Zur Elektrodynamik bewegter Körper (On the Electrodynamics of Moving Bodies)
1905 · paper · Physics
Addresses asymmetries in Maxwell's electrodynamics when applied to moving bodies. Adopts the relativity principle and constant speed of light in empty space as postulates, aiming for a consistent electrodynamics of moving bodies that makes a light ether unnecessary.
6 findings
Substance After ν' = ν√((1 − v/V)/(1 + v/V)) the text reads "für v = − ∞, ν = ∞ ist"; the formula diverges at v = −V, not v = −∞, and the diverging quantity is the observed frequency ν', not ν.
- p. 22 §7, sentence after the φ = 0 Doppler formula
für v = − ∞, ν = ∞ ist
- p. 22 §7, end of section
der sich mit der Geschwindigkeit V einer Lichtquelle näherte, diese Lichtquelle unendlich intensiv erscheinen
- p. 13 §4
Für Überlichtgeschwindigkeiten werden unsere Überlegungen sinnlos
Judge's note: The p. 22 image prints "für v = −∞, ν = ∞ ist". Reworked with the formula directly above: at v = −V the numerator is 1 − (−1) = 2 and the denominator is 1 + (−1) = 0, so ν' → ∞. As v → −∞ the ratio tends to −1 and the root is not real. §4 (p. 13) says superluminal speeds are meaningless, and the parallel intensity statement at the foot of p. 22 names "Geschwindigkeit V". The evident fix is v = −V, ν' = ∞.
Second check: another model agreed
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Substance The reflected-light frequency on p. 915 is printed as ν''' = ν(1 − 2(v/V)cos φ + (v/V)²)/(1 − v/V)²; the denominator should be 1 − (v/V)², as in the A''' line above it.
- p. 25 §8, third displayed equation at top of page (ν''')
ν''' = ν'' (1 + (v/V) cos φ'') / √(1 − (v/V)²) = ν (1 − 2(v/V) cos φ + (v/V)²) / (1 − v/V)²
- p. 25 §8, first displayed equation (A''') for comparison
A''' = ... = A (1 − 2(v/V) cos φ + (v/V)²) / (1 − (v/V)²)
- p. 25 §8, third equation at top of page (ν''')
(1 − v/V)²
- p. 25 §8, first equation at top of page (A''')
1 − (v/V)²
Only the authors can say what is right here.
Judge's note: The p. 25 image shows the A''' denominator as 1 − (v/V)² and the ν''' denominator as (1 − v/V)², with the exponent outside the large bracket. Reworked: ν'' = ν' = ν(1 − (v/V)cos φ)/√(1−(v/V)²), and cos φ'' = −(cos φ − v/V)/(1 − (v/V)cos φ), so 1 + (v/V)cos φ'' = (1 − 2(v/V)cos φ + (v/V)²)/(1 − (v/V)cos φ). The product is ν(1 − 2(v/V)cos φ + (v/V)²)/(1 − (v/V)²). Check with v/V = 0.5 and cos φ = 0: the correct form gives 1.25/0.75 ≈ 1.667, the printed form gives 1.25/0.25 = 5. At cos φ = 1 the printed form gives ν''' = ν, i.e. no Doppler shift at normal incidence. The formula is wrong as printed.
Second check: another model agreed
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Substance In the transformed equation set of §9 (p. 916) the right-hand column is printed "∂L'/∂τ = ∂Y'/∂ζ − ∂Z'/∂η" (likewise for M' and N') without the factor 1/V that appears in the untransformed set "1/V ∂L/∂t = ∂Y/∂z − ∂Z/∂y".
- p. 26 §9, second (transformed) equation block, right-hand column
∂L'/∂τ = ∂Y'/∂ζ − ∂Z'/∂η
- p. 26 §9, first equation block, right-hand column
1/V ∂L/∂t = ∂Y/∂z − ∂Z/∂y
Judge's note: On the p. 26 image the first block carries 1/V on ∂L/∂t, ∂M/∂t and ∂N/∂t. The second block carries 1/V on the left-column braces but prints bare ∂L'/∂τ, ∂M'/∂τ and ∂N'/∂τ. The corresponding transformed vacuum equations on p. 908 carry 1/V on ∂L'/∂τ etc. The factor is missing in print and the evident fix is to restore it.
Second check: another model agreed
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Reference or label §5 defines the angle as "α = arctg w_y/w_x" although the velocity components are introduced as w_ξ and w_η ("w² = w_ξ² + w_η²" on the line above); w_x and w_y are never defined.
