The parallelogram that will not close

Pound and Rebka’s measurement is usually presented as a confirmation: general relativity predicted a shift of two parts in a thousand million million across the Jefferson tower, and the shift was found.

Read the other way round, it is much stronger than a confirmation. It is a proof that spacetime is curved, obtained without assuming any of general relativity, from a laboratory result and a fact about parallelograms.

The argument #

Alfred Schild published it in 1960, the same year as the measurement, and it takes five sentences.

One. Two clocks are held at fixed heights in a field that does not change with time. Their worldlines are two curves, and each clock’s proper time is the length along its own.

Two. The lower clock emits a pulse of light, and later emits a second. Both travel to the upper clock. The two light paths are two more curves joining the two worldlines.

Three. The field is static, so nothing about the second pulse’s journey differs from the first’s: the geometry the second pulse crosses is the same geometry the first crossed, merely later. So the second path is congruent to the first — one is the other, translated in time.

Four. Suppose spacetime is flat. Then the four curves bound a parallelogram: two parallel vertical sides, two congruent slanted sides. Opposite sides of a parallelogram have equal length, so the proper time between the two emissions equals the proper time between the two receptions, and the two clocks tick at the same rate.

Five. They do not. Measured across twenty-two and a half metres, the rates differ by $2.5 \times 10^{-15}$.

The only assumption available to withdraw is the flatness.

The diagram every step of the argument is drawn on has very little in it: a straight line has a length, light travels along the diagonals, and a static arrangement looks the same at every moment. That poverty is the argument’s strength — Schild’s parallelogram needs no field equations and no metric, only the assumption that spacetime is flat and that the tower is not moving.

What is actually being measured #

The step that needs care is the fifth, because the phrase the clocks run at different rates is doing work.

The nucleus at the bottom emits at a frequency fixed by nuclear physics, and the nucleus at the top absorbs at the same frequency fixed by the same nuclear physics. Both frequencies are defined against the local clock, because that is the only clock available at either place. The finding is that light emitted at the bottom arrives at the top with a frequency that the top’s own standard calls too low — which is the statement that the two ends’ standards of time differ.

That is exactly the ratio of proper times between two pairs of events, and it is what step four forbids.

The account this replaces #

The usual explanation is that the photon loses energy climbing out of the potential well: it starts with $h\nu$, it has to do work $mgh$ against gravity with $m = h\nu/c^2$, and the fractional loss is $gh/c^2$.

That gets the number right and describes the wrong thing.

The energy account fails on inspection for three reasons. A photon has no mass, so $m = h\nu/c^2$ is a substitution with no justification behind it. A photon’s frequency is not a property it carries between two places — frequency is a rate, and a rate needs a clock, and the two ends have different clocks. And a “potential energy” for light presupposes a Newtonian field the argument is supposed to be establishing.

What survives of it is the number, and the number survives because the first-order weak-field answer is fixed by the equivalence principle whatever route is taken to it.

Three ways to try to save the flat picture, and why each fails #

An argument that reaches so large a conclusion from so little deserves to be attacked, and the standard attacks are worth working through because each identifies a real assumption.

“The light is bent, so the paths are not straight.” They need not be straight. The argument never uses straightness — only that the two paths are congruent to each other, which follows from staticity alone. Whatever curve the first pulse follows, the second follows the same curve shifted upward in time, and the figure closes whatever shape the slanted sides have.

“The clocks are not really at the same place, so comparing them is meaningless.” That objection is fatal to a great many naive arguments in this subject and is not fatal here, because nothing is compared at a distance. Each clock counts its own ticks between two of its own events, giving a proper time — a local, unambiguous number — and the argument compares two such numbers. Getting one number to the other end is what the light pulses are for, and they carry a count, not a rate.

“The atoms at the two ends are different, so the frequencies need not match.” They are the same species of nucleus, and the assumption that a given nucleus emits at a given frequency in its own rest frame is the assumption that physics is the same everywhere. Abandoning it to save flat spacetime is a much larger step than accepting curvature, and it is testable in its own right: the same nuclei have been compared at the two ends after being exchanged.

What survives all three is the structure of the argument: staticity gives congruence, congruence plus flatness gives equality, and the measurement denies the equality.

Where the parallelogram fails #

If spacetime is not flat, which side of the figure gives?

The rate of a clock at rest at radius r is slowed by $\sqrt{1 - 2GM/rc^2}$, which is the answer once a metric is available — and it is where the parallelogram fails. The two sides are not equal because the geometry is not flat, so there was never a parallelogram to close. The argument’s value is that it establishes the necessity of curvature without needing to know what the curvature is.

So what breaks is not the congruence of the light paths — that survives, and it is guaranteed by the staticity. What breaks is the flat-geometry theorem that congruent slanted sides force equal vertical ones. In a curved geometry a figure can have two congruent sides and two unequal ones, and that is precisely what curvature means for a figure of this shape.

Proper time is the length of a worldline, and lengths are what a geometry is a statement about. So Schild’s argument is a statement about lengths that do not add up in a flat geometry — which is exactly the form an argument for curvature has to take, and why the conclusion is so much stronger than the modest apparatus suggests.

