Illustrated Physics
physics, drawn
Relativity

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.
16 min read 6 figures Who is measuringWhat happens at the edge

Assumes: The ring where the two beams disagree · The speed that cannot be measured one way

Three things have already been taken away. Now is a choice of slicing and different observers slice differently. Which came first is sometimes not a fact at all, and the part that is a fact is the causal part. A rotating observer has no consistent global slicing, because a synchronisation carried around a closed path does not come back to itself. And the one-way speed of light is partly a convention, so even a single inertial observer’s now is not wholly determined by anything measurable.

What is left is not nothing, and the thing that is left is a size.

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.
Fig. 1 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. Where each clock’s resolution crosses the curve is where that clock can detect that ‘now’ is not global: a caesium clock at 785 kilometres, a quartz oscillator at twenty-five thousand, a wristwatch out past the Moon. Round the whole equator the defect is 207 nanoseconds.

The patch, and how to compute it

Take the rotating case, since it is the one where a global answer genuinely does not exist.

Carrying a synchronisation round a loop in a rotating frame accumulates a defect of 2ΩA/c22\Omega A/c^2, with AA the enclosed area. That is the Sagnac result, and that result is computed and left there: the conclusion is that no global slicing exists, which is correct and is a statement about a limit rather than about anything an observer has.

Turn it round and ask a different question. Given a tolerance ε\varepsilon — how much disagreement about “now” the purpose in hand can accept — how large a region is consistent? The answer follows at once:

2ΩL2/c2<ε⟹L<cε/2Ω2\Omega L^2/c^2 < \varepsilon \quad\Longrightarrow\quad L < c\sqrt{\varepsilon/2\Omega}

and that is a patch. Inside it a rotating observer can synchronise clocks, call the result “now”, and be wrong by less than ε\varepsilon anywhere in it. Outside it the same procedure gives answers that depend on which way round the clocks were carried.

The two facts about that expression are both worth having. The patch is enormous by ordinary standards — hundreds of kilometres on the rotating Earth for a nanosecond, which is why nobody noticed the effect for three centuries. And it is finite, which is why the statement “a rotating observer has a now” is not simply true: it has one, over a computable distance, and over a larger distance it has several that disagree.

How large a rotating observer's now is. 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.
Fig. 2 The patch’s size against how fast the observer rotates, over ten decades, for three tolerances. The Earth’s rotation gives 785 kilometres to the nanosecond; a record turntable gives 3.6 kilometres; a laboratory centrifuge at ten thousand revolutions a minute gives 207 metres. The patch shrinks only as the square root of the rotation rate and of the precision, which is why the effect went from unmeasurable to unavoidable in one step.

The square roots are what make the history intelligible. Between a pendulum clock and a caesium clock the resolution improved by about nine decades, which shrank the patch by four and a half — from a distance far larger than the Earth to a distance smaller than a continent. The effect did not get bigger; the patch got smaller than the apparatus.

Two hundred and seven nanoseconds

The hero figure’s largest number is the one with an engineering consequence.

Carry a clock synchronisation once around the equator, eastward, and it fails to close by 207 nanoseconds. That is sixty metres of light travel, and any system that determines position by timing signals has to account for it — which every satellite-navigation system does, under the name of the Sagnac correction.

The way it is handled is instructive about what this essay is claiming. The correction is not a fudge applied to a rotating calculation; the calculation is done in a non-rotating frame and then translated, precisely because a rotating frame has no consistent global simultaneity to do it in. The Earth-centred inertial frame is the one the arithmetic happens in, and the rotating frame in which everybody lives is a derived thing with a stated patch.

So the abstract conclusion about a rotating observer is load-bearing in a system a billion people use. If a rotating observer’s now were merely inconvenient rather than unavailable, the natural engineering choice would be to work in it and accept a small error; the reason nobody does is that the error is not small and is not an error.

There is a related and older measurement. Clocks flown around the world in 1971 came back disagreeing with the ground, and the comparison of an eastward and a westward flight separates the Sagnac term from the time-dilation and gravitational terms: the two flights have opposite Sagnac contributions and the same gravitational ones. That separation is the reason the experiment was flown both ways, and it is a direct measurement of the quantity the hero figure plots.

What a patch is not

It is worth being precise about what has and has not been recovered, because “local simultaneity” is easy to over-read.

