Generator

The same wire, seen twice at 0.6c

One function in the spacetime library, called 84 times across 14 essays. Below: what it draws at its defaults, what it draws at every branch an essay asks for, whether the site's own gate puts a claim to it, and everywhere it is called.

At its defaults it draws the same wire, seen twice at 0.6c. Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second.

wire-frames is one function in lib/figures/spacetime.js — worldlines, slicing and the factor that governs both. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.

The same wire, seen twice at 0.6c. Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second.

Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second.

Charge density and current, mixing like time and space

The options are the ones Charge and current are one thing passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Charge density and current, mixing like time and space. The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 0 and J = 2. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change.

The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 0 and J = 2. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change.

Charge density and current, mixing like time and space

The options are the ones Charge and current are one thing passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Charge density and current, mixing like time and space. The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 2.4 and J = 1. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change.

The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 2.4 and J = 1. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change.

Conservation, as a statement about a box in spacetime

The options are the ones Charge and current are one thing passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Conservation, as a statement about a box in spacetime. A region of space of length 1, drawn against time over an interval 1, with a current of 1.6 entering the left face and 0.6 leaving the right. The charge inside at the end differs from the charge at the start by 1.00, and that difference is exactly what crossed the two sides. Written as an equation that is ∂ρ/∂t + ∇·J = 0, and written as a four-vector statement it is that the four-divergence of (cρ, J) vanishes — one equation rather than a time part and a space part. The box makes the point that conservation is not a rule about a quantity but a rule about a flux: charge does not disappear here and appear there, it crosses a boundary, and nothing else is allowed.

A region of space of length 1, drawn against time over an interval 1, with a current of 1.6 entering the left face and 0.6 leaving the right. The charge inside at the end differs from the charge at the start by 1.00, and that difference is exactly what crossed the two sides. Written as an equation that is ∂ρ/∂t + ∇·J = 0, and written as a four-vector statement it is that the four-divergence of (cρ, J) vanishes — one equation rather than a time part and a space part. The box makes the point that conservation is not a rule about a quantity but a rule about a *flux*: charge does not disappear here and appear there, it crosses a boundary, and nothing else is allowed.

Charge density and current, mixing like time and space

The options are the ones Charge and current are one thing passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Charge density and current, mixing like time and space. The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 1.2 and J = 0. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change.

The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 1.2 and J = 0. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change.

Conservation, as a statement about a box in spacetime

The options are the ones Charge and current are one thing passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Conservation, as a statement about a box in spacetime. A region of space of length 2, drawn against time over an interval 1.5, with a current of 0.9 entering the left face and 0.9 leaving the right. The charge inside at the end differs from the charge at the start by 0.00, and that difference is exactly what crossed the two sides. Written as an equation that is ∂ρ/∂t + ∇·J = 0, and written as a four-vector statement it is that the four-divergence of (cρ, J) vanishes — one equation rather than a time part and a space part. The box makes the point that conservation is not a rule about a quantity but a rule about a flux: charge does not disappear here and appear there, it crosses a boundary, and nothing else is allowed.

A region of space of length 2, drawn against time over an interval 1.5, with a current of 0.9 entering the left face and 0.9 leaving the right. The charge inside at the end differs from the charge at the start by 0.00, and that difference is exactly what crossed the two sides. Written as an equation that is ∂ρ/∂t + ∇·J = 0, and written as a four-vector statement it is that the four-divergence of (cρ, J) vanishes — one equation rather than a time part and a space part. The box makes the point that conservation is not a rule about a quantity but a rule about a *flux*: charge does not disappear here and appear there, it crosses a boundary, and nothing else is allowed.

What checks it

physicscheck asserts something about wire-frames that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Charge and current are one thing

Ask what a boost leaves alone and the answer is charge. That answer forces the next one: a fixed charge in a contracting volume gives a density that transforms like a time component, and a current that transforms like a space one. So charge density and current density are the four parts of one object — and conservation of charge stops being an extra law and becomes the condition that makes the object exist.

Magnetism is electricity seen sideways

The force on a charge moving beside a current-carrying wire is magnetic in the laboratory and purely electrostatic in the charge's own frame. Both calculations give the same answer, and the drift speed that makes them agree corresponds to a Lorentz factor differing from one in the twenty-sixth decimal place.

Six numbers, one object

Three components of E and three of B mix into each other under a boost and never into anything else. Six numbers that transform among themselves are the independent entries of a four-by-four antisymmetric array, and writing them that way is not notation — it turns Maxwell's four equations into two, makes the two invariants the only two there could be, and shows that "electric" is a choice of axes rather than a kind of field.

