Fun With Science / Globe Deconstruction / Q2 · page 45 / Draft
Q2 of the Globe Deconstruction Review: the kilometre-long water trough is not an experiment about water. It is an experiment about the floor it sits on.
He proposes a temperature-controlled water channel a kilometre long, and asks why nobody has built it. It is a good question, better than most in the genre, and it deserves the answer it has never been given: the water is not the instrument. Whatever a kilometre of water does, it does because of the shape of the floor underneath it — and the shape of that floor is decided before a drop is poured, by whatever you used to survey it. The whole experiment lives in that decision.
“Bending hydrostatics” inverts the law it citesThe result is set by the floor, not the waterLIGO ran it at 4 km and recorded 1.25 mNobody has run his warehouse as a discriminating testHis statement of hydrostatics is four-fifths textbookHis diagram and his arithmetic agree with each otherThe experiment can be made to work
Where this lands
Hydrostatics does not say water is flat. It says a free surface is perpendicular to the net body force — which is why a tilted glass keeps a level surface, and why Archimedes’ second proposition puts the sea on a sphere. A curved ocean is not gravity overpowering hydrostatics; it is hydrostatics. So the question as phrased asks whether a law can overpower itself.
And the factual half of the question — that nobody has shown this in a controlled environment — does not survive either. LIGO built the straight kilometre four times over, had to account for 1.25 m of departure from level across each arm, and reconciled a straight datum against a level one down the same tubes. Hydrostatic levelling systems, meanwhile, are his water channel exactly, in daily use at accelerator laboratories, resolving the ground flexing under Earth tides.
Underneath the phrasing there is a real experiment, and it turns on one thing only: was the floor built level, or built straight? Level and straight are the same line on a flat Earth, and on a globe they part by 78.5 mm over one kilometre — growing with the square of the distance, so four times that over two. Build level and nothing happens in either world. Build straight and only the globe puts a wedge in the trough. Everything else on this page — capillarity, air pressure, temperature — is second order beside that, and worth one bracket so the wave-off is checkable: the meniscus is confined to within a few capillary lengths of the walls (2.7 mm for water) and does not move the mid-channel surface at all, while a one-degree end-to-end temperature difference in connected water 0.3 m deep tilts the free surface by about 60 µm. Parts per thousand of a 78 mm signal.
Two words this page needs you to keep apart.
LEVEL — perpendicular to gravity at every point; following the equipotential. What a spirit level, a plumb line, or still water gives you.
STRAIGHT — a geometric line. What light in a vacuum gives you.
On a flat Earth these are the same surface. On a globe they part by 78.5 mm over a kilometre, and by four times that over two. That gap is the entire experiment. Everything below — the four panels, LIGO, the five stations — is about how you get hold of one of them without borrowing the other.
Miller states the law carefully and, for four clauses out of five, correctly: water cannot resist shear stress, its molecules slide freely, Pascal’s law transmits pressure equally in all directions, and P = ρgh makes pressure rise linearly with depth. All true, and all of it describes what happens inside the fluid. None of it says which way the top is.
The clause that fixes the shape of the surface is the one that is missing, and it is two lines:
In equilibrium, ∇p = ρg. On a free surface the pressure is constant — it is simply atmospheric — so ∇p is perpendicular to that surface. Therefore g is perpendicular to it.
The free surface of a fluid at rest is perpendicular to the net body force. Always.
In words, for anyone who would rather not read the calculus: pressure in still water changes only as you move up or down along gravity, so a surface of constant pressure — which is what the top is — has to run square across gravity. Equivalently: it is an equipotential. That is the whole of what hydrostatics says about the shape of the top, and it says nothing whatever about flat.
Which breaks the premise the chapter opens with — “Does water always take on the shape of its container? If you answered yes, you are correct.” The free surface never takes the shape of its container, and the demonstration takes one second. Half-fill a glass and tilt it. The water does not tilt with the glass; the surface stays where it was while the container rotates underneath. In a conical flask it is sharper still — the surface is a flat disc that changes diameter as the level moves and never once follows the sloping wall. Water conforms to its vessel at the sides and the bottom. At the top it ignores the vessel and answers to the field.
So a free surface that curves over a kilometre is not gravity overpowering hydrostatics. The flat surface is the special case — what you get when the field happens to be parallel. This is not a modern reframing invented to dodge the question; it is Proposition 2 of the treatise that founded the subject. Archimedes, On Floating Bodies, Book I, around 250 BC:
“The surface of any fluid at rest is the surface of a sphere whose center is the same as that of the earth.”
