Read · Special relativity: from clock operations to electrodynamics · Results

§10 · electron force, work, and closing scope

Clocks, rods and light first, then electricity and magnetism: follow the relativity paper through its introduction and all ten sections, including the later ones most retellings skip.

The paper's results

Pages 918, 919

The longitudinal and transverse masses of the electron

Calling the force on the electron the force measured in a system that moves with it at that moment, and measuring its acceleration in K, the paper finds the longitudinal mass $\mu/(\sqrt{1-(v/V)^2})^3$ and the transverse mass $\mu/(1-(v/V)^2)$ , and warns that other definitions of force and acceleration give other numbers for the masses.

As printed

Wir fragen nun in Anlehnung an die übliche Betrachtungsweise nach der „longitudinalen“ und „transversalen“ Masse des bewegten Elektrons. Wir schreiben die Gleichungen (A) in der Form und bemerken zunächst, daß $\varepsilon X'$ , $\varepsilon Y'$ , $\varepsilon Z'$ die Komponenten der auf das Elektron wirkenden ponderomotorischen Kraft sind, und zwar in einem in diesem Moment mit dem Elektron mit gleicher Geschwindigkeit wie dieses bewegten System betrachtet. (Diese Kraft könnte beispielsweise mit einer im letzten System ruhenden Federwage gemessen werden.)
$\begin{aligned} \mu \beta^3 \frac{d^2 x}{d t^2} &= \varepsilon X = \varepsilon X' , \\ \mu \beta^2 \frac{d^2 y}{d t^2} &= \varepsilon \beta \left( Y - \frac{v}{V} N \right) = \varepsilon Y' , \\ \mu \beta^3 \frac{d^2 z}{d t^2} &= \varepsilon \beta \left( Z + \frac{v}{V} M \right) = \varepsilon Z' \end{aligned}$ Electron massLorentz factorPosition in the stationary systemTime in the stationary systemElectron chargeElectric field, stationary systemElectric field, moving systemPosition in the stationary systemElectric field, stationary systemSpeed of the moving systemSpeed of lightMagnetic field, stationary systemElectric field, moving systemPosition in the stationary systemElectric field, stationary systemMagnetic field, stationary systemElectric field, moving systemExplain this equation
Wenn wir nun diese Kraft schlechtweg „die auf das Elektron wirkende Kraft“ nennen und die Gleichung aufrechterhalten, und wenn wir ferner festsetzen, daß die Beschleunigungen im ruhenden System $K$ gemessen werden sollen, so erhalten wir aus obigen Gleichungen:
$\text{Massenzahl} \times \text{Beschleunigungszahl} = \text{Kraftzahl}$ Explain this equation
$\text{Longitudinale Masse} = \frac{\mu}{\left( \sqrt{1 - \left( \frac{v}{V} \right)^2} \right)^3} ,$ Electron massSpeed of the moving systemSpeed of lightExplain this equation
$\text{Transversale Masse} = \frac{\mu}{1 - \left( \frac{v}{V} \right)^2} .$ Electron massSpeed of the moving systemSpeed of lightExplain this equation
Natürlich würde man bei anderer Definition der Kraft und der Beschleunigung andere Zahlen für die Massen erhalten; man ersieht daraus, daß man bei der Vergleichung verschiedener Theorien der Bewegung des Elektrons sehr vorsichtig verfahren muß.

Text on pages 918–919 of the German source, § 10, paragraph 7 · Display on page 919 of the German source, § 10, display 4 · Text on page 919 of the German source, § 10, paragraph 7 · Display on page 919 of the German source, § 10, display 5 · Display on page 919 of the German source, § 10, display 6 · Display on page 919 of the German source, § 10, display 7 · Text on page 919 of the German source, § 10, paragraph 8

In modern notation

\frac{F_x}{a_x} = m\,\gamma^{3}

\frac{F_y'}{a_y} = m\,\gamma^{2}

\frac{F_y}{a_y} = m\,\gamma

Shown in modern letters; Einstein's are in §10 of the German source face.