The beam that stops pushing itself apart

A beam of electrons is a crowd of like charges, and a crowd of like charges flies apart. A slow beam does, within centimetres. A fast one hardly does at all: at a gigaelectronvolt the same current travels kilometres before its width doubles. The electric repulsion has not weakened — it has grown, by the factor γ. What has grown faster is a magnetic attraction between parallel currents, which cancels all but 1/γ² of the push. Seen from the beam itself there is no magnetism at all, only Coulomb repulsion acting while the beam's clocks run slow.

The charge that passes as a flash of light

A charge at rest has a field that no change of frame can turn into light. A charge going past at nearly the speed of light delivers a field that is, for every practical purpose, a pulse of light anyway: thin, perpendicular to the motion, with its electric and magnetic parts equal and crossed. Enrico Fermi saw in 1924 that such a field could be counted as photons, and the Large Hadron Collider now collides lead nuclei's photons with each other.

The contraction the forces work out for themselves

Relativity usually introduces length contraction as a consequence of how moving observers disagree about simultaneity, not as something that happens to matter. There is a second way to reach the same number, and it is worth knowing because it is the way the idea was first found. Take an atom, accelerate its nucleus gently, and compute what the electron does using only Maxwell's field of a moving charge and the law of motion. Nobody tells the orbit to contract, and it does, by exactly 1/γ, while each turn takes exactly γ times as long. The forces holding matter together work out the contraction for themselves.

The disc that cannot be spun

Set a disc turning and measure its circumference with rulers carried on the rim. They lie along their own direction of motion and are contracted, so more of them fit; rulers along a radius lie across the motion and are not. The ratio of circumference to radius is therefore not two pi, in a frame where nothing is happening but rotation — and Einstein said that was what set him looking for gravity in geometry.

The field nobody can transform away

A wire's magnetic field is an electric field seen from the wrong frame, and a charged plate's electric field is a magnetic one seen the same way. Neither trick works on a light wave. Two combinations of E and B are the same for every observer, and which side of one line a field sits on is a fact nothing about the observer can alter.

The gyroscope a spinning Earth turns

A moving charge makes a magnetic field, and a spinning charged ball makes a dipole field around it. Gravity has the same structure one level down: a spinning mass drags the local standards of rotation around with it, and a perfect gyroscope near the Earth turns, very slightly, in the pattern of a dipole field — with the Earth's spin over the poles and against it over the equator. Averaged round a polar orbit the turning is 41 milliarcseconds a year. It took forty-four years and a satellite carrying the roundest objects ever made to measure it.

The magnet period that comes out as an X-ray

A row of permanent magnets alternating every three centimetres, with a beam of six-gigaelectronvolt electrons sent down the middle, shines X-rays with a wavelength of an ångström and a half — two hundred million times shorter than the spacing of the magnets that made them. The factor is γ squared, and the two γ's come from two different frames. In the electron's frame the row of magnets is contracted by γ and rushes past it as a wave; the electron shakes in step and radiates; and that light, sent forward, comes back to the laboratory shortened by another factor of 2γ. Length contraction is half of every X-ray a modern synchrotron makes.

The one quantity a boost leaves alone

Energy, momentum, length, duration, density and field strength all change when the observer moves. Electric charge does not, and the whole of the field-transformation argument rests on it — so it is worth asking what the evidence is.

The orbit that opens when E reaches cB

A charge in crossed electric and magnetic fields drifts sideways at E/B, looping as it goes, and the textbook leaves it there. The textbook result has a speed in it that can be made larger than light, and something has to give. What gives is the loop. As the electric field approaches c times the magnetic one, the loops swell without limit, and at E = cB the charge stops looping and is carried off for ever. Above that ratio no frame can make the electric field vanish; one can make the magnetic field vanish instead, and in that frame the charge simply runs away. The transition from a bound orbit to a run-away is decided by a single quantity every observer agrees on, E² − c²B².

The photon a fast electron throws back

Shine an infrared laser at a beam of gigaelectronvolt electrons and gamma rays come back out of the collision, each carrying seventeen million times the energy of the laser photon it started as. The electron acts as a mirror moving at nearly the speed of light, and a moving mirror multiplies the energy of light by two Doppler factors. Push the electron's energy far enough and the mirror arithmetic promises photons more energetic than the electron itself — and the photon's own momentum, which the mirror ignored, steps in to stop it.

The spin that runs ahead of its own motion

A charged particle circling in a magnetic field has two directions that turn: the direction it is moving, and the direction its spin points. If the particle's magnetism were exactly what Dirac's equation first gave — a g-factor of exactly two — the two would turn together at every speed, the spin locked to the motion as a compass needle is locked to a car. The muon's g is not quite two, and its spin creeps ahead by a fixed rate that is the same at a crawl and at 99.94 per cent of the speed of light. That constancy is a delicate cancellation between a magnetic field made stronger by motion and a rotation that exists only because the particle is being turned, and it is what made the most precise measurements of the muon possible — at one magic speed, where the electric fields that steer the particles stop affecting the spin at all.

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