The Algebraic Decoding of Newton's Vector: Revealing the Scalar Landscape of Gravitation

 J. Rogers, SE Ohio

Revised and extended draft. Companion to the simulation “Solar System as Moving Deformations” and to “The Landscape of Time.”

Abstract

Newton's law of universal gravitation, F = G m₁ m₂ / r², is usually read as a force: a causal vector that passes between pairs of masses. This paper shows that the equation can be factored, with no change to its content, into two independent readings of a scalar landscape. Writing the masses as dimensionless ratios X in a natural unit chart, which is a gauge with the same standing as SI, the law becomes F = X₁X₂ / r². It factors as X₁s₂ = X₂s₁, where s_j = X_j / r² is the slope of body j's deformation. That slope is the derivative of the shape 1/r, so the inverse-square form is what differentiating an inverse-distance shape gives, not a separate assumption. The standard equation of motion already contains only the slope, because the body's own mass cancels. The paper then lists the assumptions the standard framework leaves unspoken, shows where each one appears in the simulation, and shows that the two models give the same numbers. They differ in what they take as primitive.

The mapping in five steps

Each step is a section of the paper, followed by the line of the simulation that does the same thing.

Step	In the paper	In the simulation
1. Strip the gauge	Write each mass as a dimensionless depth X. Units are a chart, and G, c and h live only in the Jacobians between charts.	Each object has a depth, G·m in AU³/yr². The simulation's units are just another chart.
2. Factor the equation	F = X₁X₂ / r² splits two ways, X₁s₂ = X₂s₁. That gives two independent readings, a_(1) = s₂ and a_(2) = s₁, and no force is needed.	Each object asks the landscape one thing at its own coordinates, skipping itself: downhill(x_i, y_i, z_i, skip = i). Two objects, two independent readings.
3. The slope is the derivative of a shape	A deformation of depth X and shape 1/r has height h = −X/r and slope dh/dr = X/r².	The profile is the height −depth/r, and the code computes depth/r² times the direction: w = depth[j] / (r2 * Math.sqrt(r2)); sx += dx * w.
4. Each body reads the other's slope	The force is the slope multiplied by the mass of the body that read it. That is an accounting step, and Newton's second law divides it away again.	The step changes velocity by the slope times dt, v += a * dt / 2. Nothing multiplies by mass, so no force is ever formed.
5. It is an identity, not a mechanism	Newton's law is two bodies each reading the other's slope.	The same numbers come out: closed orbits, moon periods, a straight barycenter.

Steps 1 to 3 are sections 2 to 4. Step 4 is developed in sections 5.1 and 5.2, and step 5 in sections 5.4 and 7. With many bodies, step 2 repeats for every other object and the slopes add. That is the simulation's downhill loop and Newton's many-body sum, term by term.

1. Introduction: the vector ontology and its discontents

In the standard vector ontology, the universe is treated as a collection of two-body problems. The equation F = G m₁ m₂ / r² is read as a mechanism: two objects generate a force that crosses the gap between them.

That reading takes a unit of the human coordinate chart, the newton, and treats it as a substance. As argued in earlier work (The Vulgar Error), the map is mistaken for the territory. This paper decodes the algebra of the equation and shows that the force F is an intermediate quantity in an accounting. What remains when it is removed is a scalar landscape that the equation of motion was reading all along.

2. Units are a gauge

A unit system is a choice of three numbers, a unit of mass, of length and of time, by which physical quantities are written as numbers. SI is one such choice. The Planck chart is another. The simulation's chart, with lengths in astronomical units, times in years and masses entering as depths G·m, is a third. They differ by Jacobians, the fixed factors that convert numbers in one chart into numbers in another. None of them is privileged. What carries physical content are the dimensionless ratios, which are the same in every chart.

In the Planck chart the units are m_P = √(hc/G), r_P = √(hG/c³) and t_P = r_P/c. Writing a mass as X = m/m_P and a distance as r_nat = r/r_P, Newton's law becomes

F_nat = X₁ X₂ / r_nat²,

measured in units of the force Jacobian J_F = m_P r_P / t_P², which equals c⁴/G. The SI force is F_SI = J_F · F_nat. The corresponding Jacobian for acceleration is J_a = r_P / t_P². The constants G, c and h appear only inside these Jacobians. They are conversion factors between charts, and in the natural chart they are all 1.

