noch der Erfahrung gemäß fest, daß die Größe\n\n$\\frac{2\\overline{AB}}{t'_{A}-t_{A}}=V$\n\neine universelle Konstante (die Lichtgeschwindigkeit im leeren Raume) sei.\n\nWesentlich ist, daß wir die Zeit mittels im ruhenden System\nruhender Uhren definiert haben; wir nennen die eben definierte Zeit wegen dieser Zugehörigkeit zum ruhenden System „die Zeit des ruhenden Systems“."
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      "text": "§ 2.\nÜber die Relativität von Längen und Zeiten.\n\nDie folgenden Überlegungen stützen sich auf das Relativitätsprinzip und auf das Prinzip der Konstanz der Lichtgeschwindigkeit, welche beiden Prinzipien wir folgendermaßen definieren.\n\n1. Die Gesetze, nach denen sich die Zustände der physikalischen Systeme ändern, sind unabhängig davon, auf welches von zwei relativ zueinander in gleichförmiger Translationsbewegung befindlichen Koordinatensystemen diese Zustandsänderungen bezogen werden.\n\n2. Jeder Lichtstrahl bewegt sich im „ruhenden“ Koordinatensystem mit der bestimmten Geschwindigkeit $V$, unabhängig davon, ob dieser Lichtstrahl von einem ruhenden oder bewegten Körper emittiert ist. Hierbei ist\n\nGeschwindigkeit ${=\\rm \\frac{Lichtweg}{Zeitdauer}}$,\n\nwobei „Zeitdauer“ im Sinne der Definition des § 1 aufzufassen ist.\n\nEs sei ein ruhender starrer Stab gegeben; derselbe besitze, mit einem ebenfalls ruhenden Maßstabe gemessen, die Länge $l$. Wir denken uns nun die Stabachse in die $X$-Achse des ruhenden Koordinatensystems gelegt und dem Stabe hierauf eine gleichförmige Paralleltranslationsbewegung (Geschwindigkeit $v$) längs der $X$-Achse im Sinne der wachsenden $x$ erteilt. Wir fragen nun nach der Länge des bewegten Stabes, welche wir uns durch folgende zwei Operationen ermittelt denken:\n\na) Der Beobachter bewegt sich samt dem vorher genannten Maßstabe mit dem auszumessenden Stabe und mißt direkt durch Anlegen des Maßstabes die Länge des Stabes, ebenso, wie wenn sich auszumessender Stab, Beobachter und Maßstab in Ruhe befänden.\n\nb) Der Beobachter ermittelt mittels im ruhenden Systeme aufgestellter, gemäß § 1 synchroner, ruhender Uhren, in welchen Punkten des ruhenden Systems sich Anfang und Ende des auszumessenden Stabes zu einer bestimmten Zeit $t$ befinden.\nDie Entfernung dieser beiden Punkte, gemessen mit dem schon benutzten, in diesem Falle ruhenden Maßstabe ist ebenfalls eine Länge, welche man als „Länge des Stabes“ bezeichnen kann.\n\nNach dem Relativitätsprinzip muß die bei der Operation a) zu findende Länge, welche wir „die Länge des Stabes im bewegten System“ nennen wollen, gleich der Länge $l$ des ruhenden Stabes sein.\n\nDie bei der Operation b) zu findende Länge, welche wir „die Länge des (bewegten) Stabes im ruhenden System“ nennen wollen, werden wir unter Zugrundelegung unserer beiden Prinzipien bestimmen und finden, daß sie von $l$  verschieden ist.\n\nDie allgemein gebrauchte Kinematik nimmt stillschweigend an, daß die durch die beiden erwähnten Operationen bestimmten Längen einander genau gleich seien, oder mit anderen Worten, daß ein bewegter starrer Körper in der Zeitepoche $t$ in geometrischer Beziehung vollständig durch denselben Körper, wenn er in bestimmter Lage ruht, ersetzbar sei.\n\nWir denken uns ferner an den beiden Stabenden ($A$ und $B$  ) Uhren angebracht, welche mit den Uhren des ruhenden Systems synchron sind, d. h. deren Angaben jeweilen der „Zeit des ruhenden Systems“ an den Orten, an welchen sie sich gerade befinden, entsprechen; diese Uhren sind also „synchron im ruhenden System“.\n\nWir denken uns ferner, daß sich bei jeder Uhr ein mit ihr bewegter Beobachter befinde, und daß diese Beobachter auf die beiden Uhren das im § 1 aufgestellte Kriterium für den synchronen Gang zweier Uhren anwenden. Zur Zeit [Anm: „Zeit“ bedeutet hier „Zeit des ruhenden Systems“ und zugleich „Zeigerstellung der bewegten Uhr, welche sich an dem Orte, von dem die Rede ist, befindet“.] $t_{A}$ gehe ein Lichtstrahl von $A$  aus, werde zur Zeit $t_{B}$ in $B$ reflektiert und gelange zur Zeit $t'_{A}$ nach $A$ zurück. Unter Berücksichtigung des Prinzipes von der Konstanz der Lichtgeschwindigkeit finden wir:\n\n$t_{B}-t_{A}=\\frac{r_{AB}}{V-v}$\n\nund\n\n$t'_{A}-t_{B}=\\frac{r_{AB}}{V+v},$\n\nwobei $r_{AB}$ die Länge des bewegten Stabes — im ruhenden System gemessen — bedeutet. Mit dem bewegten Stabe bewegte Beobachter würden also die beiden Uhren nicht synchron gehend finden, während im ruhenden System befindliche Beobachter die Uhren als synchron laufend erklären würden.\n\nWir sehen also, daß wir dem Begriffe der Gleichzeitigkeit keine absolute Bedeutung beimessen dürfen, sondern daß zwei Ereignisse, welche, von einem Koordinatensystem aus betrachtet, gleichzeitig sind, von einem relativ zu diesem System bewegten System aus betrachtet, nicht mehr als gleichzeitige Ereignisse aufzufassen sind."
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      "text": "§ 3.\nTheorie der Koordinaten- und Zeittransformation von dem ruhenden auf ein relativ zu diesem in gleichförmiger Translationsbewegung befindliches System.\n\nSeien im „ruhenden“ Raume zwei Koordinatensysteme, d. h. zwei Systeme von je drei von einem Punkte ausgehenden, aufeinander senkrechten starren materiellen Linien, gegeben. Die $X$-Achsen beider Systeme mögen zusammenfallen, ihre $Y$- und $Z$-Achsen bezüglich parallel sein. Jedem Systeme sei ein starrer Maßstab und eine Anzahl Uhren beigegeben, und es seien beide Maßstäbe sowie alle Uhren beider Systeme einander genau gleich.\n\nEs werde nun dem Anfangspunkte des einen der beiden Systeme ($k$) eine (konstante) Geschwindigkeit $v$ in Richtung der wachsenden $x$ des anderen, ruhenden Systems ($K$) erteilt, welche sich auch den Koordinatenachsen, dem betreffenden Maßstabe sowie den Uhren mitteilen möge. Jeder Zeit $t$ des ruhenden Systems $K$ entspricht dann eine bestimmte Lage der Achsen des bewegten Systems und wir sind aus Symmetriegründen befugt anzunehmen, daß die Bewegung von $k$  so beschaffen sein kann, daß die Achsen des bewegten Systems zur Zeit $t$  (es ist mit „$t$“ immer eine Zeit des ruhenden Systems bezeichnet) den Achsen des ruhenden Systems parallel seien.\n\nWir denken uns nun den Raum sowohl vom ruhenden System $K$ aus mittels des ruhenden Maßstabes als auch vom\nbewegten System $k$ mittels des mit ihm bewegten Maßstabes ausgemessen und so die Koordinaten $x,y,z$ bez. $\\xi,\\eta,\\zeta$ ermittelt. Es werde ferner mittels der im ruhenden System befindlichen ruhenden Uhren durch Lichtsignale in der in § 1 angegebenen Weise die Zeit $t$  des ruhenden Systems für alle Punkte des letzteren bestimmt, in denen sich Uhren befinden; ebenso werde die Zeit $\\tau$ des bewegten Systems für alle Punkte des bewegten Systems, in welchen sich relativ zu letzterem ruhende Uhren befinden, bestimmt durch Anwendung der in § 1 genannten Methode der Lichtsignale zwischen den Punkten, in denen sich die letzteren Uhren befinden.\n\nZu jedem Wertsystem $x,y,z,t$, welches Ort und Zeit eines Ereignisses im ruhenden System vollkommen bestimmt, gehört ein jenes Ereignis relativ zum System $k$ festlegendes Wert System $\\xi,\\eta,\\zeta,\\tau$, und es ist nun die Aufgabe zu lösen, das diese Größen verknüpfende Gleichungssystem zu finden.\n\nZunächst ist klar, daß die Gleichungen linear sein müssen wegen der Homogenitätseigenschaften, welche wir Raum und Zeit beilegen.\n\nSetzen wir $x'=x-vt$, so ist klar, daß einem im System $k$  ruhenden Punkte ein bestimmtes, von der Zeit unabhängiges Wertsystem $x',y,z$ zukommt. Wir bestimmen zuerst $\\tau$ als Funktion von $x',y,z$ und $t$. Zu diesem Zwecke haben wir in Gleichungen auszudrücken, daß $\\tau$ nichts anderes ist als der Inbegriff der Angaben von im System $k$ ruhenden Uhren, welche nach der im § 1 gegebenen Regel synchron gemacht worden sind.\n\nVom Anfangspunkt des Systems $k$ aus werde ein Lichtstrahl zur Zeit $\\tau_{0}$längs der $X$-Achse nach $x'$ gesandt und von dort zur Zeit $\\tau_{1}$ nach dem Koordinatenursprung reflektiert, wo er zur Zeit $\\tau_{2}$ anlange; so muß dann sein:\n\n$\\frac{1}{2}(\\tau_{0}+\\tau_{2})=\\tau_{1}$\n\noder, indem man die Argumente der Funktion $\\tau$ beifügt und das Prinzip der Konstanz der Lichtgeschwindigkeit im ruhenden Systeme anwendet:\n\n$\\frac{1}{2}\\left[\\tau(0,0,0,t)+\\tau\\left(0,0,0,\\left\\{ t+\\frac{x'}{V-v}+\\frac{x'}{V+v}\\right\\} \\right)\\right]$\n\n$=\\tau\\left(x',0,0,t+\\frac{x'}{V-v}\\right).$\n\nHieraus folgt, wenn man $x'$ unendlich klein wählt:\n\n$\\frac{1}{2}\\left(\\frac{1}{V-v}+\\frac{1}{V+v}\\right)\\frac{\\partial\\tau}{\\partial t}=\\frac{\\partial\\tau}{\\partial x'}+\\frac{1}{V-v}\\frac{\\partial\\tau}{\\partial t},$\n\noder\n\n$\\frac{\\partial\\tau}{\\partial x'}+\\frac{v}{V^{2}-v^{2}}\\frac{\\partial\\tau}{\\partial t}=0.$\n\nEs ist zu bemerken, daß wir statt des Koordinatenursprunges jeden anderen Punkt als Ausgangspunkt des Lichtstrahles hätten wählen können und es gilt deshalb die eben erhaltene Gleichung für alle Werte von $x',y,z$.\n\nEine analoge Überlegung — auf die $H$- und $Z$-Achse angewandt — liefert, wenn man beachtet, daß sich das Licht längs dieser Achsen vom ruhenden System aus betrachtet stets mit der Geschwindigkeit $\\sqrt{V^{2}-v^{2}}$ fortpflanzt:\n\n$\\frac{\\partial\\tau}{\\partial y}=0$\n\n$\\frac{\\partial\\tau}{\\partial z}=0.$\n\nAus diesen Gleichungen folgt, da $\\tau$ eine lineare Funktion ist:\n\n$\\tau=a\\left(t-\\frac{v}{V^{2}-v^{2}}x'\\right)$,\n\nwobei $a$ eine vorläufig unbekannte Funktion $\\varphi(v)$ ist und der Kürze halber angenommen ist, daß im Anfangspunkte von $k$ für $\\tau=0$ $t=0$ sei.\n\nMit Hilfe dieses Resultates ist es leicht, die Größen $\\xi,\\eta,\\zeta$ zu ermitteln, indem man durch Gleichungen ausdrückt, daß sich das Licht (wie das Prinzip der Konstanz der Lichtgeschwindigkeit in Verbindung mit dem Relativitätsprinzip verlangt) auch im bewegten System gemessen mit der Geschwindigkeit $V$ fortpflanzt. Für einen zur Zeit $\\tau=0$ in Richtung der wachsenden $\\xi$ ausgesandten Lichtstrahl gilt:\n\n$\\xi=V\\tau$,\n\noder\n\n$\\xi=aV\\left(t-\\frac{v}{V^{2}-v^{2}}x'\\right)$.\n\nNun bewegt sich aber der Lichtstrahl relativ zum Anfangspunkt\nvon $k$ im ruhenden System gemessen mit der Geschwindigkeit $V-v$, so daß gilt:\n\n$\\frac{x'}{V-v}=t.$\n\nSetzen wir diesen Wert von $t$ in die Gleichung für $\\xi$ ein, so erhalten wir:\n\n$\\xi=a\\frac{V^{2}}{V^{2}-v^{2}}x'.$\n\nAuf analoge Weise finden wir durch Betrachtung von längs den beiden anderen Achsen bewegte Lichtstrahlen:\n\n$\\eta=V\\tau=aV\\left(t-\\frac{v}{V^{2}-v^{2}}x'\\right),$\n\nwobei\n\n$\\frac{y}{\\sqrt{V^{2}-v^{2}}}=t;\\ x'=0;$\n\nalso\n\n$\\eta=a\\frac{V}{\\sqrt{V^{2}-v^{2}}}y$\n\nund\n\n$\\zeta=a\\frac{V}{\\sqrt{V^{2}-v^{2}}}z.$\n\nSetzen wir für $x'$ seinen Wert ein, so erhalten wir:\n\n$\\begin{align}\\tau & =\\varphi(v)\\beta(t-\\frac{v}{V^{2}}x),\\\\\n\\xi & =\\varphi(v)\\beta(x-vt),\\\\\n\\eta & =\\varphi(v)y,\\\\\n\\zeta & =\\varphi(v)z,\n\\end{align}$\n\nwobei\n\n$\\beta=\\frac{1}{\\sqrt{1-\\left(\\frac{v}{V}\\right)^{2}}}$\n\nund $\\varphi$ eine vorläufig unbekannte Funktion von $v$ ist. Macht man über die Anfangslage des bewegten Systems und über den Nullpunkt von $\\tau$ keinerlei Voraussetzung, so ist auf den rechten Seiten dieser Gleichungen je eine additive Konstante zuzufügen.\n\nWir haben nun zu beweisen, daß jeder Lichtstrahl sich, im bewegten System gemessen, mit der Geschwindigkeit $V$ fortpflanzt, falls dies, wie wir angenommen haben, im ruhenden\nSystem der Fall ist; denn wir haben den Beweis dafür noch nicht geliefert, daß das Prinzip der Konstanz der Lichtgeschwindigkeit mit dem Relativitätsprinzip vereinbar sei.\n\nZur Zeit $t=\\tau=0$ werde von dem zu dieser Zeit gemeinsamen Koordinatenursprung beider Systeme aus eine Kugelwelle ausgesandt, welche sich im System $K$  mit der Geschwindigkeit $V$ ausbreitet. Ist ($x,y,z$) ein eben von dieser Welle ergriffener Punkt, so ist also\n\n$x^{2}+y^{2}+z^{2}=V^{2}t^{2}$.\n\nDiese Gleichung transformieren wir mit Hilfe unserer Transformationsgleichungen und erhalten nach einfacher Rechnung:\n\n$\\xi^{2}+\\eta^{2}+\\zeta^{2}=V^{2}\\tau^{2}$.\n\nDie betrachtete Welle ist also auch im bewegten System betrachtet eine Kugelwelle von der Ausbreitungsgeschwindigkeit $V$. Hiermit ist gezeigt, daß unsere beiden Grundprinzipien miteinander vereinbar sind.\n\nIn den entwickelten Transformationsgleichungen tritt noch eine unbekannte Funktion $\\varphi$ von $v$ auf, welche wir nun bestimmen wollen.\n\nWir führen zu diesem Zwecke noch ein drittes Koordinatensystem $K'$  ein, welches relativ zum System $k$ derart in Paralleltranslationsbewegung parallel zur $\\Xi$-Achse begriffen sei, daß sich dessen Koordinatenursprung mit der Geschwindigkeit $-v$ auf der $\\Xi$-Achse bewege. Zur Zeit $t=0$ mögen alle drei Koordinatenanfangspunkte zusammenfallen und es sei für $t=x=y=z=0$ die Zeit $t'$ des Systems $K'$ gleich Null. Wir nennen $x',y',z'$ die Koordinaten, im System $K'$  gemessen, und erhalten durch zweimalige Anwendung unserer Transformationsgleichungen:\n\n$\\begin{alignat}{3}t' & =\\varphi(-v)\\beta(-v)\\left\\{ \\tau+\\frac{v}{V^{2}}\\xi\\right\\}  &  & =\\varphi(v)\\varphi(-v)t,\\\\x' & =\\varphi(-v)\\beta(-v)\\left\\{ \\xi+v\\tau\\right\\}  &  & =\\varphi(v)\\varphi(-v)x,\\\\y' & =\\varphi(-v)\\eta &  & =\\varphi(v)\\varphi(-v)y,\\\\z' & =\\varphi(-v)\\zeta &  & =\\varphi(v)\\varphi(-v)z.\\end{alignat}$\n\nDa die Beziehungen zwischen $x',y',z'$und $x,y,z$  die Zeit $t$ nicht enthalten, so ruhen die Systeme $K$  und $K'$ gegeneinander,\nund es ist klar, daß die Transformation von $K$ auf $K'$  die identische Transformation sein muß. Es ist also:\n\n$\\varphi(v)\\varphi(-v)=1$.\n\nWir fragen nun nach der Bedeutung von $\\varphi(v)$. Wir fassen das Stück der $H$-Achse des Systems $k$  ins Auge, das zwischen $\\xi=0,\\eta=0,\\zeta=0$ und $\\xi=0,\\eta=l,\\zeta=0$ gelegen ist. Dieses Stück der $H$-Achse ist ein relativ zum System $K$  mit der Geschwindigkeit $v$ senkrecht zu seiner Achse bewegter Stab, dessen Enden in $K$ die Koordinaten besitzen:\n\n$x_{1}=vt,\\ y_{1}=\\frac{l}{\\varphi(v)},\\ z_{1}=0$\n\nund\n\n$x_{2}=vt,\\ y_{2}=0,\\ z_{2}=0.$\n\nDie Länge des Stabes, in $K$ gemessen, ist also $l/\\varphi(v)$; damit ist die Bedeutung der Funktion $\\varphi$ gegeben. Aus Symmetriegründen ist nun einleuchtend, daß die im ruhenden System gemessene Länge eines bestimmten Stabes, welcher senkrecht zu seiner Achse bewegt ist, nur von der Geschwindigkeit, nicht aber von der Richtung und dem Sinne der Bewegung abhängig sein kann. Es ändert sich also die im ruhenden System gemessene Länge des bewegten Stabes nicht, wenn $v$ mit $-v$ vertauscht wird. Hieraus folgt:\n\n$\\frac{l}{\\varphi(v)}=\\frac{l}{\\varphi(-v)},$\n\noder\n\n$\\varphi(v)=\\varphi(-v)$.\n\nAus dieser und der vorhin gefundenen Relation folgt, daß $\\varphi(v)=1$ sein muß, so daß die gefundenen Transformationsgleichungen übergehen in:\n\n$\\begin{align}\\tau & =\\beta(t-\\frac{v}{V^{2}}x),\\\\\n\\xi & =\\beta(x-vt),\\\\\n\\eta & =y,\\\\\n\\zeta & =z,\n\\end{align}$\n\nwobei\n\n$\\beta=\\frac{1}{\\sqrt{1-\\left(\\frac{v}{V}\\right)^{2}}}.$"
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  "edition": "urn:eng:einstein:clir:electrodynamics-1905#EINSTEIN_1905_DE",
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  "id": "urn:eng:einstein:clir:electrodynamics-1905#EIN_DE_S4",
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      "language": "de",
      "status": "official",
      "text": "§ 4.\nPhysikalische Bedeutung der erhaltenen Gleichungen, bewegte starre Körper und bewegte Uhren betreffend.\n\nWir betrachten eine starre Kugel [Anm: Das heißt einen Körper, welcher ruhend untersucht Kugelgestalt besitzt.] vom Radius $R$, welche relativ zum bewegten System $k$ ruht, und deren Mittelpunkt im Koordinatenursprung von $k$ liegt. Die Gleichung der Oberfläche dieser relativ zum System $K$ mit der Geschwindigkeit $v$ bewegten Kugel ist:\n\n$\\xi^{2}+\\eta^{2}+\\zeta^{2}=R^{2}$.\n\nDie Gleichung dieser Oberfläche ist in $x,y,z$ ausgedrückt zur Zeit $t=0$:\n\n$\\frac{x^{2}}{\\left(\\sqrt{1-\\left(\\frac{v}{V}\\right)^{2}}\\right)^{2}}+y^{2}+z^{2}=R^{2}.$\n\nEin starrer Körper, welcher in ruhendem Zustande ausgemessen die Gestalt einer Kugel hat, hat also in bewegtem Zustande — vom ruhenden System aus betrachtet — die Gestalt eines Rotationsellipsoides mit den Achsen\n\n$R\\sqrt{1-\\left(\\frac{v}{V}\\right)^{2}},\\ R,\\ R.$\n\nWährend also die $Y$- und $Z$-Dimension der Kugel (also auch jedes starren Körpers von beliebiger Gestalt) durch die Bewegung nicht modifiziert erscheinen, erscheint die $X$-Dimension im Verhältnis $1:\\sqrt{1-(v/V)^{2}}$ verkürzt, also um so stärker, je größer $v$  ist. Für $v=V$ schrumpfen alle bewegten Objekte — vom „ruhenden“ System aus betrachtet — in flächenhafte Gebilde zusammen. Für Überlichtgeschwindigkeiten werden unsere Überlegungen sinnlos; wir werden übrigens in den folgenden Betrachtungen finden, daß die Lichtgeschwindigkeit in unserer Theorie physikalisch die Rolle der unendlich großen Geschwindigkeiten spielt.\n\nEs ist klar, daß die gleichen Resultate von im „ruhenden“ System ruhenden Körpern gelten, welche von einem gleichförmig bewegten System aus betrachtet werden. —\n\nWir denken uns ferner eine der Uhren, welche relativ zum ruhenden System ruhend die Zeit $t$, relativ zum bewegten\nSystem ruhend die Zeit $\\tau$ anzugeben befähigt sind, im Koordinatenursprung von $k$  gelegen und so gerichtet, daß sie die Zeit $\\tau$ angibt. Wie schnell geht diese Uhr, vom ruhenden System aus betrachtet?\n\nZwischen die Größen $x$, $t$ und $\\tau$, welche sich auf den Ort dieser Uhr beziehen, gelten offenbar die Gleichungen:\n\n$\\tau=\\frac{1}{\\sqrt{1-\\left(\\frac{v}{V}\\right)^{2}}}(t-\\frac{v}{V^{2}}x)$\n\nund\n\n$x=vt$.\n\nEs ist also\n\n$\\tau=t\\sqrt{1-\\left(\\frac{v}{V}\\right)^{2}}=t-\\left(1-\\sqrt{1-\\left(\\frac{v}{V}\\right)^{2}}\\right)t,$\n\nworaus folgt, daß die Angabe der Uhr (im ruhenden System betrachtet) pro Sekunde um $\\left(1-\\sqrt{1-(v/V)^{2}}\\right)$ Sek. oder — bis auf Größen vierter und höherer Ordnung um $\\frac{1}{2}(v/V)^{2}$ Sek. zurückbleibt.\n\nHieraus ergibt sich folgende eigentümliche Konsequenz. Sind in den Punkten $A$ und $B$ von $K$  ruhende, im ruhenden System betrachtet, synchron gehende Uhren vorhanden, und bewegt man die Uhr in $A$ mit der Geschwindigkeit $v$  auf der Verbindungslinie nach $B$, so gehen nach Ankunft dieser Uhr in $B$ die beiden Uhren nicht mehr synchron, sondern die von $A$ nach $B$ bewegte Uhr geht gegenüber der von Anfang an in $B$ befindlichen um $\\frac{1}{2}tv^{2}/V^{2}$ Sek. (bis auf Größen vierter und höherer Ordnung) nach, wenn $t$  die Zeit ist, welche die Uhr von $A$ nach $B$ braucht.\n\nMan sieht sofort, daß dies Resultat auch dann noch gilt, wenn die Uhr in einer beliebigen polygonalen Linie sich von $A$ nach $B$  bewegt, und zwar auch dann, wenn die Punkte $A$ und $B$  zusammenfallen.\n\nNimmt man an, daß das für eine polygonale Linie bewiesene Resultat auch für eine stetig gekrümmte Kurve gelte, so erhält man den Satz: Befinden sich in $A$ zwei synchron gehende Uhren und bewegt man die eine derselben auf einer geschlossenen Kurve mit konstanter Geschwindigkeit, bis sie wieder nach $A$ zurückkommt, was $t$ Sek. dauern möge, so geht die letztere Uhr bei ihrer Ankunft in $A$  gegenüber der unbewegt\ngebliebenen um $\\frac{1}{2}t(v/V)^{2}$ Sek. nach. Man schließt daraus, daß eine am Erdäquator befindliche Unruhuhr um einen sehr kleinen Betrag langsamer laufen muß als eine genau gleich beschaffene, sonst gleichen Bedingungen unterworfene, an einem Erdpole befindliche Uhr."
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  "edition": "urn:eng:einstein:clir:electrodynamics-1905#EINSTEIN_1905_DE",
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      "contentHash": "sha256:c78541a862c858a7e546b54b4f1b1dde1cd701545a1c0fa24b7d977e268bb88b",
      "language": "de",
      "status": "official",
      "text": "§ 5.\nAdditionstheorem der Geschwindigkeiten.\n\nIn dem längs der $X$-Achse des Systems $K$ mit der Geschwindigkeit $v$ bewegten System $k$ bewege sich ein Punkt gemäß den Gleichungen:\n\n$\\begin{align}\\xi= & w_{\\xi}\\tau,\\\\\n\\eta= & w_{\\eta}\\tau,\\\\\n\\zeta= & 0,\n\\end{align}$\n\nwobei $w_{\\xi}$ und $w_{\\eta}$ Konstanten bedeuten.\n\nGesucht ist die Bewegung des Punktes relativ zum System $K$. Führt man in die Bewegungsgleichungen des Punktes mit Hilfe der in § 3 entwickelten Transformationsgleichungen die Größen $x,y,z,t$  ein, so erhält man:\n\n$\\begin{align}x & =\\frac{w_{\\xi}+v}{1+\\frac{vw_{\\xi}}{V^{2}}}t,\\\\\ny & =\\frac{\\sqrt{1-\\left(\\frac{v}{V}\\right)^{2}}}{1+\\frac{vw_{\\xi}}{V^{2}}}w_{\\eta}t,\\\\\nz & =0.\n\\end{align}$\n\nDas Gesetz vom Parallelogramm der Geschwindigkeiten gilt also nach unserer Theorie nur in erster Annäherung. Wir setzen:\n\n$\\begin{align}U^{2} & =\\left(\\frac{dx}{dt}\\right)^{2}+\\left(\\frac{dy}{dt}\\right)^{2},\\\\\nw^{2} & =w_{\\xi}^{2}+w_{\\eta}^{2}\n\\end{align}$\n\nund\n\n$\\alpha={\\rm arctg}\\frac{w_{\\eta}}{w_{\\xi}};$\n\n$\\alpha$ ist dann als der Winkel zwischen den Geschwindigkeiten $v$  und $w$ anzusehen. Nach einfacher Rechnung ergibt sich:\n\n$U=\\frac{\\sqrt{(v^{2}+w^{2}+2v\\ w\\ \\cos\\alpha)-\\left(\\frac{v\\ w\\ \\sin\\alpha}{V}\\right){}^{2}}}{1+\\frac{v\\ w\\ \\sin\\alpha}{V^{2}}}.$\n\nEs ist bemerkenswert, daß $v$ und $w$ in symmetrischer Weise in den Ausdruck für die resultierende Geschwindigkeit eingehen. Hat auch $w$ die Richtung der $X$-Achse ($\\Xi$-Achse), so erhalten wir:\n\n$U=\\frac{v+w}{1+\\frac{vw}{V^{2}}}.$\n\nAus dieser Gleichung folgt, daß aus der Zusammensetzung zweier Geschwindigkeiten, welche kleiner sind als $V$, stets eine Geschwindigkeit kleiner als $V$ resultiert. Setzt man nämlich $v=V-\\varkappa$, $w=V-\\lambda$, wobei $\\varkappa$ und $\\lambda$ positiv und kleiner als $V$ seien, so ist:\n\n$U=V\\frac{2V-\\varkappa-\\lambda}{2V-\\varkappa-\\lambda+\\frac{\\varkappa\\lambda}{V}}<V.$\n\nEs folgt ferner, daß die Lichtgeschwindigkeit $V$ durch Zusammensetzung mit einer „Unterlichtgeschwindigkeit“ nicht geändert werden kann. Man erhält für diesen Fall:\n\n$U=\\frac{V+w}{1+\\frac{w}{v}}=V.$\n\nWir hätten die Formel für $U$ für den Fall, daß $v$  und w gleiche Richtung besitzen, auch durch Zusammensetzen zweier Transformationen gemäß § 3 erhalten können. Führen wir neben den in § 3 figurierenden Systemen $K$ und $k$  noch ein drittes, zu $k$ in Parallelbewegung begriffenes Koordinatensystem $k'$ ein, dessen Anfangspunkt sich auf der $\\Xi$-Achse mit der Geschwindigkeit $w$ bewegt, so erhalten wir zwischen den Größen $x,y,z,t$ und den entsprechenden Größen von $k'$ Gleichungen, welche sich von den in § 3 gefundenen nur dadurch unterscheiden, daß an Stelle von „$v$“ die Größe\n\n$\\frac{v+w}{1+\\frac{vw}{V^{2}}}$\n\ntritt; man sieht daraus, daß solche Paralleltransformationen — wie dies sein muß — eine Gruppe bilden.\n\nWir haben nun die für uns notwendigen Sätze der unseren zwei Prinzipien entsprechenden Kinematik hergeleitet und gehen dazu über, deren Anwendung in der Elektrodynamik zu zeigen.\n\nII. Elektrodynamischer Teil."
