Hilgenberg-OC: Primer

Project: Ott Christoph Hilgenberg (1896-1976), Berlin engineer and geoscientist, author of Vom wachsenden Erdball ("On the growing Earth", 1933) and of the idea that gravity is a flow of ether into matter, that the swallowed ether becomes new matter, and that the Earth has therefore grown from a much smaller globe. His training is given two ways: "a diploma in Mechanical Engineering" according to the folder document [read: p.2], and electrical machine engineering at the Technische Hochschule (TH) Berlin, 1919-1922, according to German Wikipedia [web: https://de.wikipedia.org/wiki/Ott_Christoph_Hilgenberg, via Investigator]. This review has not settled which is right. From May 1925 to September 1928 he worked in America as a geophysicist for an oil-prospecting company [read: p.2, p.5].

Source gap (repeated from Step 1 and the arbiter). The project folder holds one document: G. Scalera and T. Braun, "Ott Christoph Hilgenberg in twentieth-century Geophysics" (2003), a secondary summary written by Earth-expansion advocates (interested parties: Scalera edited the honour volume it appears in). It contains no equations. None of Hilgenberg's own works is in the folder. The only primary Hilgenberg text any step of this review saw is part of the 1933 book, read online by the Investigator (OCR of pp.1-18 and two page images, scan pages n20-n21, at https://archive.org/details/Hilgenberg1933). Never read by anyone in this review: the 1929 and 1931 booklets on gravity (a copy of the 1931 booklet is at the UK Met Office Library), the 1933 Magnus-effect reports (one in Z. Techn. Phys. 14, 211-214), the 1939 booklet on sink and source flow experiments, the 1965 Geologische Rundschau paper (source of the 4590 km figure), the 1974 monograph Geotektonik, neuartig gesehen, the unpublished 1967 "swallowing constant" manuscript, the 1934 paper comparing the metre prototypes with the Earth's computed growth, and the English translations of Hilgenberg's own texts printed in the 2003 honour volume (pp.178ff.). So every statement below about what Hilgenberg's physics says rests on the advocates' paraphrase or on a partial reading of one book. The verdicts on the measurable questions (radius change, mass growth) rest on independent published measurements and do not depend on the missing documents, but they test his numbers only as far as those numbers are known.

Source tags. [read] = from the folder document (page numbers are PDF pages) or from the Investigator's reading of the 1933 book ("[read: 1933 book, via Investigator]"); [web] = Internet source with URL; [memory] = general knowledge, with confidence; [calculation] = arithmetic done for this primer and reproduced in example.py. Verdicts are the arbiter's (arbiter.md); this primer adds none.

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1. The idea in one paragraph

Hilgenberg asked two questions at once: why do things fall, and why do the continents look as if they once fitted together on a smaller ball? His answer to the first was that space is not empty but is a fluid-like ether, and every lump of matter is a drain (a "sink") into which this ether pours. Being in the inflowing stream is what we feel as gravity. His answer to the second followed from the first: if ether keeps pouring into the Earth, and is turned into ordinary atoms inside it, the Earth must keep gaining mass and swelling. He tested the geometry himself by cutting continents and continental shelves out of paper and gluing them onto a ball about six-tenths the diameter of a globe, and found that they closed up with almost no gaps. Later he refined this into maps of the past Earth at smaller radii, using fossil magnetism to place the old poles. He also proposed that a fluid effect he studied in the laboratory (an "inverted" Magnus effect) keeps spinning the planets up. The fluid analogies he used are real physics, and "space flowing into a mass" even has an exact counterpart in general relativity. Where the idea runs into trouble is in the numbers it predicts: a growing Earth today, at a rate that modern satellite measurements and fossil magnetism can check, and do not find.

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2. What you need to know first

Exponential growth. If a quantity grows in proportion to itself, dM/dt = mu M, with mu a constant (the relative growth rate, units "per second"), the solution is M(t) = M0 e^(mu t), where M0 is the starting value. Taking logarithms: mu = ln(M/M0)/t. A useful habit: a relative rate per second times 3.156e7 gives the rate per year. Read: any calculus text's section on differential equations of growth and decay (for example Stewart, Calculus).

Volume, area and radius of a sphere. Area A = 4 pi R^2, volume V = (4/3) pi R^3. So if a ball shrinks to a fraction k of its radius, its area becomes k^2 and its volume k^3 of the original. Hilgenberg's whole argument rests on these two scalings (area for the continental fit, volume for the mass). Also: you cannot lay a piece of one sphere flat onto a sphere of a different radius without stretching it, because the curvature (1/R^2) differs (Gauss's theorema egregium, "remarkable theorem") [memory, high]. That matters when continents are moved from one globe to another.

Fluid flow: sources, sinks and continuity. A sink is a point where fluid disappears at a volume rate Q (cubic metres per second); a source is where it appears. For steady flow of an incompressible fluid (constant density), whatever enters a sphere around the sink must cross every larger sphere too, so the inward speed is v(r) = Q/(4 pi r^2): it falls as 1/r^2. The general rule is the continuity equation, d(rho)/dt + div(rho v) = s, where rho is the fluid density, v the velocity field and s the rate at which fluid is created (positive) or destroyed (negative) per unit volume. In steady flow, the acceleration of a fluid parcel moving along a radial line is a = v dv/dr. Read: D. J. Acheson, Elementary Fluid Dynamics (Oxford), which covers sources, sinks, potential flow and the Magnus effect at undergraduate level; or Kundu and Cohen, Fluid Mechanics.