A patch is not a preferred slicing. Inside it, an observer’s synchronisation is consistent to within ε\varepsilon — and so is a different observer’s, differently, and the two still disagree about the order of spacelike-separated events by the ordinary amount. The patch is about self-consistency, not about agreement between observers.

It is not a boundary. Nothing happens at LL; the defect grows smoothly, and the patch is wherever the tolerance was set. Two people with different purposes have different patches in the same room.

And the conventional part does not go away inside it. The one-way speed of light is still a convention at any distance, so the patch is the region in which a chosen convention is self-consistent rather than the region in which the convention becomes unnecessary. Those are different statements and the second is not true anywhere.

The two speeds move and their average does not. 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.
Fig. 3 The part that no patch removes, from the synchronisation-convention argument: the two one-way speeds against the synchronisation convention chosen, with the round-trip speed beneath them. Every choice is consistent with everything measurable, and the round trip is exactly c for all of them. Shrinking the region does not narrow that family — the convention is free at any distance, and what a patch buys is that one choice of it closes on itself.

What the patch does is convert a statement of impossibility into a number, and the number is what any actual use of the concept needs.

Where else a patch has to be quoted

The rotating case is the one needed here, and the construction is worth running over three others because each has the same shape and a quite different size.

Simultaneity at β = 0.35. 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.
Fig. 4 The inertial case for comparison: two observers’ slices through the same spacetime, tilted relative to each other by their relative motion. Here there is no patch — each observer’s slicing is globally consistent, extends everywhere, and disagrees with the other’s by the same amount at every distance. A patch appears only when an observer’s own slicing fails to close on itself.

A spacecraft. An orbiting observer is both rotating about the Earth and in a gravitational field, and the two effects give patches of quite different sizes: the orbital rotation rate is 10−310^{-3} radians per second, so the rotational patch is about six kilometres to the nanosecond, and the gravitational one is millimetres to a part in a thousand million. Which matters depends entirely on what is being measured.

A laboratory on the surface. The Earth’s rotation gives 785 kilometres and its gravity gives three millimetres, and the two answer different questions: the first is about synchronising clocks in different buildings and the second is about whether a falling apparatus counts as inertial.

And a particle accelerator. A bunch going round a ring at nearly the speed of light is a rotating frame with Ωr/c\Omega r/c of order one, which is precisely where the treatment in this essay fails — and which is why accelerator physics is done in the laboratory frame with the particles’ own proper time carried along each trajectory, rather than in any frame attached to the ring.

The pattern is that the patch has to be quoted for the quantity being measured and not for the observer, and two experiments in the same room can have patches differing by nine decades.

The gravitational patch, whose answer contains almost nothing

The same question can be asked of a gravitational field, where the corresponding claim is the equivalence principle: a freely falling frame is inertial.

The patch a falling frame is inertial in. 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.
Fig. 5 The size of the region in which a freely falling frame counts as inertial, against how precisely that is demanded, for three gravitational environments. The answer 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. On the Earth’s surface that is 3.2 millimetres at a part in a thousand million.

A falling frame is inertial at a point and not over any finite region, because the field is not uniform: the tidal acceleration across a region of size LL is 2GML/r32GML/r^3, and it is that which spoils the equivalence.

Comparing it with the local gravity GM/r2GM/r^2 gives something unexpected:

atidalg=2Lr<δ⟹L<δr2\frac{a_\text{tidal}}{g} = \frac{2L}{r} < \delta \quad\Longrightarrow\quad L < \frac{\delta r}{2}

Newton’s constant has cancelled. So has the mass. So has the speed of light. The size of an inertial patch is the fractional tolerance times the distance to the centre of the field, divided by two, and it contains no constant of nature at all.

That cancellation has a consequence which sounds wrong and is right. A patch near a neutron star is small not because its gravity is strong but because the star is small: at a hundredth of a millimetre for a part in a thousand million, against three point two millimetres at the Earth’s surface, the ratio is exactly the ratio of the radii. A frame falling towards a supermassive black hole has a patch of kilometres near its horizon, because the horizon is a long way from the centre — and an observer falling through it notices nothing locally for exactly that reason.

What the two patches have in common

The rotating patch goes as ε\sqrt{\varepsilon} and the gravitational one as δ\delta, so they are not the same function. What they share is the shape of the argument, and it is worth stating because it generalises.