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  "edition": "urn:eng:einstein:clir:electrodynamics-1905#EINSTEIN_1905_EN",
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  "id": "urn:eng:einstein:clir:electrodynamics-1905#EIN_EN_S1",
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      "contentHash": "sha256:e3058b9932f0980f3fe6d0fffbf85e524c385da8a80b7b2df3c14ef958039b0f",
      "language": "en",
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      "text": "§ 1.\nDefinition of Synchronism.\n\nLet us have a co-ordinate system, in which the Newtonian equations hold. For distinguishing this system from another which will be introduced hereafter, we shall always call it \"the stationary system.\"\n\nIf a material point be at rest in this system, then its position in this system can be found out by a measuring rod, and can be expressed by the methods of Euclidean Geometry, or in Cartesian co-ordinates.\n\nIf we wish to describe the motion of a material point, the values of its coordinates must be expressed as functions of time. It is always to be borne in mind that such a mathematical definition has a physical sense, only when we have a clear notion of what is meant by time. We have to take into consideration the fact that those of our conceptions, in which time plays a part, are always conceptions of synchronism. For example, we say that a train arrives here at 7 o'clock ; this means that the exact pointing of the little hand of my watch to 7, and the arrival of the train are synchronous events.\n\nIt may appear that all difficulties connected with the definition of time can be removed when in place of time, we substitute the position of the little hand of my watch. Such a definition is in fact sufficient, when it is required to define time exclusively for the place at which the clock is stationed. But the definition is not sufficient when it is required to connect by time events taking place at different stations, —or what amounts to the same thing,— to estimate by means of time (zeitlich werten) the occurrence of events, which take place at stations distant from the clock.\n\nNow with regard to this attempt; —the time-estimation of events, we can satisfy ourselves in the following manner. Suppose an observer —who is stationed at the origin of coordinates with the clock— associates a ray of light which comes to him through space, and gives testimony to the event of which the time is to be estimated, — with the corresponding position of the hands of the clock. But such an association has this defect, —it depends on the position of the observer provided with the clock, as we know by experience. We can attain to a more practicable result by the following treatment.\n\nIf an observer be stationed at A with a clock, he can estimate the time of events occurring in the immediate neighbourhood of A, by looking for the position of the hands of the clock, which are synchronous with the event. If an observer be stationed at B with a clock, —we should add that the clock is of the same nature as the one at A,— he can estimate the time of events occurring about B. But without further premises, it is not possible to compare, as far as time is concerned, the events at B with the events at A. We have hitherto an A-time, and a B-time, but no time common to A and B. This last time (i.e., common time) can be defined, if we establish by definition that the time which light requires in travelling from A to B is equivalent to the time which light requires in travelling from B to A. For example, a ray of light proceeds from A at A-time t_A towards B, arrives and is reflected from B at B-time t_B, and returns to A at A-time t'_A. According to the definition, both clocks are synchronous, if\n\n$t_B - t_A = t'_A - t_B$.\n\nWe assume that this definition of synchronism is possible without involving any inconsistency, for any number of points, therefore the following relations hold :—\n\n1. If the clock at B be synchronous with the clock at A, then the clock at A is synchronous with the clock at B.\n\n2. If the clock at A as well as the clock at B are both synchronous with the clock at C, then the clocks at A and B are synchronous.\n\nThus with the help of certain physical experiences, we have established what we understand when we speak of clocks at rest at different stations, and synchronous with one another ; and thereby we have arrived at a definition of synchronism and time.\n\nIn accordance with experience we shall assume that the magnitude\n\n$\\frac{2\\ \\overline{AB}}{t'_{A}-t_{A}}=c$, where c is a universal constant.\n\nWe have defined time essentially with a clock at rest in a stationary system. On account of its adaptability to the stationary system, we call the time defined in this way as \"time of the stationary system.\""
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      "text": "§ 2.\nOn the Relativity of Length and Time.\n\nThe following reflections are based on the Principle of Relativity and on the Principle of Constancy of the velocity of light, both of which we define in the following way :—\n\n1. The laws according to which the nature of physical systems alter are independent of the manner in which these changes are referred to two co-ordinate systems\nwhich have a uniform translatory motion relative to each other.\n\n2. Every ray of light moves in the \"stationary co-ordinate system\" with the same velocity c, the velocity being independent of the condition whether this ray of light is emitted by a body at rest or in motion. Therefore\n\n$\\text{velocity} = \\frac{\\text{Path of Light}}{\\text{Interval of time}},$\n\nwhere, by 'interval of time,' we mean time as defined in § 1.\n\nLet us have a rigid rod at rest ; this has a length l, when measured by a measuring rod at rest ; we suppose that the axis of the rod is laid along the X-axis of the system at rest, and then a uniform velocity v, parallel to the axis of X, is imparted to it. Let us now enquire about the length of the moving rod ; this can be obtained by either of these operations.—\n\n(a) The observer provided with the measuring rod moves along with the rod to be measured, and measures by direct superposition the length of the rod : — just as if the observer, the measuring rod, and the rod to be measured were at rest.\n\n(b) The observer finds out, by means of clocks placed in a system at rest (the clocks being synchronous as defined in § 1), the points of this system where the ends of the rod to be measured occur at a particular time t. The distance between these two points, measured by the previously used measuring rod, this time it being at rest, is a length, which we may call the \"length of the rod.\"\n\nAccording to the Principle of Relativity, the length found out by the operation a), which we may call \"the\nlength of the rod in the moving system\" is equal to the length l of the rod in the stationary system.\n\nThe length which is found out by the second method, may be called  'the length of the moving rod measured from the stationary system'. This length is to be estimated on the basis of our principle, and we shall find it to be different from l.\n\nIn the generally recognised kinematics, we silently assume that the lengths defined by these two operations are equal, or in other words, that at an epoch of time t, a moving rigid body is geometrically replaceable by the same body, which can replace it in the condition of rest.\n\nRelativity of Time.\n\nLet us suppose that the two clocks synchronous with the clocks in the system at rest are brought to the ends A, and B of a rod, i.e., the time of the clocks correspond to the time of the stationary system at the points where they happen to arrive ; these clocks are therefore synchronous in the stationary system.\n\nWe further imagine that there are two observers at the two watches, and moving with them, and that these observers apply the criterion for synchronism to the two clocks. At the time t_A, a ray of light goes out from A, is reflected from B at the time t_B, and arrives back at A at time t'_A. Taking into consideration the principle of constancy of the velocity of light, we have\n\n$t_{B}-t_{A}=\\frac{r_{AB}}{c-v}$ ,\n\nand\n\n$t'_{A}-t_{B}=\\frac{r_{AB}}{c+v}$ ,\nwhere $r_{AB}$ is the length of the moving rod, measured in the stationary system. Therefore the observers stationed with the watches will not find the clocks synchronous, though the observer in the stationary system must declare the clocks to be synchronous. We therefore see that we can attach no absolute significance to the concept of synchronism ; but two events which are synchronous when viewed from one system, will not be synchronous when viewed from a system moving relatively to this system."
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      "text": "§ 3.\nTheory of Co-ordinate and Time-Transformation from a stationary system to a system which moves relatively to this with uniform velocity.\n\nLet there be given, in the stationary system two co-ordinate systems, i.e., two series of three mutually perpendicular lines issuing from a point. Let the X-axes of each coincide with one another, and the Y and Z-axes be parallel. Let a rigid measuring rod, and a number of clocks be given to each of the systems, and let the rods and clocks in each be exactly alike each other.\n\nLet the initial point of one of the systems (k) have a constant velocity in the direction of the X-axis of the other which is stationary system K, the motion being also communicated to the rods and clocks in the system (k). Any time t of the stationary system K corresponds to a definite position of the axes of the moving system, which are always parallel to the axes of the stationary system. By t, we always mean the time in the stationary system.\n\nWe suppose that the space is measured by the stationary measuring rod placed in the stationary system, as well as by the moving measuring rod placed in the moving\nsystem, and we thus obtain the co-ordinates (x, y, z) for the stationary system, and (ξ, η, ζ) for the moving system. Let the time t be determined for each point of the stationary system (which are provided with clocks) by means of the clocks which are placed in the stationary system, with the help of light-signals as described in § 1. Let also the time τ of the moving system be determined for each point of the moving system (in which there are clocks which are at rest relative to the moving system), by means of the method of light signals between these points (in which there are clocks) in the manner described in § 1.\n\nTo every value of (x, y, z, t) which fully determines the position and time of an event in the stationary system, there correspond a system of values (ξ, η, ζ, τ) ; now the problem is to find out the system of equations connecting these magnitudes.\n\nPrimarily it is clear that on account of the property of homogeneity which we ascribe to time and space, the equations must be linear.\n\nIf we put x'=x-vt, then it is clear that at a point relatively at rest in the system K, we have a system of values (x' y z) which are independent of time. Now let us find out τ as a function of (x,y,z,t). For this purpose we have to express in equations the fact that τ is not other than the time given by the clocks which are at rest in the system k which must be made synchronous in the manner described in § 1.\n\nLet a ray of light be sent at time $\\tau_{0}$ from the origin of the system k along the X-axis towards x' and let it be reflected from that place at time $\\tau_{1}$ towards the origin of moving co-ordinates and let it arrive there at time $\\tau_{2}$ ; then we must have\n\n$\\frac{1}{2}(\\tau_{0}+\\tau_{2})=\\tau_{1}$\nIf we now introduce the condition that $\\tau$ is a function of co-ordinates, and apply the principle of constancy of the velocity of light in the stationary system, we have\n\n$\\frac{1}{2}\\left[\\tau(0,\\ 0,\\ 0,\\ t)+\\tau\\left(0,\\ 0,\\ 0,\\ \\left\\{ t+\\frac{x'}{c-v}+\\frac{x'}{c+v}\\right\\} \\right)\\right]$\n\n$=\\tau\\left(x',\\ 0,\\ 0,\\ t+\\frac{x'}{c-v}\\right)$.\n\nIt is to be noticed that instead of the origin of coordinates, we could select some other point as the exit point for rays of light, and therefore the above equation holds for all values of (x, y, z, t).\n\nA similar conception, being applied to the y- and z-axis gives us, when we take into consideration the fact that light when viewed from the stationary system, is always propagated along those axes with the velocity $\\sqrt{c^{2}-v^{2}}$, we have the questions:\n\n$\\frac{\\partial\\tau}{\\partial y}=0,\\ \\frac{\\partial\\tau}{\\partial z}=0$.\n\nFrom these equations it follows that $\\tau$ is a linear function of x'  and t. From equations (1) we obtain\n\n$\\tau=a\\left(t-\\frac{vx'}{c^{2}-v^{2}}\\right)$,\n\nwhere $a$ is an unknown function of v.\n\nWith the help of these results it is easy to obtain the magnitudes (ξ, η, ζ), if we express by means of equations the fact that light, when measured in the moving system is always propagated with the constant velocity c (as the principle of constancy of light velocity in conjunction with the principle of relativity requires). For a\ntime $\\tau = 0$, if the ray is sent in the direction of increasing ξ, we have\n\n$\\xi=c\\tau$, i.e. $\\xi=ac\\left(t-\\frac{vx'}{c^{2}-v^{2}}\\right)$.\n\nNow the ray of light moves relative to the origin of k with a velocity c-v, measured in the stationary system ; therefore we have\n\n$\\frac{x'}{c-v}=t$.\n\nSubstituting these values of t in the equation for ξ, we obtain\n\n$\\xi=a\\frac{c^{2}}{c^{2}-v^{2}}x'$.\n\nIn an analogous manner, we obtain by considering the ray of light which moves along the y-axis,\n\n$\\eta=c\\tau=ac\\left(t-\\frac{vx'}{c^{2}-v^{2}}\\right)$,\n\nwhere $\\frac{y}{\\sqrt{c^{2}-v^{2}}}=t,\\ x'=0$.\n\nTherefore $\\eta=a\\frac{c}{\\sqrt{c^{2}-v^{2}}}y,\\ \\zeta=a\\frac{c}{\\sqrt{c^{2}-v^{2}}}z$.\n\nIf for x', we substitute its value x—tv, we obtain\n\n:$\\tau=\\phi\\ (v)\\cdot\\beta\\left(t-\\frac{vx}{c^{2}}\\right)$,\n\n:$\\xi=\\phi\\ (v)\\cdot\\beta\\left(x-vt\\right)$,\n\n:$\\eta=\\phi\\ (v)\\ y$,\n\n:$\\zeta=\\phi\\ (v)\\ z$,\n\nwhere $\\beta=\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}$, and $\\phi(v)=\\frac{\\alpha c}{\\sqrt{c^{2}-v^{2}}}=\\frac{\\alpha}{\\beta}$ is a function of v.\n\nIf we make no assumption about the initial position of the moving system and about the null-point of t, then an additive constant is to be added to the right hand side.\n\nWe have now to show, that every ray of light moves in the moving system with a velocity c (when measured in the moving system), in case, as we have actually assumed, c is also the velocity in the stationary system ; for we have not as yet adduced any proof in support of the assumption that the principle of relativity is reconcilable with the principle of constant light-velocity.\n\nAt a time $\\tau = t = 0$ let a spherical wave be sent out from the common origin of the two systems of co-ordinates, and let it spread with a velocity c in the system K. If (x, y, z), be a point reached by the wave, we have\n\n$x^2 + y^2 + z^2 = c^2t^2$.\n\nwith the aid of our transformation-equations, let us transform this equation, and we obtain by a simple calculation,\n\n$\\xi^2 + \\eta^2 + \\zeta^2 = c^2\\tau^2$.\n\nTherefore the wave is propagated in the moving system with the same velocity c, and as a spherical wave. Therefore we show that the two principles are mutually reconcilable.\n\nIn the transformations we have got an undetermined function $\\phi(v)$, and we now proceed to find it out.\n\nLet us introduce for this purpose a third co-ordinate system k', which is set in motion relative to the system k, the motion being parallel to the $\\xi$-axis. Let the velocity of the origin be (—v). At the time $t = 0$, all the initial co-ordinate points coincide, and for $t=x=y=z=0$, the time t'  of the system k' = 0. We shall say that (x, y z t) are the co-ordinates measured in the system k, then by a\ntwo-fold application of the transformation-equations, we obtain\n\n$t'=\\phi(-v)\\beta(-v)\\left\\{ \\tau+\\frac{v}{c^{2}}\\xi\\right\\} =\\phi(v)\\phi(-v)t$,\n\n$x' = \\phi(v)\\beta(v)(\\xi + v \\tau) = \\phi(v)\\phi(-v)x$, etc.\n\nSince the relations between (x', y', z', t'), and (x, y, z, t) do not contain time explicitly, therefore K and k'  are relatively at rest.\n\nIt appears that the systems K and k' are identical.\n\n$\\therefore\\phi(v)\\ \\phi(-v)=1$,\n\nLet us now turn our attention to the part of the y-axis between ($\\xi = 0, \\eta = 0, \\zeta = 0$), and ($\\xi = 0, \\eta = 1, \\zeta = 0$). Let this piece of the y-axis be covered with a rod moving with the velocity v relative to the system K and perpendicular to its axis ;—the ends of the rod having therefore the co-ordinates\n\n$\\left. \\begin{array}{lll}\nx_{1}=vt, & y=\\frac{l}{\\phi(v)},     & z_{1}=0\\\\\nx_{2}=vt, & y_{2}=\\frac{l}{\\phi(v)}, & z_{2}=0\n\\end{array} \\right\\}$\n\nTherefore the length of the rod measured in the system K is $\\frac{l}{\\phi(v)}$. For the system moving with velocity (-v), we have on grounds of symmetry,\n\n$\\frac{l}{\\phi(v)}=\\frac{l}{\\phi(-v)}$\n\n$\\therefore\\phi(v)=\\phi(-v),\\ \\therefore\\phi(v)=1$"
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      "text": "§ 4.\nThe physical significance of the equations obtained concerning moving rigid bodies and moving clocks.\n\nLet us consider a rigid sphere (i.e., one having a spherical figure when tested in the stationary system) of radius R which is at rest relative to the system (K), and whose centre coincides with the origin of K then the equation of the surface of this sphere, which is moving with a velocity v relative to K, is\n\n$\\xi^{2}+\\eta^{2}+\\zeta^{2}=R^{2}$\n\nAt time t = 0 the equation is expressed by means of (x, y, z, t,) as\n\n$\\frac{x^{2}}{\\left(\\sqrt{1-\\frac{v^{2}}{c^{2}}}\\right)^{2}}+y^{2}+z^{2}=R^{2}$.\n\nA rigid body which has the figure of a sphere when measured in the moving system, has therefore in the moving condition — when considered from the stationary system, the figure of a rotational ellipsoid with semi-axes\n\n$R\\sqrt{1-\\frac{v^{2}}{c^{2}}}, R, R$.\n\nTherefore the y and z dimensions of the sphere (therefore of any figure also) do not appear to be modified by the motion, but the x dimension is shortened in the ratio $1:\\sqrt{1-\\frac{v^{2}}{c^{2}}}$ ; the shortening is the larger, the larger is v. For v = c, all moving bodies, when considered from a stationary system shrink into planes. For a velocity larger than the velocity of light, our propositions become\nmeaningless ; in our theory c plays the part of infinite velocity.\n\nIt is clear that similar results hold about stationary bodies in a stationary system when considered from a uniformly moving system.\n\nLet us now consider that a clock which is lying at rest in the stationary system gives the time t, and lying at rest relative to the moving system is capable of giving the time τ ; suppose it to be placed at the origin of the moving system k, and to be so arranged that it gives the time τ. How much does the clock gain, when viewed from the stationary system K? We have,\n\n$\\tau=\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}\\left(t-\\frac{v}{c^{2}}x\\right)$, and $x=vt$,\n\n$\\therefore\\tau-t=\\left[1-\\sqrt{1-\\frac{v^{2}}{c^{2}}}\\right]t$.\n\nTherefore the clock loses by an amount $\\frac{1}{2}\\frac{v^{2}}{c^{2}}$ per second of motion, to the second order of approximation.\n\nFrom this, the following peculiar consequence follows. Suppose at two points A and B of the stationary system two clocks are given which are synchronous in the sense explained in § 3 when viewed from the stationary system. Suppose the clock at A to be set in motion in the line joining it with B, then after the arrival of the clock at B, they will no longer be found synchronous, but the clock which was set in motion from A will lag behind the clock which had been all along at B by an amount $\\frac{1}{2}t\\frac{v^{2}}{c^{2}}$, where t is the time required for the journey.\n\nWe see forthwith that the result holds also when the clock moves from A to B by a polygonal line, and also when A and B coincide.\n\nIf we assume that the result obtained for a polygonal line holds also for a curved line, we obtain the following law. If at A, there be two synchronous clocks, and if we set in motion one of them with a constant velocity along a closed curve till it comes back to A, the journey being completed in t-seconds, then after arrival, the last mentioned clock will be behind the stationary one by $\\frac{1}{2}t\\frac{v^{2}}{c^{2}}$ seconds. From this, we conclude that a clock placed at the equator must be slower by a very small amount than a similarly constructed clock which is placed at the pole, all other conditions being identical."
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      "text": "§ 5.\nAddition-Theorem of Velocities.\n\nLet a point move in the system k (which moves with velocity v along the x-axis of the system K) according to the equation\n\n$\\xi=w_{\\xi}\\tau,\\ \\eta=w_{\\eta}\\tau,\\ \\zeta=0$,\n\nwhere $w_{\\xi}$ and $w_{\\eta}$ are constants.\n\nIt is required to find out the motion of the point relative to the system K. If we now introduce the system of equations in § 3 in the equation of motion of the point, we obtain\n\n$x=\\frac{w_{\\xi}+v}{1+\\frac{vw_{\\xi}}{c^{2}}},\\ y=\\frac{\\left(1-\\frac{v^{2}}{c^{2}}\\right)^{\\frac{1}{2}}w_{\\eta}t}{1+\\frac{vw_{\\xi}}{c^{2}}},\\ z=0$.\nThe law of parallelogram of velocities hold up to the first order of approximation. We can put\n\n$U^{2}=\\left(\\frac{\\partial x}{\\partial t}\\right)^{2}+\\left(\\frac{\\partial y}{\\partial t}\\right)^{2},\\ w^{2}=w_{\\xi}^{2}+w_{\\eta}^{2}$,\n\nand\n\n$\\alpha=\\tan^{-1}\\frac{w}{w_{\\xi}}$\n\ni.e., $\\alpha$ is put equal to the angle between the velocities v, and w. Then we have—\n\n$U=\\frac{\\left[(v^{2}+w^{2}+2vw\\ \\cos\\ \\alpha)-\\left(\\frac{vw\\ \\sin\\ \\alpha}{c}\\right)^{2}\\right]^{\\frac{1}{2}}}{1+\\frac{vw\\ \\cos\\ \\alpha}{c^{2}}}$\n\nIt should be noticed that v and w enter into the expression for velocity symmetrically. If w has the direction of the &xi;-axis of the moving system,\n\n$U=\\frac{v+w}{1+\\frac{vw}{c^{2}}}$\n\nFrom this equation, we see that by combining two velocities, each of which is smaller than c, we obtain a velocity which is always smaller than c. If we put $v = c - \\chi$, and $w = c - \\lambda$ where $\\chi$ and $\\lambda$ are each smaller than c,\n\n$U=c\\frac{2c-\\chi-\\lambda}{2c-\\chi-\\lambda+\\frac{\\chi\\lambda}{c^{2}}}<c$.\n\nIt is also clear that the velocity of light c cannot be altered by adding to it a velocity smaller than c. For this case,\n\n$U=\\frac{c+v}{1+\\frac{cv}{c^{2}}}=c$\nWe have obtained the formula for U for the case when v and w have the same direction ; it can also be obtained by combining two transformations according to section § 3. If in addition to the systems K, and k, we introduce the system k', of which the initial point moves parallel to the &xi;-axis with velocity w, then between the magnitudes, x, y, z, t and the corresponding magnitudes of k', we obtain a system of equations, which differ from the equations in §3, only in the respect that in place of v, we shall have to write,\n\n$(v+w)/\\left(1+\\frac{vw}{c^{2}}\\right)$\n\nWe see that such a parallel transformation forms a group.\n\nWe have deduced the kinematics corresponding to our two fundamental principles for the laws necessary for us, and we shall now pass over to their application in electrodynamics.\n\nII. — ELECTRODYNAMICAL PART."
