Read · Special relativity: from clock operations to electrodynamics

§7 · Doppler shift and aberration

Read this section in the whole paper →

Show me one example before the notation →

Detail

Letters

Show modern qualifications

Einstein's letters change 12 of this paper's 40 formulas. 28 stay in today's letters, with a note saying so.

A derivation

One phase fixes frequency and direction

Why must a frequency change accompany a direction change?

The 6 printed paragraphs this passage explains

§7, paragraph 1: Far from the origin of the stationary system is a source of electromagnetic waves, which near the origin are plane waves with a given amplitude, frequency and direction of the wave normal.
§7, paragraph 2: Einstein asks what these waves are like for an observer at rest in the moving system. With the transformations of sections 6 and 3 they are again plane waves, with a new frequency and a new direction.
§7, paragraph 3: From the new frequency: an observer moving at speed v relative to a very distant source, at an angle φ to the line joining them, receives light whose frequency depends on v and φ in a way Einstein gives exactly, which is Doppler's principle for any speed. When the observer moves straight away from the source the formula becomes a simple square-root ratio, and Einstein remarks that, against the usual view, the frequency becomes infinite as the observer's speed of approach reaches the speed of light, v = −V. The page misprints that speed as v = −∞.
§7, paragraph 4: The new direction gives the law of aberration in its most general form: the angle at which the moving observer receives the light differs from the angle in the source's system, and light arriving at right angles in one system arrives slanted in the other.
§7, paragraph 5: Einstein also finds how the amplitude of the waves changes for the moving observer, and how simple the change is when the observer moves along the line to the source.
§7, paragraph 6: It follows that a light source approached at the speed of light would have to appear infinitely intense.

q := \frac{\nu'}{\nu}

q = \gamma\,\left(1 - \frac{v}{c}\,\cos\left(\varphi\right)\right)

Shown in modern letters; Einstein's are in §7 of the German source face.

The frequency ratio is gamma times one minus v over c times the unprimed direction cosine.

\cos\left(\varphi'\right) = \frac{\cos\left(\varphi\right) - \frac{v}{c}}{1 - \frac{v}{c}\,\cos\left(\varphi\right)}

\cos\left(\varphi'\right) = \frac{\cos\left(\varphi\right) - \frac{v}{V}}{1 - \frac{v}{V}\,\cos\left(\varphi\right)}

The aberration direction cosine is cosine phi minus v over c, divided by one minus v over c times cosine phi.

The phase of a plane wave, $\omega t - k_x x - k_y y - k_z z$ , counts wave crests, and every observer counts the same crests at the same event. Substitute the inverse coordinate map, which writes $t$ , $x$ , $y$ and $z$ in primed coordinates. The coefficient of $t'$ is the new angular frequency and the coefficients of $x'$ , $y'$ and $z'$ are the new wave vector, so one substitution gives both formulas above. The Doppler effect of sound is an analogy only: it depends on motion through the air, and nothing here does.

At $v = 0.6c$ and $\varphi = 0$ , $q = 0.5$ . At $\varphi$ = 90 degrees, $q = 1.25$ and $\cos\varphi' = -0.6$ : light that crossed the motion at right angles in the original frame travels at about 127 degrees to the $x'$ axis in the new one. Each angle belongs to one frame, and the Doppler-and-aberration laboratory labels which.

Open the foundation: Fields and waves

Show every step here: One phase fixes frequency and direction

A tool for this step

Marks that look alike and do different jobs

What do Δ, d, a prime and a small raised number each ask of the reader?

Δ is a change by a definite amount. In the Brownian paper Δ is how far a particle moves in one short interval: an actual distance, such as 1 micrometre.

d, as in dx/dt, is a rate at one instant: how fast x changes as t changes, the value that a change in x divided by a change in t settles on as both shrink. It is not d times x divided by d times t. The curly ∂ is the same idea when other quantities are held fixed.

