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

§8 · finite light energy and moving mirrors

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 913, 914

The energy of a light complex in the moving system

Measured in the moving system k, the light enclosed by a sphere that moves with it at the speed of light has its energy per unit volume multiplied by $q^2$ and its volume divided by $q$ , where $q=(1-\frac{v}{V}\cos\varphi)/\sqrt{1-(v/V)^2}$ , so its energy changes by the factor $q$ , the same factor by which its frequency changes.

As printed

Es wäre daher $A'^2/A^2$ das Verhältnis der „bewegt gemessenen“ und „ruhend gemessenen“ Energie eines bestimmten Lichtkomplexes, wenn das Volumen eines Lichtkomplexes in $K$ gemessen und in $k$ gemessen das gleiche wäre. Dies ist jedoch nicht der Fall.
Nennt man $S$ das Volumen der Kugel, $S'$ dasjenige dieses Ellipsoides, so ist, wie eine einfache Rechnung zeigt:
$\frac{S'}{S} = \frac{\sqrt{1 - \left( \frac{v}{V} \right)^2}}{1 - \frac{v}{V} \cos \varphi} .$ Volume of the light packet, moving systemVolume of the light packet, stationary systemSpeed of the moving systemSpeed of lightAngle to the motion, stationary systemExplain this equation
Nennt man also $E$ die im ruhenden System gemessene, $E'$ die im bewegten System gemessene Lichtenergie, welche von der betrachteten Fläche umschlossen wird, so erhält man: welche Formel für $\varphi = 0$ in die einfachere übergeht:
$\frac{E'}{E} = \frac{\frac{A'^2}{8\pi} S'}{\frac{A^2}{8\pi} S} = \frac{1 - \frac{v}{V} \cos \varphi}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} ,$ Energy of the light packet, moving systemEnergy of the light packet, stationary systemLight amplitude (moving system)Volume of the light packet, moving systemLight amplitude (stationary system)Volume of the light packet, stationary systemSpeed of the moving systemSpeed of lightAngle to the motion, stationary systemExplain this equation
$\frac{E'}{E} = \sqrt{\frac{1 - \frac{v}{V}}{1 + \frac{v}{V}}} .$ Energy of the light packet, moving systemEnergy of the light packet, stationary systemSpeed of the moving systemSpeed of lightExplain this equation
Es ist bemerkenswert, daß die Energie und die Frequenz eines Lichtkomplexes sich nach demselben Gesetze mit dem Bewegungszustande des Beobachters ändern.

Text on page 913 of the German source, § 8, paragraph 1 · Text on page 913 of the German source, § 8, paragraph 2 · Display on page 913 of the German source, § 8, display 3 · Text on pages 913–914 of the German source, § 8, paragraph 2 · Display on page 913 of the German source, § 8, display 4 · Display on page 914 of the German source, § 8, display 5 · Text on page 914 of the German source, § 8, paragraph 5

In modern notation

\frac{E'}{E} = q

\frac{V_{\mathrm{packet}}'}{V_{\mathrm{packet}}} = \frac{1}{q}

\frac{u'}{u} = q^{2}

Shown in modern letters; Einstein's are in §8 of the German source face.

Premise: That $A'^2/8\pi$ is the light energy per unit volume in k is taken from the principle of relativity, as the first sentence of § 8 says.
Premise: The light is a plane wave in empty space, and the complex is the part of it inside the moving sphere; because no energy crosses that surface, the same light is compared in both systems.
A comparison, not a premise: The volume's factor $\sqrt{1-(v/V)^2}/(1-\frac{v}{V}\cos\varphi)$ depends on the direction of the light: with $\varphi=0$ and $v$ at 0.6 of the speed of light it is 2, where a rigid body at rest in K is shortened to 0.8 of its length along the motion.
A comparison, not a premise: § 8 remarks that the energy and the frequency of a light complex change by the same law, and does not connect this with the light quanta of the March paper; reading a complex's energy as $h\nu$ in every system came later.

