The principle of least action states that the evolution of a physical system - how a system progresses from one state to another- is given by a stationary point of the action. So I think this is varying the path and keeping two points fixed- the points of the initial and final state I know in classical mechanics you tend to vary with respect to time. I have no intuition on why we are taking variations with respect to the metric? The metric can be used to find the space-time displacements so I think I see that the metric must be associated with varying paths through space-time, but my thoughts are unclear. Could someone explain why this is intuitively? Thanks.