In each case a symmetry is exact at a point and is being extended over a region. Extending it produces an error that vanishes with the region’s size at some order — second order for the rotation, first for the tidal field — and the patch is where that error falls below the tolerance. A local symmetry is a symmetry plus a size, and the size is not a detail: it is what decides whether an argument that assumes the symmetry can be used on a given apparatus.

The same structure appears wherever an idealisation has been pushed past its range. A rigid body is a body over a length short compared with the time a signal takes to cross it. A thin lens is a lens over an aperture small compared with its focal length. An incompressible fluid is a fluid over a flow slow compared with its sound speed. In each case the honest form of the statement is a quantity with a size attached, and in each case the size is usually left out.

Going all the way round leaves the clocks 2.52 femtoseconds out. Accumulated offset between neighbouring clocks round the rim of a turntable of radius 1.5 m turning at 8 rad/s, with 16 stations, each pair synchronised by Einstein's own rule — send a pulse each way and call the arrival times equal. Each pair agrees perfectly with its neighbour, by 0.157 femtoseconds per step, and the sum round the whole loop is 2.517 femtoseconds rather than zero. That total is 4AΩ/c², the same quantity the two beams measure. So there is no way to assign a time to every point of a rotating platform such that neighbouring clocks agree: the procedure that works locally everywhere fails to close, and simultaneity on a turntable is not a global notion at all.
Fig. 6 The mechanism, from the rotating-ring argument: clocks set pairwise around a rotating ring, and the discrepancy that appears on coming back to the first. The defect accumulates smoothly with the enclosed area, which is what makes a patch a matter of tolerance rather than of a boundary — and it is the same quantity this whole essay has been putting numbers on.

The measurement that has closed on the millimetre

The gravitational patch has stopped being a thought experiment, and the reason is worth a section because it dates this essay.

Optical lattice clocks now compare to a fractional frequency of about a part in 101810^{18}. A clock at the Earth’s surface runs faster than one a centimetre higher by about a part in 101810^{18}, because the rate depends on the potential — so two such clocks a centimetre apart on the same bench are measurably out of step, and the difference has been measured.

That puts the gravitational patch at the centimetre for anything using such clocks, and it has an immediate consequence for the concept of simultaneity: a network of optical clocks in different buildings cannot be synchronised at all without knowing each one’s height to a centimetre, and the height is what the comparison measures. The technique has a name, chronometric levelling, and it is a way of surveying a geoid with a clock rather than with a spirit level.

So the patch is not merely computable but currently binding. An experiment that assumes a shared now across a laboratory bench is making an error of a part in 101810^{18}, which was nothing in 1990 and is a systematic today. The history of this whole subject is the same sentence repeated at smaller distances, and the distance is now smaller than the apparatus.

What survives all four demolitions

Setting those results beside each other, it is worth saying what is left standing, because the accumulated demolition can read as more sceptical than it is.

The interval is not a convention and not local. The quantity nobody argues about is invariant, exact, global, and independent of every choice in this essay. Every observer, rotating or not, accelerating or not, in a field or not, agrees about it.

The causal order is not a convention. Which of two timelike-separated events comes first is a fact, and no synchronisation convention or patch size touches it.

Proper time is not a convention. What a particular clock reads between two events on its own worldline is a measurable number with no choice in it.

What is conventional or local is the extension of one clock’s reading to distant places — and that extension is what “now” means. So the honest summary is that simultaneity is a coordinate rather than an observable, and that a coordinate can be laid out consistently over a computable patch.

Leading order, and two patches kept apart

The rotating patch is computed from the leading Sagnac term. The full expression for a rotating frame has further terms in Ωr/c\Omega r/c, and at rim speeds approaching the speed of light the whole treatment fails — the rotating frame’s coordinates become singular at r=c/Ωr = c/\Omega, and there is no consistent frame beyond it at all.

The tolerance is treated as a single number. A real measurement has a precision that depends on what is being measured and on how long for, so “the patch” is a family of sizes rather than one, and the right member of the family depends on the experiment.

The gravitational patch is computed for a spherical field and to leading order. A real field has higher multipoles, and near a rotating body the frame dragging adds a term the comparison above does not contain. The cancellation of GG and MM survives all of that in order of magnitude and the factor of two does not.

