Simple(r) Derivation of Gravitational Potential = 0.5×c²/D²
(c is the speed of light, D is time dilation in the area)
Recently we derived formula for gravitational Potential = 0.5×c²/D² from Newton-Laplace formula for wave speed “c” in any medium
c = sqrt(γ×P/ρ)
by replacing the speed of wave “c” with time dilated speed of light c/D, then by leaving out γ and ρ dependencies (i.e. simplifying formula to c/D~sqrt(P)), and by recovering missing factors later, when merging with Dr. Robinson’s result. Now, let’s try a simpler approach by keeping γ and ρ in place:
- c/D = sqrt(γ×P/ρ). Now remove sqrt by squaring both sides:
- c²/D² = γ×P/ρ
- At point X: c²/D(X)² = γ×P(X)/ρ
- At point X+Δ: c²/D(X+Δ)² = γ×P(X+Δ)/ρ
- Subtract those: c²/D(X+Δ)² – c²/D(X)² = γ×(P(X+Δ) – P(X))/ ρ
Divide both sides by Δ, and multiply and divide the right side by 1m²:
💥 (c²/D(X+Δ)² – c²/D(X)²) / Δ = γ×(P(X+Δ) – P(X)) ×1m² / (ρ×1m²×Δ)
(P(X+Δ)–P(X))×1m² = – F(X), where F(X) is resulting force acting on 1m²×Δ layer. Negative sign before F(X) indicates opposite-to-X direction of F, when P(X+Δ)>P(X):
And if P(X+Δ)<P(X), then P(X+Δ)–P(X) is negative, – F(X) is negative too, therefore, F(X) is positive, thus, resulting force F points to the same direction as X.
ρ×1m²×Δ represents mass of this layer. Force divided by mass is acceleration or derivative of potential with negative sign, thus:
- γ×(P(X+Δ) – P(X)) ×1m² / (ρ×1m²×Δ) = γ×Potential'(X).
Apostrophe ' denotes derivative by location (by X in this case).
And since (c²/D(X+Δ)² – c²/D(X)²) / Δ = (c²/D(X)²)', formula
💥 (c²/D(X+Δ)² — c²/D(X)²) / Δ = γ×(P(X+Δ) — P(X)) ×1m² / (ρ×1m²×Δ)
is simplified to:
- (c²/D(X)²)' = γ×Potential'(X)
- c²/D² = γ×Potential (in physics, +constant member is ignored for potential).
Factor γ is linked to the degrees of freedom f of the medium “particles”, as γ = 1+2/f (see an explanation in Relation with degrees of freedom). For a particle moving at the speed of light, there are only two degrees of freedom: it can oscillate only in the plane perpendicular to the velocity of the particle, because oscillation in the direction of motion will change the speed of particle (or plane wave propagation), and this speed (which should be the speed of light) cannot change. More formal explanation see in P.S. below. Thus, with f = 2 and γ = 1+2/f = 2, we have:
- c²/D² = γ×Potential = 2×Potential => Potential = 0.5×c²/D²
Gravity g depends only on D' or ∇D (derivative by location or gradient of D).
P.S. Quote from a theoretical physicist: “In summary, massless particles have only two degrees of freedom because their spin states are representations of the two-dimensional rotation group SO(2), which is Abelian…”
P.P.S. Now (if you have not yet), please read Einstein’s General Relativity Becomes Elementary in 2024 on time dilation D = exp(G×M/(R×c²)) at distance R from mass M. It removes “Event Horizon” from physics.
P.P.P.S. If interested in SR rework, read
