One Shape, Many Orders: The Inverse Square as the First Gradient of 1/r, and How It Came to Be Read as Action at a Distance

J. Rogers, SE Ohio

Draft. Companion to “The Algebraic Decoding of Newton's Vector,” “Tides Fall Out of the Landscape” and “Solar System as Moving Deformations.”

Abstract

Gravity, as usually taught, comes with a law of its own: a force that falls as the inverse square of the distance. This paper shows that the inverse square is the first gradient of a single shape, 1/r, and that the next orders of the same gradient are already known by other names. The slope 1/r² is the change of the rate of time per length, times c². The slope of the slope, 1/r³, is the tide. The slope of that, 1/r⁴, is the asymmetry between the near and far sides of the tide. We measure all three on the simulation's landscape function. We extend the ladder to its fourth and fifth rungs and to galaxy size, where the series stops converging and the sum takes over. We then trace how the 1/r² came to be read as action at a distance, a reading Newton himself called an absurdity, and show that reading the shape as primary removes the notion: an object reads the slope of the shape at the coordinate where it sits, and nothing acts where it is not.

The ladder in five steps

Step	What it is	Form	Where it appears
1. The shape	The rate of time at a coordinate	1 − X/(rc²), so the height is −X/r	The potential, the clock shift
2. First gradient	The change of the rate of time per length, times c²	X/r²	The acceleration, called the force
3. Second gradient	The change of the slope across a body	2X/r³ along, −X/r³ across	The tide
4. Third gradient	The change of the tide across a body	6X/r⁴	The asymmetry between the near and far sides of the tide
5. No new laws	Each order is one more derivative of the same shape		Taylor terms of one function

The force is the first rung, one derivative of a shape. The inverse square was never a separate law. The ladder continues past step 4, with a fourth gradient 24X/r⁵ and a fifth 120X/r⁶. Each is smaller than the last by about R/d, and each is hidden below the measurement floor until a body is a large fraction of its distance (sections 3 and 4).

1. The shape and its derivatives

Take a deformation of depth X, which is G·m in SI units. Its shape is 1/r. Read as a rate of time, the clock rate at distance r from it is

ρ(r) = 1 − X/(r c²),

which is the weak-field clock rate. The height of the landscape is the rate of time, and the shape is the shape of that rate.

First gradient. The change of the rate of time per length is

dρ/dr = X/(c² r²).

Multiplied by c², the one scale that converts a rate into a speed squared, this is X/r², which has the units of an acceleration. The lengths in the denominator are the “per length” of the gradient, and the numerator carries the units. At the Earth the Sun's clock rate is off by 9.87 × 10⁻⁹. It changes by 6.60 × 10⁻²⁰ per meter, and times c² that is 5.93 × 10⁻³ m/s², the Earth's acceleration. In the natural chart, where c is 1, the acceleration is the gradient of the rate of time with nothing added.

Second gradient. Differentiate the slope once more. Along the line to the source the slope changes by d(X/r²)/dr = −2X/r³. Across the line the slope's direction turns, which changes its component toward the line by X/r³. These are the two numbers of the tidal tensor, +2X/r³ stretching and −X/r³ compressing twice. This is what the standard theory calls the tide.

Third gradient. Differentiate again: d(−2X/r³)/dr = 6X/r⁴. This is the asymmetry of the tide. Across a body of radius R, the slope at distance d ± R expands as

X/(d − R)² − X/d² = 2XR/d³ + 3XR²/d⁴ + … X/(d + R)² − X/d² = −2XR/d³ + 3XR²/d⁴ − …

The first term is odd, equal and opposite on the two sides. The second is even and the same on both sides. Adding the two sides leaves 6XR²/d⁴ and cancels the odd part. Subtracting them leaves 4XR/d³, so half the difference is the second rung.

The ladder is the Taylor series of the slope across a body. Each rung is one more derivative of the same shape, and nothing on it is a law of its own.

2. The ladder measured on the simulation's landscape

We called the simulation's downhill function with the Moon as the only deformation and read the slope on Earth's surface, at the near and far points, and at the center. Nothing else was computed.

The third rung. Near side plus far side, relative to the center, comes out at 5.471 × 10⁻⁸ m/s², against 6GMR²/d⁴ = 5.469 × 10⁻⁸ m/s². Half the difference between the two sides is 1.100 × 10⁻⁶ m/s², which is the second rung, 2GMR/d³. The 5% asymmetry between the Moon's near-side and far-side tides, which the first-order tide formula lacks, is the third rung.

The size of each rung.

Rung	Form	Sun, at the Earth (1 AU)	Moon, at the Earth (384,400 km)
Shape: rate shift	X/(rc²)	9.871 × 10⁻⁹	1.419 × 10⁻¹³
Slope: acceleration	X/r²	5.930 × 10⁻³ m/s²	3.318 × 10⁻⁵ m/s²
Slope of the slope: tide	2X/r³	7.928 × 10⁻¹⁴ s⁻²	1.726 × 10⁻¹³ s⁻²
Third: asymmetry	6X/r⁴	1.590 × 10⁻²⁴ m⁻¹ s⁻²	1.347 × 10⁻²¹ m⁻¹ s⁻²

The Moon's tide, 1.7 × 10⁻¹³ s⁻², is more than twice the Sun's, 7.9 × 10⁻¹⁴ s⁻², although the Sun's slope at the Earth is 180 times larger. That is the 1/r³: the tide cares about how fast the slope changes across the body, and the Moon is much nearer.

