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The Schwarzschild metric (ignoring the angular parts) looks like this:

##ds^2 = (1 - \frac{2GM}{c^2 r}) c^2 dt^2 - \frac{1}{1 - \frac{2GM}{c^2 r}} dr^2##

The Rindler metric in 2 spacetime dimensions looks like this:

##ds^2 = (g r)^2 dt^2 - dr^2##

They are very different, but they have some similarities, namely that a test particle initially at "rest" (that is, ##\frac{dr}{dt} = 0##) that is unsupported will "fall" to lower values of ##r##.

One critical difference is that to keep something at rest in the Schwarzschild metric need not require any energy, while keeping something at rest in the Rindler metric does require energy.

To see that it's possible for something to be at rest in the Schwarzschild metric, imagine a spherical shell completely surrounding a black hole at a safe distance away. An object could then rest on that shell without requiring any energy.

In contrast, the Rindler metric is just flat spacetime in a different coordinate system. An object being at "rest" in the Rindler metric implies that it is accelerating in the usual inertial coordinates, which requires expending energy.

So thinking about what the two metrics mean, physically, allows you to see that it is possible in the one case to have an object at rest without expending energy in the one case, but not in the other. My question is that if you have an unfamiliar metric, is there some mathematical test for whether it is possible to have an object at rest that requires no energy?

Maybe it's simply a matter of whether there are orbits? (geodesics that return to the same spatial location after an amount of time)?

##ds^2 = (1 - \frac{2GM}{c^2 r}) c^2 dt^2 - \frac{1}{1 - \frac{2GM}{c^2 r}} dr^2##

The Rindler metric in 2 spacetime dimensions looks like this:

##ds^2 = (g r)^2 dt^2 - dr^2##

They are very different, but they have some similarities, namely that a test particle initially at "rest" (that is, ##\frac{dr}{dt} = 0##) that is unsupported will "fall" to lower values of ##r##.

One critical difference is that to keep something at rest in the Schwarzschild metric need not require any energy, while keeping something at rest in the Rindler metric does require energy.

To see that it's possible for something to be at rest in the Schwarzschild metric, imagine a spherical shell completely surrounding a black hole at a safe distance away. An object could then rest on that shell without requiring any energy.

In contrast, the Rindler metric is just flat spacetime in a different coordinate system. An object being at "rest" in the Rindler metric implies that it is accelerating in the usual inertial coordinates, which requires expending energy.

So thinking about what the two metrics mean, physically, allows you to see that it is possible in the one case to have an object at rest without expending energy in the one case, but not in the other. My question is that if you have an unfamiliar metric, is there some mathematical test for whether it is possible to have an object at rest that requires no energy?

Maybe it's simply a matter of whether there are orbits? (geodesics that return to the same spatial location after an amount of time)?

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