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2:−1:−1
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For Earth modelled as a point mass, Φ(r) = -GM/r gives a radial second derivative of magnitude 2GM/R3 and, since Φ is harmonic in vacuum, transverse second derivatives of magnitude GM/R3 with the opposite sign — a trace-free tensor in the exact ratio 2:{-1}:{-1} . With the standard gravitational parameter GM⊕ = 3.986×10^{14}\,m3s−2 and mean radius R⊕ = 6.371×10^{6}\,m , this gives GM⊕/R⊕3 ≈ 1.541×10^{-6}\,s−2 and a radial magnitude of 3.08×10^{-6}\,s−2 , matching the standard geodetic free-air gravity gradient of about 3.086×10^{-6}\,s−2 , or 0.3086 milligal per metre,…
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Published contexts (6)
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Equation 128 · Evolutionary Physics
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
For Earth modelled as a point mass, Φ(r) = -GM/r gives a radial second derivative of magnitude 2GM/R3 and, since Φ is harmonic in vacuum, transverse second derivatives of magnitude GM/R3 with the opposite sign — a trace-free tensor in the exact ratio 2:{-1}:{-1} . With the standard gravitational parameter GM⊕ = 3.986×10^{14}\,m3s−2 and mean radius R⊕ = 6.371×10^{6}\,m , this gives GM⊕/R⊕3 ≈ 1.541×10^{-6}\,s−2 and a radial magnitude of 3.08×10^{-6}\,s−2 , matching the standard geodetic free-air gravity gradient of about 3.086×10^{-6}\,s−2 , or 0.3086 milligal per metre,…
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Article →Equation 147 · Evolutionary Physics
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
Even a network that has cleared every check above owes one more: any fictitious, non-curvature contribution to a naive second difference has to be ruled out on its own terms, not merely assumed absent. A uniformly rotating platform’s centrifugal potential, -21|Ω×r|^2 , has a nonzero Hessian too — for rotation about the z -axis, diag(−Ω2,−Ω2,0) — but that Hessian is exactly constant across any laboratory-scale network, tied to a rotation axis independently known from astrometry to extraordinary precision, and entirely absent from the trace-free, 2:{-1}:{-1} , 1/r3 -falling pattern a genuine point-like or spherical mass produces. A real…
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Article →Equation 149 · Evolutionary Physics
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
Even a network that has cleared every check above owes one more: any fictitious, non-curvature contribution to a naive second difference has to be ruled out on its own terms, not merely assumed absent. A uniformly rotating platform’s centrifugal potential, -21|Ω×r|^2 , has a nonzero Hessian too — for rotation about the z -axis, diag(−Ω2,−Ω2,0) — but that Hessian is exactly constant across any laboratory-scale network, tied to a rotation axis independently known from astrometry to extraordinary precision, and entirely absent from the trace-free, 2:{-1}:{-1} , 1/r3 -falling pattern a genuine point-like or spherical mass produces. A real…
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Article →Equation 152 · Evolutionary Physics
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
Earth’s own centrifugal contribution is the concrete case worth sizing, precisely because it is not negligible enough to ignore by assumption. With Ω ≈ 7.292×10^{-5}\,rads−1 fixed by astrometry to far more figures than any clock network could hope to determine on its own [ 10 ] , the centrifugal Hessian has magnitude Ω2 ≈ 5.32×10^{-9}\,s−2 — smaller than the radial tidal magnitude computed above by a factor of nearly six hundred, but not so much smaller that a network aiming to certify the 2:{-1}:{-1} ratio to good precision can skip subtracting it. Left in, it would masquerade as roughly a fifth of one percent of the genuine tidal…
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Article →Equation 154 · Evolutionary Physics
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
Earth’s own centrifugal contribution is the concrete case worth sizing, precisely because it is not negligible enough to ignore by assumption. With Ω ≈ 7.292×10^{-5}\,rads−1 fixed by astrometry to far more figures than any clock network could hope to determine on its own [ 10 ] , the centrifugal Hessian has magnitude Ω2 ≈ 5.32×10^{-9}\,s−2 — smaller than the radial tidal magnitude computed above by a factor of nearly six hundred, but not so much smaller that a network aiming to certify the 2:{-1}:{-1} ratio to good precision can skip subtracting it. Left in, it would masquerade as roughly a fifth of one percent of the genuine tidal…
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Article →Equation 155 · Evolutionary Physics
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
The kill criterion follows directly from the derivation rather than being appended to it: if a network’s fitted tidal estimator fails to shrink as its baseline moves away from a known mass, fails to track the local vertical as the network is relocated, or survives a proper subtraction of every independently known rotational and accelerated-frame contribution without settling toward the trace-free 2:{-1}:{-1} pattern derived above, the claim that curvature has been diagnosed is false, regardless of how large the raw number looks on a screen. What survives that test is not a new theory of gravity, a new force, or a new invariant beyond what general relativity already supplies. It is a way of…
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