Fun With Science / Globe Deconstruction / Q2 · page 45
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.
Half-fill a glass and tilt it. The water does not tilt with the glass.
He proposes a temperature-controlled water channel a kilometre long and asks why nobody has built it. The answer: the water is not the instrument. Whatever a kilometre of water does, it does because of the shape of the floor under it, and that shape is decided before a drop is poured.
“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 the 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. The question asks whether a law can overpower itself.
The factual half — 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 along each 4 km arm, and reconciled a straight datum against a level one down the same tubes.
Underneath the phrasing is a real experiment, and it turns on one thing: was the floor built level, or built straight? Level is perpendicular to gravity at every point — the equipotential, the surface along which gravity does no work; what a spirit level or still water gives you. Straight is a geometric line; what light in a vacuum gives you. The two coincide on a flat plane; on a globe they part by 78.5 mm over a kilometre, four times that over two. That gap is the entire experiment. Build level and nothing happens in either world. Build straight and only the globe puts a wedge in the trough.
Conceded, once: his diagram and his three inches agree; his statement of hydrostatics is four-fifths textbook; and nobody has run his warehouse indoors as a discriminating test — the channels that exist use the water as a datum, not a proof. His scepticism, too, is aimed at spacetime curvature as the explanation of gravity — p. 57 offers “gravity = we still have no idea” as a live option — not at the downward effect itself. Nothing here needs relativity: the water conclusion follows from a downward effect and his own hydrostatics.
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, 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 missing clause is two lines:
In equilibrium, ∇p = ρg. On a free surface the pressure is constant — 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.
The top is an equipotential. That is all hydrostatics says about its shape, and it says nothing 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 does: tilt the glass and the surface stays put while the container rotates underneath; in a conical flask it is a flat disc that never 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.
The flat surface is the special case — what you get when the field happens to be parallel. The curved one 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.”
A sharp critic will say the postulate behind it — the parts of a fluid pressing toward a centre — bakes the sphere into the premises. Partly fair, and the point survives: the founding text makes the free surface a consequence of the force field, and the second thing it derives is a sphere. “The study of hydrostatics stands in complete opposition to this hypothesis” is precisely inverted.
The book states the globe’s mechanism itself at p. 89: heliocentrism says the water will not be level with a tangent container “because the downward vector of spacetime curvature changes to a greater angle as we move from left to right on the diagram.” Strike “spacetime curvature” — p. 57 is agnostic about it, and so is this page — and what remains is a downward vector whose direction changes along the trough. A free surface stands square to that vector wherever it points; there is no strength for gravity to need. Whether the direction varies over a kilometre is a question about the floor. The restatement at p. 196, whether “the effects of spacetime curvature take priority over hydrostatics on a large scale”, is the same inversion: there is no priority contest between a force and the law that says the surface follows it.
Two ways to build the floor, two possible worlds, four outcomes. Curvature is exaggerated in all four drawings; at true scale every line 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. The water contributes no information of its own: in all four panels it sits on the equipotential. It is the floor that changes.
Capillarity, air pressure and temperature are second order beside the floor: 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, 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.
Panel C has been built, at four times the span he asks for, by people with no interest in this argument. LIGO’s arms are 4 km evacuated beam tubes, built along a line of sight rather than a level surface, and the observatory’s alignment paper — 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 — 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.”
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.
And LIGO ran both references, side by side, down the same four kilometres. The tube was laid to ellipsoidal heights — heights above a mathematical reference ellipsoid, a straight-space coordinate — from dual-frequency differential GPS in WGS-84, then checked against conventional survey, which “provided orthometric heights relative to the geoid (as opposed to the ellipsoidal heights provided by GPS).” An orthometric height is height above the geoid, the level surface. A straight datum and a level datum, reconciled: panels C and A, built in Louisiana and Washington.
The evidence 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 plane straight and level are one surface and there is nothing to reconcile. “The ends sit higher off the ground” is not a way around the result — it is the result. Nor do ellipsoidal heights presuppose a globe: the ellipsoid is a coordinate convention, the physically straight line is the light itself, arriving at a mirror four kilometres away, and an independent orthometric datum carried by spirit levelling was closed against it. The amount of reconciliation is the measurement.
Cite the engineering document, not the visitor page. LIGO’s public facts page says only that “the Earth curves away by nearly a meter” and calls the work “precision leveling of the concrete slab”; the APS newsletter Matters of Gravity, number 8, autumn 1996, reported the Hanford foundation “straight and level with an accuracy of 1.5 cm.” Straight and level, about a foundation that could not be both. The two words are used interchangeably by almost everyone — this page’s thesis in a sentence.
Panel C is the rare one. Anything referenced to the equipotential we put up by the thousand, and the specifications say so.
Every long runway is a level kilometre. 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.” The standard has no curvature allowance because its datum is the equipotential. 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, 212 times sharper than the planet.
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 (the figures reported at the 2010 breakthrough; the geodesy report is on the construction-records page). 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.
A hydrostatic levelling system is a long, thermally controlled, interconnected water channel, used as the level datum for aligning particle accelerators — Miller’s apparatus, specification for specification. 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, enough to record the ground flexing under solar and lunar tides.
