Read · Special relativity: from clock operations to electrodynamics

Introduction · magnet, conductor, and the postulates

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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.

An assumption

One apparatus, two descriptions

Why should changing the observer change the explanation?

The 4 printed paragraphs this passage explains

Introduction, paragraph 1: Maxwell's electrodynamics, as it was usually understood, treats a magnet moving past a resting conductor differently from a conductor moving past a resting magnet, although the current observed depends only on their relative motion. Einstein calls this an asymmetry that the phenomena themselves do not seem to have.
Introduction, paragraph 2: Such examples, and the failed attempts to detect the earth's motion through the light medium, lead Einstein to conjecture that absolute rest has no counterpart in electrodynamics either. He raises this principle of relativity to a postulate, adds that light in empty space always travels at one speed V whatever the motion of its source, and says a light ether will prove superfluous.
Introduction, paragraph 3: Every electrodynamics rests on the kinematics of rigid bodies, since its statements concern rigid bodies used as coordinate systems, clocks and electromagnetic processes. Neglecting this, Einstein says, is the root of the present difficulties.
§6, paragraph 9: So the asymmetry noted in the introduction, between a moving magnet and a moving conductor, disappears, and the question of where the electromotive force is seated, as in unipolar machines, loses its point.

In the usual account of 1905, a magnet moving past a conductor at rest creates an electric field, and that field drives the current. A conductor moving past a magnet at rest meets no electric field at all; the charges moving with it feel a force from the magnetic field instead, and that force drives the current. The two accounts predict the same current for the same relative motion. The paper sets out to remove the asymmetry of the explanation, not to show that the two observers measure the same fields: they do not.

Einstein raises two conjectures to the status of postulates. First, the laws of electrodynamics and optics hold in the same form in every frame in which the laws of mechanics hold. Second, light in empty space always travels with a definite speed $V$ , whatever the motion of the body that emits it. The paper takes them as its starting point; the unsuccessful attempts to detect motion relative to the ether are cited as what suggests the first, not as a proof of it.

The magnet-and-conductor laboratory compares the two descriptions for the same relative motion. At low speed the electric force in one frame equals the magnetic force in the other; at higher speed they differ by the factor $\gamma$ derived in §6, which is how a force across the motion changes between frames.

Open the foundation: Dot and cross products

Open the foundation: Charges, fields and light

Show every step here: One apparatus, two descriptions

Letters: the paper writes V for the speed of light in empty space; this explanation writes c. It calls the frame in which Newton's equations hold the stationary system, K from §3 on, and a system moving uniformly relative to it k. The factor this explanation writes γ is the paper's β, which first appears in §3.

The apparatus: a magnet and a conductor, a loop of wire, in motion relative to each other, with the current in the conductor as the thing observed.
The first case, as the physics of 1905 explained it: the magnet moves and the conductor rests. Around the moving magnet an electric field arises, carrying a definite energy, and where parts of the conductor are, that field drives a current.
The second case: the magnet rests and the conductor moves. No electric field arises around the magnet. In the conductor there is instead an electromotive force, to which no energy corresponds in itself, and it drives a current.
Compare the observations: for the same relative motion, the current has the same size and the same course in both cases. What is observed depends only on the relative motion of conductor and magnet.
Compare the explanations: one case has an electric field and the other has none. The asymmetry lies in the account, not in what is observed.
Worked low-speed case: the conductor moves at 10 metres per second across a magnetic field of 0.5 tesla. On a charge q carried with it the magnetic force is qvB, and per unit charge that is $vB=10\times0.5=5$ volts per metre. In the conductor's own frame §6 gives an electric field of γ times vB, and at this speed γ − 1 is about $v^2/(2c^2)=5.6\times10^{-16}$ : no measurement could tell the two numbers apart.
At 0.6c they differ by the factor γ = 1.25. §6 derives it: a force across the motion changes between frames by that factor, so the two accounts describe one push.
The paper adds a second kind of evidence: the unsuccessful attempts to detect the earth's motion relative to the medium of light. Together these lead to the conjecture that absolute rest corresponds to no property of the phenomena, in electrodynamics as in mechanics, and that the same laws of electrodynamics and optics hold in every coordinate system in which the equations of mechanics hold, as had already been shown for quantities of the first order.
The first premise, the principle of relativity: the paper raises that conjecture to a premise. It is assumed, not proved from the examples.
The second premise: light in empty space always travels with a definite speed V, independent of the state of motion of the body that emits it. The paper says the two premises are only apparently incompatible.
What the two premises are said to give: a simple and consistent electrodynamics of moving bodies, built on Maxwell's theory for bodies at rest. A light ether becomes superfluous, since no absolutely stationary space is introduced and no velocity is assigned to a point of empty space in which electromagnetic processes take place.
Where the work starts: every such theory rests on the kinematics of rigid bodies, since its statements concern rigid bodies used as coordinate systems, clocks, and electromagnetic processes. So §1 begins with what time means.
The status of the laboratory: a simulation built on the two premises shows their consequences. It cannot prove them.