- p. 15 §5, last displayed equation on the page
α = arctg w_y / w_x
- p. 15 §5, definitions just above
w² = w_ξ² + w_η²
Judge's note: The p. 15 image shows subscripts y and x in the arctg expression, while the line above and the equations of motion use w_ξ, w_η. The evident fix is α = arctg(w_η/w_ξ). It is a wrong label on otherwise sound quantities.
Second check: another model agreed
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Reference or label The common factor of the field transformation is introduced as "Faktor ψ(v)" on p. 908, but the three conditions determining it on p. 909 are printed with φ: "φ(v)·φ(−v) = 1", "φ(v) = φ(−v)", "φ(v) = 1".
- p. 18 §6, text and relations at bottom of page
Faktor ψ(v) übereinstimmen müssen
- p. 19 §6, three displayed equations at top of page
φ(v) · φ(−v) = 1
- p. 18 §6, last paragraph and equations
abhängigen Faktor ψ(v) übereinstimmen müssen
- p. 19 §6
φ(v) = 1
Judge's note: Compared the p. 18 and p. 19 images. P. 18 prints the trident ψ in the text and in all six relations. P. 19 prints the looped φ glyph, the same as in §3 on pp. 10–11, in all three equations. φ is the §3 kinematic function, so the symbol on p. 19 should be ψ.
Second check: another model agreed
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Wording P. 899 says "wobei a eine vorläufig unbekannte Funktion φ(v) ist", but the p. 900 equations "τ = φ(v)β(t − v/V² x)", "η = φ(v)y" follow from the preceding expressions only if φ(v) = a·β, so φ denotes two different functions.
- p. 9 §3, sentence after τ = a(t − v x'/(V²−v²))
wobei a eine vorläufig unbekannte Funktion φ(v) ist
- p. 10 §3, transformation equations after substituting x'
ξ = φ(v) β (x − v t)
- p. 10 §3, transformation equations
τ = φ(v) β (t − v/V² x)
- p. 10 §3
und φ eine vorläufig unbekannte Funktion von v ist
Only the authors can say what is right here.
Judge's note: Read pp. 9–10 images and reworked the step. τ = a(t − v(x−vt)/(V²−v²)) = a·V²/(V²−v²)·(t − vx/V²) = aβ²(t − vx/V²); η = a·V/√(V²−v²)·y = aβ·y. Matching the printed τ = φβ(t − vx/V²) and η = φy gives φ = aβ, not a. The final transformation is unaffected because the function is undetermined at this point, so this is a notation inconsistency and the result stands.
Second check: another model agreed
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8 minor (typos and the like; most checked by the judge only)
Wording In §3 the symbol x' is first defined by "Setzen wir x' = x − vt" (p. 898) and later reused, without comment, for the x-coordinate in the third system K' ("Wir nennen x', y', z' die Koordinaten, im System K' gemessen", p. 901).
- p. 8 §3, definition
Setzen wir x' = x − v t
- p. 11 §3, third coordinate system
Wir nennen x', y', z' die Koordinaten, im System K' gemessen
Judge's note: Checked p. 8 and p. 11 page images. Both uses are printed as described. The auxiliary x' has been eliminated by p. 900 before the reuse, so no equation is made wrong; this is an undeclared reuse of a symbol only.
Second check: another model agreed
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Wording "Nennt man φ' den Winkel zwischen Wellennormale (Strahlrichtung) im bewegten System und der Verbindungslinie „Lichtquelle–Beobachter“" — but the formula cos φ' = (cos φ − v/V)/(1 − (v/V) cos φ) is the equation for a', the direction cosine relative to the direction of motion.
- p. 22 §7, definition of φ' before the aberration formula
Nennt man φ' den Winkel zwischen Wellennormale (Strahlrichtung) im bewegten System und der Verbindungslinie „Lichtquelle–Beobachter“
- p. 21 §7, definition of φ
mit der ... Geschwindigkeit des Beobachters den Winkel φ bildet
- p. 21 §7, definition of φ
mit der auf ein relativ zur Lichtquelle ruhendes Koordinatensystem bezogenen Geschwindigkeit des Beobachters den Winkel φ bildet
Only the authors can say what is right here.
Judge's note: Read pp. 21–22 images. On p. 911, a' = (a − v/V)/(1 − a v/V), where a and a' are direction cosines of the wave normal relative to the X/Ξ axis, the direction of motion. φ is defined as the angle between the line source–observer and the observer's velocity, so a = cos φ. The printed aberration formula therefore needs φ' to be the angle between the wave normal in k and the direction of motion. As worded, φ' is measured against the line source–observer, which does not match the formula. Only the author can say what wording was intended.