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      "text": "§ 1.\nThe problem of determining the influence exerted on electric and optical phenomena by a translation, such as all systems have in virtue of the Earth's annual motion, admits of a comparatively simple solution, so long as only those terms need be taken into account, which are proportional to the first power of the ratio between the velocity of translation w and the velocity of light c. Cases in which quantities of the second order, i.e. of the order $\\frac{w^{2}}{c^{2}}$, may be perceptible, present more difficulties. The first example of this kind is Michelson's well known interference-experiment, the negative result of which has led FitzGerald and myself to the conclusion that the dimensions of solid bodies are slightly altered by their motion through the aether.\n\nSome new experiments in which a second order effect was sought for have recently been published. Rayleigh [Note: Rayleigh, Phil. Mag. (6) 4 (1902), p. 678.] and Brace [Note: Brace, Phil. Mag. (6) 7 (1904), p. 317.] have examined the question whether the Earth's motion may cause a body to become doubly refracting; at first sight this might be expected, if the just mentioned change of dimensions is admitted. Both physicists have however come to a negative result.\n\nIn the second place Trouton and Noble [Note: Trouton and Noble, London Roy. Soc. Trans. A. 202 (1903), p. 165] have endeavoured to detect a turning couple acting on a charged condenser, whose plates make a certain angle with the direction of translation. The theory\nof electrons, unless it be modified by some new hypothesis, would undoubtedly require the existence of such a couple. In order to see this, it will suffice to consider a condenser with aether as dielectricum. It may be shown that in every electrostatic system, moving with a velocity $\\mathfrak{w}$, [Note: A vector will be denoted by a German letter, its magnitude by the corresponding Latin letter.] there is a certain amount of electromagnetic momentum. If we represent this, in direction and magnitude, by a vector $\\mathfrak{G}$, the couple in question will be determined by the vector product [Note: See my article: Weiterbildung der Maxwell'schen Theorie. Elektronentheorie. Encyclopädie V 14, § 21, a. (This article will be quoted as M.E.)]\n\n$\\left[\\mathfrak{G}.\\mathfrak{w}\\right]$. (1)\n\nNow, if the axis of z is chosen perpendicular to the condenser plates, the velocity w having any direction we like, and if U is the energy of the condenser, calculated on the ordinary way, the components of $\\mathfrak{G}$ are given [Note: M. E. § 56, c.] by the following formulae, which are exact up to the first order:\n\n$\\mathfrak{G}_{x}=\\frac{2U}{c^{2}}\\mathfrak{w}_{x},\\quad \\mathfrak{G}_{y}=\\frac{2U}{c^{2}}\\mathfrak{w}_{y},\\quad \\mathfrak{G}_{z}=0$.\n\nSubstituting these values in (1), we get for the components of the couple, up to terms of the second order,\n\n$\\frac{2U}{c^{2}}\\mathfrak{w}_{y}\\mathfrak{w}_{z},\\quad-\\frac{2U}{c^{2}}\\mathfrak{w}_{x}\\mathfrak{w}_{z},\\quad0$.\n\nThese expressions show that the axis of the couple lies in the plane of the plates, perpendicular to the translation. If α is the angle between the velocity and the normal to the plates, the moment of the couple will be $\\frac{U}{c^{2}}w^{2}\\sin\\ 2\\alpha$; it tends to turn the condenser into such a position that the plates are parallel to the Earth's motion.\n\nIn the apparatus of Trouton and Noble the condenser was fixed to the beam of a torsion-balance, sufficiently delicate to be deflected by a couple of the above order of magnitude. No effect could however be observed."
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      "text": "§ 11.\nIt is easily seen that the proposed theory can account for a large number of facts.\n\nLet us take in the first place the case of a system without translation, in some parts of which we have continually $\\mathfrak{p}=0$, $\\mathfrak{d}=0$ and $\\mathfrak{h}=0$. Then, in the corresponding state for the moving system, we shall have in corresponding parts (or, as we may say, in the same parts of the deformed system) $\\mathfrak{p}'=0$, $\\mathfrak{d}'=0$ and $\\mathfrak{h}'=0$. These equations implying $\\mathfrak{p}=0$, $\\mathfrak{d}=0$, $\\mathfrak{h}=0$, as is seen by (26) and (6), it appears\nthat those parts which are dark while the system is at rest, will remain so after it has been put in motion. It will therefore be impossible to detect an influence of the Earth's motion on any optical experiment, made with a terrestrial source of light, in which the geometrical distribution of light and darkness is observed. Many experiments on interference and diffraction belong to this class.\n\nIn the second place, if in two points of a system, rays of light of the same state of polarization are propagated in the same direction, the ratio between the amplitudes in these points may be shown not to be altered by a translation. The latter remark applies to those experiments in which the intensities in adjacent parts of the field of view are compared.\n\nThe above conclusions confirm the results I have formerly obtained by a similar train of reasoning, in which however the terms of the second order were neglected. They also contain an explanation of MICHELSONS's negative result, more general and of somewhat different form than the one previously given, and they show why RAYLEIGH and BRACE could find no signs of double refraction produced by the motion of the Earth.\n\nAs to the experiments of TROUTON and NOBLE, their negative result becomes at once clear, if we admit the hypotheses of §8. It may be inferred from these and from our last assumption (§ 10) that the only effect of the translation must have been a contraction of the whole system of electrons and other particles constituting the charged condenser and the beam and thread of the torsion-balance. Such a contraction does not give rise to a sensible change of direction.\n\nIt need hardly be said that the present theory is put forward with all due reserve. Though it seems to me that it can account for all well established facts, it leads to some consequences that cannot as yet be put to the test of experiment. One of these is that the result of MICHELSON'S experiment must remain negative, if the interfering rays of light are made to travel through some ponderable transparent body.\n\nOur assumption about the contraction of the electrons cannot in itself be pronounced to be either plausible or inadmissible. What we know about the nature of electrons is very little and the only means of pushing our way farther will be to test such hypotheses as I have here made. Of course, there will be difficulties, e.g. as soon as we come to consider the rotation of electrons. Perhaps we shall have to suppose that in those phenomena in which, if there is no translation, spherical electrons rotate about a diameter, the points of the electrons in the moving system will describe elliptic paths,\ncorresponding, in the manner specified in § 10, to the circular paths described in the other case."
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  "edition": "urn:eng:lorentz:clir:electromagnetic-phenomena-1904#LORENTZ_1904_EN",
  "fragmentKind": "article",
  "id": "urn:eng:lorentz:clir:electromagnetic-phenomena-1904#LOR_S2",
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    {
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      "language": "en",
      "status": "official",
      "text": "§ 2.\nThe experiment of which I have spoken are not the only reason for which a new examination of the problems connected with the motion of the Earth is desirable. Poincaré [Note: Poincaré, Rapports du Congrés de physique de 1900, Paris, 1, p. 22, 23] has objected\nto the existing theory of electric and optical phenomena in moving bodies that, in order to explain Michelsons's negative result, the introduction of a new hypothesis has been required, and that the same necessity may occur each time new facts will be brought to light. Surely, this course of inventing special hypothesis for each new experimental result is somewhat artificial. It would be more satisfactory, if it were possible to show, by means of certain fundamental assumptions, and without neglecting terms of one order of magnitude or another, that many electromagnetic actions are entirely independent of the motion of the system. Some years ago, I have already sought to frame a theory of this kind. [Note: Lorentz, Zittingsverslag Akad. v. Wet., 7 (1899), p. 507, Amsterdam Proc., 1898-99, p. 427] I believe now to be able to treat the subject with a better result. The only restriction as regards the velocity will be that it be smaller than that of light."
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  "fragmentKind": "article",
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  "locator": "article/3",
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    {
      "contentHash": "sha256:0f68f1555934edbbae5693be94745ef9df17fccb0e945ea9e450cb1a43118020",
      "language": "en",
      "status": "official",
      "text": "§ 3.\nI shall start from the fundamental equations of the theory of electrons. [Note: M. E., § 2.]  Let $\\mathfrak{d}$ be the dielectric displacement in the aether, $\\mathfrak{h}$ the magnetic force, $\\varrho$ the volume-density of the charge of an electron, $\\mathfrak{v}$ the velocity of a point of such a particle, and $\\mathfrak{f}$ the electric force, i.e. the force, reckoned per unit charge, which is exerted by the aether on a volume-element of an electron. Then, if we use a fixed system of coordinates,\n\n$\\begin{cases}\ndiv\\ \\mathfrak{d}=\\varrho,\\quad div\\ \\mathfrak{h}=0,\\\\\nrot\\ \\mathfrak{h}=\\frac{1}{c}\\left(\\dot{\\mathfrak{d}}+\\varrho\\mathfrak{v}\\right),\\\\\nrot\\ \\mathfrak{d}=-\\frac{1}{c}\\dot{\\mathfrak{h}},\\\\\n\\mathfrak{f}=\\mathfrak{d}+\\frac{1}{c}\\left[\\mathfrak{v.h}\\right].\\end{cases}$. (2)\n\nI shall now suppose that the system as a whole moves in the direction of x with a constant velocity w, and I shall denote by $\\mathfrak{u}$ any velocity a point of an electron may have in addition to this, so that\n\n$\\mathfrak{v}_{x}=\\mathfrak{w}+\\mathfrak{u}_{x},\\quad\\mathfrak{v}_{y}=\\mathfrak{u}_{y},\\quad\\mathfrak{v}_{z}=\\mathfrak{u}_{z}$.\n\nIf the equations (2) are at the same time referred to axes moving with the system, they become\n\n$div\\ \\mathfrak{d}=\\varrho,\\quad div\\ \\mathfrak{h}=0$,\n\n$\\frac{\\partial\\mathfrak{h}_{z}}{\\partial y}-\\frac{\\partial\\mathfrak{h}_{y}}{\\partial z}=\\frac{1}{c}\\left(\\frac{\\partial}{\\partial t}-w\\frac{\\partial}{\\partial x}\\right)\\mathfrak{d}_{x}+\\frac{1}{c}\\varrho\\left(w+\\mathfrak{u}_{x}\\right)$,\n\n$\\frac{\\partial\\mathfrak{h}_{x}}{\\partial z}-\\frac{\\partial\\mathfrak{h}_{z}}{\\partial x}=\\frac{1}{c}\\left(\\frac{\\partial}{\\partial t}-w\\frac{\\partial}{\\partial x}\\right)\\mathfrak{d}_{y}+\\frac{1}{c}\\varrho\\mathfrak{u}_{y}$,\n\n$\\frac{\\partial\\mathfrak{h}_{y}}{\\partial x}-\\frac{\\partial\\mathfrak{h}_{x}}{\\partial y}=\\frac{1}{c}\\left(\\frac{\\partial}{\\partial t}-w\\frac{\\partial}{\\partial x}\\right)\\mathfrak{d}_{z}+\\frac{1}{c}\\varrho\\mathfrak{u}_{z}$,\n\n$\\frac{\\partial\\mathfrak{d}_{z}}{\\partial y}-\\frac{\\partial\\mathfrak{d}_{y}}{\\partial z}=-\\frac{1}{c}\\left(\\frac{\\partial}{\\partial t}-w\\frac{\\partial}{\\partial x}\\right)\\mathfrak{h}_{x}$,\n\n$\\frac{\\partial\\mathfrak{d}_{x}}{\\partial z}-\\frac{\\partial\\mathfrak{d}_{z}}{\\partial x}=-\\frac{1}{c}\\left(\\frac{\\partial}{\\partial t}-w\\frac{\\partial}{\\partial x}\\right)\\mathfrak{h}_{y}$,\n\n$\\frac{\\partial\\mathfrak{d}_{y}}{\\partial x}-\\frac{\\partial\\mathfrak{d}_{x}}{\\partial y}=-\\frac{1}{c}\\left(\\frac{\\partial}{\\partial t}-w\\frac{\\partial}{\\partial x}\\right)\\mathfrak{h}_{z}$,\n\n$\\mathfrak{f}_{x}=\\mathfrak{d}_{x}+\\frac{1}{c}\\left(\\mathfrak{u}_{y}\\mathfrak{h}_{z}-\\mathfrak{u}_{z}\\mathfrak{h}_{y}\\right)$,\n\n$\\mathfrak{f}_{y}=\\mathfrak{d}_{y}-\\frac{1}{c}w\\mathfrak{h}_{z}+\\frac{1}{c}\\left(\\mathfrak{u}_{z}\\mathfrak{h}_{x}-\\mathfrak{u}_{x}\\mathfrak{h}_{z}\\right)$,\n\n$\\mathfrak{f}_{z}=\\mathfrak{d}_{z}+\\frac{1}{c}w\\mathfrak{h}_{y}+\\frac{1}{c}\\left(\\mathfrak{u}_{x}\\mathfrak{h}_{y}-\\mathfrak{u}_{y}\\mathfrak{h}_{x}\\right)$."
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  "edition": "urn:eng:lorentz:clir:electromagnetic-phenomena-1904#LORENTZ_1904_EN",
  "fragmentKind": "article",
  "id": "urn:eng:lorentz:clir:electromagnetic-phenomena-1904#LOR_S4",
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  "locator": "article/4",
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    {
      "contentHash": "sha256:b01d79b1b07af31fd03043d46f616b5ba869005ce5f6f440724244b46528a2da",
      "language": "en",
      "status": "official",
      "text": "§ 4.\nWe shall further transform these formulae by a change of variables. Putting\n\n$\\frac{c^{2}}{c^{2}-w^{2}}=k^{2}$, (3)\n\nand understanding by l another numerical quantity, to be determined further on, I take as new independent variables\n\n$x'=klx,\\quad y'=ly,\\quad z'=lz$, (4)\n\n$t'=\\frac{l}{k}t-kl\\frac{w}{c^{2}}x$, (5)\n\nand I define two new vectors $\\mathfrak{d}'$ and $\\mathfrak{h}'$ by the formulae\n\n$\\mathfrak{d}_{x}^{'}=\\frac{1}{l^{2}}\\mathfrak{d}_{x},\\quad\\mathfrak{d}_{y}^{'}=\\frac{k}{l^{2}}\\left(\\mathfrak{d}_{y}-\\frac{w}{c}\\mathfrak{h}_{z}\\right),\\quad\\mathfrak{d}_{z}^{'}=\\frac{k}{l^{2}}\\left(\\mathfrak{d}_{z}+\\frac{w}{c}\\mathfrak{h}_{y}\\right)$,\n\n$\\mathfrak{h}_{x}^{'}=\\frac{1}{l^{2}}\\mathfrak{h}_{x},\\quad\\mathfrak{h}_{y}^{'}=\\frac{k}{l^{2}}\\left(\\mathfrak{h}_{y}+\\frac{w}{c}\\mathfrak{d}_{z}\\right),\\quad\\mathfrak{h}_{z}^{'}=\\frac{k}{l^{2}}\\left(\\mathfrak{h}_{z}-\\frac{w}{c}\\mathfrak{d}_{y}\\right)$,\n\nfor which, on account of (3), we may also write\n\n$\\begin{cases}\n\\mathfrak{d}_{x}=l^{2}\\mathfrak{d}_{x}^{'},\\quad\\mathfrak{d}_{y}=kl^{2}\\left(\\mathfrak{d}_{y}^{'}+\\frac{w}{c}\\mathfrak{h}_{z}^{'}\\right),\\quad\\mathfrak{d}_{z}=kl^{2}\\left(\\mathfrak{d}_{z}^{'}-\\frac{w}{c}\\mathfrak{h}_{y}^{\\mathfrak{'}}\\right),\\\\\n\\mathfrak{h}_{x}=l^{2}\\mathfrak{h}_{x}^{'},\\quad\\mathfrak{h}_{y}=kl^{2}\\left(\\mathfrak{h}_{y}^{'}-\\frac{w}{c}\\mathfrak{d}_{z}^{'}\\right),\\quad\\mathfrak{h}_{z}=kl^{2}\\left(\\mathfrak{h}_{z}^{'}+\\frac{w}{c}\\mathfrak{d}_{y}^{\\mathfrak{'}}\\right),\\end{cases}$. (6)\n\nAs to the coefficient l, it is to be considered as a function of w, whose value is 1 for w=0, and which, for small values of w, differs from unity no more than by an amount of the second order.\n\nThe variable t'  may be called the local time; indeed, for k=1, l=1 it becomes identical with what I have formerly understood by this name.\n\nIf, finally, we put\n\n$\\frac{1}{kl^{3}}\\varrho=\\varrho'$, (7)\n\n$k^{2}\\mathfrak{u}_{x}=\\mathfrak{u}_{x}^{'},\\quad k\\mathfrak{u}_{y}=\\mathfrak{u}_{y}^{'},\\quad k\\mathfrak{u}_{z}=\\mathfrak{u}_{z}^{'}$, (8)\n\nthese latter quantities being considered as the components of a new vector $\\mathfrak{u}'$, the equations take the following form:\n\n$\\left.\\begin{align}\n & div'\\ \\mathfrak{d}'=\\left(1-\\frac{wu_{x}'}{c^{2}}\\right)\\varrho',\\quad div'\\ \\mathfrak{h}'=0,\\\\\n & rot'\\ \\mathfrak{h'}=\\frac{1}{c}\\left(\\frac{\\partial\\mathfrak{d}'}{\\partial t'}+\\varrho'\\mathfrak{u}\\right),\\\\\n & rot'\\ \\mathfrak{d}'=-\\frac{1}{c}\\frac{\\partial\\mathfrak{h}'}{\\partial t'},\n\\end{align}\\right\\}$ (9)\n\n$\\left.\\begin{align}\n & \\mathfrak{f}_{x}=l^{2}\\mathfrak{d}_{x}^{'}+l^{2}\\frac{1}{c}\\left(\\mathfrak{u}_{y}^{'}\\mathfrak{h}_{z}^{'}-\\mathfrak{u}_{z}^{'}\\mathfrak{h}_{y}^{'}\\right)+l^{2}\\frac{w}{c^{2}}\\left(\\mathfrak{u}_{y}^{'}\\mathfrak{d}_{y}^{'}-\\mathfrak{u}_{z}^{'}\\mathfrak{d}_{z}^{'}\\right),\\\\\n & \\mathfrak{f}_{y}=\\frac{l}{k}^{2}\\mathfrak{d}_{y}^{'}+\\frac{l}{k}^{2}\\frac{1}{c}\\left(\\mathfrak{u}_{z}^{'}\\mathfrak{h}_{x}^{'}-\\mathfrak{u}_{x}^{'}\\mathfrak{h}_{z}^{'}\\right)-\\frac{l}{k}^{2}\\frac{w}{c^{2}}\\mathfrak{u}_{x}^{'}\\mathfrak{d}_{y}^{'},\\\\\n & \\mathfrak{f}_{z}=\\frac{l}{k}^{2}\\mathfrak{d}_{z}^{'}+\\frac{l}{k}^{2}\\frac{1}{c}\\left(\\mathfrak{u}_{x}^{'}\\mathfrak{h}_{y}^{'}-\\mathfrak{u}_{y}^{'}\\mathfrak{h}_{x}^{'}\\right)-\\frac{l}{k}^{2}\\frac{w}{c^{2}}\\mathfrak{u}_{x}^{'}\\mathfrak{d}_{z}^{'}.\n\\end{align}\\right\\}$ (10)\n\nThe meaning of the symbols div'  and rot'  in (9) is similar to that of div and rot in (2); only, the differentiations with respect to x, y, z are to be replaced by the corresponding ones with respect to x', y', z' ."
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  "edition": "urn:eng:lorentz:clir:electromagnetic-phenomena-1904#LORENTZ_1904_EN",
  "fragmentKind": "article",
  "id": "urn:eng:lorentz:clir:electromagnetic-phenomena-1904#LOR_S8",
  "kind": "fragment",
  "locator": "article/8",
  "package": "urn:eng:lorentz:clir:electromagnetic-phenomena-1904",
  "texts": [
    {
      "contentHash": "sha256:3747dd9e09d39caa7141a933b6c4c5151f82141ed4bd67b463ba320fbf88bb65",
      "language": "en",
      "status": "official",
      "text": "§ 8.\nThus far we have only used the fundamental equations without any new assumptions. I shall now suppose that the electrons, which I take to be spheres of radius R in the state of rest, have their dimensions changed by the effect of a translation, the dimensions in the direction of motion becoming kl times and those in perpendicular direction l times smaller.\n\nIn this deformation, which may be represented by $\\left(\\frac{1}{kl},\\ \\frac{1}{l},\\ \\frac{1}{l}\\right)$, each element of volume is understood to preserve its charge.\n\nOur assumption amounts to saying that in an electrostatic system Σ, moving with a velocity w, all electrons are flattened ellipsoids with their smaller axes in the direction of motion. If now, in order to apply the theorem of §6, we subject the system to the deformation (kl, l, l), we shall have again spherical electrons of radius R.\nHence, if we alter the relative position of the centres of the electrons in Σ by applying the deformation (kl, l, l), and if, in the points thus obtained, we place the centres of electrons that remain at rest, we shall get a system, identical to the imaginary system Σ' , of which we have spoken in § 6. The forces in this system and those in Σ will bear to each other the relations expressed by (21).\n\nIn the second place I shall suppose that the forces between uncharged particles, as well as those between such particles and electrons, are influenced by a translation in quite the same way as the electric forces in an electrostatic system. In other terms, whatever be the nature of the particles composing a ponderable body, so long as they do not move relatively to each other, we shall have between the forces acting in a system (Σ' ) without, and the same system (Σ) with a translation, the relation specified in (21), if, as regards the relative position of the particles, Σ'  is got from Σ by the deformation (kl, l, l), or Σ from Σ'  by the deformation $\\left(\\frac{1}{kl},\\ \\frac{1}{l},\\ \\frac{1}{l}\\right)$.\n\nWe see by this that, as soon as the resulting force is 0 for a particle in Σ' , the same must be true for the corresponding particle in Σ. Consequently, if, neglecting the effects of molecular motion, we suppose each particle of a solid body to be in equilibrium under the action of the attractions and repulsions exerted be its neighbours, and if we take for granted that there is but one configuration of equilibrium, we may draw the conclusion that the system Σ' , if the velocity w is imparted to it, will of itself change into the system Σ. In other terms, the translation will produce the deformation $\\left(\\frac{1}{kl},\\ \\frac{1}{l},\\ \\frac{1}{l}\\right)$.\n\nThe case of molecular motion will be considered in § 12.\n\nIt will easily be seen that the hypothesis that has formerly been made in connexion with MICHELSON'S experiment, is implied in what has now been said. However, the present hypothesis is more general because the only limitation imposed on the motion is that its velocity be smaller than that of light."
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{
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  "edition": "urn:phys:clir:classical-ether#CLASSICAL_ETHER_RU",
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  "kind": "fragment",
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    {
      "contentHash": "sha256:452a727ea75848b80ccfc741e860f2a07a55b81bec8339509ffce8059a31dc24",
      "language": "ru",
      "status": "official",
      "text": "ПРОСТАЯ МОДЕЛЬ НЕПОДВИЖНОГО СВЕТОНОСНОГО ЭФИРА: АВТОРСКАЯ РЕКОНСТРУКЦИЯ\n\nНастоящий документ является АВТОРСКОЙ РЕКОНСТРУКЦИЕЙ, составленной для целей сравнения. Он не воспроизводит и не переводит ни одной публикации, не приписывается ни одному физику и не излагает взглядов какого-либо лица или эпохи. Он собирает В ОДНУ НАЗВАННУЮ МОДЕЛЬ тот минимальный набор допущений, относительно которого имеет смысл считать ожидаемый эффект опыта Майкельсона и Морли и сопоставлять его с ответами теории Лоренца 1904 года и специальной теории относительности.\n\nНастоящее изложение не является нормативным актом и не имеет силы источника права ни в одной юрисдикции."
    }
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}

