General Relativity/Tidal forces

Radially from an object of mass m, the tidal acceleration is $\frac{-2Gm}{r^3}*\mathrm{\Delta }X$ and perpendicular to the radial line, the acceleration is $\frac{Gm}{r^3}*\mathrm{\Delta }X$ (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 $\frac{Gm}{r^2}$. When you consider two test particles separated by a distance of $\mathrm{\Delta }X$, 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 $\frac{Gm}{(r+\mathrm{\Delta }X{)}^2}-\frac{Gm}{r^2}=\frac{Gm{r}^2-Gm(r+\mathrm{\Delta }X{)}^2}{r^2(r+\mathrm{\Delta }X{)}^2}=\frac{-2Gmr\mathrm{\Delta }X+Gm\mathrm{\Delta }{X}^2}{r^2(r+\mathrm{\Delta }X{)}^2}\sim \frac{-2Gmr\mathrm{\Delta }X}{r^4}=\frac{-2Gm}{r^3}\mathrm{\Delta }X$

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