A Derivation of Newton’s Law of Universal Gravitation
from the McGucken Principle of the Fourth Expanding Dimension

Dr. Elliot McGucken

Light Time Dimension Theory — elliotmcguckenphysics.com

April 2026

I.  Introduction

Newton’s law of universal gravitation,

F = −GMm / r²  r̂

(1)

is among the most precisely verified laws in the history of physics [1, 2]. Yet Newton himself acknowledged that his law described the what of gravity while leaving the why entirely unresolved. His famous declaration — hypotheses non fingo — was an admission that the mechanism producing the inverse square force remained unknown [3].

General relativity [4] provided a geometric restatement of gravity as spacetime curvature, from which Newton’s law emerges as a weak-field limit. But general relativity does not explain why the force falls off as 1/r² rather than some other power. This remains a feature of the geometry, not a theorem derived from a deeper principle.

The McGucken Principle [5] proposes such a deeper principle. It states that the fourth coordinate of Minkowski spacetime, x₄ = ict, is not a mathematical convenience but a physical geometric axis expanding at the invariant rate

dx4/dt = ic

(2)

In prior work [5], it was shown that from this single postulate the master equation u^μu_μ = −c², all of special relativity, the second law of thermodynamics, quantum nonlocality, the Schrödinger equation, the Principle of Least Action, and Huygens’ Principle all follow as mathematical theorems. The present paper derives Newton’s inverse square law as a further theorem of dx₄/dt = ic.

II.  The McGucken Principle and the Master Equation

Minkowski showed in 1907–1908 that Einstein’s special relativity admits a four-dimensional geometric interpretation [6]. In Minkowski’s original notation a spacetime event has coordinates (x₁, x₂, x₃, x₄) where

x4 = ict

(3)

The imaginary factor i ensures the four-dimensional Pythagorean sum reproduces the spacetime interval:

ds² = dx1² + dx2² + dx3² + dx4² = dx² − c²dt²

(4)

This notation was subsequently treated as an archaic convention. The McGucken Principle reverses this dismissal: differentiating x₄ = ict with respect to t yields immediately

dx4/dt = ic

(5)

x₄ is a genuine geometric axis advancing at the fixed rate ic per unit coordinate time — invariant in velocity, wavelength, and frequency, regardless of position or the presence of matter. From this, the four-velocity constraint yields the master equation [5]:

uμuμ = −c²

(6)

Every object’s total rate of traversal through four-dimensional spacetime is the fixed constant c. The four-speed budget is always fully spent:

(dx/dt)² + (dy/dt)² + (dz/dt)² + (dx4/dt)² = c²

(7)

A particle at rest in three dimensions directs its entire budget into x₄. As three-dimensional speed v increases, more budget is redirected into spatial motion and less into x₄. At v = c — the photon case — the x₄ component is exactly zero: dτ = 0.

III.  Mass Stretches the Three Spatial Dimensions

The McGucken Principle establishes that x₄ expands at the invariant rate ic throughout the universe. What mass does is deform the three spatial dimensions in its vicinity. The metric near a spherically symmetric mass M, in the weak-field limit (v ≪ c, GM/rc² ≪ 1), is the Schwarzschild metric [4, 7]:

ds² = −(1 − 2GM/rc²)c²dt² + (1 + 2GM/rc²)(dx² + dy² + dz²)

(8)

The spatial factor (1 + 2GM/rc²) means the proper distance between two events at coordinate separation δr at distance r from the mass is not δr but

δℓ ≈ (1 + GM/rc²) δr

(9)

Three-dimensional space is physically stretched near the mass. The closer to the mass, the greater the stretching. Far from the mass (r → ∞), space is asymptotically flat and proper distances equal coordinate distances. The x₄ axis expands at the invariant rate ic, indifferent to the mass — but it must bridge different amounts of proper spatial distance at different values of r.

IV.  Gravitational Time Dilation from the Invariant Expansion of x₄

IV.1  The Photon Clock Argument

A photon clock consists of two mirrors separated by proper distance δℓ, with a photon bouncing between them. Each round trip takes coordinate time

δtclock = 2δℓ / c

(10)

In the McGucken framework, the photon clock counts ticks of x₄’s expansion: each tick is the time for x₄ to advance across one proper length δℓ at rate c. Now place one clock at coordinate radius r from mass M, and another at r + Δr. Both have the same coordinate mirror separation δr. By equation (9) their proper separations differ:

δℓ(r) = (1 + GM/rc²)δr  >  δℓ(r + Δr) = (1 + GM/(r+Δr)c²)δr

(11)

The clock closer to the mass has a longer proper mirror separation and therefore a longer tick period. The clock deeper in the gravitational well runs more slowly. This is gravitational time dilation, derived entirely from the invariance of dx₄/dt = ic meeting the stretched spatial geometry near M. Since all physical processes are governed by the rate at which x₄ advances, all clocks slow down by the same factor — the universality of gravitational time dilation is a theorem, not a postulate.

