Illustrated Physics
physics, drawn
Concept

Clock synchronisation — where it appears

The convention by which separated clocks are said to agree, which no measurement can establish without first assuming something about the speed of light. Einstein's choice — that light takes equal times out and back — is what makes simultaneity depend on the frame, and a different choice would move the disagreement elsewhere rather than remove it.

Named by 4 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 Two corrections, opposite in sign and different in size. How fast a clock in a circular orbit runs compared with one on the ground, in microseconds a day, against the height of the orbit — with the two effects drawn apart rather than added. Being high speeds a clock up, by an amount that saturates: the potential term is bounded because there is only so much potential to climb out of. Moving slows it down, and a higher orbit is a slower one, so that term shrinks toward zero. They cancel at 3186 km — a radius of exactly 1.5 Earth radii, which follows from setting the sum to zero and contains neither G, nor the Earth's mass, nor the speed of light. At 20200 km the gravitational term is 45.7 µs a day and the speed term −7.2, leaving 38.5. Left uncorrected, that is 11.5 km of position error a day, growing without limit, from a clock that is working perfectly.

The clock that is wrong in two directions

A satellite clock loses 7.2 microseconds a day to its speed and gains 45.9 to its height. The two effects have opposite signs, different sizes and different dependence on the orbit, so there is exactly one altitude where they cancel — and 38.6 microseconds a day, left alone, is eleven and a half kilometres of position error.

relativity · Time dilation 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.

Coordinate timeRelativity of simultaneitySimultaneityEquivalence principleProper timeReference frameRotating frameBlock universeCausalityEscape velocityGravitational redshiftGravitational time dilation

All concepts