3. The deeper rungs

The gradients of the shape continue. The n-th gradient of −X/r has magnitude n!·X/rⁿ⁺¹: 6X/r⁴ for the third, 24X/r⁵ for the fourth and 120X/r⁶ for the fifth. For ordinary bodies they are too small to see, so none of them has been named a law.

Across a body of radius R at distance d, the slope readings at the near and far points expand as

X/(d ∓ R)² − X/d² = (X/d²) Σ (k+1)(±R/d)ᵏ.

The terms of that sum are the rungs. In units of X/d², the second gradient (the tide) contributes 2(R/d), the third 3(R/d)², the fourth 4(R/d)³ and the fifth 5(R/d)⁴. The odd rungs are equal and opposite on the two sides of the body. The even rungs are the same on both.

R/d	Third, relative to the tide	Fourth	Fifth
0.0166 (the Moon, seen from the Earth)	2.5%	0.055%	0.0011%
0.1 (a star with a close planet)	15%	2%	0.25%
0.3 (close binary stars, a galaxy beside its neighbor)	45%	18%	6.8%

Measured. We read the near and far points from the simulation's landscape. The odd part matches the series 2ε + 4ε³ + 6ε⁵, with ε = R/d, to 1.000000 at R/d = 0.0166 and 1.000004 at 0.1. At 0.3 it matches to 1.0027, where the series stops at the sixth gradient. At R/d = 0.1, stopping after the second, third, fourth, fifth and sixth gradients leaves errors in the near-side reading of 15%, 1.9%, 0.24%, 0.029% and 0.0034%. Each rung improves the reading about tenfold.

Their standard names. On the side of the body being deformed, these are the higher-degree tides. The third-degree terms of the Moon's tide are part of the standard tide-generating potential. On the side of the source, they are the higher harmonics of its shape, which satellite orbits measure. A dumbbell of two equal deformations at ±a, read from a distance d with a/d = 0.1, gives a slope 3.05% stronger on the axis and 1.48% weaker on the equator. That is a ratio of 2.06 to 1, the quadrupole pattern, and oblateness appears without any oblateness formula.

Hidden, not absent. For the Earth and the Moon, the fourth and fifth rungs are 0.055% and 0.0011% of the tide. That is below what tide gauges measure. They sit below the noise floor and are not absent, and they become visible where R/d is large.

4. At galaxy size

At galaxy size, R/d is no longer small. The Milky Way's disk, about 15 kpc from the center to the edge, sits 50 kpc from the LMC, so R/d is about 0.3. Reading across a body of that relative size, with a single point deformation as the perturber:

R/d	Near-side reading vs far side
0.3	2.55 to 1
0.5	5.4 to 1
0.8	34.7 to 1

A symmetric tide would be 1 to 1. At these sizes the odd rungs dominate, and the tide is lopsided: a strong near-side pull and a weak far-side one. That is what stretches a bridge toward a perturber and leaves a tail on the far side. Toomre and Toomre (1972) obtained bridges and tails in close galaxy encounters from test particles around point masses, with no tide formula.

The series in R/d stops converging as R approaches d. The sum of heights does not, so galaxy simulations add point deformations directly, and every rung arrives at once.

A galaxy as the source. We built extended sources from 20,000 point deformations of total depth 1 and radius 1, and read the slope at different radii. The inside values carry a few percent of sampling noise.

Source	Inside	Outside
Uniform sphere	slope proportional to r (slope/r from 0.98 to 1.17)	slope · r² = 1.00 beyond r = 1.5
Halo whose enclosed depth grows linearly with r	slope proportional to 1/r (slope · r from 0.96 to 1.02)	slope · r² = 1.00 beyond r = 1.5

The 1/r² is the exterior of a compact source. The slope's dependence on distance is the derivative of whatever shape the sum makes. The halo's 1/r slope is the flat-rotation profile that the simulation uses for the Milky Way's outer part.

5. How the inverse square came to be read as action at a distance

Newton derived the inverse square from Kepler's laws and wrote it as a relation between pairs of bodies. In the Principia (1687) he was explicit that he used the word “attraction” in a general sense, without deciding what physical cause lay behind it, and in the General Scholium added in 1713 he wrote that he had not been able to find the cause of gravity from the phenomena and framed no hypotheses.

He went further in a letter to Richard Bentley in 1693. He wrote that the idea that gravity could act between bodies across a vacuum, with nothing between them to carry it, was “so great an absurdity” that no one capable of thinking in philosophy could fall into it. He left the question of the agent open, to his readers.

