§7 · Doppler shift and aberration · Special relativity: from clock operations to electrodynamics

§7 · Doppler shift and aberration

One phase fixes frequency and direction

Why must a frequency change accompany a direction change?

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

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

\cos\varphi'=\frac{\cos\varphi-v/c}{1-(v/c)\cos\varphi}

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.

Return to why a change of frequency comes with a change of direction. (included in this file)

Return to ν′ and φ′, the other observer's frequency and direction. (included in this file)

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

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

\cos\varphi'=\frac{\cos\varphi-v/c}{1-(v/c)\cos\varphi}

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.

Return to why a change of frequency comes with a change of direction. (included in this file)

Aberration: the direction changes too

Modern teaching equation, not the equation as printed.

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

The Doppler factor

Modern teaching equation, not the equation as printed.

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

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.

Name the frequency ratio

Modern teaching equation, not the equation as printed.

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

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.

Assumptions

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.

Limits

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.

Foundations included in this file

Reading a graph

How does a curve show the relationship between two physical quantities?

Walk for a minute and note how far you have gone every ten seconds: 0 m at the start, 14 m after ten seconds, 28 m after twenty, and so on. The first drawing below puts time along the bottom and distance up the side, with a dot for each pair. That is a graph: each dot is one moment, read across for the time and up for the distance.

The dots of a steady walk fall on a straight line, and the steeper the line, the faster the walk. This one climbs 14 metres for every 10 seconds: 1.4 metres per second. A flat line would mean standing still, and a line that bends upward would mean speeding up.

The papers draw other pairs the same way. Along the bottom goes what you choose or wait for, such as a position or a time; up the side goes what you then find, such as how many particles sit there. A bell-shaped curve, high in the middle and falling away on both sides, says that most particles are near the centre and fewer are farther out.

Worked example: Four particles that average to zero

Four particles end at −3, −1, +1 and +3 micrometres from where they started.
The second drawing puts those positions along the bottom, with a bar one particle tall above each.
The bars stand two on each side of zero, at the same distances out. The picture is symmetric, so the average position is 0.
Yet no bar stands at zero: every particle moved. The graph shows at a glance what the average hides.

Where this lesson stops

Before reading a curve, read its two labels and their units, and ask whether its height is a count, a density or a running total.

This lesson builds on

A sign records direction

Return to the chapter

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.

Return to the chapter

A sign records direction

How can movement add up to zero?

On the ruler below, the starting point is zero. A final mark three units to the right has displacement +3; one three units to the left has displacement −3. Adding the signed displacements gives zero. Adding the distances from the start gives six.

Displacement compares where something ends with where it started, whatever it did in between. A walker who goes three units right and comes back has displacement zero after walking six, which is the lower pair of arrows in the drawing.

Worked example: Two walkers, opposite directions, average zero

Two walkers both finish one unit from their start, one at −1 and the other at +1. The signed average is zero. The average distance is one.

Where this lesson stops

The sign names a direction relative to a chosen axis; it does not mean a negative distance.

This lesson builds on

Adding and averaging

Return to the chapter

Adding and averaging

What does an average keep, and what does it lose?

Four jars hold 3, 1, 1 and 3 marbles. Pour them together and there are 8. Share the 8 equally between the four jars and each holds 2. Two is the average, although no jar held 2 to begin with.

The average keeps the total and the count and nothing else, so 3, 1, 1, 3 and 2, 2, 2, 2 have the same average. Listing every value twice doubles both the total and the count, and the average stays at 2.

Worked example: Three mornings with the same average

One morning the four jars hold 3, 1, 1 and 3 marbles. Add them: 8. Divide by the four jars: an average of 2.
Another morning they hold 8, 0, 0 and 0. Add them: 8 again. Divide by four: 2 again, and now three jars are empty.
A third morning every jar holds 2, which is the average itself. Add them: 8. Divide by four: 2.
Write any of those three lists out twice over. The total doubles to 16 and the count doubles to 8 jars, and 16 divided by 8 is still 2.
Four descriptions and one average. It keeps the total and the count, and the differences between the mornings are what it leaves behind.

Where this lesson stops

An average is the total shared equally among the observations, and two very different lists can have the same one.

Return to the chapter

Rates of change and derivatives

How can a body have a speed at a single instant?

A speed needs two readings of position and the time between them: divide the distance covered by the time taken, and you have the average speed over that interval. An instant has no duration, so that recipe cannot be used at an instant as it stands.

So shorten the interval and watch what happens. A ball rolls so that after t seconds it has gone 3t² metres: 3 m after one second, 12 m after two. Start at the one-second mark and average over shorter and shorter stretches:

Over the next second it goes from 3 m to 12 m: 9 m in 1 s, an average of 9 m/s.
Over the next tenth of a second it reaches 3.63 m: 0.63 m in 0.1 s, an average of 6.3 m/s.
Over the next hundredth it reaches 3.0603 m: 0.0603 m in 0.01 s, an average of 6.03 m/s.
Over the next thousandth it reaches 3.006003 m: 0.006003 m in 0.001 s, an average of 6.003 m/s.

The averages close in on 6 m/s. That number is the derivative: the ball's speed at the instant t = 1 s. The interval is never set to zero, which would mean dividing zero distance by zero time. It only shrinks, and the averages settle.