Second proposition, first book, derived from a postulate about how the parts of a fluid press on one another — a postulate in which they press toward a centre, which a sharp critic will say bakes the sphere into the premises. Partly fair, and the point survives it: hydrostatics has never contained a law that water is flat. The founding text of the subject makes the shape of a free surface a consequence of the force field, and the second thing it derives is a sphere. No telescope, no satellite, no agency. “The study of hydrostatics stands in complete opposition to this hypothesis” is precisely inverted: the study of hydrostatics opens by asserting it.
To be fair to the phrase rather than the argument: “bending the laws of hydrostatics” is a reasonable thing to write if you carry hydrostatics as a rule that says water is flat, which is how it is usually taught and how nearly everyone remembers it. It is not that rule. It is a rule about perpendicularity, and the flatness people remember is the perpendicular answer in a field that happens to be uniform.
Two ways to build the floor, two possible worlds, four outcomes. Curvature is exaggerated in all four drawings; at true scale every line here would be indistinguishable from straight.
Three of the four produce nothing, and the one that produces a signal is the one where the floor was built straight on a globe. That is the entire experiment. The water contributes no information of its own: in all four panels it does the identical thing, which is to sit on the equipotential. It is the floor that changes.
Panel C is not hypothetical. It has been built — at four times the span he asks for, by people with no interest in this argument, and the result is in the public record with a document number.
LIGO’s arms are 4 km evacuated beam tubes. They had to be built along a line of sight, not along a level surface, and the observatory’s own alignment paper — Althouse, Hand, Jones, Lazzarini & Weiss, LIGO-P000006-C — states the consequence outright:
“The beam tubes needed to be aligned along the propagation direction of light in vacuum and not along the direction perpendicular to local gravity on the surface of the Earth. The curvature of the Earth will cause the Earth’s surface to deviate from the straight line propagated by light in vacuum by 1.25 meters over a 4 km path if the line starts out level with the surface. The alignment was, therefore, not the same as that for a level highway or pipeline.”
We have not reproduced a footnote marker that sits in the original at this point. It is worth knowing what the footnote says: the gravitational deflection of light over the four-kilometre path is negligible. A pleasing thing to have had to rule out, on a page about gravity and straight lines.
W. E. Althouse, S. D. Hand, L. K. Jones, A. Lazzarini & R. Weiss, Precision alignment of the LIGO 4 km arms using the dual-frequency differential global positioning system, LIGO-P000006-C. Fetched and checked against the paper itself, not quoted from a summary.
That 1.25 m is d²/2R at R = 6,371 km to within half a per cent — the same tangent-departure formula as Miller’s three inches, sixteen times the span. The paper puts it a second way in passing: “a straight line in space varies in ellipsoidal height by ~1.25 m over a 4 km baseline.”
If you go looking, LIGO’s own outreach page undersells this. Their public facts page says only that “over the 4km length of each arm, the Earth curves away by nearly a meter,” and describes the work as “precision leveling of the concrete slab” — wording that suggests a level floor when the technical paper is explicit that the slab had to follow the straight beam line instead. Cite the engineering document, not the visitor page.
And it is not only the outreach page. The APS gravitation newsletter Matters of Gravity, number 8, autumn 1996, reporting Hanford construction inside the project’s own community: “The final survey of the foundation along the two arms indicates that they are straight and level with an accuracy of 1.5 cm.” Straight and level, about the very foundation the alignment paper is explicit could not be both. Nobody is being careless with the public there. The two words are used interchangeably by almost everyone, almost always — which is this page’s whole thesis in a sentence.
And LIGO did not simply choose one reference. It ran both, side by side, down the same four kilometres. The tube was laid to ellipsoidal heights from dual-frequency differential GPS in WGS-84 — a straight line in space — and then checked against conventional survey, which the paper notes “provided orthometric heights relative to the geoid (as opposed to the ellipsoidal heights provided by GPS).” A straight datum and a level datum, over kilometres, reconciled against each other. That is panel C and panel A of the table above, built in Louisiana and Washington and paid for by the National Science Foundation.
It is worth being exact about what the evidence here is, because it is not “they built a straight tube.” A straight tube is buildable on any world, and the standing flat-earth reply to LIGO — that you would simply aim the beam slightly upward at end mirrors mounted higher off the ground — correctly observes as much. The evidence is that two independent height systems were run down the same four kilometres and disagreed, by 1.25 m, in exactly the d²/2R profile, with both closing on themselves. On a flat Earth straight and level are one surface and there is nothing to reconcile. And “the ends sit higher off the ground” is not a way around the result — it is the result. The levelled ground falls away from the straight tube.
