I (think I) understand that:(adsbygoogle = window.adsbygoogle || []).push({});

1. With the Schwarzschild metric, the ratio of proper time to coordinate time decreases ("clock runs slower") with decreasing radial distance. (And ratio of proper distance to coordinate distance increases.)

2. The geodesic path followed by a freely falling object maximizes the magnitude of the spacetime interval between two points. (Books all say interval is "extremal" because sometimes it's a minimum rather than maximum, but no one ever explains the minimal case to me.)

I am confused because these two ideas seem contradictory. It seems that when your clock is ticking more slowly, your spacetime interval will be smaller than if it ran up a larger number of ticks.

Of course if I am comparing a worldline that bends toward the gravitational mass to a worldline that doesn't, I don't have the same endpoints, so it's apples and oranges. The principle of maximizing the interval ("principle of extremal aging") only tells me that if I picked an alternative route with the same endpoints, the interval would be smaller because it would involve more distance, and the interval is time-squared minus distance-squared.

Nevertheless I try to compare an object released from rest (apple just beginning to fall from tree) to one that stays where it is (thus not following a geodesic): I say to myself, the apple that refuses to fall is racking up more clock ticks while traveling zero distance; surely its interval (if I could find a way to compare the different worldlines) must be larger than that of the apple that falls.

I've seen all the embedding diagrams and visual aids, showing that time is "stretched out" by gravity and that the geodesics bend into the more stretched-out region. I focus in particular on the representation where a sheet with one time and one space dimension is rolled into a cylinder, with the time dimension running around it. Spacetime curvature is then shown as a fattening of the cylinder, and the geodesics bend in the direction of this fattening. Here an object at rest in a region without curvature would just follow a circle around the cylinder perpendicular to its axis; but as soon as it encounters a bulging of the cylinder, it bends into the bulge.

Here my confusion is the same: I expect the trajectory to bend away from the bulge, not toward it.

The bulging cylinder picture at first seemed to help me, because I thought, yes, it's traveling a longer distance in spacetime. But then I said wait, this "longer distance" through stretched-out time involves fewer clock ticks, not more. The time is dilated so that there is "more of it" between ticks, but it's the ticks that matter. Isn't it?

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# How does gravitational time dilation increase the spacetime interval?

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