Premise: The force is the one measured in the system moving with the electron at that moment, and the acceleration is measured in K.
A comparison, not a premise: With force and acceleration both measured in K, the transverse ratio is $\mu\beta$ , today's convention, while the longitudinal ratio is $\mu\beta^3$ under both. The paper says as much: other definitions give other numbers, and theories of the motion of the electron must be compared with great care.
An inference beyond the argument: § 10 extends the masses to ponderable material points, since one becomes an electron in the paper's sense when an arbitrarily small charge is added to it.

Decoder

$\mu$ , $\varepsilon$: The electron's mass, as long as it moves slowly, and its charge.
$\beta$: $1/\sqrt{1-(v/V)^2}$ , the factor written $\gamma$ today. The longitudinal mass is $\mu\beta^3$ and the transverse mass $\mu\beta^2$ .
$\mu\beta^3$ in the third row of (A): As printed. The paper's own steps give $\mu\beta^2$ for the $z$ row, as for the $y$ row: both directions are across the motion, and the transverse mass two displays below is $\mu\beta^2$ . Every face keeps the printed $\beta^3$ , and the modern layer has $\gamma^2$ .
$\varepsilon X'$ , $\varepsilon Y'$ , $\varepsilon Z'$: The force on the electron measured in the system that moves with it at that moment, which a spring balance at rest in that system could measure, as § 10 says.
"Massenzahl × Beschleunigungszahl = Kraftzahl": Mass number times acceleration number equals force number: the equation the paper keeps, with force and acceleration defined as it has just stated.

Try it

SR-13: An electron at 0.6 of the speed of light: what ratio of force to acceleration does each convention give, along the motion and across it? Choose the preset “Force convention comparison at 0.6c” there.

Where this stops: The two masses are ratios of force to acceleration under stated conventions; the electron’s own mass does not change with the observer’s speed. The electron is treated as a point driven slowly, without the effects of its own radiation or field. Read the argument

Argument: A consequence of the equations of motion (A), with force and acceleration defined as the paper states
Model: Exact within the paper's dynamics of a charged point; the numbers depend on the definition of force
Evidence: A derivation, not evidence
History: Longitudinal and transverse mass were the usual terms of the theory of the electron, as the paper's question says; these values follow from its kinematics

Pages 919, 920

The kinetic energy of the electron

An electron accelerated slowly from rest along the X-axis by an electrostatic force takes from the field the energy $W=\mu V^2\{1/\sqrt{1-(v/V)^2}-1\}$ , which grows without bound as $v$ approaches $V$ , so that, as with the paper's earlier results, speeds greater than that of light have no possibility of existence; the same expression must hold for ponderable masses.

As printed

Wir bestimmen die kinetische Energie des Elektrons. Bewegt sich ein Elektron vom Koordinatenursprung des Systems $K$ aus mit der Anfangsgeschwindigkeit 0 beständig auf der $X$ -Achse unter der Wirkung einer elektrostatischen Kraft $X$ , so ist klar, daß die dem elektrostatischen Felde entzogene Energie den Wert $\int \varepsilon X d x$ hat. Da das Elektron langsam beschleunigt sein soll und infolgedessen keine Energie in Form von Strahlung abgeben möge, so muß die dem elektrostatischen Felde entzogene Energie gleich der Bewegungsenergie $W$ des Elektrons gesetzt werden. Man erhält daher, indem man beachtet, daß während des ganzen betrachteten Bewegungsvorganges die erste der Gleichungen (A) gilt:
$W = \int \varepsilon X d x = \int_0^v \beta^3 v d v = \mu V^2 \left\{ \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} - 1 \right\} .$ Kinetic energyElectron chargeElectric field, stationary systemPosition in the stationary systemSpeed of the moving systemLorentz factorElectron massSpeed of lightExplain this equation
$W$ wird also für $v = V$ unendlich groß. Überlichtgeschwindigkeiten haben — wie bei unseren früheren Resultaten — keine Existenzmöglichkeit.
Auch dieser Ausdruck für die kinetische Energie muß dem oben angeführten Argument zufolge ebenso für ponderable Massen gelten.