The Magnus effect and its inversion. A spinning ball or cylinder in a stream feels a sideways force (lift) perpendicular to the stream: a topspin tennis ball dips, a spinning cylinder in a wind tunnel is pushed to one side. In real (viscous) fluids, in a narrow range of conditions the force reverses sign: the "inverse" or "negative" Magnus effect. It happens near Reynolds number Re about 1e5 (Re = U D / nu, flow speed times body size divided by the fluid's kinematic viscosity; it measures how important viscosity is), when the thin boundary layer of fluid next to the surface turns turbulent on one side first [web: https://arxiv.org/pdf/2302.11260 ; https://cambridge.org/core/journals/journal-of-fluid-mechanics/article/inverse-magnus-effect-on-a-rotating-sphere-when-and-why/0E9FEF2880AA45C8B12587374C6F10EE]. Read: Acheson, as above.

Newtonian gravity and the equivalence principle. Newton: a mass M pulls with acceleration g = GM/r^2 at distance r, with G = 6.674e-11 N m^2 kg^-2. The mass that resists being pushed (inertial mass) equals the mass that gravity pulls on (gravitational or "heavy" mass), to better than one part in 10^13 [memory, high]. Einstein turned that equality into the equivalence principle: being at rest in a gravitational field cannot be told apart, locally, from being in an accelerating rocket. Hilgenberg started from the same equality (see section 4). Read: J. R. Taylor, Classical Mechanics (gravitation and non-inertial frames); for the principle in plain language, Einstein's own Relativity: The Special and the General Theory.

Plate tectonics and how the Earth's size is measured. The mainstream picture: the outer shell of the Earth is a set of rigid plates moving on a sphere of fixed size; new ocean floor is made at mid-ocean ridges and destroyed by sinking ("subduction") at deep-sea trenches. Two independent ways of checking the Earth's size: paleomagnetism (rocks record the direction of the Earth's magnetic field when they form; the inclination I of that field relates to latitude L by tan I = 2 tan L for a dipole field [memory, high]; comparing the latitude difference with the measured distance between two sites of the same age gives the radius at that time) and space geodesy (satellite laser ranging, very-long-baseline radio interferometry, GPS and the DORIS radio system, combined to track the Earth's shape to a fraction of a millimetre per year). Read: W. Lowrie, Fundamentals of Geophysics (Cambridge); C. M. R. Fowler, The Solid Earth.

A little history. Before plate tectonics won in the late 1960s, the expanding Earth was a respectable minority view held by serious geologists (S. W. Carey, L. Egyed) and connected to P. Dirac's and P. Jordan's idea of a slowly weakening G [memory, high]. The idea is older than Hilgenberg. The folder document (by advocates of Earth expansion) names the earlier figures [read: p.3-5]: - R. Mantovani (1889; a popular science magazine piece in 1909) is called "the first to defend the idea of Earth's expansion as the cause of drift". Wegener later acknowledged Mantovani's drift ideas, but judged expansion "a form of extremism to be avoided" [read: p.3]. - I. O. Yarkovski (1888) proposed an ether theory of gravity. The authors suggest that Russian circles "beginning from Yarkovski" may have influenced the Germans: "many of them, including Hilgenberg, knew the language" [read: p.4]. No document is cited to show that Hilgenberg read him. - B. Lindemann's 1927 book Kettengebirge, kontinentale Zerspaltung und Erdexpansion (whose title includes "Erdexpansion", Earth expansion) "was known and quoted by Hilgenberg in 1933" [read: p.4]. - Bogolepov (Bogolepow in the reference list) wrote articles in Russian on planetary expansion from 1922, and H. W. Hixon wrote in a popular magazine in 1920 [read: p.4-5].

The authors leave a tension unresolved. They write that in 1933 Hilgenberg "may very well not have been aware that the expansion theory had come earlier" [read: p.3], yet they also say he quoted Lindemann's expansion book. They call the history of his influences "unsolved" [read: p.3]. Read: H. R. Frankel, The Continental Drift Controversy (Cambridge, 4 vols), which treats the expansion debate [memory, medium on which volume]; N. Oreskes, The Rejection of Continental Drift.

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3. The textbook view

3.1 Gravity as a flowing medium: what the mathematics allows

Take the idea literally and ask what a sink flow does to a body carried along with it.

(a) A plain sink in an incompressible fluid. Volume conservation gives v = -k/r^2 (minus means inward; k = Q/(4 pi)). The acceleration of a parcel carried by the flow is

a = v dv/dr = (-k/r^2)(2k/r^3) = -2k^2/r^5.

That falls off as r^-5, not as r^-2. Make it match the surface gravity g = 9.82 m/s^2 at the Earth's radius R = 6.371e6 m: then v(R) = sqrt(gR/2) = 5.59 km/s, and at the Moon's distance (60 R) the pull would be g/60^5 = 1.3e-8 m/s^2 instead of the observed g/60^2 = 2.7e-3 m/s^2, wrong by a factor 60^3, about 2e5 [calculation; example.py part B1]. So "being carried along by an ordinary sink flow" cannot be gravity.

(b) The flow that does give Newton. Let space flow inward at the free-fall speed, v = -sqrt(2GM/r). Then a = v dv/dr = d(v^2/2)/dr = d(GM/r)/dr = -GM/r^2, exactly Newton [calculation, part B2]. At the Earth's surface that speed is 11.19 km/s, the escape speed. But this flow is not incompressible: its divergence is not zero ((1/r^2) d(r^2 v)/dr = (3/2) v/r), so "space" must disappear all along the way in, not only inside the body.

