To put this question in context, special relativity defines the relationship between the 1st derivative with respect to time (velocity) and the first derivative with respect to space (slope). This is the essence of γv = c tan(tilt angle) = c sinh(rapidity). General relativity defines the relationship between the 2nd derivative with respect to time (acceleration, which is force/mass) and the 2nd derivative with respect to space (curvature). The 3rd derivative with respect to time is jerk. But that's where this question comes in.

In a function of a single space dimension, slope is just the pitch of a tangent line at a point on the function. Curvature is the rate of change of slope, and if we plot the slope against the space axis, the curvature is the slope of a tangent line to this new curve. In terms of the original function, it describes how much the tangent is bent. We can apply the same algorithm to the 3rd derivative, and it is the tangent line to a point on the 2nd derivative. It describes how much the curvature changes at a point. I am not sure what to call it, but the situation is worse in 3D space functions. The 1st derivative is now a gradient in 3 directions, and reflects the tilt of a plane at a point on the function. The 2nd derivative describes the rate of change of each of the 3 components of the gradient with respect to each of the 3 coordinates. This is a composite of divergence and curl. Extending this to a 3rd order derivative implies a tensor of rank 3 with 27 components, and, again, I'm not sure what to call it, or what its physical significance is.

I am inclined to identify it as a measure of "bumpiness" in space. It seems as if there is a connection to the concept of "action", but it isn't obvious. In the first place, the principle of least action, which defines a geodesic curve, is not always a straight line, so it does not mean least curvature. It actually refers to maximizing smoothness, which seems to relate to the 3rd derivative with respect to space, a property of space which is independent of jerkiness in time, which even a smooth spatial trajectory can have. Similarly, space can have a bumpiness that is independent of the smoothness of time. Of course, bumpy space and jerky time can both be present, and can in fact be inverses so that the trajectory is actually smooth (or not). Can anyone shed any light on this subject?