An alternative way to derive the geodesic deviation equation for rapidly diverging geodesics

Abstract

We present a derivation of the equation of geodesic deviation under the assumption that the geodesics are adjacent in some neighbourhood, but their rate of separation is arbitrary. The resulting modified equation of geodesic deviation is nonlinear, it reduces to the ordinary linear geodesic deviation equation when the changes of position of corresponding points on the two geodesics as well as the changes of directions of the corresponding tangents are small. Our derivation is straightforward but shorter and more lucid than the earlier ones. Some of the consequences of the modified equation are also discussed.