(c) General relativity has exactly this picture, as a choice of coordinates. The Schwarzschild solution of general relativity (GR; the field outside a non-rotating mass) can be written in Painleve-Gullstrand coordinates, where space is flat and "flows" inward at v = sqrt(2GM/r), reaching the speed of light c at the black-hole horizon r = 2GM/c^2 (8.9 mm for a body of the Earth's mass) [web: https://arxiv.org/abs/gr-qc/0411060, Hamilton and Lisle, "The river model of black holes", published in American Journal of Physics 76, 519 (2008) [memory, high, per QE]]. This is an exact result, not an analogy. But it is a re-description: nothing is consumed, nothing becomes matter, and it predicts nothing GR does not already predict.

(d) A sink pushed by a stream. A third reading, and the one Hilgenberg's 1933 wording supports ("a body accelerates in a field because a sink sits in a stream" [read: 1933 book, via Investigator]), is that a sink placed in a moving fluid feels a force. In nineteenth-century hydrodynamics, C. A. and V. Bjerknes showed that pulsating spheres in an ideal fluid attract or repel with a force falling as the inverse square of distance [web: https://mathshistory.st-andrews.ac.uk/DSB/Bjerknes_Vilhelm.pdf, per the arbiter]. The mathematician reconstructs the force on a sink of volume strength m in a stream of speed u as F = rho_e m u (rho_e the ether density), which between two sinks gives F = rho_e m1 m2/(4 pi r^2), inverse square and attractive [memory, medium; not web-confirmed, and the exact sign rule is also unconfirmed]. Treat this as a reviewer's reconstruction, not as Hilgenberg's law (section 4).

3.2 The Earth's size

Mainstream geophysics holds the Earth's radius effectively constant over the last few hundred million years. The two key measurements:

• Space geodesy (present rate). Wu et al. (2011, Geophys. Res. Lett. 38, L13304), combining four techniques: the mean radius changes by 0.1 mm/yr with a one-standard-deviation (1 sigma) uncertainty of 0.2 mm/yr, i.e. no change detected [web: https://www.sciencedaily.com/releases/2011/08/110817120527.htm]. This bounds the present rate only.
• Paleomagnetism (past radius). McElhinny, Taylor and Stevenson (1978, Nature 271, 316-321): over the last 400 million years (Myr) the radius changed by less than 0.8%; the Moon's surface has been static for 4 billion years (limit 0.06%) [web: https://www.nature.com/articles/271316a0, abstract only]. Expansion advocates dispute the reconstruction assumptions behind such estimates: they argue that the 1978 test assumes fixed-radius reconstructions [physicist review]. Their published replies (Scalera; Maxlow 2002, cited by Scalera 2021) are records no step of this review has read, and no assessment of the dispute was found [contrarian review; arbiter QE revision].

Other evidence the reviewers cite: seismic tomography shows cold slabs sinking under trenches to about 700 km and deeper [read: p.11-12, the advocates' own statement; mainstream reading, memory high]; the oldest ocean floor in place is about 180 Myr old [memory, high]; the length of the day is increasing by a net 1.7 ms per century (tidal braking minus rebound effects), and Devonian corals record about 400 days per year, days of about 21.9 hours, around 400 Myr ago [web: https://astro.uni-bonn.de/~pbrosche/aa/acta/vol03/acta03_100.html ; https://scienceblog.com/?p=600265].

3.3 Mass, energy and conservation

Mainstream physics conserves mass-energy and baryon number (the count of protons and neutrons) in every observed process. Any process making 1 kg of matter needs E = mc^2 = 9e16 J from somewhere. Planetary ephemerides (the orbit tables fitted to decades of radar and spacecraft ranging) bound the Sun's fractional change of mass to about 1e-13 per year (EPM2019: between -13.4e-14 and -8.7e-14 per year, consistent with mass lost as sunlight and wind) [web: https://arxiv.org/pdf/2201.09804]. Lunar laser ranging bounds a change in G to order 1e-13 per year [web, per arbiter: https://ilrs.gsfc.nasa.gov/lw18/docs/presentations/Session9/13-0402-Mueller.pdf]. No such direct limit on the Earth's own GM (G times mass) was found in this review. The advocates have their own treatment of this test, Scalera et al., "Are artificial satellites orbits influenced by an expanding Earth?" (Earth-Prints, via Investigator); only its title is known.

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4. Where the inventor departs

4.1 His real starting points

Be fair to what Hilgenberg got right first:

1. The equality of heavy and inertial mass is a real, deep fact, and Hilgenberg built on it [read: 1933 book, via Investigator]. The folder authors state his version: "if a force is needed to accelerate a mass in space, then a force manifests itself if space is accelerating with respect to a mass" [read: p.8]. This is close in spirit to Einstein's elevator argument. (The authors' judgement that it is "even more solid and symmetrical than that formulated by Einstein" [read: p.8] is theirs, not his.)
2. Sink and source effects are real hydrodynamics. Two sinks attract, a sink and a source repel, a sink in a stream is carried with it [read: 1933 book, via Investigator]; this is the Bjerknes family of effects (section 3.1d). He did his own experiments (1931 booklet pp.14-15) and credited Westphal with earlier one-dimensional ones [read: 1933 book, via Investigator].
3. "Space flowing into a mass" is exact mathematics in GR (the river model, 3.1c), and so is "inflow at the speed of light means a black hole" (section 4.3).
4. The shelf-sphere geometry is a real observation. Continental outlines taken at the shelf edge fit together far better than coastlines; on today's sphere this is the famous Bullard (1965) fit of the Atlantic continents [memory, high]. Hilgenberg found that on a ball about 0.6 of the diameter the shelves close all round. He invited readers to repeat it with paper and glue on a globe of at least 20 cm ("nachzupruefen") [read: 1933 book, via Investigator].
5. He was candid. He calls constant density "not true, but simplifying", and lists as unanswered how ether enters atoms and how swallowed ether becomes mass [read: 1933 book, via Investigator]. He made predictions that can be checked: a growth rate, a rising internal temperature, shortening days.
6. Paleomagnetic poles on globes of varying radius were, by the advocates' account, used first by him (1962-65) [read: p.12-13; the authors' claim].