A prime marks another observer's value. In §7 of the relativity paper ν is the frequency of a light wave for one observer and ν′ the frequency of the same wave for an observer moving relative to the first; φ and φ′ are its direction for each.

A small raised number is repeated multiplication. In the light-quanta paper's §5, W = (v/v₀)ⁿ multiplies v/v₀ by itself n times: with v/v₀ = ½ and n = 4 that is 1/16.

The papers of 1905 print lg for the natural logarithm, which a modern text writes ln. Today lg often means the logarithm to base 10; in these papers it never does.

Worked example: Five marks, five jobs

A particle at 3 μm moves to 4 μm: Δx = 1 μm, a definite change.
A ball that has gone 3t² metres after t seconds has average speeds of 9, 6.3 and 6.03 m/s over shorter and shorter intervals from t = 1 s; they settle on dx/dt = 6 m/s.
An observer at rest finds a frequency ν; one moving away from the source finds ν′, a smaller number for the same wave.
(½)⁴ = ½ × ½ × ½ × ½ = 1/16.
lg 2 in the 1905 papers is ln 2 = 0.693, not the base-10 value 0.301.

Where this lesson stops

Each mark has one job where it appears. A formula that uses several at once, such as the diffusion equation, is read one mark at a time in its own lesson.

Letters: in §7 a source very far away in K sends plane waves whose phase is $\Phi=\omega\left(t-\frac{ax+by+cz}{V}\right)$ , where ω is the angular frequency, a, b, c the direction cosines of the wave normal, and V the speed of light. In k the phase is written with $\omega'$ , $a'$ , $b'$ , $c'$ and the moving coordinates ξ, η, ζ, τ. ν is the frequency, φ the angle in K between the line from the source to the observer and the observer's velocity, and $\varphi'$ the corresponding angle in k.

The phase counts wave crests, and every observer counts the same crests at the same event, so the phase at an event is the same number in both systems.
Write K's coordinates in k's with the inverse map: $t=\beta(\tau+vξ/V^2)$ , $x=\beta(ξ+v\tau)$ , y = η, z = ζ.
Substitute into Φ: $\omega\left[\beta\left(\tau+\frac{vξ}{V^2}\right)-\frac{a\beta(ξ+v\tau)+bη+cζ}{V}\right]$ .
Collect the τ terms: $\omega\beta(1-av/V)\tau$ . That coefficient is the new angular frequency, $\omega'=\omega\beta(1-av/V)$ , as the paper prints it.
Collect the ξ terms: $-\frac{\omega\beta}{V}(a-v/V)ξ$ . Divide by $\omega'$ to get the new direction cosine: $a'=\frac{a-v/V}{1-av/V}$ . The η and ζ terms give $b'=\frac{b}{\beta(1-av/V)}$ and $c'=\frac{c}{\beta(1-av/V)}$ .
With a = cos φ and ν = ω/2π, the frequency ratio and the new direction are the two formulas below. The paper calls the first the Doppler principle for arbitrary speeds, and the second the law of aberration in its most general form.

q := \frac{\nu'}{\nu}

q = \gamma\,\left(1 - \frac{v}{c}\,\cos\left(\varphi\right)\right)

Shown in modern letters; Einstein's are in §7 of the German source face.

The frequency ratio is gamma times one minus v over c times the unprimed direction cosine.

\cos\left(\varphi'\right) = \frac{\cos\left(\varphi\right) - \frac{v}{c}}{1 - \frac{v}{c}\,\cos\left(\varphi\right)}

\cos\left(\varphi'\right) = \frac{\cos\left(\varphi\right) - \frac{v}{V}}{1 - \frac{v}{V}\,\cos\left(\varphi\right)}

The aberration direction cosine is cosine phi minus v over c, divided by one minus v over c times cosine phi.

