Simultaneity at β = 0.5

At its defaults it draws simultaneity at β = 0.5. Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

simultaneity 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.

Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

By how much a synchronisation carried around a region of the rotating Earth fails to come back to itself, against the size of that region — from a metre to the whole planet, both axes logarithmic. The three horizontal lines are what three kinds of clock can resolve, and where each crosses the curve is where that clock can detect that 'now' is not a global notion: a good wristwatch at 785 thousand km, a quartz oscillator at 25 thousand km, a caesium clock at 785 km. Carried the whole way round the equator the defect is 207 nanoseconds, which is sixty metres of light travel and is the correction every satellite-navigation system applies. The effect is not small and not exotic; it is a routine engineering term, and the reason it was not an engineering term before 1955 is that nothing could measure it. A wristwatch's now is global out past the Moon; a caesium clock's reaches about the width of a large country.

The size of the largest region in which a rotating observer's synchronisation is consistent to within a given tolerance, against how fast the observer is rotating — ten decades of rotation rate and nine of size, both logarithmic, for tolerances of a picosecond, a nanosecond and a microsecond. Carrying a synchronisation round a closed path in a rotating frame does not return it to itself: the defect is twice the rotation rate times the enclosed area divided by the square of the speed of light, which is the Sagnac result established computes, and a patch is consistent when that defect is below what is being demanded. the Earth's rotation gives a nanosecond patch 785.0 km across; a record turntable gives a nanosecond patch 3.6 km across; a laboratory centrifuge gives a nanosecond patch 207 m across. The relation is inverted by bisection rather than substituted. Two things follow. The patch is enormous compared with any apparatus, which is why the effect is invisible without an interferometer. And it is finite, which is why a global 'now' is not merely inconvenient for a rotating observer but unavailable — and the honest question is not whether such an observer has a now but over what distance.

The one-way speeds of light as a function of the synchronisation convention, with the round-trip average beneath them. The outward speed is c/2ε and the return speed c/2(1−ε), so choosing ε below a half makes light faster one way and slower the other. The lower line is the harmonic mean of the two, which is the round-trip speed, and it is exactly c for every choice — computed across five conventions the round-trip times agree to 0e+0 seconds. That constancy is what is measured, in every experiment from Michelson and Morley onward. The one-way speed is not measured in any of them, and cannot be: a one-way measurement requires two clocks that agree, agreement requires a synchronisation, and every synchronisation procedure — radar, slow clock transport, a rigid rod — either assumes an answer or reproduces the assumption made in setting it up. The invariance of c is a statement about round trips, and the isotropy of the one-way speed is a convention chosen because it makes the equations simple.

Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

The size of the region in which a freely falling frame counts as inertial, against how precisely that is demanded — nine decades of tolerance and twelve of size, both logarithmic, for three gravitational environments. A falling frame is not inertial over any finite region: the tidal acceleration across it grows with its size, and the patch is where that acceleration falls below what is being asked for. Comparing the two accelerations gives an answer with almost nothing in it — the patch is the tolerance times the distance to the centre, divided by two, with Newton's constant, the mass and the speed of light all cancelling, which the figure carries out rather than takes on trust. the Earth's surface allows 3.19 mm; low Earth orbit allows 3.39 mm; a neutron star's surface allows 0.01 mm at a part in a thousand million. So the equivalence principle is exactly true at a point and approximately true over a size that can be quoted: a few millimetres on the Earth's surface for a nanoscale measurement, a few kilometres for a coarse one, and micrometres near anything compact. Every statement special relativity makes in a gravitational field is a statement about a region of that size.

physicscheck asserts something about simultaneity 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.