Is there a simple geometric interpretation of the Einstein tensor ? I know about its algebraic definitions ( i.e. via contraction of Riemann's double dual, as a combination of Ricci tensor and Ricci scalar etc etc ), but I am finding it hard to actually understand it geometrically. Specifically, what I am looking for is an interpretation along the lines of what Misner/Thorne/Wheeler do in "Gravitation" for the Riemann tensor, in terms of the "slots" of tensor understood as being a linear "machine". You take the tangent vector to a reference geodesic, insert it into both slots 1 and 3 of Riemann, and insert the separation vector to some neighbouring geodesic into slot 2; the result is a vector which signifies the ( covariant ) rate of change of your separation vector ( relative acceleration ) between the geodesics, with respect to your chosen time coordinate. Geodesic deviation, in other words. Can something similar be done with the Einstein tensor ? Apologies in advance if this question turns out to be either meaningless, imprecise, or trivial. Hopefully you can see what I am trying to get at.