How big the departure is, and why nobody noticed for so long #

The measurement is a part in $10^{15}$, and it is worth asking why a discrepancy that small counts as a proof of anything rather than as a systematic error.

The answer is that the effect is not being compared against zero with an error bar. It is being compared against a number — $gh/c^2$, computed from the local gravitational acceleration and the height of a shaft, with no adjustable parameter — and the agreement was ten per cent in 1960 and one per cent in 1964. A systematic error mimicking a predicted number to one per cent, at a size fixed by nothing the experimenters controlled, is not a plausible account.

The reason it could not have been found earlier is instrumental rather than conceptual. Before 1958 there was no way to compare two frequencies at a part in $10^{15}$, because every emission line was broadened by the recoil of the emitting atom and by its thermal motion, both of which swamp the effect by many orders of magnitude. The Mössbauer effect removed both at once: a nucleus locked into a crystal lattice recoils as part of the whole crystal, so the recoil energy is divided by Avogadro’s number and disappears, and the line is left at its natural width of a part in $10^{12}$.

That is the pattern behind most confirmations of this kind. The prediction had been on paper since 1907; what arrived in 1958 was a line narrow enough to measure it with, and it arrived from solid-state physics rather than from anybody looking for a test of gravity.

What the argument does not deliver #

It establishes that no flat spacetime can contain two static clocks with a redshift between them. It does not say what the geometry is, how much curvature there is, or what produces it. Those need a field equation, and Schild’s argument is entirely silent about them.

What a uniform acceleration cannot imitate is the tidal term — a second derivative of the field, and what curvature actually is. That is what this argument does not deliver: it establishes that a flat geometry is inconsistent with a static gravitational field, and it says nothing whatever about how much curvature there is or what equation determines it. Necessity is all it gives, and necessity is what it was built for.

That division of labour is worth naming, because the two halves of general relativity are usually presented as one. Geometry — that spacetime is curved, that free fall follows its straightest lines, that clocks measure lengths along worldlines — is forced by measurements of this kind. Dynamics — that the curvature is related to the stress–energy by a particular equation with a particular constant — is a separate proposal, and one that could have been wrong while the geometry stayed right.

The Shapiro delay measures the same geometric fact at a larger scale, with the clock and the light exchanging roles: there a signal takes longer to cross a region than flat space allows, and here two signals take different times to climb the same tower. Both are the same metric coefficient, measured in two arrangements.

The flat-space theories the argument was aimed at #

Schild was not attacking a straw man. In the 1950s there was serious work on treating gravity as a field on a flat background — a tensor field propagating on ordinary Minkowski spacetime, in the way that electromagnetism is a vector field propagating on it — and the appeal is obvious: the machinery of field theory is built for flat spacetime, and quantising a field on a fixed background is a solved problem in a way that quantising a geometry is not.

The argument in this essay is aimed exactly there, and it lands. If the spacetime is literally flat, with the ordinary parallelogram theorem holding on it, then two static clocks cannot disagree — whatever field is propagating on top of it and whatever that field does to rods and clocks. The measured redshift therefore rules out a flat spacetime in which clocks measure the flat metric’s intervals.

The escape, and it is an honest one, is to let the field affect the clocks. If the tensor field couples to matter in such a way that every rod and every clock is influenced by it, then what a clock measures is not the flat background’s interval but a modified one — and the modification is exactly a metric. Push that through consistently, requiring the field to couple to its own energy as well as to everybody else’s, and the resulting theory turns out to be general relativity, with the flat background surviving as an unobservable scaffolding that no measurement can reach.

That result — worked out in the 1960s and 1970s and usually credited to Deser’s version of it — is the proper resolution and it does not weaken Schild’s argument. What it says is that a flat-spacetime theory of gravity either contradicts the measurement or becomes the curved theory with an extra unobservable structure attached. Either way the geometry that clocks and rulers report is not flat.

The episode is a good example of what an experimental result of this kind is for. It does not select a theory; it eliminates a whole style of theory, and it does so with an argument short enough to check.

The centimetre that shows it now #

The measurement Schild used was at the edge of what was possible in 1960 and is now routine, and the progress is worth a paragraph because it changes what the effect is used for.

Pound and Rebka needed twenty-two metres of tower and the narrowest nuclear line available, and got ten per cent. Optical atomic clocks — which count the oscillations of a laser locked to a narrow optical transition rather than a microwave one — reach fractional accuracies below a part in $10^{18}$, and the gravitational shift at that level corresponds to a height difference of about a centimetre.

The demonstration was made in 2010 with two aluminium-ion clocks, one raised by a third of a metre relative to the other, and the predicted shift was resolved cleanly. It is the same experiment as Pound and Rebka’s with the tower replaced by a laboratory jack.

Two things follow from that. The first is that the effect has stopped being a test and become a systematic: any comparison of two clocks anywhere on Earth has to state their heights, because a centimetre of elevation difference is a measurable frequency difference, and the definition of what “the same time” means across a laboratory now requires a survey.

