The geodesic deviation equation (GDE) describes the tendency of objects to accelerate towards or away from each other due to spacetime curvature. The GDE assumes that nearby geodesics have a small rate of separation, which is formally treated as the same order in smallness as the separation itself. This assumption is discussed in various papers but is not articulated in any standard textbooks on general relativity. Relaxing this assumption leads to the generalized geodesic deviation equation (GGDE). We elucidate the distinction between the GDE and the GGDE by explicitly computing the relative acceleration between timelike geodesics in two-dimensional de Sitter spacetime. We do this by considering a fiducial geodesic and a secondary geodesic (both timelike) that cross with nonzero speed. These geodesics are spanned by a spacelike geodesic, whose tangent evaluated at the fiducial geodesic defines the separation. The second derivative of the separation describes the relative acceleration between the fiducial and secondary geodesics. Near the crossing point, where the separation between the timelike geodesics is small but their rates of separation can be large, we show that the GGDE holds but the GDE fails to apply.

中文翻译：

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德西特时空中的广义测地偏差

测地偏差方程 (GDE) 描述了物体由于时空曲率而相互加速或远离的趋势。GDE 假设附近的测地线具有小的分离率，正式地被视为与分离本身相同的小阶。这个假设在各种论文中都有讨论，但在任何关于广义相对论的标准教科书中都没有明确阐述。放宽此假设可得出广义测地偏差方程 (GGDE)。我们通过明确计算二维德西特时空中类时测地线之间的相对加速度来阐明 GDE 和 GGDE 之间的区别。我们通过考虑以非零速度交叉的基准测地线和辅助测地线（均时）来做到这一点。这些测地线被类似空间的测地线所跨越，其在基准测地线处计算的切线定义了分离。分离的二阶导数描述了基准测地线和二次测地线之间的相对加速度。在交叉点附近，时间类测地线之间的分离很小但它们的分离率可能很大，我们表明 GGDE 成立但 GDE 不适用。