Worked at v = 0.6c. Light travelling the same way as the observer, φ = 0: q = 1.25 × (1 − 0.6) = 0.5. The paper's form for φ = 0, $\sqrt{(1-v/V)/(1+v/V)}=\sqrt{0.4/1.6}$ , gives the same 0.5.
Light meeting the observer head on, φ = 180 degrees: q = 1.25 × 1.6 = 2.0.
Light crossing the motion at right angles in K, φ = 90 degrees: q = 1.25, and $\cos\varphi'=-0.6$ , the paper's $-v/V$ , so the light travels at about 126.9 degrees to the moving x axis. Check the new direction: $a'$ = −0.6 and $b'=1/1.25$ = 0.8, and $0.36+0.64=1$ , a unit direction as it must be.
Each angle belongs to one frame. Light that crosses the motion at right angles in one frame does not in general do so in the other.
The paper also transforms the amplitude: $A'^2=A^2\frac{(1-(v/V)\cos\varphi)^2}{1-(v/V)^2}$ , so the amplitude changes by the same factor q as the frequency. It remarks that an observer approaching a source at the speed V would find it infinitely intense; no observer moves at V, so that is a limit, not a frame.
The Doppler effect of sound is an analogy only: it depends on motion relative to the air, and nothing here depends on motion relative to a medium.

A tool for this step

Fields and waves

What is a wave in a field, and what does a light detector measure?

A field assigns a value to every place at every moment. The temperature in a room is a field of single numbers. The electric field is a field of arrows, each with a size and a direction.

In a wave, the field at each place rises and falls in a repeating cycle, and the pattern travels. For the simplest wave the field goes as cos(kx − ωt). The wavelength λ is the distance from one crest to the next; the frequency ν is the number of crests passing a point each second; the crests travel at c = λν. The quantities in the cosine are k = 2π/λ and ω = 2πν.

E\left(x,\,t\right) = E_0\,\cos\left(k\,x - \omega\,t\right)

c = \lambda\,\nu

The field at x and t is E nought times the cosine of k x minus omega t, and the speed c equals the wavelength times the frequency.

Visible light oscillates about 6 × 10¹⁴ times a second, far too fast for any detector to follow. A detector responds to the energy delivered, which goes as the square of the field, averaged over many cycles. Over a whole cycle the average of cos² is exactly one half:

\left\langle \cos^{2}\left(\omega\,t\right)\right\rangle = \frac{1}{2}

The average of cosine squared of omega t, over a whole cycle, is one half.

A small lamp sending power P out equally in all directions spreads it over spheres of area 4πr². The intensity, the power arriving on each square metre, at distance r is therefore P/(4πr²): at twice the distance, a quarter of the intensity.

The Doppler shift of sound is a useful analogy for a change of frequency that comes with motion, but it has a limit. Sound travels in air, so a moving source and a moving listener are different situations, told apart by motion relative to the air. §7 of the relativity paper derives the Doppler shift of light from the relative velocity of source and observer alone; no medium enters the calculation.

Worked example: A 600-terahertz wave and a one-watt lamp

Wavelength of light at 600 terahertz, 6.0 × 10¹⁴ Hz: λ = c/ν = 299 792 458 ÷ 6.0 × 10¹⁴ = 4.997 × 10⁻⁷ m, which is 499.7 nanometres, blue-green light.
A lamp giving out 1 W equally in all directions. At 1 m the sphere has area 4π × 1² = 12.57 m², so the intensity is 1 ÷ 12.57 = 0.0796 W/m².
At 2 m the sphere has area 4π × 2² = 50.27 m², so the intensity is 0.0199 W/m², a quarter of the value at 1 m.

I = \frac{P}{4\,\pi\,r^{2}}

The intensity equals the power divided by four pi r squared.

Where this lesson stops

A field is a value assigned to every place and time. A light detector records the energy delivered averaged over many cycles, not the field at one instant.

Explore the equations in this step

Read each operation, check its units and assumptions, or open the mathematical step behind it. These are modern teaching equations, not the equations as printed.