The obvious rejoinder is that ellipsoidal heights presuppose a globe, so LIGO assumed its conclusion. They do not, and it did not. The ellipsoid is a coordinate convention; the physically straight line is the light itself, in an evacuated tube, and it demonstrably arrives at a mirror four kilometres away. What the survey then did was close an independent orthometric datum, carried by spirit levelling, against that straight one. The circularity charge dissolves in the reconciliation: two datums resting on different assumptions were made to agree, and the amount by which they had to be reconciled is the measurement.
Panel C is the rare one. The other three — anything referenced to the equipotential — we put up by the thousand, and the specifications say so in as many words.
The question “has anyone ever built a level kilometre?” has a dull answer: every long runway is one. ICAO Annex 14 defines runway elevations as orthometric heights above the geoid — “the equipotential surface in the gravity field of the Earth … the direction of gravity is perpendicular to the geoid at every point.” There is no curvature allowance anywhere in the standard, and there does not need to be one: the datum is the equipotential, so curvature is already inside it. The claim is not that a runway is geometrically flat, or even at constant elevation — it plainly is not. It is that its height datum is the equipotential, so the allowance everyone expects to find is absent because the datum absorbed it, not because the engineers forgot. For scale, a code-4 runway may carry an effective slope of 1 per cent — 40 m over 4 km, against a curvature term of 1.25 m — and ICAO permits vertical curves as tight as 30,000 m radius, some 212 times sharper than the planet.
Long tunnels are the same story from the other end. The Gotthard Base Tunnel was driven to a geoid-referenced control network and broke through with 8 cm of transverse and 1 cm of vertical error over 57 km. A straight chord between its portals would sit about 64 m below the level line at mid-tunnel — not a near miss, a different object.
This is the part worth sitting with. A hydrostatic levelling system is a long, thermally controlled, interconnected water channel, used as the level datum for aligning particle accelerators. That is Miller’s apparatus, specification for specification. They are standard equipment at accelerator laboratories, and they work: the free surfaces in the connected vessels form “a part of the same equipotential surface… if the density of the water is the same in all containers”, to better than a micron. They are sensitive enough to record the ground flexing under solar and lunar tides.
So the honest answer to “why has nobody built the kilometre of water?” is that they did, they use it every day, and they use it as an instrument rather than as a proof — because, as the four panels show, a water channel referenced to itself finds the equipotential and stops there. It cannot tell you the shape of the surface it just found. For that you need the straight line, which is why LIGO bought GPS and SLAC specified its accelerator housing to be “within ¼ in. of a straight line for the entire length” while noting, in the same design book, that “survey crews follow the curvature of the earth with their level readings and sightings.”
Two ways of saying where the top is, known to be different, written into the specifications of both machines. Neither laboratory was settling a shape-of-the-Earth argument. They were building something that had to work.
Two towers, 693 ft tall, standing 4,260 ft apart in the water between Brooklyn and Staten Island. Each was built plumb — perpendicular to local gravity, which is what “vertical” means to a bridge engineer. Because the two verticals are not parallel, the tops are farther apart than the bases, and the amount was a stated design criterion rather than an afterthought.
| Quantity | Value | Where it comes from |
|---|---|---|
| Base spacing | 4,260 ft = 1,298.4 m | Main span, as built |
| Tower height | 693 ft = 211.2 m | Above mean high water |
| Predicted splay | 43.05 mm (1.695 in) | baseline × height ÷ R |
| Recorded design | 41.275 mm (1 5/8 in) | Design criterion |
| Flat-Earth prediction | 0.000 mm | Parallel verticals cannot splay |
The splay is recorded in the literature on the bridge and its designer, Othmar Ammann — Rastorfer’s Six Bridges: The Legacy of Othmar H. Ammann and Caro’s The Power Broker among them — as a stated design criterion rather than an afterthought or a later calculation. We should be straight about the provenance, though: both are books about the bridge and the man, not engineering documents. Ammann’s own ASCE paper on the design is behind a paywall we could not get through, and no drawing or datum statement we could reach says what height the 1 5/8 in was computed from. That matters because the arithmetic below lands 4 per cent away from it, and a pier-top datum would close the gap exactly — but a pier-top datum is our inference, not a sourced fact.
Our computed figure sits 1.77 mm above the recorded one, about 4 per cent. We are not going to pretend that gap away. Rounding to the nearest eighth of an inch covers ±1.59 mm of it and no more. The likeliest remainder is the height datum: a tower height of 664 ft reproduces 1 5/8 in exactly, which is roughly what you get measuring the steel from the top of its pier rather than from mean high water. We have not confirmed which datum the design figure used, and until we do, treat the agreement as four per cent rather than two millimetres.
