What puts a scale on a tilted axis
At its defaults it draws what puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.
interval-hyperbola 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.
A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.
A horizon 0.97 light years behind, made by nothing but the motion
The options are the ones Nothing is allowed to be rigid 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 and time up, in units where light travels at 45°, for a rocket holding a constant proper acceleration of 1 gravity. The worldline is the hyperbola x² − c²t² = (c²/a)², asymptotic to the light line it never crosses. Three light signals are drawn: one released at x = 0.55 catches up at t = 0.63, one released at x = 0 never arrives, one released at x = -0.6 never arrives. The dividing line is the asymptote itself. Everything at or behind it is permanently out of reach, and for one gravity that boundary sits 0.97 light years behind the rocket's starting point. Nothing is there — no mass, no field, no surface. The horizon is a consequence of never stopping.
The turnaround, made gentler and gentler, and the difference that does not move
The options are the ones The clock that does not feel the turn 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 round trip to a star 4 light-years away at 0.6c, with the turnaround done at nine different accelerations from a tenth of a gravity to a thousand. The upper curve is the age difference between the twins and the lower one is how much of that difference the turnaround itself contributes. At 0.1 g the turn accounts for 51 per cent of it; at 1000 g it accounts for 0.00 per cent, and it keeps falling. The total does not follow it down: it tends to 2.67 years, which is what the instantaneous-turnaround cartoon gives. So the acceleration is not what makes the twins differ. It is what makes one twin's path the bent one, and a bent path through spacetime is shorter for the same reason a bent path on a map is longer — but the *amount* is in the legs, not in the corner, and the corner's contribution can be made as small as anyone likes without the difference going away.
The same speed, seven decades of acceleration, and one clock rate
The options are the ones The clock that does not feel the turn 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 clock carried round a circle at 0.9994c, at radii from a millimetre to a kilometre. The speed is the same in every case and so is the rate: 3.464e-2 of a stationary clock's, because proper time is the integral of dt/γ and γ contains the speed and nothing else. The centripetal acceleration meanwhile runs from 7.6e+15 to 7.6e+21 times gravity. That the flat line *is* flat is an assumption, not a theorem — the clock hypothesis — and the rising curve is what a term proportional to the acceleration would look like at the smallest size that has been ruled out, ξ = 0.001. The muon storage ring at CERN held muons at this speed and about 10¹⁸ g and found their dilated lifetime agreeing with the speed-only prediction to a part in a thousand, which is where the bound comes from. Nothing derives it: the hypothesis says an ideal clock is small enough that the tidal stresses on it do not matter, and whether a real clock is ideal is a question about the clock.
The longer line on the page, 80.0% of the time lived
The options are the ones The clock that does not feel the turn passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Two worldlines between the same two events, drawn to scale. The straight one is at rest; the bent one goes out at 0.6 of the speed of light and comes back. On the page the bent line is 1.166 times as long. The time actually lived is the proper time, summed here along each drawn path as √(dt² − dx²) leg by leg: 2.0000 for the one who stays, 1.6000 for the one who leaves, a ratio of 0.8000 against the leg-by-leg closed form's 0.8000, which for a turn at the halfway point is √(1−β²). In this geometry the straight path between two events is the longest rather than the shortest, and the minus sign in the interval is the only reason why.
What puts a scale on a tilted axis
The options are the ones The invariant that survives a boost 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 spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.
What checks it
physicscheck asserts something about interval-hyperbola 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.
Nothing is allowed to be rigid
A rigid body would move its far end at the instant its near end was pushed, which is a signal at infinite speed. Relativity forbids it — not approximately, and not as a limit that a hard enough material approaches. What follows is a ceiling on how stiff matter may be, and that ceiling caps the mass of every neutron star.
RelativityThe clock that does not feel the turn
Proper time is the integral of dt over gamma, which presumes that a clock's rate depends on its speed and on nothing else — not on its acceleration, not on how long it has been accelerating. That is an assumption about clocks rather than a theorem about spacetime, and the twin result is empty without it.
RelativityThe invariant that survives a boost
Energy and momentum are both answers to the question "how fast is it going, and according to whom". One combination of them is not, and that combination is the mass — which is why two photons of 511 keV can be a thing of mass 1.022 MeV or a thing of no mass at all, depending only on the angle between them.
RelativityThe longest way round is the shortest clock
Of all the routes between two events, the one with no acceleration in it carries the most time on its own clock. That is the opposite of the Euclidean statement about straight lines, it comes entirely from one minus sign, and in a gravitational field it is why a thrown ball follows the path it does.
RelativityThe orbit that ages less than a throw
A clock in orbit and a clock thrown straight up leave the same point at the same moment and meet there again one period later. Both fall freely the whole way, so both follow paths of stationary proper time — and the thrown clock comes back 4.1 microseconds older. Even a clock held still by a rocket, which is not falling at all, beats the orbit. Free fall picks out a path that is stationary, not one that is longest.
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 ship that never arrives at c
Accelerate at one gravity and never stop. The speed creeps toward light and never reaches it, and meanwhile the galactic centre is twenty shipboard years away and Andromeda twenty-nine. What makes the journey survivable is that rapidity has no ceiling; what makes it impossible is that the fuel goes as the exponential of the same quantity.
RelativityThe string that breaks between two rockets
Two rockets a metre apart, given identical acceleration programmes, stay a metre apart in the laboratory for ever. A string tied between them breaks anyway. Nothing pulls on it, nothing in the laboratory moves relative to anything else, and the string is stretched — because the distance it has to span is measured on the rockets' slices of simultaneity and not on the laboratory's.
RelativityThe temperature of an acceleration
Empty space is empty for an observer who is not accelerating. For one who is, the same state of the same field is a thermal bath at a temperature proportional to the acceleration — and the constant of proportionality is 4 × 10⁻²¹ kelvin for every metre per second squared, which is why nobody has felt it. What the effect changes is not what can be measured but what a particle is.
RelativityThe wall of silence behind a rocket that never stops
Hold a constant acceleration and the worldline is a hyperbola asymptotic to a light ray — so there is a light ray that never catches it. An observer who never stops accelerating has a horizon a distance c²/a behind, made by nothing but the motion, and at one gravity it sits 0.97 light years back. Nothing is there. No mass, no surface, no field.
RelativityWhich came first, and who decides
Two events far apart can happen in either order, depending on who is asked, and both answers are correct. That is not a loophole in causality but the reason causality survives at all — because the pairs whose order is negotiable are exactly the pairs neither of which could have caused the other.