{
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  "edition": "urn:phys:clir:classical-ether#CLASSICAL_ETHER_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:classical-ether#AUTH_S02",
  "kind": "fragment",
  "locator": "section/2",
  "package": "urn:phys:clir:classical-ether",
  "texts": [
    {
      "contentHash": "sha256:ca22c1ca0e539348a145cdfc9fa2281f89e53ec74e6c9ff9378906b73dae97f2",
      "language": "ru",
      "status": "official",
      "text": "Раздел 2. Выделенная система отсчёта.\nМодель принимает: существует система отсчёта, в которой светоносная среда покоится. Эта система называется системой эфира и является выделенной: движение относительно неё есть абсолютное движение, а покой в ней есть абсолютный покой."
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{
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  "edition": "urn:phys:clir:classical-ether#CLASSICAL_ETHER_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:classical-ether#AUTH_S03",
  "kind": "fragment",
  "locator": "section/3",
  "package": "urn:phys:clir:classical-ether",
  "texts": [
    {
      "contentHash": "sha256:ec80b831241ac42fe75ca1e9d3c6fc78973798cc048c477ef5b4933a6a35edf8",
      "language": "ru",
      "status": "official",
      "text": "Раздел 3. Абсолютное время.\nМодель принимает: промежуток времени между двумя событиями один и тот же во всех системах отсчёта и не зависит от движения наблюдателя. Одновременность абсолютна: два события, одновременные в одной системе, одновременны в любой."
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}