IV.2  The Clock Rate Gradient as the Gravitational Potential

The proper time rate at coordinate radius r follows from the temporal component of metric (8):

dτ/dt = √(1 − 2GM/rc²) ≈ 1 − GM/rc² ≡ 1 + Φ(r)/c²

(12)

where the Newtonian gravitational potential is

Φ(r) = −GM/r

(13)

The radial gradient of the clock rate is

∂(dτ/dt)/∂r = GM/r²c² = −(1/c²) ∂Φ/∂r

(14)

Proper time increases with r: clocks run faster farther from the mass. The gradient of the clock rate is precisely (up to the factor −1/c²) the gradient of the Newtonian potential. This gradient is not a mysterious action-at-a-distance field; it is the geometric consequence of the invariant expansion of x₄ across a spatially inhomogeneous metric.

V.  Newton’s Inverse Square Law from the Principle of Least Action

V.1  The Principle of Least Action as a Theorem of dx₄/dt = ic

In prior work [5], the Principle of Least Action was derived as a theorem of the McGucken Principle. The relativistic action for a free particle is the unique Lorentz-invariant scalar associated with its worldline:

S = −mc² ∫ dτ

(15)

The master equation (6) fixes the four-speed at c, making proper time τ the natural geometric measure of worldline length. Varying S with fixed endpoints yields the geodesic equation

duμ/dτ = 0     (free particle)

(16)

which in the non-relativistic limit v ≪ c reduces to the Principle of Least Action [5, 7]. A free particle follows the path of maximal proper time through curved spacetime.

V.2  The Geodesic Equation in a Weak Gravitational Field

For a slowly moving particle in the weak field (8), the geodesic equation (16) evaluated via the Christoffel symbols of the metric reduces to [4, 7]:

d²xi/dt² = −∂Φ/∂xi,    i = 1, 2, 3

(17)

Substituting the potential (13):

∂Φ/∂r = ∂/∂r(−GM/r) = +GM/r²

(18)

The equation of motion in the radial direction is:

d²r/dt² = −GM/r²

(19)

In vector form:

d²r/dt² = −(GM/r²)r̂

(20)

By Newton’s third law, the mutual force on masses m and M is:

V.3  Physical Interpretation

The particle is not attracted by a mysterious action-at-a-distance force. The gradient of clock rates (14) means that proper time accumulated by any worldline depends on its radial position. The particle, obeying the geodesic principle — a theorem of dx₄/dt = ic — follows the path of maximal proper time. Because proper time varies with r as encoded by Φ = −GM/r, the geodesic is deflected toward the mass. Gravity is not a force; it is the four-dimensional least-action path of a particle navigating a universe in which x₄ expands invariantly while the three spatial dimensions are stretched by the presence of mass.

VI.  The Inverse Square Law from Spherical Symmetry and Gauss’s Theorem

VI.1  The McGucken Sphere and Isotropic Expansion

The McGucken Principle states that x₄ expands at rate c in a spherically symmetric manner from every point in space [5], since dx₄/dt = ic contains no preferred spatial direction. This expansion generates the McGucken Sphere — a spherical shell of radius R = ct centred on any event, identical to the forward light cone. The area of this sphere at radius r is

A(r) = 4πr²

(22)

VI.2  Gauss’s Theorem Applied to Gravitational Flux

The mass M acts as a point source of gravitational influence. The invariant spherically symmetric expansion of x₄ distributes this influence uniformly over every spherical shell surrounding M. The total integrated gravitational flux through any closed spherical surface surrounding M is conserved — Gauss’s law for gravity [1, 8]:

&oiint; g · dA = −4πGM

(23)

For a spherical surface of radius r, the field is uniform over the surface by the spherical symmetry of x₄’s expansion. Therefore:

|g| · 4πr² = 4πGM   ⇒   |g| = GM/r²

(24)

VI.3  Why 1/r² and Not Some Other Power

The surface area of a sphere in n-dimensional space scales as rⁿ⁻¹. In three spatial dimensions (n = 3) the area scales as r², and flux per unit area falls as 1/r². There are exactly three spatial dimensions perpendicular to x₄, making the boundary sphere two-dimensional with area 4πr². The inverse square law is the statement that gravitational influence is an isotropic flux from a point source distributed over the boundary of a ball in the three spatial dimensions perpendicular to x₄ — the same reasoning that explains Coulomb’s law and the inverse square fall-off of light from a point source.

