= Gravity magnitude versus tidal gradient
{title2=$g=\mu/R^2,\quad \lambda_r=2\mu/R^3$}
For a fixed source <point mass>, the magnitude $g=\mu/R^2$ of the <gravitational acceleration> measures the local pull; the radial <eigenvalue> $\lambda_r=2\mu/R^3$ of the <tidal tensor> measures how that pull changes across space. Doubling $R$ divides $g$ by four but $\lambda_r$ by eight. For a fixed small radial separation $L$, the linear relative <acceleration> is $\Delta a\simeq\lambda_r L$, so
$$
\frac{\Delta a}{g}\simeq\frac{2L}{R}.
$$
A small body can therefore experience strong common <gravitational acceleration> with a comparatively small <tidal force>. A measurement of $g$ alone does not determine the <tidal tensor>: sources at different distances can have the same $g$ but different spatial variation.
\Image[/tidal-gravity-distance-scaling.png]
{title=Gravity falls as inverse distance squared, while point-mass tidal stretching falls as inverse distance cubed}
{description=The left panel compares independently normalized <gravitational acceleration> and radial <tidal tensor> <eigenvalue> for a fixed source. The right panel compares exact and linear relative <accelerations> for two radially separated <test particles> at a fixed midpoint, showing the finite-separation correction as their separation grows.}
The local stretching time scale is $\lambda_r^{-1/2}=\sqrt{R^3/(2\mu)}$: it marks when an initially resting separation could change by order itself if the <tidal tensor> stayed fixed. During <free fall>, $R$ changes, so this scale is not a complete trajectory or a universal disruption time. <Tidal disruption> additionally compares the <tidal force> with binding by <self-gravity> and material <forces>; $g$ by itself supplies no such comparison.
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