{
  "contentHash": "sha256:b02f60234121f7a7d027a5addc99d6629bd9b55d5cbf98bb2e41ff20a6f38a51",
  "edition": "urn:phys:clir:classical-ether#CLASSICAL_ETHER_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:classical-ether#AUTH_S04",
  "kind": "fragment",
  "locator": "section/4",
  "package": "urn:phys:clir:classical-ether",
  "texts": [
    {
      "contentHash": "sha256:58ba383b5f74246ac73cd01c42e07abbeb7154b4f7dfe5cb1071cbaf76764e35",
      "language": "ru",
      "status": "official",
      "text": "Раздел 4. Галилеевы преобразования.\nМодель принимает: при переходе от системы, движущейся со скоростью v вдоль оси x, координата преобразуется как разность координаты и пройденного пути, а время не преобразуется вовсе. Отсюда прямо следует правило сложения скоростей: скорость тела относительно первой системы есть сумма его скорости относительно второй и скорости второй относительно первой."
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}

{
  "contentHash": "sha256:be8aed0f2a3b482d75a86788c185c0258c05ca9489db98efefcacb433fb6aeb1",
  "edition": "urn:phys:clir:classical-ether#CLASSICAL_ETHER_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:classical-ether#AUTH_S05",
  "kind": "fragment",
  "locator": "section/5",
  "package": "urn:phys:clir:classical-ether",
  "texts": [
    {
      "contentHash": "sha256:d512bc1d35da18a1f480d47c14dfc74f13b42c9b7c0232028bf116e1983dccf1",
      "language": "ru",
      "status": "official",
      "text": "Раздел 5. Скорость света относительно среды.\nМодель принимает: свет распространяется с одной и той же скоростью относительно СРЕДЫ, независимо от движения источника. Относительно наблюдателя, движущегося сквозь среду, скорость света складывается галилеевски и потому зависит от направления. Это утверждение существенно отличается от утверждения теории, в которой скорость света фиксирована относительно ИСТОЧНИКА: последнюю (эмиссионную) теорию настоящая реконструкция не выражает и не описывает."
    }
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}

