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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