Geodesic Deviation Equation

Question

When we derive the Geodesic equation, we want to actually understand the geometrical meaning of Riemann tensor. We see from the geodesic equation that the second derivative of the deviation vector is proportional to the Riemann tensor. This tells that whether our spacetime is curved or not.

Here we start by considering two geodesics and define a deviation vector between these geodesic. We then derive the evolution equation for this vector. So we derive the acceleration of deviation vector along the geodesic, that is, we take the second derivative of the deviation vector with respect to the parameter along the geodesic.

Why do we go on to take the second derivative of the deviation vector? The first derivative of the deviation vector can also say that how the deviation vector evolves as we move along one of the geodesic. So what is the need of taking the second derivative which ends up being proportional to the Riemann tensor?

Answer 1

In Minkowski (flat) spacetime free particles describe straight lines, however not necessarily parallel. For instance if the worldlines are in the same plane they may be at an angle; in that case the velocity of the deviation vector is not zero, but linear with respect to some affine parameter. Instead the acceleration is nil. This is simply the Newton first law of motion: in an inertial frame of reference, a free object either remains at rest or continues to move at a constant velocity.
If you consider a curved spacetime you apply the same logic, that is you look for the acceleration of the deviation vector between two geodesics. The physical interpretation is the manifestation of gravitational tidal forces.