The relative acceleration of neighboring freely falling bodies is measured by geodesic deviation, comparing their infinitesimal separation in a parallel transport frame along a reference timelike geodesic with tangent . With defined by the covariant derivative commutator, it has the displayed form. In the Newtonian limit the corresponding tidal tensor is the spatial gradient of the gravitational acceleration. A single freely falling body can remove local gravitational acceleration by its frame choice, whereas relative tidal acceleration measures curvature.
Tidal acceleration refers to the gradual increase in the rotational period of a celestial body, such as a planet or moon, due to the gravitational interaction with another nearby body. This phenomenon is primarily observed in the context of the Earth-Moon system and the Earth-Sun system. In the case of the Earth and Moon, the gravitational pull of the Moon creates tides in the Earth's oceans.
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