Composing a boost with a speed, and never passing one
At its defaults it draws composing a boost with a speed, and never passing one. The speed one observer measures when a body moving at v is seen from a frame already moving at u, for u = 0.4, 0.6, 0.9, 1 times the speed of light. Every curve ends at one and none crosses it. The straight dashed line is the Galilean answer, u + v, which reaches 1.4c and is wrong. The flat line at the top is light: composing c with anything gives c back.
velocity-addition 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 speed one observer measures when a body moving at v is seen from a frame already moving at u, for u = 0.4, 0.6, 0.9, 1 times the speed of light. Every curve ends at one and none crosses it. The straight dashed line is the Galilean answer, u + v, which reaches 1.4c and is wrong. The flat line at the top is light: composing c with anything gives c back.
Composing a boost with a speed, and never passing one
The options are the ones Everything from an exchange of pulses passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The speed one observer measures when a body moving at v is seen from a frame already moving at u, for u = 0.4, 0.6, 0.9, 1 times the speed of light. Every curve ends at one and none crosses it. The straight dashed line is the Galilean answer, u + v, which reaches 1.4c and is wrong. The flat line at the top is light: composing c with anything gives c back.
In rapidity, boosts simply add
The options are the ones Everything from an exchange of pulses passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Velocity as a fraction of the speed of light, against rapidity. Composing 0.75c with 0.75c means adding their rapidities, 0.973 and 0.973, to get 1.946 — and the velocity at that rapidity is 0.9600c, which is what the velocity addition formula gives. The curve flattens towards one, which is why no amount of adding reaches it.
Composing a boost with a speed, and never passing one
The options are the ones Speeds that refuse to add, and the quantity that does passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The speed one observer measures when a body moving at v is seen from a frame already moving at u, for u = 0.4, 0.6, 0.9, 1 times the speed of light. Every curve ends at one and none crosses it. The straight dashed line is the Galilean answer, u + v, which reaches 1.4c and is wrong. The flat line at the top is light: composing c with anything gives c back.
In rapidity, boosts simply add
The options are the ones Speeds that refuse to add, and the quantity that does passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Velocity as a fraction of the speed of light, against rapidity. Composing 0.75c with 0.75c means adding their rapidities, 0.973 and 0.973, to get 1.946 — and the velocity at that rapidity is 0.9600c, which is what the velocity addition formula gives. The curve flattens towards one, which is why no amount of adding reaches it.
How fast something looks as it moves across the sky
The options are the ones Speeds that refuse to add, and the quantity that does passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The apparent transverse speed of a source, in units of the speed of light, against the angle between its motion and the line of sight, at β = 0.8, β = 0.95, β = 0.99. Every curve rises above one over a range of angles, reaching 1.33 at 36.8°, 3.04 at 18.4°, 7.02 at 8.1° — and those maxima are γβ at arccos β, found by searching the drawn curves rather than put into them. Nothing is moving faster than light. What has happened is that the source has come closer between the two observations, so the second flash had less far to travel and arrived sooner than it would have done; dividing the transverse distance by the interval between arrivals therefore gives too large a speed. Below β = 1/√2 no angle produces the illusion at all, so seeing it is a measurement: it puts a floor under the speed and a ceiling on the angle at once.
What checks it
physicscheck asserts something about velocity-addition 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.
Everything from an exchange of pulses
Send a flash every second and ask how often the far observer receives them. That one measured ratio generates time dilation, the composition of velocities and the twin result, with no coordinate transformation written down anywhere and no convention chosen about what "at the same time" means far away.
RelativitySpeeds that refuse to add, and the quantity that does
Run at half the speed of light, throw something forward at half the speed of light, and the result is not the speed of light. It is four-fifths of it, and there is a variable in which the arithmetic is still simple addition.
RelativityThe brightness that is not the same for everyone
A moving source is not merely shifted in colour. Its light is concentrated forwards, so the same lamp is enormously brighter seen coming than seen going — by the fourth power of one number, which for a jet at ninety-nine per cent of light speed is a factor of a thousand million. The reason is that only one combination of intensity and frequency is the same for every observer, and everything else follows from it.
RelativityThe contraction no photograph shows
Every textbook picture of a relativistically moving object drawn between 1905 and 1959 showed it squashed. A camera would have shown it turned. The contraction is entirely real in the measurement that defines it, and a photograph is not that measurement.
RelativityThe drag that was only an addition
Light in moving water is carried along by it, but only partly — by a fraction of the water's speed that depends on the refractive index in a way nobody could account for. Fresnel invented the coefficient to save a theory, Fizeau measured it in 1851, and it sat unexplained for half a century. It is the first term of the relativistic velocity addition and nothing else.
RelativityThe motion that measures faster than light
Take two photographs of a jet a year apart, measure how far a blob moved across the sky, divide by a year, and the answer can be seven times the speed of light. Nothing has broken. The blob came closer between the two pictures, so the second flash had less far to travel and arrived early, and the interval between arrivals is not the interval between departures.
RelativityThe push that does not point where the body goes
Newton's second law survives relativity in the form F = dp/dt and in no other. Written as F = ma it fails outright, and not merely by a factor — at high speed a body pushed at forty-five degrees accelerates at eighty, because the same force is divided by γ³ along the motion and by γ across it.
RelativityThe space that speeds live in
Speeds do not add, and the reason is that the set of all possible velocities is not a flat space. It is a hyperbolic plane of curvature minus one in rapidity — and the rotation two boosts leave behind is exactly the area of the triangle they make in it.
RelativityThe turn that two pushes leave behind
Two boosts in different directions do not compose to a boost. The product carries a rotation, so a frame carried once round a closed path comes back turned — and the size of that turn was the factor of two standing between the calculated and the measured splitting of a spectral line.