In symbols, with Δt for the short interval, the same recipe reads:

\frac{dx}{dt} = \lim_{\Delta t \to 0} \frac{x(t+\Delta t) - x(t)}{\Delta t}

d x by d t is the value that the change in position divided by the change in time settles on as delta t shrinks towards zero.

A derivative carries units: those of the quantity that changes, divided by those of the quantity you vary. Position in metres, varied over time in seconds, gives metres per second. On a graph of position against time, the derivative at a point is the steepness of the curve there.

Einstein's Brownian paper needs rates like this for a concentration of particles, which changes along a tube and in time at once. With two things changing, a rate has to say which one it follows, so the paper writes a curly ∂ and holds the other fixed:

\frac{\partial f}{\partial t} = D\,\frac{\partial^2 f}{\partial x^2}

The partial derivative of f with respect to t equals D times the second partial derivative of f with respect to x.

The lesson on partial derivatives, listed at the end of this page, reads that equation one piece at a time.

Worked example: Why the averages settle on 6 m/s

The list of averages can be explained, not only watched. Start at any time t and let the interval be Δt. In that interval the ball moves 3(t + Δt)² − 3t² metres. Since (t + Δt)² is t² + 2tΔt + (Δt)², that distance is:

3(t+\Delta t)^2 - 3t^2 = 6t\,\Delta t + 3(\Delta t)^2

Three times t plus delta t, squared, minus three t squared, equals six t delta t plus three delta t squared.

Divide that distance by the time Δt to get the average speed over the interval:

\frac{6t\,\Delta t + 3(\Delta t)^2}{\Delta t} = 6t + 3\,\Delta t

Six t delta t plus three delta t squared, divided by delta t, equals six t plus three delta t.

The 3Δt term shrinks with the interval and leaves 6t metres per second. At t = 1 s that is 6 m/s. The extra 3, 0.3, 0.03 and 0.003 in the list above are that 3Δt term, for Δt of 1, 0.1, 0.01 and 0.001 seconds.

Where this lesson stops

A derivative is the value an average rate settles on as the interval shrinks, in units of one quantity per unit of the other. The interval shrinks but is never set to zero, so nothing is ever divided by zero.

This lesson builds on

Functions and graphs

Return to the chapter

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(x,t) = E_0 \cos(kx - \omega t), \qquad 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:

\langle \cos^2(\omega t) \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.

This lesson builds on

Functions and graphs
Rates of change and derivatives

Return to the chapter

Functions and graphs

How does a graph record the way one quantity depends on another?

A bath fills from a tap. After one minute it holds 10 litres, after two 20, after three 30. For each time there is exactly one amount of water, and that pairing, one output for each input, is a function. Plot the pairs with time along the bottom and litres up the side and they fall on a straight line that climbs 10 litres for every minute. The slope, in litres per minute, is how fast the bath fills.

In the papers the pairs are physical: the mean square displacement of a Brownian particle for each elapsed time, or the concentration for each position along a tube. A function says how one thing responds when another changes.

A graph puts the input along the horizontal axis and the output up the vertical one, so each point is one pair. Both axes carry units, and so does everything read off the graph: the slope of mean square displacement against time is in square micrometres per second, and the area under a probability density is a plain probability.

A formula such as ⟨x²⟩ = 2Dt describes an average over many particles. One particle's squared displacement scatters widely around the line; the average over many lies close to it. The Brownian paper predicts the average, and ends by hoping that a researcher will soon decide the question by observation.

Worked example: Mean square displacement plotted against time

\langle x^2\rangle = 2Dt

The mean square displacement equals two times the diffusion coefficient times the time.

Take D = 0.5 square micrometres per second, so 2D = 1 square micrometre per second.
At t = 1 s the mean square displacement is 1 µm²; at t = 4 s it is 4 µm²; at t = 9 s, 9 µm².
Plotted against time, these points lie on a straight line through the origin whose slope is 2D, 1 µm² per second.
Their square roots, the root-mean-square displacements, are 1, 2 and 3 µm: four times the time gives twice the distance.

Where this lesson stops

A graph is a record of paired quantities, each with its units. A slope read off it has units too, and a slope stated without them explains nothing.

This lesson builds on

Reading a graph

Return to the chapter

Static laboratory examples

sr-09: a built-in worked result

Static host-calculated example, not a running experiment or an observation. Canonical units and owner-supplied statuses are retained. Open sr-09 online

Build-time scalar readouts
Quantity	Value	Unit	Status
frameSpeed	0.6	c	value
propagationAngleStationary	0	deg	value
propagationAngleMoving	0	deg	value
waveFrequencyStationary	500000000000000	Hz	value
waveFrequencyMoving	250000000000000	Hz	value
dopplerFactor	0.5	1	value
wavePhase	-3.141592653589793	rad	value
lorentzFactor	1.25	1	value
classicalObserverDopplerFactor	0.4	1	value
classicalSourceDopplerFactor	0.625	1	value
recedingDopplerFactor	0.5	1	value
approachingDopplerFactor	2	1	value

0 array outputs are not reproduced. This table does not claim to reproduce the interactive plot.

Settings (canonical units)

beta: 0.6
countingWindowCycles: 10
detectorMotion: rest-in-k
detectorSpeed: 0
frequencyTHz: 500
propagationAngleDeg: 0
secondOrderSpeed: 0.005
selectedEventId: sr-09-event-origin-tick
showCovectorNote: false

Evaluator: source:sha256:1a61c807c88e8809f94b01283476550a319d4703dd56141f63516844ce2fcf8f

Source 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).