A note on h and ħ. This paper uses Planck's original units, built on h. The reduced chart built on ħ changes every X and every r_nat by a factor of √(2π), and the Jacobians change to compensate, so no physical acceleration changes. In the author's view the reduced chart hides a 2π geometric ratio inside the gauge, which obscures the underlying geometry. That is a statement about which chart is cleaner, not about the physics.

3. The symmetric factoring, with the names fixed

The notation below replaces the earlier draft's a₁ and a₂, which were used in two different senses.

• X_j is the depth of body j: dimensionless in the Planck chart, G·m_j in the simulation's chart.
• s_j(r) = X_j / r_nat² is the slope of body j's deformation at distance r. It depends only on the source, body j.
• a_(i) is the acceleration of body i. The subscript says who accelerates, and s_j says whose deformation is being read.

The natural force factors two ways:

F_nat = X₁ · s₂ = X₂ · s₁.

Body 1 sits at distance r from body 2 and reads body 2's slope, so a_(1) = s₂ = X₂ / r_nat². Body 2 reads body 1's slope, so a_(2) = s₁ = X₁ / r_nat². Then F_nat = X₁ a_(1) = X₂ a_(2).

The force is what you get by multiplying a slope by the mass of the body that read it. It is formed from each reading separately, and the two products happen to be equal.

For the reader of the earlier draft:

Earlier draft	This draft
a₂ in section 3 (acceleration of mass 1)	a_(1) = s₂
a₁ in section 4 (X₁ / r²)	s₁
a₂ in section 5 (acceleration of mass 2)	a_(2) = s₁
F_nat = X₁X₂ m_P / (r_nat t_P)²	F_SI = J_F · X₁X₂ / r_nat², with J_F = m_P r_P / t_P² written out

4. Shape and slope: why 1/r² is the derivative of 1/r

Take a deformation of depth X and shape 1/r. Its height at distance r is

h(r) = −X / r,

a dimple that is zero far away and deeper toward the source. Its slope is

dh/dr = d(−X r⁻¹)/dr = X r⁻² = X / r².

So the inverse-square form is what differentiation does to an inverse-distance shape. The exponent 2 is the 1 from the shape plus the 1 that the derivative adds. The slope is positive outward, so downhill points toward the source.

The reverse also holds. Integrating the slope inward from infinity gives −X/r back. The shape and the slope carry the same information, and the shape is the primitive one. This is the gravitational potential per unit mass, φ = −GM/r, which textbooks introduce after the force and treat as bookkeeping. Here the order is reversed: the shape comes first and the slope is read from it.

There is also a geometric reason for the 1/r shape. In three dimensions, a point source that spreads its deformation evenly over a sphere of area 4πr² produces exactly this shape, which is the statement that 1/r solves the Laplace equation away from the source.

One number in four charts. The acceleration of the Earth toward the Sun:

Chart	Depth	Distance	Slope at that distance	Acceleration in SI
SI	G·M = 1.3271 × 10²⁰ m³/s²	1.4960 × 10¹¹ m	GM/r²	5.9303 × 10⁻³ m/s²
Planck (h)	X = 3.6449 × 10³⁷	r_nat = 3.6925 × 10⁴⁵	X/r_nat² = 2.6732 × 10⁻⁵⁴, times J_a = 2.2184 × 10⁵¹ m/s²	5.9303 × 10⁻³ m/s²
Reduced (ħ)	X = 9.1364 × 10³⁷	r_nat = 9.2558 × 10⁴⁵	X/r_nat² = 1.0665 × 10⁻⁵⁴, times J_a = 5.5607 × 10⁵¹ m/s²	5.9303 × 10⁻³ m/s²
Simulation	depth = 4π² AU³/yr²	1 AU	39.478 AU/yr²	5.9303 × 10⁻³ m/s²

The four charts give different X and different r_nat, and one acceleration.

5. How the standard framework works the same as the simulation

5.1 The equation of motion already contains only the slope

In the standard framework, the second law and the gravitational force combine as

m₁ a_(1) = G m₁ m₂ / r², so a_(1) = G m₂ / r².

The mass m₁ cancels. What is left contains only the other body's G m, which is its depth, and the distance. That is the slope of body 2's 1/r shape at body 1's coordinate. The force F was formed on the way, multiplied by m₁, and then divided by m₁ again. The standard framework already computes a slope reading. The decoding shows that the intermediate step is not needed.

5.2 Many bodies: the same sum, term by term

For many bodies the standard framework uses superposition:

a_(i) = Σ over j ≠ i of G m_j (x_j − x_i) / |x_j − x_i|³.