{
  "contentHash": "sha256:5c8117dd4e1b8593b9ff60c8b026c24bce6478ee36f2afaf725f211030405d30",
  "edition": "urn:phys:clir:classical-ether#CLASSICAL_ETHER_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:classical-ether#AUTH_S07",
  "kind": "fragment",
  "locator": "section/7",
  "package": "urn:phys:clir:classical-ether",
  "texts": [
    {
      "contentHash": "sha256:f1f7ea28d4e26ded0b90df09a2105176bf4f78b106f56b26656ed5443cd2252c",
      "language": "ru",
      "status": "official",
      "text": "Раздел 7. Отсутствие сокращения тел.\nМодель принимает: размеры тел не изменяются от движения сквозь среду. Плечо интерферометра, направленное вдоль движения, имеет ту же длину, что и плечо поперёк. Именно эта идеализация отделяет настоящую реконструкцию от теории Лоренца 1904 года, где сокращение постулировано."
    }
  ]
}

{
  "contentHash": "sha256:ab138f4288f4fb61891ca6a9a225e8d113b73cc67b9e379fefe0f9ffc52edaec",
  "edition": "urn:phys:clir:classical-ether#CLASSICAL_ETHER_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:classical-ether#AUTH_S09",
  "kind": "fragment",
  "locator": "section/9",
  "package": "urn:phys:clir:classical-ether",
  "texts": [
    {
      "contentHash": "sha256:b732fdaf75d91aa2fa43e580636441178c9b585276372556fc76a0a6ec8c7214",
      "language": "ru",
      "status": "official",
      "text": "Раздел 9. Что модель предсказывает в общих постановках.\nПромежуток времени между двумя событиями во второй системе равен промежутку в первой: время абсолютно. Длина стержня, измеренная в движущейся системе, равна его собственной длине: сокращения нет. Скорость тела относительно первой системы равна сумме скоростей: сложение галилеево. Скорость светового сигнала относительно наблюдателя, движущегося сквозь среду, зависит от направления и в общем случае не равна скорости света относительно среды."
    }
  ]
}

{
  "contentHash": "sha256:875de5839043d99f2190bf3d3f3d9bee58acbeec79fa50f7d84bc8984ecfadab",
  "edition": "urn:phys:clir:classical-ether#CLASSICAL_ETHER_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:classical-ether#AUTH_S10",
  "kind": "fragment",
  "locator": "section/10",
  "package": "urn:phys:clir:classical-ether",
  "texts": [
    {
      "contentHash": "sha256:049ab65dde8ed89729ad69d723730a7bcd218a661e43402c9a7ec7f18bdc91dd",
      "language": "ru",
      "status": "official",
      "text": "Раздел 10. Границы применения реконструкции.\nМодель применяется к инерциальным системам, к распространению света в пустоте и к скоростям, много меньшим скорости света, — там, где первый неисчезающий порядок по квадрату отношения скоростей достаточен. Выводы для прибора, в котором свет идёт сквозь вещество или среду с показателем преломления, отличным от единицы, требуют дополнительных условий и настоящей реконструкцией не делаются."
    }
  ]
}

{
  "contentHash": "sha256:09bd0897f4fd4ad61b617462557b54b9923c4420e9946530ac22fd6efa160bc9",
  "edition": "urn:phys:clir:relativity-core#RELATIVITY_PROTOCOL_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:relativity-core#AUTH_S05",
  "kind": "fragment",
  "locator": "section/5",
  "package": "urn:phys:clir:relativity-core",
  "texts": [
    {
      "contentHash": "sha256:68d0254ee3bb17c4854cd471d5d12b6cdbec990450f47e90d67fa2185682db12",
      "language": "ru",
      "status": "official",
      "text": "Раздел 5. Отношение теории к принципу: четыре значения.\nОтношение теории к принципу может быть четверояким: принцип ПРИНЯТ; принцип ЯВНО ОТВЕРГНУТ; принцип НЕ НУЖЕН — теория обходится без него и не утверждает ни его, ни его отрицания; отношение НЕ УСТАНОВЛЕНО настоящим описанием. Четыре значения не сводятся к трём и тем более к двум. «Не нужен» и «отвергнут» различаются существенно: теория, которой не нужна выделенная система отсчёта, не утверждает, что среды не существует. «Не установлено» есть УТВЕРЖДЕНИЕ пакета о своей границе и подаётся фактом так же, как остальные три.\n\nОтсутствие всякого факта об отношении не есть ни одно из четырёх значений. У принципа, о котором теория не сказала ничего, все четыре вывода не установлены, и это правильный ответ, а не отвержение. Замыкание перечня принципов здесь не ставится сознательно: принципов столько, сколько названо источниками, и утверждать полноту перечня настоящий словарь не может."
    }
  ]
}

{
  "contentHash": "sha256:c16ae1c462797fc6c34d0670f179b367afb75eca680894837dd478e81d7b9b96",
  "edition": "urn:phys:clir:relativity-core#RELATIVITY_PROTOCOL_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:relativity-core#AUTH_S06",
  "kind": "fragment",
  "locator": "section/6",
  "package": "urn:phys:clir:relativity-core",
  "texts": [
    {
      "contentHash": "sha256:d96d43cf9bd13570a8b77b803abe00c639b8f26b8bbfcdf87cd7bc855ca548ca",
      "language": "ru",
      "status": "official",
      "text": "Раздел 6. Постановка.\nПостановкой называется описание ОПЕРАЦИОННЫХ условий опыта или мысленного опыта: какие тела и сигналы участвуют, какими процедурами измерены расстояния и промежутки времени, какая система отсчёта названа лабораторной и с какой скоростью относительно неё движется вторая. Постановка не принадлежит ни одной теории: она общая, и именно поэтому предсказания разных теорий на ней сопоставимы.\n\nПостановка сообщает то, что можно предъявить процедурой измерения, и ничего сверх того. Величины, которые имеют смысл только внутри модели, — абсолютное время, местное время, скорость относительно среды — в постановку не входят; они входят в модельные посылки прогона."
    }
  ]
}

{
  "contentHash": "sha256:f9751ca7e60bf2d5d164fb20e2128bf990b0dcf46ccbdd5a9a85207167cd147e",
  "edition": "urn:phys:clir:relativity-core#RELATIVITY_PROTOCOL_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:relativity-core#AUTH_S07",
  "kind": "fragment",
  "locator": "section/7",
  "package": "urn:phys:clir:relativity-core",
  "texts": [
    {
      "contentHash": "sha256:902370ac4565b21c9ff573d7285b6b1a221874d04415f657b7de9810b91d04dc",
      "language": "ru",
      "status": "official",
      "text": "Раздел 7. Наблюдаемая величина.\nНаблюдаемой величиной называется то, о чём спрашивают предсказание: промежуток времени между двумя событиями во второй системе, длина стержня, измеренная в движущейся системе, скорость тела относительно лабораторной системы, сдвиг интерференционных полос. У наблюдаемой есть род: РАЗМЕРНАЯ, у которой предсказание есть величина с единицей, и БЕЗРАЗМЕРНАЯ, у которой предсказание есть точное отношение — например, отношение скорости к скорости света.\n\nРазделение не техническое. Отношение скорости к скорости света при сложении двух скоростей по шесть десятых скорости света равно пятнадцати семнадцатым; десятичной записи у этого числа нет, и требовать её значило бы либо округлить ответ без объявленной политики, либо отказаться от точного сравнения. Поэтому безразмерное предсказание хранится точной дробью, а размерное — величиной с единицей."
    }
  ]
}

{
  "contentHash": "sha256:2dba673cb888207d582e910d6fc751159a27ea81c8a57a285637cbb31d30bd2a",
  "edition": "urn:phys:clir:relativity-core#RELATIVITY_PROTOCOL_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:relativity-core#AUTH_S12",
  "kind": "fragment",
  "locator": "section/12",
  "package": "urn:phys:clir:relativity-core",
  "texts": [
    {
      "contentHash": "sha256:1e2942424833a05c3fa084dccf53569096d6516d7db9d225f67508e53dcab3e8",
      "language": "ru",
      "status": "official",
      "text": "Раздел 12. Различие предпосылок.\nДве теории различаются предпосылкой, когда одна принимает принцип, а другая его явно отвергает. Различие ВЫВОДИТСЯ из двух отношений, а не объявляется третьим фактом: сравнение обязано называть конкретный принцип и показывать, из чего взята каждая сторона. Молчание одной из сторон различия не даёт.\n\nСогласие в предпосылке выводится тем же способом и требует двух фактов: обе теории принимают один принцип."
    }
  ]
}

{
  "contentHash": "sha256:051961b1d7a56600811c1c5739ac603c755e7eb2fb2582e6505c7960f8f819f0",
  "edition": "urn:phys:clir:relativity-core#RELATIVITY_PROTOCOL_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:relativity-core#AUTH_S13",
  "kind": "fragment",
  "locator": "section/13",
  "package": "urn:phys:clir:relativity-core",
  "texts": [
    {
      "contentHash": "sha256:6726285e3cbe2391edd8a8108617ed89350473728cf0b7f4de9faef559d00904",
      "language": "ru",
      "status": "official",
      "text": "Раздел 13. Различие предсказаний.\nДва прогона одной постановки различаются предсказанием, когда наблюдаемая одна и та же, а значения разные. Сравнивать разрешено только предсказания ОДНОЙ наблюдаемой на ОДНОЙ постановке: совпавшие числа при разных наблюдаемых, разных единицах или разных условиях совпадением предсказаний не являются.\n\nСовпадение предсказаний выводится положительно — из равенства значений одной наблюдаемой на одной постановке. Различие выводится тогда, когда обе стороны ответили, а совпадение не установлено. Правило различия поражаемо, и это существенно: на постановке, где ответила одна теория, ни совпадение, ни различие не выводятся."
    }
  ]
}

{
  "contentHash": "sha256:cbd136202c6b5bc6512ef03183dba84c64fcdb2219511b22bd4023baf2c7859f",
  "edition": "urn:phys:clir:relativity-core#RELATIVITY_PROTOCOL_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:relativity-core#AUTH_S15",
  "kind": "fragment",
  "locator": "section/15",
  "package": "urn:phys:clir:relativity-core",
  "texts": [
    {
      "contentHash": "sha256:e6d9712bde0abd9b02b53ca1f6cc8902e5d42629d8cda8b1d0941dc26bf558b8",
      "language": "ru",
      "status": "official",
      "text": "Раздел 15. Согласие с наблюдением.\nПредсказание согласуется с наблюдением, когда наблюдение предъявлено, допуск объявлен и разность значений не превосходит допуска. Предсказание несовместимо с наблюдением, когда разность допуск превосходит. Если наблюдение не предъявлено либо допуск не объявлен, не выводится ни согласия, ни несовместимости, и это третий ответ, а не разновидность первых двух.\n\nНаблюдение, установившее лишь ВЕРХНЮЮ ГРАНИЦУ величины, не есть измерение значения. Верхняя граница опровергает предсказание, превосходящее её, и не подтверждает ни одного предсказания, лежащего под ней. Читать верхнюю границу как точное равенство нулю запрещено."
    }
  ]
}