Explore the equation · Modern model notationEinstein's letters

Aberration: the direction changes too

\cos\left(\varphi'\right) = \frac{\cos\left(\varphi\right) - \frac{v}{c}}{1 - \frac{v}{c}\,\cos\left(\varphi\right)}

\cos\left(\varphi'\right) = \frac{\cos\left(\varphi\right) - \frac{v}{V}}{1 - \frac{v}{V}\,\cos\left(\varphi\right)}

The cosine of is the cosine of minus over , divided by one minus v over c times that cosine.

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Explain this equation

Each term and operation, in words

No live values

This equation states a relation. No instrument on this page computes its terms.

Read the equation aloud in words

The aberration direction cosine is cosine phi minus v over c, divided by one minus v over c times cosine phi.

The same substitution that gives the frequency ratio gives the light's direction in the moving system. At v = 0.6c, light crossing the motion at 90 degrees in the stationary system has cos phi′ = −0.6 in the moving one.

Model assumptions and every term’s meaning

The moving system moves along x at speed v relative to the stationary system, with |v| < c. Modern teaching notation: gamma is the paper's beta, c its V.
A plane light wave whose direction makes the angle phi with the direction of motion, in the stationary system.
The same wave phase, written in both systems' coordinates, fixes frequency and direction together.

Cosine in the stationary system: 1 along the motion, 0 across it. Read the prerequisite
Cosine in the stationary system: The same cosine as in the numerator. Read the prerequisite
Cosine in the moving system: What the formula gives; the angle itself is its inverse cosine. Read the prerequisite
One minus that: Keeps the result between −1 and 1, as a cosine must be. Read the prerequisite
Cosine minus v over c: Shifts the direction against the motion. Read the prerequisite
v over c times the cosine: The part of the motion along the light. Read the prerequisite
The quotient: The moving system's cosine. Read the prerequisite
What it asserts: Two systems in relative motion see the same light arrive from different directions. Read the prerequisite
v over c: The speed as a fraction of the speed of light. Read the prerequisite
v over c: The same fraction, in the denominator. Read the prerequisite
The light's angle, stationary system: Measured in the stationary system. Read the prerequisite
The same angle again: The stationary-system angle, entering the denominator. Read the prerequisite
The light's angle, moving system: The same light's direction relative to the motion, measured in the moving system. Read the prerequisite
Speed of light: The paper's V. Read the prerequisite
Speed of the moving system: Signed along x: the sign says which way the system moves. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Explore the equation · Modern model notation

The Doppler factor

q = \gamma\,\left(1 - \frac{v}{c}\,\cos\left(\varphi\right)\right)

Shown in modern letters; Einstein's are in §7 of the German source face.

is times one minus over times the cosine of .

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Explain this equation

Each term and operation, in words

No live values

This equation states a relation. No instrument on this page computes its terms.

Read the equation aloud in words

The frequency ratio is gamma times one minus v over c times the unprimed direction cosine.

How the frequency changes depends on the speed and on the light's direction. At v = 0.6c, light travelling along the motion (phi = 0) gives q = 0.5; light across it (phi = 90 degrees) gives q = 1.25.

Model assumptions and every term’s meaning

The moving system moves along x at speed v relative to the stationary system, with |v| < c. Modern teaching notation: gamma is the paper's beta, c its V.
A plane light wave whose direction makes the angle phi with the direction of motion, in the stationary system.
The same wave phase, written in both systems' coordinates, fixes frequency and direction together.

