Illustrated Physics
physics, drawn
Concept

Relativity of simultaneity — where it appears

The dependence of which events count as at the same time on the observer's motion, and the source of every other relativistic effect. Length contraction, the ordering of distant events and the resolution of the twin paradox are all the same disagreement about slicing, seen from different angles.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

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.

Now 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.

relativity · Simultaneity 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.

The 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.

relativity · Length contraction Simultaneity at β = 0.866. 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 pole that fits and does not fit

A twenty-metre ladder is carried through a ten-metre barn at 0.866 of light speed, and both doors shut behind it. In the ladder'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.

relativity · Length contraction Which happened first, asked of several observers. Two events on a spacetime diagram: one at the origin and one 3 light-seconds away and 1 second later, so that light leaving the first cannot reach the second. Through the second event runs a family of lines, each one the set of events some observer calls simultaneous with it; an observer moving at a fraction β of the speed of light has such a line of slope β on these axes. Where a line meets the vertical axis is the time that observer assigns to the second event. For a slow observer that meeting point is above the origin and the second event happens later; for a fast one it is below and the second event happens EARLIER. The changeover is at β = 0.3333, which is the time separation divided by the space separation, and it is a legal speed only because the separation is spacelike. So the order of these two events is not a property of the events. What every observer does agree on is that neither could have caused the other, because the two lie outside each other's light cones — drawn here as the diagonals — and that agreement is what causality rests on rather than on any shared notion of before.

Which 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.

relativity · Simultaneity One expression over 3 decades of area times rate. Sagnac time difference against the product of enclosed area and rotation rate, both logarithmic. The relation Δt = 4AΩ/c² is linear in that product — the fitted slope of the drawn points is 1.0000 — and it contains no refractive index, no shape of the loop and no position of the axis inside it. a 1 km fibre gyroscope on a 10 cm coil: 1.62e-19 s, 3.14e-5 fringes; Sagnac's own ring, 1913: 4.84e-17 s, 0.0666 fringes against a reported 0.07; a laboratory turntable at one revolution a second: 6.99e-17 s, 0.0331 fringes; Michelson and Gale, 1925: 4.50e-16 s, 0.2364 fringes against a reported 0.23. Michelson and Gale's rectangle in Illinois is the one that carries the check: 0.236 fringes predicted from its own dimensions and the vertical component of the Earth's rotation at its latitude, and 0.230 reported.

The ring where the two beams disagree

Send light both ways round a closed loop on a turntable and the two beams come back at different times, by 4AΩ/c² — an expression with no refractive index in it, no shape of the loop and no position of the axis. The same number is what a set of clocks round the rim fails to close by, which is the sharper statement — on a rotating platform there is no global simultaneity to be had.

relativity · Simultaneity How large now is, on this planet. 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.

How big now is

Three earlier arguments have established that a global now is a choice, that part of the choice is convention, and that for a rotating observer no consistent global choice exists at all. What survives is a size. Every observer has a local now, and how local is computable: on the rotating Earth it is 785 kilometres to the nanosecond, and a freely falling frame is inertial over the tolerance times the distance to the centre, divided by two.

relativity · Simultaneity

Named alongside it

The objects these essays reach for when they reach for this one.

CausalityClock synchronisationThe Lorentz transformationReference frameSimultaneityCoordinate timeInvarianceInvariant intervalLength contractionLight coneThe Lorentz factorRotating frame

All concepts