A spacetime diagram at β = 0.5
At its defaults it draws a spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.
spacetime 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.
Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.
One factor, defined by an experiment rather than by a transformation
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.
A observer A stays at x = 0 and flashes a light every 1 second by their own clock. B recedes at 0.6c. The flashes are the diagonal lines; where each meets B's worldline is where B receives it. B's clock reads a longer gap between arrivals than A's read between departures, by the factor k = 2.0000, and it is the same factor between every consecutive pair — measured here off the drawn meetings rather than assumed. That single number is the whole apparatus. Nobody has written down a coordinate transformation, chosen a convention for distant simultaneity, or drawn a tilted axis; the only thing used is that light travels on the diagonals and that neither observer is special, so B's flashes reach A stretched by the same k. From it: γ = (k + 1/k)/2 = 1.2500, and β = (k² − 1)/(k² + 1) = 0.6000.
Everything special relativity says, as one ratio and its reciprocal
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 Bondi factor against speed, with its reciprocal, their arithmetic mean and their product. Four facts are visible at once and none of them needs a transformation. The product is one at every speed — approaching and receding shifts undo each other exactly — which is why a round trip to a mirror and back is a clean square and why the twin calculation is a multiplication. The mean is the Lorentz factor: at β = 0.2, k = 1.225 and (k + 1/k)/2 = 1.021, at β = 0.4, k = 1.528 and (k + 1/k)/2 = 1.091, at β = 0.6, k = 2.000 and (k + 1/k)/2 = 1.250, at β = 0.8, k = 3.000 and (k + 1/k)/2 = 1.667, at β = 0.95, k = 6.245 and (k + 1/k)/2 = 3.203. The two Doppler factors are what an observer *measures*, and the Lorentz factor is a quantity assembled from them; the usual order of presentation is the other way round, and the reversal is Bondi's contribution. As β approaches one, k runs away and 1/k goes to nothing, which is the same statement as a horizon.
Everything special relativity says, as one ratio and its reciprocal
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 Bondi factor against speed, with its reciprocal, their arithmetic mean and their product. Four facts are visible at once and none of them needs a transformation. The product is one at every speed — approaching and receding shifts undo each other exactly — which is why a round trip to a mirror and back is a clean square and why the twin calculation is a multiplication. The mean is the Lorentz factor: at β = 0.2, k = 1.225 and (k + 1/k)/2 = 1.021, at β = 0.4, k = 1.528 and (k + 1/k)/2 = 1.091, at β = 0.6, k = 2.000 and (k + 1/k)/2 = 1.250, at β = 0.8, k = 3.000 and (k + 1/k)/2 = 1.667, at β = 0.95, k = 6.245 and (k + 1/k)/2 = 3.203. The two Doppler factors are what an observer *measures*, and the Lorentz factor is a quantity assembled from them; the usual order of presentation is the other way round, and the reversal is Bondi's contribution. As β approaches one, k runs away and 1/k goes to nothing, which is the same statement as a horizon.
A spacetime diagram at β = 0.6
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.
Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.6 of the speed of light.
Everything special relativity says, as one ratio and its reciprocal
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 Bondi factor against speed, with its reciprocal, their arithmetic mean and their product. Four facts are visible at once and none of them needs a transformation. The product is one at every speed — approaching and receding shifts undo each other exactly — which is why a round trip to a mirror and back is a clean square and why the twin calculation is a multiplication. The mean is the Lorentz factor: at β = 0.2, k = 1.225 and (k + 1/k)/2 = 1.021, at β = 0.4, k = 1.528 and (k + 1/k)/2 = 1.091, at β = 0.6, k = 2.000 and (k + 1/k)/2 = 1.250, at β = 0.8, k = 3.000 and (k + 1/k)/2 = 1.667, at β = 0.95, k = 6.245 and (k + 1/k)/2 = 3.203. The two Doppler factors are what an observer *measures*, and the Lorentz factor is a quantity assembled from them; the usual order of presentation is the other way round, and the reversal is Bondi's contribution. As β approaches one, k runs away and 1/k goes to nothing, which is the same statement as a horizon.
What checks it
physicscheck asserts something about spacetime 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.
RelativityNow is a choice of slicing
Two events happening at the same time is not a fact about the events. It is a fact about who is asking, and different observers slice spacetime at different angles.
RelativityThe clock that has to slow, and why no clock can refuse
One constant speed and one right-angled triangle force a moving clock to tick slower. The argument is Pythagoras, which is what makes it inescapable rather than merely surprising.