It does not matter, and that is the point. The competing prediction is not 41 mm or 43 mm. It is zero. On a flat Earth two plumb lines are parallel, parallel lines do not diverge, and a bridge whose tower tops are an inch and five eighths wider than its feet has nothing to explain. Four per cent is noise against a prediction of exactly nothing.
The version you can carry in a bucket. Go and stand at the top of one tower with a bucket of water and a spirit level. Confirm the surface is square to that tower. Do it again at the other. Both buckets read level against their own tower — and the two towers are not parallel. To be clear what that is: a picture of the geometry, not a measurement protocol. No bubble instrument compares the tilt of two verticals 1.3 km apart without a transfer reference. The checkable chain is the one this page rests on — plumbness verifiable at each tower, the splay in the construction record, and one line of arithmetic joining them.
Two level surfaces, 1,298 m apart, tilted relative to one another by 42 arcseconds. That is Miller’s experiment, already built, in steel, carrying I-278.
Every link in that chain is checkable without trusting anyone: the tower is plumb, and you can put a level on it; the splay is in the construction record, and the documents are public; the geometry is one line of arithmetic. There is no agency in it, no satellite, and no photograph.
None of the above stops anyone running the kilometre of water for themselves, and it is worth setting out what that would take — both because the design is instructive, and because the objection it has to survive is one a sceptic should be raising.
Suppose we levelled a floor with a theodolite, then set the trough on struts that grew taller toward one end, poured the water, and photographed the wedge.
Would a single flat-earther accept that as proof of a globe?
Of course not, and they would be right not to. We made the wedge with the struts. The water reported our carpentry back to us.
Now notice that this is the same objection, exactly, in the other direction. If the pillars are specified as heights above the ground — 0, 0.19, 0.77, 1.74, 3.09 inches, the parabola a straight beam requires — those numbers were computed from R = 6,371 km. The answer was assumed to lay the foundations. Build the same schedule on a flat Earth and you get the same wedge, because the tilt came from the strut list either way.
And the mirror case is no better. “Make every pillar the same height” does not build a flat trough. It builds a trough on the equipotential — whatever shape the equipotential happens to be — because the bubble in the level is a plumb device and the plumb direction is the thing in dispute. Neither side gets a free reference out of a spirit level.
So the experiment reduces to a single question, and it is not about water at all: can you establish a straight line a kilometre long without borrowing gravity to do it? The floor is only the version of that reference you can stand on. It need not be a floor at all — an evacuated sight tube above open water does the same job, which is why the answer, when it came, came from a physics laboratory rather than a warehouse.
One more thing rescues the design, and it is hiding in the factor of four from earlier. A reference that is straight but tilted gives a depth profile that is linear in distance. Curvature gives one that is quadratic. Sample the depth at five stations instead of two and the two are trivially separable:
| Station | Curvature, d²/2R | A straight but tilted reference, matched at the far end |
|---|---|---|
| 0 m | 0.00 mm | 0.00 mm |
| 250 m | 4.91 mm | 19.62 mm |
| 500 m | 19.62 mm | 39.24 mm |
| 750 m | 44.15 mm | 58.86 mm |
| 1000 m | 78.48 mm | 78.48 mm |
They agree at the ends and differ by 19.6 mm in the middle. Curvature puts a quarter of the total at mid-span; a tilt puts half. Fit depth = h₀ − a·s − b·s² and a absorbs any tilt while b returns 1/2R untouched. So the reference must be straight, but it need not be level, and you do not need to know its tilt — which matters, because level is the property you can only obtain from the very field under dispute, while straight is a property light will supply for you. With one condition, and it is the one a sceptic should raise first: the light has to be in a vacuum.
This is the objection we would lead with if we were arguing the other side, and the page was weaker for not carrying it. A sight line through air is not straight. Air’s refractive index falls with height, so a horizontal ray bends downward, and surveyors have carried a coefficient for it since the nineteenth century: k, the ray’s curvature as a fraction of the Earth’s. Outdoors it runs about 0.13–0.17.
The uncomfortable part is what happens in a temperature-controlled building, which is what the experiment asks for. To first order
with P in hPa, T in kelvin and the lapse rate in K/m. Hold the air perfectly isothermal — dT/dz = 0, which is exactly what good climate control achieves — and at 1013 hPa and 20 °C that gives k = 0.20. The gradient that would null it is −0.0342 K/m, the autoconvective lapse rate, which no building maintains and nobody would think to ask for.