4.2 Side by side

4.3 The exact points of departure

Departure 1: from "space flows" to "space is used up". Up to "gravity can be described as a flow of space into a mass" Hilgenberg stands on solid ground (3.1b-c). The step that makes his theory different from GR is that the ether is consumed and becomes matter. That step has three consequences that can be worked out:

• A plain incompressible sink flow gives the wrong force law (r^-5; section 3.1a). The arbiter rates that reading Refuted by calculation but notes it is not Hilgenberg's reading as far as seen; it is how a reader may take the folder document's phrase "accelerated flow of space towards the planetary body".
• If gravity comes from sinks pushed by streams (3.1d), every body must swallow ether in proportion to its mass, so that G comes out universal. If the swallowed ether becomes mass, then dM/dt = mu M for every body: exponential growth is not an extra assumption but forced by the model, and G = mu^2/(4 pi rho_e) [mathematician's reconstruction]. Hilgenberg's 1933 phrase that "the gravitational constant in the sense of the ether-stream hypothesis" is the volume of ether swallowed per unit mass per unit time [read: 1933 book, via Investigator] fits this. On this reconstruction the Sun would also gain 1.6e-9 of its mass per year, against the ephemeris bound of about 1e-13 per year, a factor of order 10^4; and the implied ether inflow speed at the Earth's surface comes out far above c. The arbiter records both as consequences of a reconstruction, "a red flag, not a finding", and not as Hilgenberg's stated prediction.
• Energy. Whatever the force law, making mass costs rest energy. At his rate it is 2.7e25 W for the Earth alone (section 5, step 8). That is not a refutation (the ether is posited as the reservoir), but it is a requirement nobody has detected a source for. The folder authors argue by analogy that an ether energy density is not excluded. They quote the cosmologist Michael Turner (2000) on the energy of empty space: "estimates range from the absurdly large to the simply insignificant" [read: p.8]. The puzzle they point to is real: it is the cosmological-constant problem of mainstream physics [memory, high]. But an uncertain vacuum energy does not show where 2.7e25 W would come from, or how it would be delivered as new matter.

Departure 2: from "the shelves fit on a small ball" to "the Earth was that small". The fit is geometry; whether the Earth actually had that size is a separate, measurable question. The fit has free parameters: he used 0.62-0.65 rather than the sqrt(0.35) = 0.59 the area arithmetic gives, which leaves 10-21% area slack; he cut North America along two lines (a Mackenzie-Florida shear with about 2000 km of slip, and an Alaska line); and moving a continent between spheres of different radius forces a choice of how to distort it (about 7% tangential stretch at 3000 km from the centre, going from 0.6 R to R) [calculation, mathematician; confirmed by arbiter]. So a good fit is weak evidence by itself. The measurement that decides is the radius, sections 3.2 and 5.

Departure 3: the numbers do not form one law. The folder document's "60% of its present size", "4590 km" and "4 mm per year" disagree as printed. The Investigator's reading of the 1933 book explains the first: 0.6 is a diameter ratio of the shelf sphere, not a dated Permian radius. The rest is in section 5, step 7.

Departure 4: from a laboratory Magnus effect to planetary spin-up. The inverse Magnus effect is real but is a viscous boundary-layer effect at a particular Reynolds number. A planet in a radial ether stream has no stated viscosity, boundary layer or Reynolds number, and a Magnus force is a sideways lift, not a torque about the spin axis; spin-up needs an argument nobody supplies [physicist, mathematician]. Which of his "four new fluid-dynamic effects" was "completely proven by experiment (see the report of 1933)" [read: p.9, his curriculum quoted] is not said; linking it to the Magnus inversion is an inference from titles.

Departure 5: the black hole and the GR analogy. "Once a star has reached a critical mass, such that the incoming velocity of the ether equals that of light, the star becomes an object that no longer emits anything" [read: p.14-15]. In the river model, inflow at c is exactly where the horizon sits, so this is a genuine correspondence. The advocates' "equivalent in every way" is stronger than shown, because the river-model flow carries none of the features (consumption, conversion, growth) that make Hilgenberg's theory distinctive. The reported Riemann-to-GR step ("set the ether energy density to zero, carry out a simple mathematical conversion, add the fourth time coordinate" [read: p.14]) is not shown and is not how Einstein's field equations arise [memory, high].

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5. Worked example: Hilgenberg's 1933 growth law, followed to numbers we can measure

We start from Hilgenberg's own construction and his own assumptions, do exactly what he did, and then carry the result one step further, to the present-day rate of radius growth that satellites measure. Every number is reproduced by example.py (part A).

Inputs. Present radius R = 6.371e6 m; mass M = 5.972e24 kg; c = 2.998e8 m/s; one year = 3.156e7 s [memory, high]. From the 1933 book [read: 1933 book, via Investigator]: shelves are about 35% of the Earth's surface; his shelf sphere has 0.62-0.65 of the globe's diameter, rounded to 0.6 in the law; constant uniform density; growth over t = 1 billion years, which he writes as 3e16 s.