RelativityThe diagram a ruler cannot read
A spacetime diagram is drawn on flat paper, and the geometry it depicts is not flat. The tick marking one second on a moving observer's axis sits further from the origin than the stationary observer's, by an amount that is not the Lorentz factor and means nothing at all.
RelativityThe five places infinity turns out to be
Flat spacetime goes on for ever in every direction, and it can still be drawn whole on a page. Squeeze each family of light rays with a function that keeps their order and the infinite plane becomes a diamond with every light cone still at 45°. The price is distance, which the picture no longer shows. What it shows instead is that infinity is not one place: observers slower than light all end at a single point, instants end at another, and light ends along a whole edge of its own.
RelativityThe length that depends on when, and is not really about length
A moving object is measured shorter. The contraction is real, it is not an illusion of light travel time, and it turns out to be a disagreement about simultaneity wearing a different costume.
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 pole that fits and does not fit
A twenty-metre pole is carried through a ten-metre barn at 0.866 of light speed, and both doors shut behind it. In the pole's own frame the barn is five metres long and there is plainly no room. Both accounts are correct, and the doors' closings are 57.8 nanoseconds apart in one of them.
RelativityThe quantity nobody argues about
Relativity takes away the length of a rod and the duration of an event and hands back exactly one thing in their place. Its hyperbolae are what put a scale on the tilted axes of a spacetime diagram — without which the diagram is a picture with no units on it.
RelativityThe satellites launched into the wrong orbit
In August 2014 two European navigation satellites were left on the wrong orbit by a frozen fuel line in their rocket's upper stage: an ellipse instead of a circle. For navigation they were nearly useless. For relativity they were a gift. A clock on an elliptical orbit dips deeper into the Earth's gravity and speeds up as it falls, then climbs and slows, and both effects make it lose time near the bottom of the ellipse and gain it back near the top — a wobble of 367 nanoseconds every 12.9 hours, set by the orbit alone. Watching that wobble for three years checked the gravitational slowing of clocks five times more closely than anything had in the forty years before.
RelativityThe transformation that never mentions light
Assume space and time are homogeneous, that space is isotropic, that two changes of frame compose into a third, and that the relativity principle holds. Those four leave exactly one free constant — and three possible worlds, one of them Galileo's and one of them Einstein's. Light appears nowhere in the derivation; it enters only when the constant has to be measured.
RelativityThe turns a muon makes at any energy
A muon lives two microseconds, and in a storage ring it lives longer the faster it goes, because its clock runs slow. Its orbit runs slow too: a faster muon carries more momentum, bends less in the same field and takes longer to go round. The two slowings are the same factor of gamma, so they cancel. A muon in a 1.45 tesla ring makes 431 turns in a mean life whether it is moving at nine-tenths of the speed of light or at nine nines, and the only way to more turns is a stronger magnet. The same cancellation is why the first cyclotrons stopped working near twenty million electronvolts.
RelativityThe twin who comes back younger
If motion slows a clock, and motion is relative, each twin should find the other younger — and yet when they meet, one of them has aged less. The asymmetry is not in the speed and not in the acceleration; it is in which worldline is straight.
RelativityThe twin who never turns round
The twin paradox is usually settled by pointing at the turn: the travelling twin is the one who fired the engines and came back, and so the one whose worldline is not straight. Put the twins in a universe whose space closes on itself, like the surface of a cylinder, and the traveller can come home by going straight on. Neither twin ever accelerates, both worldlines are straight, and the traveller still comes back younger. What decides it is not any motion but the shape of space, which picks out one state of motion as special — and hands every observer a way to measure it.
RelativityThe universe one explosion makes in empty space
Throw particles out of a single event at every speed below light's and let them coast. Drawn on a spacetime diagram, nothing curves and nothing expands — it is special relativity in an empty box. Yet the particles, slicing their history by their own clocks, find themselves in a universe that is the same everywhere, expanding, obeying an exact Hubble law with no speed limit, and filling a space of constant negative curvature. Edward Milne built a cosmology on this in 1933, and it shows that 'space expanding' and 'galaxies moving' are two slicings of one set of worldlines.
RelativityTwo axes, one speed, and a diagram that does the arguing
Put position across and time up, insist that light travels at forty-five degrees for everyone, and nearly every result in special relativity becomes something to read off rather than derive.
RelativityWhat the light cones alone can decide
Keep nothing of spacetime but its light cones — which events could influence which — and ask how much geometry survives. With one dimension of space, almost none: any pair of increasing stretches of the two families of light lines preserves every cone and bends every straight worldline. With two or more, almost all of it: the only maps that keep every cone are Lorentz transformations, shifts and a uniform stretch, and nothing about straightness has to be assumed.