{
  "contentHash": "sha256:789a87ede5d3dea13864e6cf1274687cad46190399a0746dbeff6e8f01177ff7",
  "edition": "urn:phys:clir:relativity-core#RELATIVITY_PROTOCOL_RU",
  "fragmentKind": "section",
  "id": "urn:phys:clir:relativity-core#AUTH_S16",
  "kind": "fragment",
  "locator": "section/16",
  "package": "urn:phys:clir:relativity-core",
  "texts": [
    {
      "contentHash": "sha256:61625f90f3f65073f3e1fc2ec182181f27fdba4f3b48c45f93498bc2636e7c6d",
      "language": "ru",
      "status": "official",
      "text": "Раздел 16. Полнота сравнения.\nСравнение считается полным, когда по каждому объявленному вопросу сравнения получен ответ: названы отношения обеих теорий к принципу, получены предсказания обеих на постановке и, если наблюдение предъявлено, дан ответ о согласии. Полнота НЕ требует, чтобы теории разошлись: сравнение, показавшее совпадение предсказаний при разных предпосылках, полно ровно в той же мере.\n\nТребование различающего принципа, уместное при сравнении логических систем, к физическому сравнению не переносится: две теории могут давать одно и то же наблюдаемое следствие, оставаясь разными теориями, и назвать это неполнотой значило бы объявить неполным именно тот результат, ради которого сравнение производится."
    }
  ]
}

{
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  "edition": "urn:openstax:clir:relativity#M58565_MAIN",
  "fragmentKind": "section",
  "id": "urn:openstax:clir:relativity#REL_CONTRACTION_EQUATION",
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  "locator": "module/m58565#fs-id1167794072104",
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    {
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      "language": "en",
      "status": "official",
      "text": "<para id=\"fs-id1167794072104\"><term id=\"term-00002\">Length contraction</term> is the decrease in the measured length of an object from its proper length when measured in a reference frame that is moving with respect to the object:</para>"
    }
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}

{
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  "edition": "urn:openstax:clir:relativity#M58563_MAIN",
  "fragmentKind": "section",
  "id": "urn:openstax:clir:relativity#REL_LORENTZ_FACTOR",
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  "locator": "module/m58563#fs-id1167794063710",
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    {
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      "language": "en",
      "status": "official",
      "text": "<para id=\"fs-id1167794063710\">where <m:math><m:mi>γ</m:mi></m:math> is the relativistic factor (often called the <term class=\"no-emphasis\" id=\"term-00002\">Lorentz factor</term>) given by</para>"
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{
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  "fragmentKind": "defn",
  "id": "urn:openstax:clir:relativity#REL_LORENTZ_TRANSFORMATION",
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  "locator": "module/m58568#fs-id1167793277662",
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      "language": "en",
      "status": "official",
      "text": "<definition id=\"fs-id1167793277662\">\n<term>Lorentz transformation</term>\n<meaning id=\"fs-id1167793277667\">relation between position and time coordinates of the same events as seen in different reference frames, according to the special theory of relativity</meaning>\n</definition>"
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{
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      "text": "<definition id=\"fs-id1167793241040\">\n<term>second postulate of special relativity</term>\n<meaning id=\"fs-id1167793383391\">light travels in a vacuum with the same speed <emphasis effect=\"italics\">c</emphasis> in any direction in all inertial frames</meaning>\n</definition>"
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      "language": "en",
      "status": "official",
      "text": "<definition id=\"fs-id1167794070887\">\n<term>time dilation</term>\n<meaning id=\"fs-id1167793924861\">lengthening of the time interval between two events when seen in a moving inertial frame rather than the rest frame of the events (in which the events occur at the same location)</meaning>\n</definition>"
    }
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}

{
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  "id": "urn:openstax:clir:relativity#REL_VELOCITY_ADDITION",
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  "locator": "module/m58569#fs-id1167793583897",
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    {
      "contentHash": "sha256:c2066f7066f75d9c0b17d11b160dab7a72e200be1dec4006cfdb9aa863c2f6be",
      "language": "en",
      "status": "official",
      "text": "<definition id=\"fs-id1167793583897\">\n<term>relativistic velocity addition</term>\n<meaning id=\"fs-id1167794293139\">method of adding velocities of an object moving at a relativistic speeds</meaning>\n</definition>"
    }
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{
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  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_8_EN",
  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI8_DEF_CANDELA",
  "kind": "fragment",
  "locator": "section/2.1.1.7/candela",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:e0fb4dd0e599bac1b91546c9af5aa1a6650e0e66ac8a2c9d1a028b686f7c309c",
      "language": "en",
      "status": "official",
      "text": "The candela is the luminous intensity, in a given direction, of a source\nthat emits monochromatic radiation of frequency 540 × 1012 hertz and that\nhas a radiant intensity in that direction of 1/683 watt per steradian."
    }
  ]
}

{
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  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI8_DEF_METRE",
  "kind": "fragment",
  "locator": "section/2.1.1.1/metre",
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    {
      "contentHash": "sha256:aa913b40afb141ef278181a4252b925f790d5fd49e42b79b79c96a5945dd70fa",
      "language": "en",
      "status": "official",
      "text": "The metre is the length of the path travelled by light in vacuum during a\ntime interval of 1/299 792 458 of a second. The symbol, c0 (or"
    }
  ]
}

{
  "contentHash": "sha256:5987b64556dc5e96b5008b603d59f9752e7bdf8daedda7b3e16f7e0dc8196f97",
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  "locator": "section/2.1.1.3/second",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
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      "language": "en",
      "status": "official",
      "text": "The second is the duration of 9 192 631 770 periods of the radiation\ncorresponding to the transition between the two hyperfine levels of the\nground state of the caesium 133 atom."
    }
  ]
}

{
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    {
      "contentHash": "sha256:6a98194b74b8b00e83c16010c51563c02e10c281acd51dde2e1bf1ce725026f7",
      "language": "en",
      "status": "official",
      "text": "Table 1. SI base units\nBase quantity SI base unit\n_________________________________ ___________________________\nName Symbol Name Symbol\nThe symbols for quantities\nlength l, x, r, etc. metre m are generally single letters\nmass m kilogram kg of the Latin or Greek\ntime, duration t second s alphabets, printed in an\nelectric current I, i ampere A italic font, and are\nthermodynamic temperature T kelvin K recommendations.\namount of substance n mole mol\nThe symbols for units are\nluminous intensity Iv candela cd mandatory, see chapter 5."
    }
  ]
}

{
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    {
      "contentHash": "sha256:70c7c475f65a3e98d87c1ddf205e4d9809a2f2061473863c9e6f689fc1145e09",
      "language": "en",
      "status": "official",
      "text": "Table 3. Coherent derived units in the SI with special names and symbols\nSI coherent derived unit (a)\n——————————————————————————\nExpressed Expressed\nin terms of in terms of\nDerived quantity Name Symbol other SI units SI base units\nplane angle radian (b) rad 1 (b) m/m\nsolid angle steradian (b) sr (c) 1 (b) m2/m2\nfrequency hertz (d) Hz s−1\nforce newton N m kg s−2\npressure, stress pascal Pa N/m2 m−1 kg s−2\nenergy, work, joule J Nm m2 kg s−2\namount of heat\npower, radiant flux watt W J/s m2 kg s−3\nelectric charge, coulomb C sA\namount of electricity\nelectric potential difference, volt V W/A m2 kg s−3 A−1\nelectromotive force\ncapacitance farad F C/V m−2 kg−1 s4 A2\nelectric resistance ohm Ω V/A m2 kg s−3 A−2\nelectric conductance siemens S A/V m−2 kg−1 s3 A2\nmagnetic flux weber Wb Vs m2 kg s−2 A−1\nmagnetic flux density tesla T Wb/m2 kg s−2 A−1\ninductance henry H Wb/A m2 kg s−2 A−2\nCelsius temperature degree Celsius (e) o\nC K\nluminous flux lumen lm cd sr (c) cd\nilluminance lux lx lm/m2 m−2 cd\nactivity referred to becquerel (d) Bq s−1\na radionuclide (f)\nabsorbed dose, gray Gy J/kg m2 s−2\nspecific energy (imparted),\nkerma\ndose equivalent, sievert (g) Sv J/kg m2 s−2\nambient dose equivalent,\ndirectional dose equivalent,\npersonal dose equivalent\ncatalytic activity katal kat s−1 mol"
    }
  ]
}

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  "texts": [
    {
      "contentHash": "sha256:033ee040cd3b4a114dbe4e5fa1373cbbe6336dfe99324ee9ae51b65f9c7c871e",
      "language": "en",
      "status": "official",
      "text": "Table 5. SI prefixes Telecommunications and\nelectronics. The names and\nsymbols for the prefixes\nFactor Name Symbol Factor Name Symbol\ncorresponding to 210, 220,\n230, 240, 250, and 260 are,\n101 deca da 10−1 deci d respectively: kibi, Ki; mebi,\n102 hecto h 10−2 centi c Mi; gibi, Gi; tebi, Ti; pebi,\n103 kilo k 10−3 milli m Pi; and exbi, Ei. Thus, for\n106 mega M 10−6 micro µ example, one kibibyte\nwould be written:\n109 giga G 10−9 nano n\n1 KiB = 210 B = 1024 B,\n1012 tera T 10−12 pico p where B denotes a byte.\n1015 peta P 10−15 femto f Although these prefixes are\n1018 exa E 10−18 atto a not part of the SI, they\n1021 zetta Z 10−21 zepto z should be used in the field\n1024 yotta Y 10−24 yocto y of information technology"
    }
  ]
}

{
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  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "note",
  "id": "urn:bipm:clir:si-brochure#SI9_APP1_2018_INDEX",
  "kind": "fragment",
  "locator": "appendix/1/index/26th-cgpm-2018",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:db245688b99edbdfcbd6564bfe291cefaa4af217ce12e0885bce05255ce5af8b",
      "language": "en",
      "status": "official",
      "text": "26th CGPM, 2018: revision of the International System of Units, the SI 194\n(to enter into force on 20 May 2019)"
    }
  ]
}

{
  "contentHash": "sha256:9f5d40e085f96932d4856b7a346773d77abcd80d7e2c924767e98697f2b0fde7",
  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "resolution",
  "id": "urn:bipm:clir:si-brochure#SI9_APP1_2022_RESOLUTION_3",
  "kind": "fragment",
  "locator": "appendix/1/27th-cgpm-2022/resolution-3",
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  "texts": [
    {
      "contentHash": "sha256:da9a754f29e9b1412a8e324261700aa844550bca5e2efbe04d4362790bc2c8b3",
      "language": "en",
      "status": "official",
      "text": " On the extension of the range of SI prefixes (CR, 397 and Metrologia, 2023, 60,\n013001)\nResolution 3\nThe General Conference on Weights and Measures (CGPM), at its 27th meeting,\nrecalling\nthat decisions were made at previous meetings when it was considered timely to extend the\nrange of SI prefixes including Resolution 12 (paragraph 3) adopted by the CGPM at its\n11th meeting (1960), Resolution 8 adopted by the CGPM at its 12th meeting (1964),\nResolution 10 adopted by the CGPM at its 15th meeting (1975), and Resolution 4 adopted by\nthe CGPM at its 19th meeting (1991),\nconsidering\n− the essential role of the International System of Units (SI) in providing confidence in the\naccuracy and global comparability of measurements needed for international trade,\nmanufacturing, human health and safety, protection of the environment, global climate\nstudies and scientific research,\n− the benefits of encouraging the use of SI units by providing new SI prefixes for scientific\ncommunities that depend on measurements that are not covered by the current range,\nAppendix 1 • 197\n− the needs of data science in the near future to express quantities of digital information\nusing orders of magnitude in excess of 1024,\n− the importance of timely action to prevent unofficial prefix names being de facto adopted in\nother communities,\ndecides\nto add to the list of SI prefixes to be used for multiples and submultiples of units the\nfollowing prefixes:\nMultiplying factor Name Symbol\n1027 ronna R\n10−27 ronto r\n1030 quetta Q\n10−30 quecto q"
    }
  ]
}

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  "fragmentKind": "provision",
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  "texts": [
    {
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      "language": "en",
      "status": "official",
      "text": "The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed\nnumerical value of the elementary charge, e, to be 1.602 176 634 × 10−19 when expressed\nin the unit C, which is equal to A s, where the second is defined in terms of ∆νCs."
    }
  ]
}

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  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_DEF_CANDELA",
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  "locator": "section/2.3.1/candela",
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  "texts": [
    {
      "contentHash": "sha256:e0ef37a23e1dd6bb5def7debca8079b9612fe024405e5bef9b40d84e7ecb37ec",
      "language": "en",
      "status": "official",
      "text": "The candela, symbol cd, is the SI unit of luminous intensity in a given direction. It is\ndefined by taking the fixed numerical value of the luminous efficacy of monochromatic\nradiation of frequency 540 × 1012 Hz, Kcd, to be 683 when expressed in the unit lm W−1,\nwhich is equal to cd sr W−1, or cd sr kg−1 m−2 s3, where the kilogram, metre and second\nare defined in terms of h, c and ∆νCs."
    }
  ]
}

{
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  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_DEF_KELVIN",
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  "locator": "section/2.3.1/kelvin",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:e7d9ee6b2f490124223d1accc13282a3b5738bb04df8ca36916189010fd2388d",
      "language": "en",
      "status": "official",
      "text": "The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by\ntaking the fixed numerical value of the Boltzmann constant, k, to be 1.380 649 × 10−23\nwhen expressed in the unit J K−1, which is equal to kg m2 s−2 K−1, where the kilogram,\nmetre and second are defined in terms of h, c and ∆νCs."
    }
  ]
}

{
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  "id": "urn:bipm:clir:si-brochure#SI9_DEF_KILOGRAM",
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  "locator": "section/2.3.1/kilogram",
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  "texts": [
    {
      "contentHash": "sha256:96dae548d302b9211c1c70f6e206bf377de3b0c9fbb7327d36e875da4063d572",
      "language": "en",
      "status": "official",
      "text": "The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed\nnumerical value of the Planck constant, h, to be 6.626 070 15 × 10−34 when expressed in\nthe unit J s, which is equal to kg m2 s−1, where the metre and the second are defined in\nterms of c and ∆νCs."
    }
  ]
}

{
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  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_DEF_METRE",
  "kind": "fragment",
  "locator": "section/2.3.1/metre",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:9776d44ec57958c405378e1585c171bf44eebe01426be027f959a77064d4c8e0",
      "language": "en",
      "status": "official",
      "text": "The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical\nvalue of the speed of light in vacuum, c, to be 299 792 458 when expressed in the unit m\ns−1, where the second is defined in terms of the caesium frequency ∆νCs."
    }
  ]
}

{
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  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_DEF_MOLE",
  "kind": "fragment",
  "locator": "section/2.3.1/mole",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:a11e8e01b96d54037f81044afa61838d441ef84f48bcdf4855c3a7ebf94bac16",
      "language": "en",
      "status": "official",
      "text": "The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly\n6.022 140 76 × 1023 elementary entities. This number is the fixed numerical value of the\nAvogadro constant, NA, when expressed in the unit mol−1 and is called the Avogadro\nnumber."
    }
  ]
}

