# Is the zero acceleration path also the shortest path between two points?

In flat, free, Euclidean space, the shortest path and the zero acceleration path are the same path, which is a straight line. However, in general relativity, is the zero acceleration path also the shortest path between two points? I am assuming that free fall is zero acceleration.

• – Qmechanic Sep 12 '14 at 20:54
• The answer to this question is a bit subtle, and depends on what you mean by "two points". General relativity is a theory of spacetime, and the points in spacetime are places in space <b>at</b> an exact instant. The geometry of spacetime says, then, that two events fall into three categories, being spacelike seperated if the distance between them is greater than zero, timelike seperated if the distance between them is less than zero, and null seperated if it is zero. I'm going to come back and write a proper answer to this later, but you have to consider each of these cases, – Jerry Schirmer Sep 12 '14 at 20:56
• as well as the commonsense idea of "distance between two points", which becomes "the distance between two timelike curves through spacetime" – Jerry Schirmer Sep 12 '14 at 20:56