Decoder

"Lichtkomplex": A light complex: the part of a plane wave inside a spherical surface whose centre moves with the speed of light along the wave normal. No energy crosses that surface, so it encloses the same light at every moment.
$A^2/8\pi$ , $A'^2/8\pi$: The energy of the light per unit volume, in the paper's Gaussian units, measured in K and in k. $A$ and $A'$ are the amplitudes of its electric or magnetic force, transformed in § 7.
$S$ , $S'$: The volume the surface encloses, measured in K, where it is a sphere, and in k at the time $\tau=0$ , where it is an ellipsoid.
$E$ , $E'$: The light energy the surface encloses, measured in K and in k.
$\varphi$: The angle between the wave normal of the light and the direction in which k moves, measured in K. At $\varphi=0$ the light travels the way k moves.
$q$: The modern layer's name for the factor by which the light's frequency changes, § 7's Doppler factor for this direction of the light.

Try it

SR-10: k moves at 0.6 of the speed of light in the direction the light travels: by what factors do the complex's energy, volume and energy per unit volume change? Choose the preset “Receding light complex along axis (0.6c, phi = 0 deg)” there.

SR-10: The same speed, with the light travelling against the motion of k: does the energy grow or shrink, and by the same factor as the frequency? Choose the preset “Approaching light complex along axis (0.6c, phi = 180 deg)” there.

Where this stops: A light complex is not a rod, and light has no rest frame. The result needs neither light quanta nor the mass-energy relation it later supports. Read the argument

Where this is used later

The mass-energy paper of September 1905 starts from this transformation, citing § 8 of this paper.

Argument: A consequence of the transformed amplitude of § 7 and of the volume of a surface that moves with the light
Model: Exact within the paper's electrodynamics, for a plane wave in empty space
Evidence: A derivation, not evidence
History: Introduced in this paper; the remark that energy and frequency transform alike is Einstein's, and the paper does not connect it with light quanta

Pages 914, 915

Light reflected by a moving mirror, and its pressure

Light of frequency $\nu$ and energy per unit volume $A^2/8\pi$ , falling at the angle $\varphi$ on a perfect mirror that moves with the velocity $v$ along its normal, comes back with the frequency $\nu\,(1-2\frac{v}{V}\cos\varphi+(v/V)^2)/(1-(v/V)^2)$ by the paper's own steps, and presses on the mirror with $P=2\frac{A^2}{8\pi}(\cos\varphi-\frac{v}{V})^2/(1-(v/V)^2)$ , which to a first approximation is $2\frac{A^2}{8\pi}\cos^2\varphi$ , the pressure on a mirror at rest, obtained, the paper says, "in agreement with experience and with other theories".

As printed

Es sei nun die Koordinatenebene $\xi = 0$ eine vollkommen spiegelnde Fläche, an welcher die im letzten Paragraph betrachteten ebenen Wellen reflektiert werden. Wir fragen nach dem auf die spiegelnde Fläche ausgeübten Lichtdruck und nach der Richtung, Frequenz und Intensität des Lichtes nach der Reflexion.
$\nu''' = \nu'' \frac{1 + \frac{v}{V} \cos \varphi''}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} = \nu \frac{1 - 2 \frac{v}{V} \cos \varphi + \left( \frac{v}{V} \right)^2}{\left( 1 - \frac{v}{V} \right)^2} .$ Reflected frequencyReflected light frequency (moving system)Speed of the moving systemSpeed of lightReflected propagation angle (moving system)Frequency, stationary systemAngle to the motion, stationary systemExplain this equation
Die Differenz dieser beiden Ausdrücke ist nach dem Energieprinzip die vom Lichtdrucke in der Zeiteinheit geleistete Arbeit. Setzt man die letztere gleich dem Produkt $P \cdot v$ , wobei $P$ der Lichtdruck ist, so erhält man:
$P = 2 \frac{A^2}{8\pi} \frac{\left( \cos \varphi - \frac{v}{V} \right)^2}{1 - \left( \frac{v}{V} \right)^2} .$ Radiation pressure on a mirrorLight amplitude (stationary system)Angle to the motion, stationary systemSpeed of the moving systemSpeed of lightExplain this equation
In erster Annäherung erhält man in Übereinstimmung mit der Erfahrung und mit anderen Theorien
$P = 2 \frac{A^2}{8\pi} \cos^2 \varphi .$ Radiation pressure on a mirrorLight amplitude (stationary system)Angle to the motion, stationary systemExplain this equation

