Imagine a 2D plane where x is space and y is time. Let’s say the Earth is stationary at x=0, so its trajectory is the y axis, and the metric of spacetime is “curved” near it.
We can represent the metric visually by sprinkling a bunch of sand near the y axis. Then lines of inertial movement (“geodesics”) can be understood in two ways:
Given a pair of points, a geodesic is the line between them with the least sand (this represents the line being shortest according to the metric).
Given a starting point and velocity vector, keep moving so as to keep equal amounts of sand on your nearby left vs. nearby right—in other words, curve toward more sand.
Surprisingly, these two views are equivalent! For example, consider the geodesic from (1,0) to (1,1). It will bulge slightly away from the y axis, to avoid sand, and so at each point it will be curving toward more sand.
Now we can answer your original question. Place an object at (1,0) with velocity vector (0,1) (zero spatial velocity) and let it go. It will keep moving in the positive y direction, but curve toward the y axis where there’s more sand, and eventually cross it at an angle. Then it will curve back by symmetry, and so on, oscillating back and forth in the x coordinate while moving forward in time.
Can that really be a shortest line between two points? Why not. Say the object makes one full oscillation, traveling from (1,0) to (-1,1) to (1,2). If you try to “straighten” the line by pulling on the endpoints, the midpoint will be pulled toward the y axis and catch more sand. So it might well be a local minimum.
Imagine a 2D plane where x is space and y is time. Let’s say the Earth is stationary at x=0, so its trajectory is the y axis, and the metric of spacetime is “curved” near it.
We can represent the metric visually by sprinkling a bunch of sand near the y axis. Then lines of inertial movement (“geodesics”) can be understood in two ways:
Given a pair of points, a geodesic is the line between them with the least sand (this represents the line being shortest according to the metric).
Given a starting point and velocity vector, keep moving so as to keep equal amounts of sand on your nearby left vs. nearby right—in other words, curve toward more sand.
Surprisingly, these two views are equivalent! For example, consider the geodesic from (1,0) to (1,1). It will bulge slightly away from the y axis, to avoid sand, and so at each point it will be curving toward more sand.
Now we can answer your original question. Place an object at (1,0) with velocity vector (0,1) (zero spatial velocity) and let it go. It will keep moving in the positive y direction, but curve toward the y axis where there’s more sand, and eventually cross it at an angle. Then it will curve back by symmetry, and so on, oscillating back and forth in the x coordinate while moving forward in time.
Can that really be a shortest line between two points? Why not. Say the object makes one full oscillation, traveling from (1,0) to (-1,1) to (1,2). If you try to “straighten” the line by pulling on the endpoints, the midpoint will be pulled toward the y axis and catch more sand. So it might well be a local minimum.