Second check: another model agreed
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Wording Kinetic-energy equation in §10 (p. 920) omits the factor μ in the middle expression: "W = ∫ εX dx = ∫₀^v β³ v dv = μV² { 1/√(1 − (v/V)²) − 1 }"; the middle term should be μ∫₀^v β³ v dv.
- p. 30 §10, displayed equation for W
W = ∫ εX dx = ∫₀^v β³ v dv = μV² { 1/√(1 − (v/V)²) − 1 }
- p. 30 §10, equation for W
W = ∫ ε X dx = ∫ β³ v dv = μ V² {
- p. 30 §10, equation for W
W = ∫ε X dx = ∫β³ v dv = μ V² {…−1}
Only the authors can say what is right here.
Judge's note: Viewed the page image for PDF page 30: no μ is printed before the middle integral, while μV² appears on the right. Reworked the step: the first of equations (A) on p. 919 is μβ³ d²x/dt² = εX, so εX dx = μβ³ (dv/dt) dx = μβ³ v dv. Since d/dv [V²(1 − v²/V²)^(−1/2)] = v(1 − v²/V²)^(−3/2) = β³v, the integral ∫₀^v β³ v dv equals V²(β − 1). The middle expression as printed is therefore V²(β − 1), not μV²(β − 1), and equals neither neighbour (it is also dimensionally short of a mass). No convention μ = 1 is stated, and μ is explicit throughout §10. The final stated result μV²{…−1} is correct, so the faulty intermediate step leaves the result standing: S3.
Second check: another model agreed
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Typo Part heading printed as "II. Eektrodynamischer Teil." — the letter 'l' of 'Elektrodynamischer' is missing.
- p. 17 Heading of Part II
II. Eektrodynamischer Teil.
Judge's note: The p. 17 image shows the letter-spaced heading "II. Eektrodynamischer Teil." with no 'l'. It is in the print, not an extraction loss.
Second check: none (judge only)
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Typo Missing comma in the vector "((X', Y' Z') und (L', M', N'))" on p. 908.
- p. 18 §6, sentence before the primed Maxwell equations
((X', Y' Z') und (L', M', N'))
Judge's note: The p. 18 image shows "(X', Y' Z')" with no comma between Y' and Z', while the parallel "(L', M', N')" has both commas.
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Typo Doppler's name is printed with an extra 'e': "Theorie des Doppelerschen Prinzips und der Aberration" (§7 heading, p. 910) and "Dies ist das Doppelersche Prinzip" (p. 911).
- p. 20 §7 heading
Theorie des Doppelerschen Prinzips
- p. 21 §7, last line of page
Dies ist das Doppelersche Prinzip
- p. 20 §7 heading
Theorie des Doppelerschen Prinzips und der Aberration
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the principle is named after Christian Doppler, whose surname has no 'e' before the final 'r'. The p. 20 and p. 21 images both show the letter-spaced "Doppeler" spelling, so it is in the print and is used consistently.
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Typo 'Koordinatenursprung' is printed across the page break as "sich im Koordinaten-" (end of p. 917) / "sprung befinde" (start of p. 918), omitting the syllable 'ur'.
- p. 27 §10, last line of page
sich im Koordinaten-
- p. 28 §10, first line of page
sprung befinde und sich längs der X-Achse
Judge's note: The p. 27 image ends with "sich im Koordinaten-" and the p. 28 image begins with "sprung befinde und sich längs der X-Achse". The syllable 'ur' is missing in print. The listed crop was not opened; the two full pages were enough.
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Typo Subject-verb number disagreement in §10 (p. 919): "Wir bemerken, daß diese Resultate über die Masse auch für die ponderabeln materiellen Punkte gilt" — plural subject "diese Resultate" with singular verb "gilt" (should be "gelten").
- p. 29 §10, paragraph beginning 'Wir bemerken'
daß diese Resultate über die Masse auch für die ponderabeln materiellen Punkte gilt
Judge's note: Viewed the page image for PDF page 29. The print clearly reads "diese Resultate ... gilt"; the text layer (main.txt line 1281-1282) agrees apart from OCR noise, so this is not an extraction artefact. Elsewhere the paper uses the plural correctly ("die gleichen Resultate ... gelten", line 564-565). Purely grammatical; the meaning is unaffected.
Second check: none (judge only)