Newtonian Tidal Acceleration

$$\Delta a = \frac{2GMR}{d^3}$$

Status: Proven.

This is the standard first-order (leading-term) approximation for the differential gravitational acceleration across an extended body of radius $R$, caused by an external mass $M$ at distance $d$. It follows directly from a Taylor expansion of Newton’s law of gravitation, $F=GMm/r^2$, comparing the gravitational acceleration at the near and far sides of the body — the difference, not the acceleration itself, is what raises tides, distorts orbits, and can shred a body that ventures inside another’s Roche limit.

Worth noting why that difference is the interesting quantity rather than a technicality. A uniform gravitational acceleration can be transformed away entirely by falling freely along with it — that is the equivalence principle, and it is why astronauts in orbit are weightless. What free fall cannot erase is the variation in that acceleration from one side of a body to the other. Tidal acceleration is therefore the part of gravity that survives free fall, which is why it does real, visible work on oceans and stars. In general relativity the same quantity appears as geodesic deviation, encoded in the Riemann curvature tensor; the formula above is its Newtonian limit.

Real, independent confirmation: this is the textbook derivation behind ocean tides (the Moon’s and Sun’s differential gravitational acceleration across Earth), tidal locking, and tidal disruption events observed in astronomy (e.g., stars torn apart by black holes). It’s standard, undergraduate-level celestial mechanics, not a contested or novel result.

Used in: Paper 1, applied to estimate the Moon’s tidal bulge on Earth (~0.7 m). This is a direct, unmodified application of the real formula — nothing PBT-specific about the equation itself here.