Text on page 914 of the German source, § 8, paragraph 6 · Display on page 915 of the German source, § 8, display 12 · Text on page 915 of the German source, § 8, paragraph 10 · Display on page 915 of the German source, § 8, display 13 · Text on page 915 of the German source, § 8, paragraph 10 · Display on page 915 of the German source, § 8, display 14

In modern notation

\frac{\nu_{\mathrm{ref}}}{\nu} = \frac{1 - 2\,\frac{v}{c}\,\cos\left(\varphi\right) + \left(\frac{v}{c}\right)^{2}}{1 - \left(\frac{v}{c}\right)^{2}}

Shown in modern letters; Einstein's are in §8 of the German source face.

Premise: The mirror is ideal: perfectly reflecting, its motion prescribed, with no recoil, absorption or thickness.
An inference beyond the argument: The pressure comes from the energy principle: the difference between the energy arriving on the mirror and the energy leaving it is the work done by the pressure of the light.
Conditional: The light reaches the mirror only while $V\cos\varphi$ exceeds $v$ .
Approximation: The form $2\frac{A^2}{8\pi}\cos^2\varphi$ neglects the terms in $v/V$ . The formula before it is exact within the model.
A comparison, not a premise: The first approximation is the pressure Maxwell's theory gave for a mirror at rest, which Bartoli had also argued for from thermodynamics; Lebedev, and independently Nichols and Hull, measured light pressure in 1901. Those are measurements of the world, separate from this calculation.

Decoder

$A$ , $\varphi$ , $\nu$: The amplitude, the angle of incidence and the frequency of the incident light, measured in K. The mirror is the plane $\xi=0$ of k, so its normal is the direction of motion, and $\varphi$ is the angle between the wave normal and that direction.
$A'''$ , $\varphi'''$ , $\nu'''$: The same quantities for the reflected light, measured in K. The double-primed quantities describe it in k, where the mirror is at rest and reflects it as a mirror at rest does.
$(1-\frac{v}{V})^2$ in the $\nu'''$ display: As printed. The paper's own steps give $1-(v/V)^2$ , the denominator of the $A'''$ display above it; with the printed square, a mirror moving along the line of the light would return it with its frequency unchanged. Every face keeps the printed reading, and the modern layer has $1-(v/V)^2$ .
$P$: The pressure of the light on the mirror, force per unit area. It is found from the work it does: the energy arriving on unit area of the mirror in unit time, minus the energy leaving it, equals $P\cdot v$ .
$2\frac{A^2}{8\pi}$: Twice the energy per unit volume. With the energy per unit volume written $u$ , in any system of units the pressure is $2u(\cos\varphi-v/c)^2/(1-v^2/c^2)$ .

Try it

SR-11: A mirror receding at 0.6 of the speed of light, the light falling along its normal: what frequency comes back, and what pressure does the light exert? Choose the preset “Receding mirror at 0.6c (normal incidence)” there.

SR-11: The mirror receding at 0.6 of the speed of light and the light falling at the angle whose cosine is 0.6: does the light reach the mirror at all? Choose the preset “Interception limit (cos phi <= 0.6)” there.

Where this stops: Where the light only just fails to catch the mirror, a number from the calculation is not a reflection. A reflected ray can still move towards +x in the laboratory, and a mirror receding faster moves away from it. Read the argument

Argument: A consequence of transforming the light to the mirror's rest system, reflecting it there, transforming it back, and applying the energy principle
Model: Exact within the paper's electrodynamics, for a plane wave and an ideal mirror; the $\cos^2\varphi$ form is a first approximation
Evidence: A derivation, not evidence
History: The first approximation was Maxwell's pressure on a mirror at rest; the exact formula for a moving mirror is derived in this paper

§8 · finite light energy and moving mirrors

Density is not the energy of the whole packet

In the new frame the light’s energy density changes by the factor q squared, while the volume of the same light complex, measured at one instant of the new time, changes by 1/q. Its total energy therefore changes by q, the same factor as its frequency.

Read the argument

Reflection starts with interception

First check that the light can catch the mirror: its speed along the mirror’s normal, c cos φ, must exceed the mirror’s speed away from it. Then change to the mirror’s rest frame, reflect the light there, and change back.

Read the argument