{
  "contentHash": "sha256:f2066c4fd9e09488f47dc798c470461fbaa8d100064252187dc123bccc50f4c2",
  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_DEF_SECOND",
  "kind": "fragment",
  "locator": "section/2.3.1/second",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:a2b330de77b8fb6a298ffbbc5e22952db51f936a67c6a911f6df4afc0a5df01b",
      "language": "en",
      "status": "official",
      "text": "The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical\nvalue of the caesium frequency, ∆νCs, the unperturbed ground-state hyperfine\ntransition frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the\nunit Hz, which is equal to s−1."
    }
  ]
}

{
  "contentHash": "sha256:9ba3706d38178f40e0a639e30a842ab4eff445c422b385b7380cb0a6946832fb",
  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_S2_2_DEFINITION_OF_SI",
  "kind": "fragment",
  "locator": "section/2.2/definition",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:186bea9a7e8e45241bf08305fc7f7d05fee96c1d1df4b8b841043bb2a9deeb52",
      "language": "en",
      "status": "official",
      "text": "The International System of Units, the SI, is the system of units in which\n• the unperturbed ground state hyperfine transition frequency of the caesium\n133 atom, ∆νCs, is 9 192 631 770 Hz,\n• the speed of light in vacuum, c, is 299 792 458 m/s,\n• the Planck constant, h, is 6.626 070 15 × 10−34 J s,\n• the elementary charge, e, is 1.602 176 634 × 10−19 C,\n• the Boltzmann constant, k, is 1.380 649 × 10−23 J/K,\n• the Avogadro constant, NA, is 6.022 140 76 × 1023 mol−1,\n• the luminous efficacy of monochromatic radiation of frequency\n540 × 1012 Hz, Kcd, is 683 lm/W,"
    }
  ]
}

{
  "contentHash": "sha256:878e6a234d826c3cbf3aa2bb6053e15147ae7f977e1dd54d510cfc2e3eae129d",
  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_S2_3_1_CELSIUS",
  "kind": "fragment",
  "locator": "section/2.3.1/celsius",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:81773254636a642ed6da3aec77e32cafb409eae407b17017686a27135b328f9a",
      "language": "en",
      "status": "official",
      "text": "The unit of Celsius temperature is the degree Celsius, symbol °C, which is by definition equal\nin magnitude to the unit kelvin. A difference or interval of temperature may be expressed in\nkelvins or in degrees Celsius, the numerical value of the temperature difference being the\nsame in either case. However, the numerical value of a Celsius temperature expressed in\ndegrees Celsius is related to the numerical value of the thermodynamic temperature expressed\nin kelvins by the relation\nt/°C = T/K − 273.15"
    }
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{
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  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_S2_3_4_COHERENT",
  "kind": "fragment",
  "locator": "section/2.3.4/coherent",
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  "texts": [
    {
      "contentHash": "sha256:b9e6a117312c189b30c283ce54aee8fdfdea00c8fe550f1c0236e766dc9186cb",
      "language": "en",
      "status": "official",
      "text": "Derived units are defined as products of powers of the base units. When the numerical factor\nof this product is one, the derived units are called coherent derived units. The base and\ncoherent derived units of the SI form a coherent set, designated the set of coherent SI units.\nThe word “coherent” here means that equations between the numerical values of quantities\ntake exactly the same form as the equations between the quantities themselves."
    }
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}

{
  "contentHash": "sha256:cc29da09e81842f8c76adea8e9b1ed7c0e488a2fdfdb2c87473ae723f82a6c28",
  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_S2_3_4_COMPLETE_SET",
  "kind": "fragment",
  "locator": "section/2.3.4/complete-set",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:ba07f1b3c6f8c3772a950b6e0b74315712b3c468206b9b83cc70ec6b0ee93720",
      "language": "en",
      "status": "official",
      "text": "The seven base units and 22 units with special names and symbols may be used in\ncombination to express the units of other derived quantities. Since the number of quantities\nis without limit, it is not possible to provide a complete list of derived quantities and derived\nunits. Table 5 lists some examples of derived quantities and the corresponding coherent\nderived units expressed in terms of base units. In addition, Table 6 lists examples of coherent\nderived units whose names and symbols also include derived units. The complete set of SI\nunits includes both the coherent set and the multiples and sub-multiples formed by using the\nSI prefixes."
    }
  ]
}

{
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  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_S2_3_4_PREFIX_EXCEPTION",
  "kind": "fragment",
  "locator": "section/2.3.4/prefix-exception",
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  "texts": [
    {
      "contentHash": "sha256:c093af4db37dce32abaf0950fef702d7f2a024f35700fa39bfc5a3875e4db083",
      "language": "en",
      "status": "official",
      "text": "The CGPM has adopted a series of prefixes for use in forming the decimal multiples and sub-\nmultiples of the coherent SI units (see chapter 3). They are convenient for expressing the\nvalues of quantities that are much larger than or much smaller than the coherent unit.\nHowever, when prefixes are used with SI units, the resulting units are no longer coherent,\nbecause the prefix introduces a numerical factor other than one. Prefixes may be used with\nany of the 29 SI units with special names with the exception of the base unit kilogram, which\nis further explained in chapter 3."
    }
  ]
}

{
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  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_S2_3_4_SPECIAL_NAMES",
  "kind": "fragment",
  "locator": "section/2.3.4/special-names",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:515a61fe3b5e76008a87be79f0c0d988a935aee92c2d22abefed37564313eba0",
      "language": "en",
      "status": "official",
      "text": "Some of the coherent derived units in the SI are given special names. Table 4 lists 22 SI units\nwith special names. Together with the seven base units (Table 2) they form the core of the\nset of SI units. All other SI units are combinations of some of these 29 units."
    }
  ]
}

{
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  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_S3_COMPOUND",
  "kind": "fragment",
  "locator": "section/3/compound",
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    {
      "contentHash": "sha256:7271a37ae1b8b3f1bdc28994ba3fc71acb0ddbeb089df35c4c5052706b44e846",
      "language": "en",
      "status": "official",
      "text": "Compound prefix symbols, i.e. prefix symbols formed by the juxtaposition of two or more\nprefix symbols, are not permitted. This rule also applies to two or more compound prefix\nnames."
    }
  ]
}

{
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  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_S3_INSEPARABLE",
  "kind": "fragment",
  "locator": "section/3/inseparable",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:f100b1f76283e004d1b8c8b32f818afc84c62ffb98e7a0416a2057b8fbaec48b",
      "language": "en",
      "status": "official",
      "text": "The grouping formed by a prefix symbol attached to a unit symbol constitutes a new\ninseparable unit symbol (forming a multiple or sub-multiple of the unit concerned) that can\nbe raised to a positive or negative power and that can be combined with other unit symbols\nto form compound unit symbols."
    }
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{
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  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_S3_KILOGRAM",
  "kind": "fragment",
  "locator": "section/3/kilogram",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:c745c0e5afd6eb4cc7d18f307b66172592a9256bc528d5e54d8997168a615dca",
      "language": "en",
      "status": "official",
      "text": "For historical reasons, the kilogram is the only coherent SI unit that includes a prefix in its\nname and symbol. Names and symbols for decimal multiples and sub-multiples of the unit of\nmass are formed by attaching prefix names and symbols to the unit name “gram” and the unit\nsymbol “g” respectively. For example, 10−6 kg is written as milligram, mg, not as\nmicrokilogram, µkg."
    }
  ]
}

{
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  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "provision",
  "id": "urn:bipm:clir:si-brochure#SI9_S4_INTRO",
  "kind": "fragment",
  "locator": "section/4/intro",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:29b0a429548094c549dbe93554a0d57bb24f970ccc798244745252608e18d734",
      "language": "en",
      "status": "official",
      "text": "It is recognized that some non-SI units are widely used and that this is expected to continue\nfor many years. It is therefore important to recall the values of these non-SI units in terms of\nSI units, because the SI is the internationally agreed reference with respect to which all other\nunits are defined. A non-exhaustive list of non-SI units is given in Table 8, grouped into\nindicative unit categories to aid explanation."
    }
  ]
}

{
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  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "table",
  "id": "urn:bipm:clir:si-brochure#SI9_TABLE_1",
  "kind": "fragment",
  "locator": "section/2.2/table-1",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:7af45fae4cb7a313e6c6425db7389d026287af5edadb25837e7fd871ec1e9fbb",
      "language": "en",
      "status": "official",
      "text": "Table 1. The seven defining constants of the SI and the seven corresponding units\nthey define\n___________________________________________________________________________\nDefining constant Symbol Numerical value Unit\n___________________________________________________________________________\nhyperfine transition\nfrequency of Cs ∆νCs 9 192 631 770 Hz\nspeed of light in vacuum c 299 792 458 m s−1\nPlanck constant h 6.626 070 15 × 10−34 Js\nelementary charge e 1.602 176 634 × 10−19 C\nBoltzmann constant k 1.380 649 × 10 −23\nJ K−1\nAvogadro constant NA 6.022 140 76 × 1023 mol−1\nluminous efficacy Kcd 683 lm W−1"
    }
  ]
}

{
  "contentHash": "sha256:977e769c7e60e7dbdb9e8436c2ec83d1460fbe077b96d9b0ac6dc52012688787",
  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "table",
  "id": "urn:bipm:clir:si-brochure#SI9_TABLE_2",
  "kind": "fragment",
  "locator": "section/2.3.1/table-2",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:bdd9a6b3aadcc3e36539d3972cbe501a92940a3328bf1920d89660a3acc9e77b",
      "language": "en",
      "status": "official",
      "text": "Table 2. SI base units\n_____________________________________________________________________________\nBase quantity Base unit\n_____________________________________________________________________________\nName Typical symbol Name Symbol\n_____________________________________________________________________________\ntime t second s\nlength l, x, r, etc. metre m\nmass m kilogram kg\nelectric current I, i ampere A\nthermodynamic temperature T kelvin K\namount of substance n mole mol\nluminous intensity Iv candela cd"
    }
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}

{
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  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "table",
  "id": "urn:bipm:clir:si-brochure#SI9_TABLE_4",
  "kind": "fragment",
  "locator": "section/2.3.4/table-4",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:fd0d2e3ba1b4f130bcf6fbeb2bc54f86b671e4e4cab5a195f3a165f8c72d8070",
      "language": "en",
      "status": "official",
      "text": "Table 4. The 22 SI units with special names and symbols\n_____________________________________________________________________________________________\nSpecial name Unit expressed in Unit expressed in\nDerived quantity of unit Symbol terms of base units (a) terms of other SI units\n______________________________________________________________________________________________\nplane angle radian (b) rad (b) 1\nsolid angle steradian (c) sr (c) 1\nfrequency hertz (d) Hz s−1\nforce newton N kg m s−2\npressure, stress pascal Pa kg m−1 s−2 N/m2\nenergy, work, joule J kg m2 s−2 Nm\namount of heat\npower, radiant flux watt W kg m2 s−3 J/s\nelectric charge coulomb C As\nelectric potential difference (e) volt V kg m2 s−3 A−1 W/A\ncapacitance farad F kg−1 m−2 s4 A2 C/V\nelectric resistance ohm Ω kg m2 s−3 A−2 V/A\nelectric conductance siemens S kg −1 m−2 s3 A2 A/V\nmagnetic flux weber Wb kg m2 s−2 A−1 Vs\nmagnetic flux density tesla T kg s−2 A−1 Wb/m2\ninductance henry H kg m 2 s−2 A−2 Wb/A\n134 • The International System of Units\nCelsius temperature degree Celsius (f) °C K\nluminous flux lumen lm cd sr (g) cd sr\nilluminance lux lx cd sr m−2 lm/m2\nactivity referred to becquerel Bq s−1\na radionuclide (d, h)\nabsorbed dose, kerma gray Gy m2 s−2 J/kg\ndose equivalent sievert (i) Sv m2 s−2 J/kg\ncatalytic activity katal kat mol s −1"
    }
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}

{
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  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "table",
  "id": "urn:bipm:clir:si-brochure#SI9_TABLE_7",
  "kind": "fragment",
  "locator": "section/3/table-7",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:12e43dffddcc8f179c629bcd2a1de930f970f632899c8a4da3f5ff3643c05559",
      "language": "en",
      "status": "official",
      "text": "Table 7. SI prefixes\n_____________________________________________________________________________\nFactor Name Symbol Factor Name Symbol\n101 deca da 10−1 deci d\n10 2\nhecto h 10−2 centi c\n10 3\nkilo k 10 −3\nmilli m\n106 mega M 10−6 micro µ\n109 giga G 10−9 nano n\n10 12\ntera T 10 −12\npico p\n1015 peta P 10−15 femto f\n1018 exa E 10−18 atto a\n1021 zetta Z 10−21 zepto z\n10 24\nyotta Y 10 −24\nyocto y\n10 27\nronna R 10 −27\nronto r\n1030 quetta Q 10−30 quecto q"
    }
  ]
}

{
  "contentHash": "sha256:438b979940ce804c0e451ac8027a09be0d7d49ad1757f238bb5ddb1e5b69aaae",
  "edition": "urn:bipm:clir:si-brochure#SI_BROCHURE_9_EN",
  "fragmentKind": "table",
  "id": "urn:bipm:clir:si-brochure#SI9_TABLE_8",
  "kind": "fragment",
  "locator": "section/4/table-8",
  "package": "urn:bipm:clir:si-brochure",
  "texts": [
    {
      "contentHash": "sha256:57b8a23b0a6cc65a2174b77ede19272cc5689173c83bf8a600ed04f5a8699743",
      "language": "en",
      "status": "official",
      "text": "Table 8. Non-SI units\nSymbol\nUnit category Quantity Name of unit Value in SI units\nfor unit\nLong-standing time minute min 1 min = 60 s\nunits of time and hour h 1 h = 60 min = 3600 s\nangle day d 1 d = 24 h = 86 400 s\nplane and degree ° 1° = (π/180) rad\nphase angle minute ′ 1′ = (1/60)° = (π/10 800) rad\nsecond (a) ″ 1″ = (1/60)′ = (π/648 000) rad\nHistorical names area are (b) a 1 a = 1 dam2 = 102 m2\nfor decimal hectare (b) ha 1 ha = 1 hm2 = 104 m2\nmultiples and barn (c) b 1 b = 100 fm2 = 10−28 m2\nsubmultiples\nof SI units volume litre (d) l, L 1 l = 1 L = 1 dm3 = 10−3 m3\nmass tonne (e) t 1 t = 1 Mg = 103 kg\nlength angstrom (f) Å 1 Å = 0.1 nm = 10−10 m\nacceleration gal (g) Gal 1 Gal = 1 cm/s2 = 10−2 m/s2\npressure bar (h) bar 1 bar = 0.1 MPa = 105 Pa\nInternationally mass dalton (i) Da 1 Da =\nrecognised units 1.660 539 068 92(52) × 10−27 kg\nthat are not\ndecimal multiples length astronomical unit (j) au 1 au = 149 597 870 700 m\nor submultiples nautical mile (k) 1 nautical mile = 1852 m\nof SI units speed knot (k) 1 nautical mile per hour =\n(1852/3600) m/s\nenergy electronvolt (l) eV 1 eV = 1.602 176 634 × 10−19 J\nUnits used in logarithmic neper (m) Np (m)\nspecialized ratio quantities bel (m) B (m)\ntechnical decibel (m) dB (m)\ndisciplines\nreactive power var (n) var 1 var = 1 V A = 1 W"
    }
  ]
}