A tool for this step

Dot and cross products

How much of one arrow lies along another, and which way does a magnetic force push?

The dot product multiplies two arrows into a single number: $\mathbf{a}\cdot\mathbf{b} = a_x b_x + a_y b_y + a_z b_z$ , which also equals the two lengths times the cosine of the angle between them. With an arrow of length 1 along a chosen direction, the dot product is how much of the other arrow lies along that direction: its projection. Arrows at right angles have a dot product of zero.

\begin{aligned} \mathbf{a}\cdot\mathbf{b} &= a_x b_x + a_y b_y + a_z b_z \\ &= |\mathbf{a}|\,|\mathbf{b}|\cos θ \end{aligned}

a dot b equals a x b x plus a y b y plus a z b z, which equals the length of a times the length of b times the cosine of the angle between them.

For a boost along x, the part of an electric field along the motion is its dot product with an arrow of length 1 along x. A field of 5 units pointing 3 along x and 4 along y has 3 along the boost; that part is the same in both frames, and only the rest changes.

The cross product $\mathbf{a}\times\mathbf{b}$ is an arrow at right angles to both a and b. Its length is the area of the parallelogram they span, the two lengths times the sine of the angle between them, and its direction follows the right-hand rule: curl the fingers of the right hand from a toward b, and the thumb points along the product. Swapping the order reverses the arrow.

\mathbf{F} = q\,\mathbf{v}\times\mathbf{B}

The force F equals q times v cross B.

The magnetic force on a charge q moving with velocity v through a field B is $q\,\mathbf{v}\times\mathbf{B}$ . It is at right angles to the motion, so it changes the charge's direction and not its speed. For a positive charge moving along +x through a field along +z, $\mathbf{v}\times\mathbf{B}$ points along −y. In the frame moving with the charge the same push is electric: the relativity paper's §6 gives that frame an electric field along −y.

One combination of the two fields, $\mathbf{E}\cdot\mathbf{B}$ , comes out the same in every frame. That is a modern check on the field transformation, useful for catching an error in a calculation. It is not a premise of the 1905 argument, which derives the transformation from the equations of electrodynamics.

Worked example: A projection, a force direction and a quantity that stays put

Projection: $\mathbf{E} = (3, 4, 0)$ and the boost direction $(1, 0, 0)$ give $\mathbf{E}\cdot(1, 0, 0) = 3$ , the part along the boost. The field's length is $\sqrt{3^2 + 4^2} = 5$ .
Direction: with v along +x and B along +z, $(1, 0, 0)\times(0, 0, 1) = (0, -1, 0)$ , so a positive charge is pushed along −y.
Invariance, measuring fields so that c = 1: take $\mathbf{E} = (0, 0.6, 0.8)$ and $\mathbf{B} = (0, 0, 1)$ , so $\mathbf{E}\cdot\mathbf{B} = 0.8$ .
Seen from a frame moving at 0.6 along x, where γ = 1.25 (the factor 1/√(1 − v²/c²), which the relativity paper prints as β), the §6 rules give $\mathbf{E}' = (0, 0, 1)$ and $\mathbf{B}' = (0, 0.6, 0.8)$ . Both fields changed, and $\mathbf{E}'\cdot\mathbf{B}' = 1 \times 0.8 = 0.8$ did not.

JavaScript is off, so the controls below cannot change anything. The construction shows its first setting, and its closing paragraph, “What it shows, in words”, describes the rest.

Try it: a projection and an area

Arrow a is 4 long along x. Arrow b is 3 long; turn it and watch how much of it lies along a, and how much area the two arrows span.

Angle from a to b, in degrees

Type an angle and press Enter, or drag the slider from −180° to 180°. A positive angle turns b counterclockwise from a.

a·b = 4 × 3 × cos 60° = 6. b's projection on a is 1.5: that much of b lies along a.

a × b has size 10.392, the area of the parallelogram, and points out of the page.

What it shows, in words

The dot product 4 × 3 × cos φ measures how much of b lies along a: 6 at 60°, zero at 90°, −12 at 180°. The cross product has size 4 × 3 × sin φ, the area of the parallelogram the arrows span, largest at 90° and zero when they line up. By the right-hand rule it points out of the page when b is counterclockwise from a and into the page when it is clockwise; swapping the order of a and b reverses it. The figure shows a thick, b thinner, the parallelogram with a dashed outline, and a dotted line from b's tip to its projection on a.