Text on pages 919–920 of the German source, § 10, paragraph 10 · Display on page 920 of the German source, § 10, display 8 · Text on page 920 of the German source, § 10, paragraph 11 · Text on page 920 of the German source, § 10, paragraph 12

In modern notation

K = \int_{0}^{v} m\,\gamma\left(u\right)^{3}\,u\,\mathrm{d}u

K = m\,c^{2}\,\left(\gamma - 1\right)

Shown in modern letters; Einstein's are in §10 of the German source face.

Premise: The electron starts from rest at the origin of K and moves along the X-axis under an electrostatic force. It is accelerated slowly, so that it gives up no energy as radiation, and the energy taken from the field is then its kinetic energy.
Premise: The first equation of (A), with the longitudinal mass, holds throughout the motion.
Approximation: For small $v/V$ the expression becomes $\frac{1}{2}\mu v^2$ , the kinetic energy of Newton's mechanics.
An inference beyond the argument: It holds for ponderable masses by the argument the paper gives for the masses: a small charge makes a ponderable point an electron in its sense.

Decoder

$W$: The kinetic energy of the electron, set equal to the energy taken from the electrostatic field.
$\int_0^v\beta^3 v\,dv$ , as printed: The middle member lacks the factor $\mu$ , which the first equation of (A) carries; with it, the three members are equal. The result on the right is correct as printed, and every face keeps the printed middle member.
$\mu V^2$: The electron's mass times the square of the speed of light. The paper does not read it as an energy the electron has at rest; the relation between a body's energy and its inertia is the subject of the paper of September 1905.

Try it

SR-13: An electron brought from rest to 0.95 of the speed of light: how much kinetic energy has it taken from the field, and how far is that from $\frac{1}{2}\mu v^2$ ? Choose the preset “Relativistic kinetic energy at 0.95c” there.

Where this stops: Losses by radiation and interactions with matter are left out. The kinetic energy formula alone does not give the September paper’s result that a body’s mass changes with its energy. Read the argument

Common wrong turns

The relativity paper derives E = mc².

Argument: A consequence of the longitudinal equation of motion and the energy principle
Model: Exact within the paper's dynamics, for slow acceleration without radiation
Evidence: A derivation, not evidence
History: Introduced in this paper; the relation between a body's energy and its inertia is taken up in the mass-energy paper of September 1905

Pages 920, 921

Three relations for experiment on the electron

From the equations of motion (A) the paper draws three relations open to experiment: the ratio of an electron's magnetic to its electric deflectability is $v/V$ at any speed, the potential difference it has passed through fixes its speed by $P=\frac{\mu}{\varepsilon}V^2\{1/\sqrt{1-(v/V)^2}-1\}$ , and a magnetic force $N$ at right angles to its motion bends its path with the radius $R=V^2\frac{\mu}{\varepsilon}\frac{v/V}{\sqrt{1-(v/V)^2}}\frac{1}{N}$ ; these three, it says, express completely the laws by which the electron must move on its theory.

As printed

Wir wollen nun die aus dem Gleichungssystem (A) resultierenden, dem Experimente zugänglichen Eigenschaften der Bewegung des Elektrons aufzählen.
1. Aus der zweiten Gleichung des Systems (A) folgt, daß eine elektrische Kraft $Y$ und eine magnetische Kraft $N$ dann gleich stark ablenkend wirken auf ein mit der Geschwindigkeit $v$ bewegtes Elektron, wenn $Y = N \cdot v/V$ . Man ersieht also, daß die Ermittelung der Geschwindigkeit des Elektrons aus dem Verhältnis der magnetischen Ablenkbarkeit $A_m$ und der elektrischen Ablenkbarkeit $A_e$ nach unserer Theorie für beliebige Geschwindigkeiten möglich ist durch Anwendung des Gesetzes:
$\frac{A_m}{A_e} = \frac{v}{V} .$ Magnetic deflectabilityElectric deflectabilitySpeed of the moving systemSpeed of lightExplain this equation
Diese Beziehung ist der Prüfung durch das Experiment zugänglich, da die Geschwindigkeit des Elektrons auch direkt, z. B. mittels rasch oszillierender elektrischer und magnetischer Felder, gemessen werden kann.
2. Aus der Ableitung für die kinetische Energie des Elektrons folgt, daß zwischen der durchlaufenen Potentialdifferenz und der erlangten Geschwindigkeit $v$ des Elektrons die Beziehung gelten muß:
$P = \int X d x = \frac{\mu}{\varepsilon} V^2 \left\{ \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} - 1 \right\} .$ Accelerating voltageElectric field, stationary systemPosition in the stationary systemElectron massElectron chargeSpeed of lightSpeed of the moving systemExplain this equation
3. Wir berechnen den Krümmungsradius $R$ der Bahn, wenn eine senkrecht zur Geschwindigkeit des Elektrons wirkende magnetische Kraft $N$ (als einzige ablenkende Kraft) vorhanden ist.
$R = V^2 \frac{\mu}{\varepsilon} \cdot \frac{\frac{v}{V}}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} \cdot \frac{1}{N} .$ Radius of the circular pathSpeed of lightElectron massElectron chargeSpeed of the moving systemMagnetic field, stationary systemExplain this equation
Diese drei Beziehungen sind ein vollständiger Ausdruck für die Gesetze, nach denen sich gemäß vorliegender Theorie das Elektron bewegen muß.