So the controlled warehouse is close to the worst case, and the error has the same shape as the signal. A ray with k = 0.20 sags 15.7 mm over the kilometre, quadratically, against a curvature signal of 78.5 mm — twenty per cent, in the same parabola. The five-station fit above does not save you: the sag lands in b, the coefficient that is supposed to return 1/2R, not in a, the one that harmlessly absorbs tilt. You would measure a planet twenty per cent too large and see nothing wrong in your own residuals.
Which is why the successful versions of this measurement were not done with a laser across a room. They were done in vacuum, and the people who built them said so in as many words. SLAC’s two-mile alignment ran its laser down a dedicated evacuated pipe, and chapter 22 of the project book gives the reason without embellishment: the support girder “is evacuated to about 10⁻² torr to prevent air refraction effects from distorting or deflecting the alignment image.” LIGO’s straight reference is a beam in a four-kilometre vacuum tube, for this reason among others. If you want a straight kilometre you either take the air out of the path, or you cancel the bend with simultaneous reciprocal observations from both ends — which is, not coincidentally, what the modern repeats of the classic open-air experiments had to do.
Miller was right to specify a controlled environment, and right for a better reason than this page previously gave him credit for. He was wrong only about which variable needed controlling. It is not the water. It is the light.
A two-point measurement cannot do this. Any two points lie on both a line and a curve, so his design — middle against ends, or one end against the other — discards the information that separates them. That is not a mistake anyone should be blamed for; it is simply that the discriminating signal lives in the shape of the profile and you have to sample it to see it. Five stations turn an experiment that can be argued about into one that cannot.
Small, and worth clearing before the main argument. The p. 89 figure draws a straight container on pillars of increasing height, tangent to the ground at one end. For that construction the eight-inches-per-mile-squared rule is exactly right, and 8 × (1000/1609.344)² = 3.089 in = 78.5 mm. His three inches is correct for the experiment he has drawn.
The sentence beside it says something else: “the water in the middle is 3 inches higher compared to the left and right edges.” Middle against both ends is the sagitta of a chord, not the drop from a tangent, and it is the same rule at half the span — d²/8R = (d/2)²/2R — giving 8 × (500/1609.344)² = 0.772 in = 19.6 mm. So the full end-to-end drop has been applied to the midpoint, and the two descriptions differ by a factor of four. The number matches the picture; the description matches neither. It is the sort of slip that appears constantly in the surveying literature, and it does not touch the rest of his argument — but a builder would need it settled before ordering pillars.
The Verrazzano datum. As above — we compute 43.05 mm against a recorded 41.275 mm and we cannot yet account for the last 1.8 mm. If the design figure turns out to rest on a height we have mis-assigned, this section needs correcting, though not withdrawing.
Whether the warehouse has been built. In one sense yes — hydrostatic levelling systems are exactly that apparatus and are in daily use. What we could not find is one run as a discriminating test in a controlled environment: a long liquid surface compared against an independently realised straight reference, indoors or in vacuum, with the result published as a measurement of curvature rather than used as a working datum.
In the open air it has been run, repeatedly, and the page should say so. The Old Bedford canal is exactly this experiment — six miles of still water against an optical line, run by Rowbotham in 1838 and again in 1870 as the Wallace–Hampden wager, published, argued over and eventually litigated. Wallace’s three-marker design is the primitive form of the five-station argument above: the middle marker stands above the line joining the outer two by the sagitta. The modern rigorous repeat is the Rainy Lake experiment, roughly ten kilometres across a frozen lake with surveyor-grade levels, a theodolite and GPS.
The instructive part is why the 1870 result stayed arguable, and why the modern repeat needed simultaneous reciprocal observations. Both are sight lines through air, and air bends light into the same quadratic shape as the signal. Which is the real content of Miller’s “controlled environment”. Read as a demand for a reference that is not itself refracting, it is a good demand; it has been met; and the places it has been met are LIGO and SLAC rather than a canal bank. A counterexample — the indoor version, published as a curvature measurement — would still have to lead this page. We also could not confirm the total channel length of any single installation, so we quote no figure for one.
Which of his two experiments he means. The diagram draws a tangent at one end and his three inches is right for it; the sentence describes the mid-span sagitta at 19.6 mm. We have taken the diagram as the intended design.
The geoid is not a sphere. Local mass anomalies deflect the vertical by a few arcseconds per kilometre. Against a 32-arcsecond signal that is a percent-level systematic, and no amount of careful building removes it. It is a floor on the experiment even in the ideal case.