The second is that Schild’s argument can be run in a room. The proof that spacetime is not flat needs a non-zero redshift between two static clocks, and the tower has become a bench — which is an unusually direct route from an apparatus anybody can point at to a conclusion about the geometry of the world.

Why both derivations give the same number #

There is a small puzzle in this essay that is worth resolving explicitly. The energy account is dismissed as describing the wrong thing, and it gets the right answer; the equivalence-principle account is endorsed, and gets the same answer. Why should a wrong description agree?

Because at first order neither derivation contains any physics beyond the equivalence principle, and the equivalence principle fixes the answer.

The energy argument uses $E = h\nu$, a mass $h\nu/c^2$, and a work $mgh$ — and the middle step is the equivalence principle in disguise, since assigning a gravitational mass equal to the inertial one is exactly what that principle asserts. The Doppler argument uses the indistinguishability of a static field from an acceleration, which is the same principle stated directly. Both are therefore computing $gh/c^2$ from the same input, by different routes, and neither could have got anything else.

What separates them is what happens beyond first order and what each says is happening. The Doppler argument generalises: it can be carried through in a properly curved geometry and gives the exact metric factor. The energy argument does not: the “mass” of the photon has no meaning to carry forward, and the “potential” belongs to a Newtonian field the exact theory does not have.

That is the useful test of a derivation whose answer is right. Ask what it says when the small parameter is no longer small — and if it stops making sense rather than becoming difficult, its agreement at first order was a consequence of the constraint rather than of the reasoning.

What it costs #

Staticity is essential and is an idealisation. The Earth rotates, the Moon and Sun move, and the field in a tower is not exactly time-independent. The corrections are far below the effect being measured, and the argument would fail entirely without staticity — in a time-varying field the two light paths are not congruent and the parallelogram never had a reason to close.

The clocks must be at fixed heights. Two clocks in relative motion have a redshift for kinematic reasons that has nothing to do with geometry, and the whole point of the shaft experiment is that neither end moves.

And a “flat spacetime” has been assumed to mean one with the ordinary parallelogram theorem. That is exactly what flatness means, so the assumption is a definition rather than a hidden step — but it is worth saying, because the argument reads as though it were about drawings and is in fact about which theorems the geometry supports.

The extreme version of the same disagreement has the ratio of rates diverging rather than differing in the fifteenth decimal place. That is what it costs to take the argument seriously: the effect Schild’s parallelogram establishes as nonzero is the same one that becomes unbounded at a horizon, and there is no version of the theory in which it is nonzero here and bounded there.

Where the model stops #

Nothing here is a measurement of curvature. The redshift measures a difference in a metric coefficient between two heights, which is a first derivative of the field. Curvature is a second derivative, and a uniform field — one whose gradient is the same everywhere — produces a redshift and no curvature at all. The resolution is that a strictly uniform static field over an extended region is not something flat spacetime can contain either, which is what the argument shows; but the quantity measured and the quantity called curvature are not the same quantity.

The Pound–Rebka result is not the strongest test. Gravity Probe A in 1976 flew a hydrogen maser to ten thousand kilometres and confirmed the shift to seventy parts per million; optical clocks now resolve the shift over a height difference of a centimetre. What has not changed is the argument, which needed the effect to be nonzero and nothing more.

And the whole thing is first order in the field. The weak-field expression $gh/c^2$ is the first term of a series; near anything compact the exact metric is needed and the “tower” language stops applying.

A horizon has a temperature containing the same metric coefficient, taken to the end. That is where this model stops: the parallelogram argument is entirely classical and establishes that clocks at different heights disagree, and following the same coefficient to a horizon produces a quantity — a temperature — that no classical argument could have predicted.

What the picture cannot show #

The hero figure draws a parallelogram and says it is impossible, which is an awkward thing for a drawing to do. What is on the page is a flat-spacetime diagram — a piece of the plane, with the ordinary Euclidean parallelogram theorem holding on it — and the argument is that no such diagram can represent the situation. So the figure is a picture of the false hypothesis rather than of the world, and its correctness as a drawing is what makes it a refutation.

Nor can any diagram of this kind show the curvature that replaces it. A curved spacetime drawn on a flat page acquires coordinates, and every coordinate choice makes some feature look like a property of the geometry when it is a property of the labelling. The one honest thing a diagram can carry is a length, and lengths on a curved surface drawn on paper are exactly what a flat page cannot represent.

Where this ladder goes next #

Two rungs stand on gravitational-redshift. The first derived the shift from a photon and a conservation law and measured it in a lift shaft. This one runs the implication backwards and finds that the same measurement, taken as given, is already incompatible with a flat spacetime.

The habit worth carrying away is about the direction of an argument. A prediction confirmed is usually also a premise refuted, and the refutation is often the stronger statement. Deriving the redshift from general relativity makes it a consequence; deriving the impossibility of flatness from the redshift makes the whole geometrical picture a requirement rather than a proposal, and it does so with a measurement any university could repeat.

What is left on this ladder is the coefficient’s other half. The metric that fixes clock rates also fixes rulers, and the two do not scale together — which is why the geometry of space around a mass has a circumference that is not $2\pi$ times its measured radius, and why a shell of fixed area encloses more volume than Euclid allows.