Step 1. The shelf sphere. For the shelves (35% of today's area) to cover a whole smaller sphere, (R'/R)^2 = 0.35, so R'/R = sqrt(0.35) = 0.592. His choice of 0.62 gives area ratio 0.384, and 0.65 gives 0.423: 10% and 21% more area than strictly needed. He says that at 0.62 "hardly any gap remained".

Step 2. Mass ratio. At constant density mass goes as volume, so M_then/M_now = 0.6^3 = 0.216. The Earth has grown by a factor 1/0.216 = 4.63 in mass, and 1/0.6 = 1.67 in radius (this is the folder document's "nearly doubling").

Step 3. The growth constant. From M_now = M_then e^(mu t):

mu = ln(4.63)/t = 1.532/(3e16 s) = 5.11e-17 per second = 1.61e-9 per year.

Hilgenberg printed "about 5 x 10^-17" with the unit "g/sec"; the unit is a slip, since mu is grams per gram per second, i.e. just "per second". (With an exact 1 billion years, 3.156e16 s, mu = 4.86e-17 per second, 5% lower; nothing below depends on that.) His illustration also checks: 1 g of hydrogen gas holds 6.022e23/2.016 = 2.99e23 H2 molecules, times 5.1e-17 per second = 1.5e7, his "15 million new molecules per second".

Step 4. Present rate of radius growth. Hilgenberg stopped at mu. Carry it one step further. With M = (4/3) pi rho R^3 and rho constant, dM/dt = 4 pi rho R^2 dR/dt = mu M gives

dR/dt = mu R/3 = (1.61e-9 per yr)(6.371e6 m)/3 = 3.42e-3 m/yr = 3.4 mm per year.

This is close to the folder document's "4 mm per year". Note that exponential growth is fastest now: the rate is proportional to R, so the present is the strictest place to test the law.

Step 5. Compare with satellites. Space geodesy measures 0.1 +/- 0.2 mm/yr (Wu et al. 2011, section 3.2). The law's 3.4 mm/yr is 17 times the 1-sigma uncertainty, and 17 sigma away from the measured value. (With the 1965 figures the gap is larger; with a 540 Myr Paleozoic instead of his 1 Gyr, the same 0.6 ratio gives mu = 2.84e-9 per year and 6.0 mm/yr now, 30 times the uncertainty [calculation, contrarian; confirmed by arbiter].)

Step 6. Run the law backwards. R(past) = R_now e^(-mu t/3).

• 250 Myr ago (Permian): R = 6371 km x e^(-0.134) = 5570 km, 0.874 of today.
• 400 Myr ago: R = 5139 km, 19.3% smaller than today. The paleomagnetic limit for that interval is under 0.8% (McElhinny et al. 1978): the law exceeds it by a factor of about 24 [calculation]. (The arbiter's factor of 35-55 uses the 1965 Permian radius of 4590 km instead.)

Step 7. What the 1965 caption needs. The folder document's Permian map says 4590 km and 4 mm/yr [read: page image p.13]. 4590/6371 = 0.720. To reach it 250 Myr ago with an exponential law needs mu = 3 ln(6371/4590)/250 Myr = 3.94e-9 per year (1.25e-16 per second) and a present rate of 8.4 mm/yr; a straight-line law needs 7.1 mm/yr; and 4 mm/yr for 250 Myr gives only 1000 km of growth, a Permian radius of 5371 km. So 4590 km and 4 mm/yr cannot both come from one constant-rate or exponential law. The 1933 law matches the 4 mm/yr; the 4590 km belongs to another construction (the unread 1965 paper, whose time base and law are unknown). This is why the arbiter rates the printed set Internally inconsistent while treating 1933 and 1965 as two models.

Step 8. What the ether must deliver today. Mass gain dM/dt = mu M = 5.11e-17 x 5.972e24 = 3.05e8 kg per second (9.6e15 kg per year). Rest energy of that mass: 3.05e8 x c^2 = 2.7e25 W. The sunlight the Earth intercepts is 1361 W/m^2 x pi R^2 = 1.74e17 W, so this is 1.6e8 times the sunlight on Earth, or 7.2% of the Sun's entire output (3.83e26 W). That is about 1.8e35 new protons or neutrons per second. The mass flux per square metre of surface is fixed by the growth rate: rho_e v = dM/dt/(4 pi R^2) = 6.0e-7 kg/(m^2 s). If the ether came in at the escape speed, 11.19 km/s, its density would have to be 5.3e-11 kg/m^3; but nothing in the record fixes the split between density and speed, so this is a requirement, not a prediction [calculation; physicist 6.2e-11 at 4 mm/yr and contrarian 5e-11 agree within input differences].

What the example shows, and what it does not. It shows that Hilgenberg's own construction and assumptions, followed honestly, predict a present growth of about 3.4 mm/yr, which is measurable and is not seen; the law also overshoots the paleomagnetic radius limit by a factor of order 20-30. It does not show that ether is impossible, that some much slower change is excluded below the measurement limits, or that the shelf fit is a coincidence. It tests his numbers, which is what a falsifiable prediction is for; Hilgenberg deserves credit for having made one.

Two short side calculations, also in example.py part B: the incompressible sink flow's r^-5 law and the Moon test (section 3.1a); and the break-up spin for ring detachment, where the centrifugal pull at the equator equals gravity, omega = sqrt(g/R) = 1.24e-3 rad/s, a period of 84 minutes. The Earth turns once in 1436 minutes, about 17 times too slowly to shed a ring, which agrees with the advocates' own concession [read: page image p.10].

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6. Evidence table

Verdicts are the arbiter's final verdicts (arbiter.md). "Tested" means a neutral, controlled test or an independent measurement exists on record.