VII.  The McGucken Framework and Verlinde’s Entropic Gravity

Verlinde [9] proposed that gravity is an entropic force arising from changes in the information content of holographic screens. The proposal attracted criticism on four main grounds: ambiguity of holographic screen locations; the difficulty of producing a conservative force from an entropic (dissipative) mechanism; absence of any mechanism for quantum nonlocality; and inconsistency with dark matter observations [10, 11]. The McGucken derivation addresses each:

Holographic screen ambiguity. The McGucken framework replaces holographic screens with the McGucken Sphere — the forward light cone surface at radius r = ct — which is frame-independently and unambiguously defined at every event.

The conservative force problem. Gravity in the McGucken framework is not an entropic force. It is a geometric force — the geodesic of least action in a spacetime whose clock rate varies with position. Geometric forces are inherently conservative, being gradients of scalar potentials. The conservatism of gravity follows directly from Φ = −GM/r.

Irreversibility and time’s arrows. The second law of thermodynamics follows from x₄’s irreversible advance as a geometric necessity [5]. Gravity follows from the conservative gradient of clock rates and requires no dissipation. The two phenomena have separate but unified origins in dx₄/dt = ic.

Quantum nonlocality. The McGucken Equivalence [5] identifies the hidden variable underlying quantum entanglement as x₄ itself: photons share an x₄ coordinate from emission to absorption because dτ = 0 along null worldlines, making their four-dimensional separation null regardless of spatial distance. This mechanism is entirely absent from Verlinde’s framework.

VIII.  Discussion: The Complete Derivation Chain

No new postulates are introduced. The derivation uses only: (i) the McGucken Principle dx₄/dt = ic; (ii) the weak-field Schwarzschild metric; and (iii) Gauss’s theorem. The inverse square law is not fitted to observation; it emerges from the geometry.

The mechanism is explicit. The question “why does gravity fall off as 1/r²?” receives a complete mechanistic answer: because the invariant, spherically symmetric expansion of x₄ distributes the mass’s geometric influence uniformly over spherical shells of area 4πr², and the flux per unit area therefore falls as 1/r².

The universality of free fall is explained. Every particle, regardless of mass or composition, follows the same geodesic because the geodesic is a property of the spacetime geometry — of the clock rate gradient produced by x₄’s invariant expansion in stretched space — and not of the particle itself. The equivalence principle is a theorem, not a postulate.

The connection to thermodynamics is preserved. The second law (dS/dt > 0) and Newton’s law (|g| ∝ 1/r²) both follow from dx₄/dt = ic, unifying the thermodynamic and gravitational descriptions of nature within a single geometric framework.

IX.  Conclusion

We have derived Newton’s law of universal gravitation, including its inverse square dependence on distance, from the McGucken Principle dx₄/dt = ic. The derivation proceeds through four conceptually transparent steps: (1) the invariant expansion of x₄ at rate ic; (2) the stretching of spatial distances near a mass; (3) the resulting gradient of clock rates equal to the Newtonian potential; (4) the geodesic of least action deflected toward the mass by that gradient. The 1/r² dependence follows from the spherical symmetry of x₄’s expansion and Gauss’s theorem on the McGucken Sphere.

Newton’s declaration hypotheses non fingo — I frame no hypotheses — was an acknowledgement that the mechanism of gravity lay beyond his reach. The McGucken Principle provides the mechanism Newton sought: gravity is the four-dimensional least-action path of matter navigating a universe in which the fourth geometric axis expands at the invariant rate ic, and the three spatial dimensions are dilated in the vicinity of mass. The inverse square law is not a brute empirical fact but a geometric theorem: the Pythagorean distribution of an isotropic flux over the spherical surface of the McGucken Sphere.

Future work will extend this derivation to the full Einstein field equations, post-Newtonian corrections, and the strong-field regime.

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© Dr. Elliot McGucken 2026 — Light Time Dimension Theory — elliotmcguckenphysics.com