In modern notation

|q|\,U = K

r_B = \frac{\gamma\,m\,v}{|q|\,B}

Shown in modern letters; Einstein's are in §10 of the German source face.

Premise: The first and third relations follow from the second equation of (A); the second follows from the kinetic energy derived before it.
Premise: The electron has passed through the potential difference from rest, slowly enough to radiate nothing, as the kinetic energy requires.
A comparison, not a premise: The paper names the test: the speed of an electron can also be measured directly, for example with rapidly oscillating electric and magnetic fields. Kaufmann's deflection measurements of fast electrons, made from 1901 to 1906, seemed at first to favour Abraham's rival theory, while Bucherer's of 1908 favoured the Lorentz-Einstein formulas, and the question stayed open for some years. Those are measurements of the world, separate from this calculation.

Decoder

$A_m$ , $A_e$: The magnetic and the electric deflectability of the electron. An electric force $Y$ and a magnetic force $N$ deflect it equally when $Y=N\,v/V$ .
$P$: Here the potential difference the electron has passed through from rest. In § 8 the same letter is the pressure of light.
$R$ , $N$: The radius of curvature of the path, and a magnetic force at right angles to the electron's velocity, the only deflecting force. $N$ is a magnetic component here, not a number of molecules.
$\frac{\mu}{\varepsilon}V^2$: In the paper's Gaussian units. In SI, with the size of the charge $|q|$ and of the voltage $U$ , the second relation reads $|q|U=mc^2(\gamma-1)$ and the third $R=\gamma mv/(|q|B)$ , the modern layer's forms.

Try it

SR-13: An electron at 0.6 of the speed of light in a magnetic field of 0.01 tesla at right angles to its motion: what is the radius of its path? Choose the preset “Transverse magnetic deflection at B = 0.01 T” there.

SR-13: The same electron in an electric field of 100,000 volts per metre across its motion: what is the radius of its path now? Choose the preset “Transverse electric deflection at E = 10^5 V/m” there.

Argument: Consequences of the equations of motion (A) and the kinetic energy, stated by the paper as open to experiment
Model: Exact within the paper's dynamics of a charged point in given fields
Evidence: Predictions for experiment; the paper names the tests and reports no data
History: Kaufmann's measurements of 1901 to 1906 and Bucherer's of 1908 tested such relations, and the question between the theories of the electron stayed open for some years

§10 · electron force, work, and closing scope

A force-to-acceleration ratio needs two frame labels

Einstein’s transverse mass, m times gamma squared, divides the force measured in the electron’s momentary rest frame by the acceleration measured in the laboratory. Measure both in the laboratory and the ratio is m times gamma. The two numbers answer two different questions; the electron has one mass m.

Read the argument

The work integral has an observable endpoint

Adding up the work of the force along the motion, from rest to speed v, gives the electron’s kinetic energy, mc²(γ − 1). Einstein then gives two more relations a laboratory can test: the voltage that accelerates the electron, and how sharply electric and magnetic fields bend its path.

Read the argument