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7. What a learner should take away

Solid. - The equality of heavy and inertial mass, which Hilgenberg used as his starting point, is one of the best-tested facts in physics, and turning it into "gravity is like being in an accelerating stream" is a legitimate move: it is essentially how the equivalence principle is taught, and GR's river model makes it exact for a non-rotating mass. - Sinks and sources in a fluid really do attract and repel in patterns that mimic gravity and electricity (the Bjerknes analogies), and the inverse Magnus effect is real. - The arithmetic of his 1933 growth law is correct (mu = 5.1e-17 per second; "15 million molecules per gram per second"), apart from the unit slip. - The shelf-edge fit is a real geometric observation that needs explaining, and his variable-radius paleopole method was an original technique.

Wrong, and why the mistakes are instructive. - An analogy is not a mechanism. A fluid can imitate an inverse-square attraction, but only special flows do it (3.1). A plain sink gives r^-5. The lesson: when a picture "explains" a force law, write down the flow and differentiate. - A fit with free parameters is weak evidence. 10-21% area slack, two free cut lines and a free choice of distortion law make a "nearly gap-free" fit much less surprising than it looks. The lesson: before celebrating a fit, ask how well a random or null configuration would do with the same freedom. - A falsifiable law gets tested where it is most extreme. An exponential law grows fastest now, so the present-day rate is its weakest point. Hilgenberg's own numbers give 3.4 mm/yr; satellites see less than 0.2 mm/yr. The fault is not in the arithmetic but in the premise that the Earth grows at that rate. - Several numbers from different constructions do not make one theory. The folder document mixes a 1933 diameter ratio, a 1965 Permian radius and a growth rate; they do not fit one law (section 5, step 7). - Predicting a direction is a test too. He predicted shortening days; the fossil and astronomical record shows lengthening days.

Genuinely open (in this record). - Did Hilgenberg derive a force law and an ether density in the 1931 or 1939 booklets? If so, does it imply the same growth rate for every body, and how does he treat the Sun and Moon? (Decides whether his gravity is a theory or an analogy.) - What law and time base lie behind 4590 km and 4 mm/yr in the 1965 paper? - What did his 1933 laboratory experiment actually measure (apparatus, Reynolds number, sign of the lift)? - What value did he give for the "swallowing constant" in the 1967 manuscript sent to S. K. Runcorn? - What did the 1934 metre-prototype comparison claim, and on what data? Hilgenberg compared the recorded lengthening of the standard metre bars with his computed growth of the Earth's diameter, in a refereed journal (Gerlands Beitr. Geophys. 42, 409-412) [read: p.16]. Only the title is known. It is unclear how he reasoned, because if all matter grew in step, a metre bar would show no change against its own copies. A slow drift in bar length caused by the metal relaxing is a candidate explanation for any drift he saw [memory, medium], but that is not a finding. - What do the advocates' replies to the decisive tests say? Records exist but no step of this review has read them: Scalera et al., "Are artificial satellites orbits influenced by an expanding Earth?" (Earth-Prints, via Investigator), which treats the satellite test of the Earth's GM (G times mass); Maxlow (2002), cited by Scalera 2021; and the argument that the 1978 paleomagnetic test assumes fixed-radius reconstructions. - What do the TU Berlin archive files say about the rejection of his first thesis (1937/38) and about the refused chair? - Would a neutral, blind fit test favour the 0.62 sphere over today's sphere once cuts and slack are matched? - Could a constant-mass Earth squeezed to 0.72 of its radius exist (an equation-of-state question nobody in this review answered)? It bears only on the no-new-matter variant.

Where the record goes silent (analysed by the arbiter, none of the explanations established). - The thesis rejection and the refused chair. Three published accounts say his first doctoral thesis, on sink and source flow experiments, was rejected (1937 or 1938, Faculty of Mechanical Engineering, mainly for conflict with relativity: German Wikipedia; Thielmann 2015; Scalera and Braun, who add political reasons) [web: https://de.wikipedia.org/wiki/Ott_Christoph_Hilgenberg ; https://doi.org/10.1515/9783839406106-003; read: p.8]. Three accounts, tracing to the same advocacy circle, say he refused to join the NSDAP (Nazi party) to obtain a chair prepared for him [read: p.7; Jacob 2001 and Wikipedia, via Investigator]. None cites the faculty file. Against a lasting ban: he received a Dr.-Ing. from TU Berlin in 1948 (on fault structure) and was given a workroom for decades by geology professors Quiring and Zeil [web, via Investigator]. Deciding record: TU Berlin university archive (not reached). - His experiments (1929-1939) were never repeated by anyone on record. Competing explanations: unremarkable fluid dynamics nobody needed to repeat; the gravity link was never accepted; wartime Germany; self-published booklets. Decider: citations of the 1933 Z. Techn. Phys. paper; the 1931 booklet at the Met Office Library; the 1934 Gerlands Beitr. Geophys. paper. - No reply to the 1974 open letter to Wilson, Beloussov and van Bemmelen. Not sent, answered privately, or ignored once plate tectonics had consolidated; decider: those scientists' correspondence and the Hilgenberg estate (held by his daughter Helge [read: p.11]). - The 1967 swallowing-constant manuscript, the one place he may have supplied the missing number; lost, in Runcorn's papers, or in the estate. - Publication was not blocked in geology: his 1960s papers appeared in refereed journals (Springer, Schweizerbart, Italian journals) and the 1974 monograph filled a whole 194-page issue of Geotektonische Forschungen [read: p.13-17]. The ether-gravity work was self-published.