One minus that: Below 1 for light along the motion, 1 for light across it. Read the prerequisite
Cosine of the angle: 1 for light along the motion, 0 for light across it, −1 for light against it. Read the prerequisite
How much of the motion is along the light: v over c times the cosine: zero when the light crosses the motion at right angles. Read the prerequisite
What it asserts: The frequency ratio for this light, from the transformation of its phase. Sound gives a similar-looking rule by a different argument; the analogy is not the derivation. Read the prerequisite
Times gamma: Gamma alone survives for light at right angles: a frequency change with no motion along the light. Read the prerequisite
v over c: The speed as a fraction of the speed of light. Read the prerequisite
The light's angle, stationary system: Between the light's direction and the direction of motion, measured in the stationary system. Name the frame: the angle is different in the other one. Read the prerequisite
Lorentz factor: One over the square root of one minus v squared over c squared. At 0.6c it is 1.25. Read the prerequisite
Speed of light: The paper's V. Read the prerequisite
The frequency ratio: The moving system's frequency over the stationary system's. Read the prerequisite
Speed of the moving system: Signed along x: the sign says which way the system moves. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Explore the equation · Modern model notation

Name the frequency ratio

q := \frac{\nu'}{\nu}

Shown in modern letters; Einstein's are in §7 of the German source face.

is divided by .

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Explain this equation

Each term and operation, in words

No live values

This equation states a relation. No instrument on this page computes its terms.

Read the equation aloud in words

q is defined as the moving frequency over the stationary frequency.

q is a name for how much the frequency changes: the moving system's frequency divided by the stationary system's, for the same light.

Model assumptions and every term’s meaning

The moving system moves along x at speed v relative to the stationary system, with |v| < c. Modern teaching notation: gamma is the paper's beta, c its V.
A plane light wave whose direction makes the angle phi with the direction of motion, in the stationary system.
The same wave phase, written in both systems' coordinates, fixes frequency and direction together.

Divide one by the other: Both are frequencies of the same light, so the units cancel. Read the prerequisite
A definition: This line names q; the next one says what q is. Read the prerequisite
Frequency in the stationary system: Crests per second as the stationary system's clocks count them. Read the prerequisite
Frequency in the moving system: Crests per second as the moving system's clocks count them. Read the prerequisite
The frequency ratio: A pure number. Greater than 1 means the moving system counts more crests per second. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Modern qualifications

An observer moving at the speed of light appears in these formulas only as a limit; no inertial frame moves at $c$ . Stellar aberration had been observed since Bradley’s report of 1729, and the second-order Doppler shift was first measured by Ives and Stilwell in 1938. Both are measurements of the world, separate from this ideal calculation.

Assumptions and limits: One phase fixes frequency and direction

Assumed here

The light is a plane wave in empty space, and both observers describe the same events.
The angle φ is measured in the original frame, between the direction of the light and the direction of motion.

What this does not establish

Light that crosses the motion at right angles in one frame does not, in general, do so in the other.
No observer can move at the speed of light, so the formulas’ limit at v = c describes no frame.

Earlier step: Light constraints leave a scale to determine Electric and magnetic components mix together

Source context: German source · English · Interlinear gloss · Facsimile

References

A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik (4), 17, 132–148 (1905).

A. Einstein, On the motion of particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Annalen der Physik (4), 17, 549–560 (1905), §§1–5. Bibliographic pointer; this preview is not a source transcription or translation.

A. Einstein, Zur Elektrodynamik bewegter Körper. Annalen der Physik (4), 17, 891–921 (1905).

A. Einstein, Does the inertia of a body depend upon its energy content? Annalen der Physik (4), 18, 639–641 (1905). External 1923 Perrett–Jeffery translation, electronically transcribed by John Walker; its notation was modernized. A reference for this explanatory preview, not this edition’s translation or its pinned facsimile.

28 foundation readings sit behind this argument.

Read this offline

Each chapter is one HTML file that opens in any browser without a connection. It holds the explanation at every level of detail, the lessons it links to, its source references and the numbers worked out when the site was built. It has no notes of yours and no running simulations. It is an explanation, not an edition of the paper.

Download §7 · Doppler shift and aberration HTML, 513 KB

Links to online sources still need a connection, and the interactive plots are not included. Every chapter you can keep

Original journal scan