Where this lesson stops

The dot product is how much of one arrow lies along another; the cross product is an arrow for the area two arrows span. This lesson stops at three dimensions.

A tool for this step

Charges, fields and light

What does an electric or a magnetic field do to a charge?

A comb run through dry hair lifts small scraps of paper before it touches them. The charge on the comb fills the space around it with an electric field, and the field pushes on any charge placed in it, harder for a larger charge. For a charge q in a field E the force is qE, along the field for a positive charge and against it for a negative one.

A magnet does nothing to a charge at rest. A charge moving with velocity v through a magnetic field B feels a push at right angles both to its motion and to the field, which is what turns an electric motor. Together these are the Lorentz force:

\mathbf{F} = q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)

The force equals q times the sum of the electric field and the cross product of the velocity with the magnetic field.

Carrying a charge q across a potential difference ΔV changes its energy by qΔV. The electron's charge has magnitude e = 1.602 × 10⁻¹⁹ coulombs, so one volt changes an electron's energy by 1.602 × 10⁻¹⁹ joules. That amount is called one electron volt, 1 eV. §8 of the light-quanta paper states the energy of electrons that light frees from a metal in these terms: by the potential needed to stop them.

An electron held near a point by a force proportional to its displacement, like a mass on a spring, vibrates at a natural frequency of its own, and it absorbs and emits light near that frequency. §1 of the light-quanta paper fills a closed box with such bound electrons, which it calls resonators, together with free electrons, gas molecules and radiation, and asks what equilibrium requires. Using Planck's condition of 1900 for resonators and radiation, the answer gives the radiation an infinite total energy. That is the difficulty in the section's title.

Maxwell's equations tie fields to charges and currents in four statements: charges are the sources of electric field; there are no magnetic charges; a changing magnetic field drives a circulating electric field, which is Faraday's induction; and currents and changing electric fields drive circulating magnetic fields. Together they allow waves of field that cross empty space at the speed of light, c.

The relativity paper opens with a case from this theory. Move a magnet past a conductor, and the theory describes an electric field that drives a current. Move the conductor past the magnet instead, and it describes no electric field, only a magnetic force on the conductor's charges. The current is the same either way. The paper counts this difference of description, with no difference in what is observed, among its grounds for taking only relative motion to matter.

Worked example: An electron across one volt

\Delta W = q\,\Delta V

The change in energy equals the charge times the potential difference.

An electron starts at rest and crosses a potential difference of 1 V.
It gains energy e × 1 V = 1.602 × 10⁻¹⁹ joules: one electron volt.
At low speed that energy is ½mv², with the electron's mass m = 9.109 × 10⁻³¹ kg. So v² = 2 × 1.602 × 10⁻¹⁹ ÷ 9.109 × 10⁻³¹ = 3.52 × 10¹¹ m²/s².
The square root gives v ≈ 5.93 × 10⁵ m/s, about 600 kilometres per second.

That is fast, but only 0.2 per cent of the speed of light, so the low-speed formula ½mv² is safe here. The charge and mass used are modern values.

Where this lesson stops

Charge is the property that makes electric forces. Fields carry the force between separated charges, without action at a distance.

Modern qualifications

This page reconstructs the argument in modern words; it is not a translation of the paper’s opening. Whether the Michelson–Morley experiment shaped Einstein’s own route is a historical question with its own evidence; the paper itself cites only the unsuccessful attempts to detect the earth’s motion through the ether.

Assumptions and limits: One apparatus, two descriptions

Assumed here

Both descriptions concern the same apparatus and the same events, seen from two inertial frames.
The principle of relativity and the constancy of the speed of light are assumptions, and stated as such.

What this does not establish

A simulation built on the postulates cannot prove them.
The model of a magnet and a conductor loop does not cover every induction experiment.

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.

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Introduction · magnet, conductor, and the postulates

Original journal scan

One apparatus, two descriptions

Modern qualifications

What this relies on

Assumed here

What this does not establish

References

Reading a graph

Fractions and ratios

A letter stands for a quantity

Marks that look alike and do different jobs

A sign records direction

Powers of ten and physical units

Squares and square roots

Adding and averaging

Rates of change and derivatives

Density is not probability

Dot and cross products

Charges, fields and light

Exponential change

Fields and waves

Counting what crosses a boundary

Events and distant clocks

Functions and graphs

Hyperbolic functions and rapidity

Adding continuously

Logarithms: turning products into sums

Matrices and linear maps

The push of light

Partial derivatives and held-fixed quantities

Quantities and units

Approximating a curve near a point

The units of 1905

Vectors and their components

Energy of motion and inertia

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