Kept in the record whatever the verdict. Yarkovsky's 1888 Hypothese cinetique de la gravitation universelle, the same principle (ether inflow causing gravity and turning into matter) 45 years earlier [web: https://wiki.naturalphilosophy.org/index.php?title=Ivan_Osipovich_Yarkovsky, an advocacy wiki; also discussed in the folder document, with the remark that Hilgenberg knew Russian, read: p.3-4, p.18]; the lineage Mantovani, Lindemann, Bogolepov and Hixon, and the authors' unresolved tension about what Hilgenberg knew (section 2); the two opposite descriptions of Dayton Miller's ether-drift results (the advocates: "flawed by spurious effects" [read: p.7]; Thielmann 2015: a successful detection [web, via Investigator]), beside the mainstream reanalysis by Shankland et al. 1955 (temperature effects and statistical fluctuation; a published characterisation by physicists defending the standard view, not a replication of Miller's apparatus) [web: https://en.wikipedia.org/wiki/Dayton_Miller, per arbiter]; the authors' vacuum-energy argument (section 4.3); Hilgenberg's other works in the folder's list, not weighed by any step: the 1932 booklet on the vortex-ring nature of cyclones, the 1939 paper on sunspot numbers and geophysical maxima (Ann. Hydrographie), the 1940 paper on continental drift and the permanence of oceans, and the 1959 vortex-ring atom model [read: p.16-17]; Hilgenberg's citation of Fricke (his ref. 69) for a numerical relation between temperature and gravity on celestial bodies, which supports his prediction that the Earth's own temperature rises as it grows [read: 1933 book, scan n21, via Investigator]; Krafft's Ether and Matter telling readers to read Hilgenberg's 1931 and 1939 booklets, evidence that they circulated (sites.md); the authors' statements that "the Technical University was created by him, taking advantage of the confusion in Berlin" (hedged: its documented content is that he recovered the library's books from the Soviet sector and was briefly a director in library work; the source is his daughter's paper) [read: p.15], and that copies of his globes were "damaged irreparably in 1968" by students [read: p.11]; Riemann's ether-sink idea, which Hilgenberg says he reached independently [read: 1933 book, via Investigator]; C. F. Krafft's vortex ether, translated by Hilgenberg (1953); Jordan's weakening-G expansion; Herndon's decompression model (an interested party with a rival model); Scalera's 2021 first attempt at an ether density, speed and flux, with geoneutrino data offered as "corroboration" (Scalera's characterization; Scalera is an interested party; not read by anyone in this review) [web: https://www.earth-prints.org/handle/2122/14738]; the 2001 TU Berlin forum, with witnesses who knew him (R. Fiedler-Volmer, who describes his persistence against "sometimes even insulting ignorance"; H. H. Oppermann) and named critics (G. Bruhn: conservation of mass, tomography, meteorite input far too small; R. Seiler; U. Thorweihe) [web: https://web.archive.org/web/20021223154811/http://www.tu-berlin.de/presse/tui/01mai/hilgenb_lb.htm]; the 2001 Lautenthal colloquium and Scalera's replica globes (advocates); the arte/ZDF documentary "Und sie bewegt sich doch!" (2007); J. P. Farrell's popular accounts of Hilgenberg's vortex physics and 1959 vortex-ring atom models (later accounts by an author with a speculative thesis; not read). The mainstream tests (McElhinny 1978, Wu 2011) tested expansion in general, not Hilgenberg specifically.

Small slips noted in the documents, for the record. Hilgenberg's "g/sec" for a pure rate; his "1 g hydrogen" means hydrogen gas, H2; his 1 Gyr Paleozoic base (from 1933 radioactive dating) is about twice the modern 540 Myr [memory, high]; the folder authors date the Penzias-Wilson microwave background discovery to 1968 (it was 1964-65 [memory, high]); the folder document names Giessmann and Bartsch as the 1933 publisher, while the title page says Verlag O. Hilgenberg, printed by Giessmann and Bartsch [web: https://archive.org/details/Hilgenberg1933].

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8. Check yourself

1. Hilgenberg's shelf sphere has 0.62 of the globe's diameter. What fraction of today's surface area does it have, and how much more area is that than the 35% the shelves occupy?
2. Show that if density stays constant and dM/dt = mu M, then dR/dt = mu R/3. Using mu = 1.61e-9 per year, what is dR/dt today, and how does it compare with 0.1 +/- 0.2 mm/yr?
3. A fluid flows into a point sink with v = -k/r^2. Show that a parcel carried by the flow accelerates as r^-5. What flow speed v(r) would give an inverse-square acceleration instead, and why is that flow not an ordinary incompressible sink?
4. Why can the 1965 Permian radius (4590 km at 250 Myr ago) and a growth of 4 mm/yr not both hold under a constant-rate law? What constant rate would 4590 km need?
5. Hilgenberg predicted that days get shorter as the Earth grows. Devonian corals show about 400 days per year. Which way do the corals point, and what does that tell you about the prediction?

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9. Further reading

Mainstream. - D. J. Acheson, Elementary Fluid Dynamics (Oxford University Press): sources, sinks, potential flow, the Magnus effect, boundary layers. - J. R. Taylor, Classical Mechanics (University Science Books): gravitation, non-inertial frames. - A. J. S. Hamilton and J. P. Lisle, "The river model of black holes", arXiv gr-qc/0411060, American Journal of Physics 76, 519 (2008): gravity as flowing space, exactly. - W. Lowrie, Fundamentals of Geophysics (Cambridge University Press): paleomagnetism, the Earth's figure, plate tectonics. - M. W. McElhinny, S. R. Taylor and D. J. Stevenson, "Limits to the expansion of Earth, Moon, Mars and Mercury and to changes in the gravitational constant", Nature 271, 316-321 (1978) [title from memory, medium; reference web-checked by the arbiter]. - X. Wu et al., "Accuracy of the International Terrestrial Reference Frame origin and Earth expansion", Geophys. Res. Lett. 38, L13304 (2011) [title from memory, medium]. - H. R. Frankel, The Continental Drift Controversy (Cambridge University Press), for the expansion debate in its historical setting. - V. Bjerknes, Fields of Force (1906), the hydrodynamic analogies of pulsating spheres.