And the two patches are treated separately. An observer on the rotating Earth’s surface is in both situations at once, and the two defects add — which is exactly the accounting a satellite-navigation system performs, and which no figure here draws together.

A defect that is not a clock running wrong

The hero figure draws a defect against a size and cannot show what the defect is. It is not a clock running wrong: every clock in the region keeps perfect proper time. What fails is the bookkeeping that assigns a single number to “the time” at many places at once, and the failure appears only when the assignment is carried around a loop and compared with itself. A plot of a duration is a plot of a discrepancy in a convention, and nothing physical is drifting.

The patch figures draw sizes against tolerances as though a tolerance were a dial. In practice the precision of a synchronisation depends on the clocks, the links between them, the temperature of the cables and the history of the calibration, and most of those are not under anybody’s control to within an order of magnitude. A curve drawn against tolerance is a family of answers rather than a prediction.

And none of the figures can draw the object this whole essay is about, which is a slice. A simultaneity is a three-dimensional surface in a four-dimensional spacetime; a patch is a region of that surface over which the surface can be defined consistently; and every drawing here is a scalar summary of a geometrical fact that needs four dimensions to state and cannot be reduced to a length without losing what makes it strange.

Still open: whether the patch is the right way to think about it

The construction in this essay is a practical one and it is not the only available attitude to the difficulty, which is worth saying because the alternatives have adherents.

One position is that a patch is the whole content of locality and that physics is properly written as a theory of local quantities with the global bookkeeping treated as a convenience. That is close to how general relativity is normally taught and it accounts for the mathematics being written in terms of tensors at points and connections between neighbouring points rather than global coordinates.

A second is that the patch is an artefact of insisting on coordinates at all, and that the right formulation dispenses with them — that what exists is a causal structure, a conformal geometry and a set of worldlines, and that simultaneity never needed to be defined. Causal set approaches take that seriously, and the difficulty is that recovering ordinary geometry from a causal order alone is technically hard and has not been fully done.

A third, which is not popular and is not refuted, is that there is a preferred slicing after all — that the cosmic microwave background picks out a frame, that the universe has a definite age, and that the relativity of simultaneity is an exact symmetry of local physics sitting inside a cosmology that breaks it. Nothing in local experiment distinguishes that from the standard view, and its status is a question about what a symmetry of the laws commits one to about the world.

The habit worth carrying away is the one this whole essay is. When something turns out to be impossible, ask over what range it is nearly possible. A global now does not exist for a rotating observer, which is a clean result and is useless by itself. The patch — 785 kilometres to the nanosecond, three millimetres for a falling frame at a part in a thousand million — is what the result becomes when it is asked to do any work, and computing it takes one line from a defect that was already on the page.

Part 5 of 5

This essay is one argument about Simultaneity. The others:

Where to put the far clock's zero. Two clocks three light-seconds apart, synchronised by radar: a pulse leaves the near clock at 0, bounces off the far one, and returns at 6 seconds. The far clock must be set to some time between those, and every choice is drawn. Einstein's convention puts it at 3 — halfway — and gives the same speed of light in both directions. Any other value is equally consistent with every measurement that can be made, because everything measurable involves a round trip and the round trip takes 6 seconds under every one of them: computed here across the five conventions, the round-trip times differ by 0e+0 seconds. The lines are the resulting surfaces of simultaneity, which fan out from the halfway choice. What each choice fixes is the one-way speed of light — ε = 0.25 makes it 2.00c outward and 0.67c back, ε = 0.4 makes it 1.25c outward and 0.83c back, ε = 0.5 makes it 1.00c outward and 1.00c back, ε = 0.6 makes it 0.83c outward and 1.25c back, ε = 0.75 makes it 0.67c outward and 2.00c back — and no experiment distinguishes them, because measuring a one-way speed requires two synchronised clocks and synchronising them requires the answer. The speed that cannot be measured one way Part 4 — Every measurement of the speed of light ever made has sent it out and brought it back. Measuring it one way needs two clocks that agree, and making two distant clocks agree needs a rule about when the far one should read what — which is a choice, not a discovery. The constancy of c is a fact about round trips; its isotropy is a convention, chosen because it makes the equations simple.

Where this goes next

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. 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. 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. The 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.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Clock synchronisationCoordinate timeEquivalence principleMeasurementReference frameRelativity of simultaneityRotating frameSimultaneitySynchronisationTidal forces