The simulation's landscape does the same, in one function that every object calls with its own coordinates:

const dx = x[j] - qx, dy = y[j] - qy, dz = z[j] - qz, r2 = dx*dx + dy*dy + dz*dz + EPS;
w = depth[j] / (r2 * Math.sqrt(r2));
sx += dx * w;  sy += dy * w;  sz += dz * w

The weight w times dx is (depth / r²) × (dx / r): the slope X/r² times the unit direction toward the source. With depth_j = G m_j, this is the standard term, one for one. No object forms a force and no object multiplies by its own mass.

5.3 The unspoken assumptions

The decoding exposes assumptions that the standard framework leaves unspoken. Each one has a place in the simulation.

Unspoken in the standard framework	Where it shows in the simulation
The mass that sources the field equals the mass that resists. The same X plays both roles, and this is why F₁₂ = −F₂₁	An object's own depth never enters its own motion. The depth-weighted barycenter stays straight to better than 1e-8 AU when the outside deformations are off
Contributions add linearly (superposition)	Heights add in the landscape, and the slope is their sum
An exact 1/r shape from a point source	One profile for compact deformations, softened at very short range
Distance is flat Euclidean r	Coordinates are flat and measured from the Sun
One shared “now” for every pair	Every object is read at the same instant
A straight line is the default motion (the first law)	The step carries each object's velocity forward
A unit chart, with G as its constant	Depth is G·m in AU³/yr²

5.4 Why the two models give the same numbers

The factoring in section 3 is an identity, so any quantity computed from F = m a is also computed from the slope, and nothing else is added. The simulation is the numerical demonstration. It reproduces closed orbits, equal areas, the period and distance relation, and the periods of the twenty-one moons to within about 0.25% (the Moon to about 1%, which is the size of the Sun's own effect on it). The depth-weighted barycenter stays straight, and Mercury's advance from the other planets comes out at 519 arcseconds per century, against about 532 expected from them. These are the standard Newtonian results, as the identity says they must be.

5.5 What differs

The two models differ in what is primitive. The standard framework takes the force as basic and the potential as a convenient integral of it. This framework takes the shape as basic and the force as the slope of the shape times a mass. In the Newtonian limit nothing observable separates them. They begin to separate where one of the unspoken assumptions stops holding.

6. Where the unspoken assumptions stop holding

The list in section 5.3 is a checklist for where the Newtonian numbers will miss. Each relaxed assumption corresponds to a known correction, and the simulation lets each one be tested alone. For Mercury's relativistic advance of 42.98 arcseconds per century, the results are:

Relaxation	Advance	Share of 42.98
Motion follows the clock rate of a moving body, linear form of the rate	21.49	1/2
The same, full form of the rate, √(1 + 2H/c²)	28.66	2/3
Full form, plus radial length measured by delay (dr/ρ)	42.98	1

The last step multiplies the previous one by exactly 3/2, which is the same 3/2 as in the GPS clock formula. The flat Euclidean r assumption therefore supplies the last third of Mercury's advance. Superposition and the shared “now” are the other two places to look. A nonlinear combination of heights has not been tried here, and the correction for a delayed “now” is known to cancel at first order. These are the next tests, and the details are in the companion paper.

7. Conclusion

Newton's equation is not a causal mechanism. It is an identity that can be read as two bodies each reading the slope of the other's 1/r deformation, with the slope multiplied by the reader's mass to form a force that the equation of motion immediately divides away. The 1/r² in the law is the derivative of the 1/r shape. The constants G, c and h live in the Jacobians between unit charts, and the physical content is in the dimensionless ratios. The standard framework already contains the landscape. It simply treats the slope as primary, and the shape as bookkeeping.

Appendix: names used

• X_j: depth of body j. Dimensionless in the Planck chart, G·m_j in the simulation's chart.
• s_j(r): slope of body j's deformation at distance r, equal to X_j / r².
• a_(i): acceleration of body i. The subscript says who accelerates.
• F: the product of a slope and the mass of the body that read it. It is formed only in the standard accounting.
• J_m, J_r, J_t, J_F, J_a: the gauge Jacobians between a unit chart and the natural one. In the Planck chart, J_F = c⁴/G.

References

• Newton, I. (1687). Philosophiae Naturalis Principia Mathematica.
• Planck, M. (1899). Über irreversible Strahlungsvorgänge. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, 440–480.
• Companion: “The Landscape of Time: Clock Rates on the Slope Planets Fall Down.”
• Companion: “Solar System as Moving Deformations.”