Hilgenberg's own key works (none in the project folder) [read: p.16-17]. - Das Raetsel Gravitation geloest (Kassel, 1929, 14 pp.). - Ueber Gravitation, Tromben und Wellen in bewegten Medien (Berlin, 1931, 78 pp.): the full gravity theory and his sink/source experiments; copy at the Met Office Library, UK [web: https://library.metoffice.gov.uk/portal/Default/en-GB/RecordView/Index/280883]. - Vom wachsenden Erdball (Berlin, 1933, 56 pp.), online at https://archive.org/details/Hilgenberg1933 and http://www.nuclearplanet.com/Hilgenberg.html (reproduced with his daughter's permission). - "Ueber die Umkehrung des Magnuseffekts", Z. Techn. Phys. 14, 211-214 (1933), and Ueber den Magnus-Effekt und seine Umkehrung (Berlin-Charlottenburg, 1933, 28 pp.). - "Vergleich der Laengenzunahme der Meterprototypen mit der errechneten Zunahme des Erddurchmessers" ("Comparison of the lengthening of the metre prototypes with the computed increase of the Earth's diameter"), Gerlands Beitr. Geophys. 42, 409-412 (1934) [read: p.16]: an empirical claim no step of this review has read (section 7). - Ueber Stroemungsversuche mit Senken und Quellen, die das Wesen der Schwerkraft grundlegend erklaeren (Berlin-Charlottenburg, 1939, 72 pp.). - "Die Palaeogeographie der expandierenden Erde vom Karbon bis zum Tertiaer nach palaeomagnetischen Messungen", Geol. Rundschau 55, 878-924 (1965). - Geotektonik, neuartig gesehen, Geotektonische Forschungen 45, 1-194 (1974), with the English open letter "Debate about the Earth".

Hilgenberg's own words in English, and writing about him (interested parties). G. Scalera and K.-H. Jacob (eds.), Why expanding Earth? A book in honour of Ott Christoph Hilgenberg (INGV, the Istituto Nazionale di Geofisica e Vulcanologia, Rome, 2003). Besides the Scalera and Braun chapter in the project folder, it prints English translations of Hilgenberg's own texts: his unpublished manuscript ("this volume pag.178-28", the page range cut off in the OCR) [read: p.9], and selected parts of his last manuscript, Erdentstehung und Entwicklung (1976, 57 pp.), translated at INGV [read: p.15]. It also includes his daughter Helge Hilgenberg's biographical paper [read: p.15]. The volume is the first place to look for his flow law and swallowing constant in his own words. It is not in the folder, and no step of this review has read it.

Try it yourself. Run example.py: .venv/bin/python review/Hilgenberg-OC/example.py Change t_H to 5.4e8 x 3.156e7 (a 540 Myr Paleozoic) or k to 0.65 and watch the present-day rate move. Or do it with paper: Hilgenberg's own invitation was to cut out the shelves and glue them on a ball 0.62 the size of a globe.

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Answers

1. Area scales as the square: 0.62^2 = 0.384 of today's surface. Compared with 0.35 that is 0.384/0.35 = 1.10, so about 10% more area than the shelves need (at 0.65 it is 21%). A perfect closure would need sqrt(0.35) = 0.59.
2. M = (4/3) pi rho R^3, so dM/dt = 4 pi rho R^2 dR/dt. Setting this equal to mu M = mu (4/3) pi rho R^3 and cancelling gives dR/dt = mu R/3. Today: 1.61e-9 x 6.371e6 m / 3 = 3.4e-3 m, i.e. 3.4 mm per year. That is 17 times the 0.2 mm/yr uncertainty of the measurement, and the measured value is consistent with zero.
3. dv/dr = 2k/r^3, so a = v dv/dr = (-k/r^2)(2k/r^3) = -2k^2/r^5. For inverse square you need v^2/2 = GM/r, i.e. v = -sqrt(2GM/r); then a = d(v^2/2)/dr = -GM/r^2. Its divergence (1/r^2) d(r^2 v)/dr is (3/2) v/r, not zero, so the fluid volume is not conserved along the way: "space" would have to vanish everywhere outside the body, not only in a sink inside it. In general relativity this flow is the river model, a coordinate description with nothing consumed.
4. 4 mm/yr for 250 Myr is 1000 km, giving a Permian radius of 5371 km, not 4590 km. Reaching 4590 km needs (6371 - 4590) km / 250 Myr = 7.1 mm/yr at a constant rate, or 8.4 mm/yr today under an exponential law. So the two printed numbers come from different constructions (the 1933 law gives about 3.4 mm/yr and a Permian radius of 5570 km).
5. 400 days per year means each day was about 365.25 x 24/400 = 21.9 hours long: days were shorter in the past and have lengthened since. Hilgenberg's growth predicts the opposite (days shortening over time, so longer in the past). The prediction has the wrong sign, which agrees with tidal braking in the mainstream picture. (If instead you hold angular momentum fixed while the Earth grows at constant density, the past day comes out far too short, 4.7 hours at a radius of 0.72; that conflicts with the corals in the other direction.)