General Relativity/Tidal forces

Radially from an object of mass m, the tidal acceleration is $${-2Gm \over r^3}*\Delta X$$ and perpendicular to the radial line, the acceleration is $${Gm \over r^3}*\Delta X$$ (Where G is the gravitational constant and deltaX is the seperation distance of two test particles.

To calculate those, you must remember that the acceleration due to gravity is $${Gm \over r^2}$$. When you consider two test particles separated by a distance of $$\Delta X$$, the results vary depending on if they are along a common radius from the center of the earth or perpendicular to it. (Any other cases can be decomposed into a combination of those two cases). Where the two test particles are radial, the two effective radii are r and (r+\Delta X). If you look at the relative acceleration between the two particles, you get $${Gm \over (r+\Delta X)^2} - {Gm \over r^2} = { {Gmr^2-Gm(r+\Delta X)^2} \over r^2(r+\Delta X)^2} = {-2Gmr \Delta X + Gm \Delta X^2 \over r^2(r+\Delta X)^2} \sim {-2Gmr \Delta X \over r^4} = {-2Gm \over r^3}\Delta X $$

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