geodesic deviation

Release two balls side by side high above the Earth and let them fall. Because both are drawn toward the planet's center, their paths are not quite parallel; they slowly converge as they drop. Release one above the other and they instead stretch apart, the lower one falling faster. This relative acceleration between neighbouring free-fallers, which no jump into a single falling frame can erase, is geodesic deviation, and it is gravity's true, unremovable signature.

The effect is governed by the geodesic deviation equation, D^2 xi^a / dtau^2 = -R^a_bcd u^b xi^c u^d, where xi is the separation vector between two nearby geodesics, u is the four-velocity along them, and R^a_bcd is the Riemann curvature tensor. In words: the relative acceleration of neighbouring free-falling particles is driven directly by curvature. Where the Riemann tensor vanishes, all geodesics that start parallel stay parallel; where it does not, they accelerate toward or away from one another, and the pattern of that acceleration is exactly the tidal field.

This equation is the deep resolution of an apparent tension in general relativity. The equivalence principle says a single freely-falling observer can always transform gravity away locally, so how can gravity be real? The answer is that gravity is not felt by one particle but revealed by the relative motion of two: a uniform field is fictitious and removable, but its variation, the tides, is genuine curvature that geodesic deviation makes measurable. It is also the physical content of a passing gravitational wave, which stretches and squeezes the separations between free test masses as it goes by.