The doctrine that he declined to hold came in anyway. In his preface to the second edition of the Principia (1713), Roger Cotes argued that gravity should be counted among the primary properties of bodies. The inverse square, written as a force between pairs, then looked like exactly the unmediated influence that Newton had called absurd. Leibniz objected that this made gravity an occult quality and a miracle, and the objection runs through his correspondence with Samuel Clarke (1715–1716).

The mathematics was already pointing the other way. Laplace (1780s) showed that the potential outside matter obeys a local differential equation, and Poisson (1813) generalized it to ∇²V = −4πGρ. In this form the shape is primary and the force is its gradient. Faraday's field, and Maxwell's equations for electricity and magnetism, took the same step for those forces. For gravity, a theory without action at a distance arrived only with Einstein's field equations in 1915. In textbooks, the pair-sum of inverse squares remained the way gravity was first introduced.

6. The reading in this paper: nothing acts where it is not

The landscape reading takes the shape as primary. A deformation of depth X has a shape 1/r that exists at every coordinate. An object sitting at a coordinate reads the slope of the shape there. It reads what is at its own coordinates, and nothing acts across a gap.

The pair-sum, adding up inverse squares from each source, is the integral form of the same fact. It is a way of calculating the shape from its sources, like adding up the Green's function of Poisson's equation. In the differential form, the relation between the shape at a point and the sources nearby is local. The two forms agree, so “action at a distance” is a feature of how the formula is written, not of the physics it describes. The simulation computes the shape by summing over deformations, as a method, and each object reads only the slope at its own coordinates.

This is also why the sequence of orders matters. If 1/r² were a separate law, there would be no reason for 1/r³ to follow from it. In the shape reading, the tide and its asymmetry are simply the same shape differentiated again.

7. What this does not settle, and the limits

• Propagation. If a deformation moves, how fast does the shape at a distant coordinate change? The simulation reads every object at the same instant. In the standard theory, the pull points at the source's present position to first order and shows delay only at second order, through acceleration. In the landscape reading, length is a delay in now, so the delay is built into how a coordinate is defined. That is not yet derived.
• The origin of the shape. The depth X and the profile 1/r are inputs. In three dimensions, 1/r is the shape a point source makes when its deformation spreads evenly, but this paper does not derive why deformations spread that way.
• Other orders. The ladder shows the point-source case. Extended bodies bring in more terms, and sections 3 and 4 only show a dumbbell and two kinds of extended cloud.

The limits of the relative-time picture:

• The range of the ladder. The series in R/d converges only for R < d. Beyond that the sum of heights takes over from the series.
• The rate of time. The rate 1 − X/(rc²) is the first-order form. The full form, √(1 − 2X/(rc²)), reaches zero at r = 2X/c², about 2.95 km for the Sun and 9 mm for the Earth. Every rung in this paper is a weak-field rung. At the Earth the Sun's X/(rc²) is 1 × 10⁻⁸.
• The noise floor. Each deeper rung hides below the measurement floor until a body is a large fraction of its distance. That is a statement about measurement and not about existence.

8. Conclusion

The inverse square is the first gradient of the shape 1/r. It is the change of the rate of time per length, times c². The tide, 1/r³, is the second gradient, and the asymmetry of the tide, 1/r⁴, is the third. Each was measured on the simulation's landscape. None of them is a law of its own, and the force is the first rung of the ladder. The reading of 1/r² as action at a distance came from writing the shape as a pair-sum and then treating the sum as a mechanism, which Newton himself refused to do.

Appendix: names used

• X: the depth of a deformation, G·m in SI units.
• Shape: the height of the landscape, −X/r. Read as a rate of time it is ρ = 1 − X/(rc²).
• Slope: the change of the rate of time per length, times c², X/r².
• Tide: the change of the slope across a body, 2X/r³ along and −X/r³ across.
• Rung: one order of derivative of the shape.

References

• Newton, I. (1687; 2nd ed. 1713). Philosophiae Naturalis Principia Mathematica, including the Scholium to Book I, Section XI, the General Scholium, and Cotes's preface to the second edition.
• Newton, I. (1693). Third letter to Richard Bentley, 25 February 1692/3.
• Alexander, H. G. (ed.) (1956). The Leibniz–Clarke Correspondence. Manchester University Press.
• Poisson, S. D. (1813). Remarques sur une équation qui se présente dans la théorie des attractions des sphéroïdes. Nouveau Bulletin des Sciences par la Société Philomatique de Paris, 3, 388–392.
• Einstein, A. (1915). Die Feldgleichungen der Gravitation. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, 844–847.
• Companion: “The Algebraic Decoding of Newton's Vector.”
• Companion: “Tides Fall Out of the Landscape.”
• Companion: “Solar System as Moving Deformations.”

• Cartwright, D. E. and Tayler, R. J. (1971). New computations of the tide-generating potential. Geophysical Journal of the Royal Astronomical Society, 23, 45–74.
• Toomre, A. and Toomre, J. (1972). Galactic bridges and tails. Astrophysical Journal, 178, 623–666.