So the kilometre of water has been built and is used every day — as an instrument, not a proof, because a water channel referenced to itself finds the equipotential and stops there. It cannot tell you the shape of what it found. For that you need the straight line, which is why LIGO bought GPS and SLAC specified its accelerator housing “within ¼ in. of a straight line for the entire length” while noting 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 both machines by people building something that had to work.
Two towers, 693 ft tall, 4,260 ft apart, each built plumb. Because the two verticals are not parallel, the tops are farther apart than the bases, and the amount was a stated design criterion. The full audit is on the construction-records page.
| 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-plane prediction | 0.000 mm | Parallel verticals cannot splay |
The splay is recorded in Rastorfer’s Six Bridges and Caro’s The Power Broker — books, not engineering documents; Ammann’s ASCE paper is paywalled, and no drawing we could reach says what height the 1 5/8 in was computed from. Our figure sits 1.77 mm (4 per cent) above the recorded one; rounding to the nearest eighth of an inch covers ±1.59 mm of it. A tower height of 664 ft — roughly the steel above its pier — reproduces 1 5/8 in exactly, but that datum is our inference: treat the agreement as four per cent, not two millimetres.
The competing prediction is not 41 mm or 43 mm. It is zero. On a flat plane two plumb lines are parallel and tower tops an inch and five eighths wider than their feet have nothing to explain. Two level surfaces 1,298 m apart, tilted by 42 arcseconds (an arcsecond is 1/3600 of a degree): the geometry of his experiment in steel — resting on two books and a 4 per cent residual, which is why it is a supporting case and not the main one.
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: 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 (the rise of the arc above the straight line joining its ends), the same rule at half the span, d²/8R = (d/2)²/2R: 8 × (500/1609.344)² = 0.772 in = 19.6 mm. The two descriptions differ by a factor of four; the number matches the picture. The slip is common in the surveying literature and does not touch his argument, but a builder would need it settled before ordering pillars. This page takes the diagram as the design.
Suppose we levelled a floor with a theodolite, set the trough on struts that grew taller toward one end, and photographed the wedge. Would a single flat-earther accept that as proof of a globe?
Of course not, and they would be right: we made the wedge with the struts. The same objection runs the other way: 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 plane and you get the same wedge. And “make every pillar the same height” builds a trough on the equipotential, whatever shape that is, because the bubble in a 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 one question, and it is not about water: 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; an evacuated sight tube above open water does the same job, which is why the answer came from a physics laboratory rather than a warehouse.
One more thing rescues the design, hiding in the factor of four above. A reference that is straight but tilted gives a depth profile linear in distance; curvature gives one that is quadratic. Sample the depth at five stations instead of two and the two separate:
| 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 need not be level — and level can only be had from the field under dispute, while straight is a property light will supply. A two-point measurement cannot do this: any two points lie on both a line and a curve, so his design discards the information that separates them. With one condition, the one a sceptic should raise first: the light has to be in a vacuum.
A sight line through air is not straight. Air’s refractive index falls with height, so a horizontal ray bends downward, and surveyors carry a coefficient for it: 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. To first order
with P the air pressure in hPa, T its temperature in kelvin and dT/dz the change of temperature with height in K/m. Hold the air isothermal — dT/dz = 0, which is 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 (the cooling-with-height at which air density stops changing), which no building maintains.
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 does not save you: the sag lands in b, the coefficient that is supposed to return 1/2R, not in a. You would measure a planet twenty per cent too large and see nothing wrong in your own residuals.
Which is why the successful versions were done in vacuum. SLAC’s two-mile alignment ran its laser down an evacuated pipe, and chapter 22 of the project book gives the reason: 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. A straight kilometre means taking the air out of the path, or cancelling the bend with simultaneous reciprocal observations — sighting each way at the same moment, so the same bend enters both readings and drops out.
Miller was right to specify a controlled environment. He was wrong only about which variable needed controlling. It is not the water. It is the light.
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 litigated. Wallace’s three-marker design is the primitive five-station argument: the middle marker stands above the line joining the outer two by the sagitta. On a flat plane it stands above that line by 0. On a globe the sagitta of a six-mile chord is, by the same eight-inch rule as his three inches, 8 × 3² = 72 in, 1.83 m in geometry alone, and about 1.5–1.6 m once the outdoor k of 0.13–0.17 bends the sight line partway with the water. Wallace’s middle marker stood above the line; the argument that followed was about refraction. The modern rigorous repeat is the Rainy Lake experiment, roughly ten kilometres across a frozen lake with surveyor-grade levels, a theodolite and GPS. Both are sight lines through air, which is why the 1870 result stayed arguable and the modern repeat needed reciprocal observations — and that is the real content of Miller’s “controlled environment”: a reference that is not itself refracting, a good demand, met at LIGO and SLAC rather than on a canal bank.
Whether the warehouse has been built. What we could not find is a hydrostatic levelling system run as a discriminating test in a controlled environment — a long liquid surface compared against an independently realised straight reference, indoors or in vacuum, published as a measurement of curvature rather than used as a working datum. A published example would lead this page. We also could not confirm the total channel length of any single installation, so we quote no figure for one.
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.