The Foucault pendulum is able to measure earth’s rotation in part because of the frame established by the rest of the universe. But in the scenario I described, the frame dragging effect of one or both planets blows up your ability to use the standard equations. Would the corrections introduced by including frame-dragging show a solution that varies depending on which of the planets is “really” moving?
I must admit I’m a little baffled by this. I’m pretty ignorant of GR, but I was strongly under the impression that
(a) the frame dragging effect was miniscule, and
(b), that Foucault’s pendulum works simply because there is no force acting on the pendulum to change the plane of its rotation. Thus, a perfect polar pendulum on a planet in a universe with no other bodies in it will never have any force exerted on it other than gravity and will continue to swing in the same plane. If the planet is rotating, an observer on the planet will be able to tell this by observing the pendulum, even in the absence of any other body in the universe. Similarly, in the above paradox, an observer can tell whether their planet is revolving around the other planet while remaining oriented towards it because the pendulum will rotate over the course of a “year”.
To appreciate how differently things are when you remove the rest of the universe, consider this: what if the universe is just one planet with the people on it? How will a Foucault pendulum behave in that universe? Shouldn’t it behave quite differently, given that the rotation of the planet means the rotation of the entire universe, which is meaningless?
To appreciate how differently things are when you remove the rest of the universe, consider this: what if the universe is just one planet with the people on it?
As Prase said above, that depends on the boundary conditions. As the clearest example, if you imagine a flat empty Minkowski space and then add a lightweight sphere into it, then special relativity will hold and observers tied to the sphere’s surface would be able to tell whether it’s rotating by measuring the Coriolis and centrifugal forces. There would be a true anti-Machian absolute space around them, telling them clearly if they’re rotating/accelerating or not. This despite the whole scenario being perfectly consistent with GR.
I must admit I’m a little baffled by this. I’m pretty ignorant of GR, but I was strongly under the impression that
(a) the frame dragging effect was miniscule, and
(b), that Foucault’s pendulum works simply because there is no force acting on the pendulum to change the plane of its rotation. Thus, a perfect polar pendulum on a planet in a universe with no other bodies in it will never have any force exerted on it other than gravity and will continue to swing in the same plane. If the planet is rotating, an observer on the planet will be able to tell this by observing the pendulum, even in the absence of any other body in the universe. Similarly, in the above paradox, an observer can tell whether their planet is revolving around the other planet while remaining oriented towards it because the pendulum will rotate over the course of a “year”.
To appreciate how differently things are when you remove the rest of the universe, consider this: what if the universe is just one planet with the people on it? How will a Foucault pendulum behave in that universe? Shouldn’t it behave quite differently, given that the rotation of the planet means the rotation of the entire universe, which is meaningless?
As Prase said above, that depends on the boundary conditions. As the clearest example, if you imagine a flat empty Minkowski space and then add a lightweight sphere into it, then special relativity will hold and observers tied to the sphere’s surface would be able to tell whether it’s rotating by measuring the Coriolis and centrifugal forces. There would be a true anti-Machian absolute space around them, telling them clearly if they’re rotating/accelerating or not. This despite the whole scenario being perfectly consistent with GR.
Rotation of the planets doesn’t mean rotation of the universe, don’t forget there are not only the planets, but also the gravitational field.