Imagine a simplified scenario: only one planet. Is the planet rotating or not? You could construct a Foucault pendulum and see. It will show you a definite answer: either its plane of oscillation moves relatively to the ground or not. This doesn’t depend on distant stars.
Sure it does. If the rest of the objects in the universe were rotating in unison around the earth while the earth was still, that would be observationally indistinguishable from the earth rotating. The GR equations (so I’m told[1]) account for this in that, if the rest of the universe were treated as rotating, that would send gravitaitonal waves that would jointly cause the earth to be still in that frame of reference.
Remove that external mass, and you’ve removed the gravity waves. Nothing cancels the gravity wave generated by the motion of the planets.
It is possible that more versions of boundary conditions are acceptable in the absence of distant objects and the question whether the planet is rotating is then less defined.
Yes, I think that agrees with my answer to the question.
Einstein’s theory further had the property that moving matter would generate gravitational waves, propagating curvatures. Einstein suspected that if the whole universe was rotating around you while you stood still, you would feel a centrifugal force from the incoming gravitational waves, corresponding exactly to the centripetal force of spinning your arms while the universe stood still around you. So you could construct the laws of physics in an accelerating or even rotating frame of reference, and end up observing the same laws—again freeing us of the specter of absolute space.
Let me write one more reply since I think my first one wasn’t entirely clear.
Let’s put all this into a thought experiment like this: Universe A contains only a light observer with a round bottle half full of water. Universe B contains all that, and moreover a lot of uniformly isotropically distributed distant massive stars. In both universes the spacetime region around the observer can be described by Minkowski metric. At the beginning, the observer sees that the water is spread near the walls of the bottle with a round vacuum bubble in the middle; this minimises the energy due to surface tension. Now, the observer gives the bottle some spin. Will the observation in universe A be different from that in universe B?
If GR is right, then no, it wouldn’t. In both, the observers will see the water concentrated in regions most distant from a specific straight line, which is reasonable to call the axis of rotation. To see that, it is enough to realise that the distant stars influence the bottle only by means of the gravitational field, and it remains almost the same in both cases—approximately Minkowskian, assumed that the bottle and the observer aren’t of black hole proportions.
Of course one can then change the coordinates to those in which the bottle is static. With respect to these coordinates, the stars in universe B would rotate, and in universe A, well, nothing much can be said. But in both universes, we will find a gravitational field which creates precisely the effects of the rotation of the now static bottle. The stars are there only to distract the attention.
We can almost do the coordinate change in the Newtonian framework: it amounts to use of centrifugal force, which can be thought of as a gravitational force (it is universal in the same way as the gravitational force; of course, this is the equivalence principle). There are only two “minor” problems in Newtonian physics: first, orthodox Newtonianism recognises only gravitational force emanating from massive objects in the way described by Newton’s gravitational law, which is why the centrifugal force has to be treated differently, and second, there is the damned velocity dependent Coriolis force.
Okay, I give up. I don’t know the math well enough to speak confidently on this issue. I was just taking the Machian principles in the article I linked and extrapolating them to the scenario I envisioned, using some familiarity with frame-dragging effects.
Still, I think it’s an interesting exercise in finding the implications of a universe without the background mass, and not as easy to answer as some initially assumed.
Yes, it’s interesting, I was confused for quite a while, still the answer is simpler than what I initially assumed, which makes it a good brain teaser.
if the rest of the universe were treated as rotating, that would send gravitaitonal waves that would jointly cause the earth to be still in that frame of reference
This is not so simple. The force of the gravitational waves depends on the mass of the rest of the universe. One can easily imagine the same observable rest of the universe with a very different mass (just remove all the dark matter or so). Both can’t generate the same gravitational waves, but there would be no significant observable effect on Earth. The metric around here would be still more or less Schwarzschild (or Kerr). The fact that steady state can be interpreted as rotation whose effects are cancelled by gravitational waves has not necessarily much to do with the existence of other objects in the universe. Even in empty space, the gravitational waves can come from infinity.
So, while it’s true that there is no absolute space with respect to which one measures the acceleration, there are still Foucault pendula. Because there is no absolute space, to define what constitutes rotation using any particular coordinates would be absurd. But we can still quite reasonably define rotation (extend our present definition of rotation) by use of the pendulum, or bucket, or whatever similar device. Even in single-planet universes, there can be buckets with both flat and parabolic surfaces.
Sure it does. If the rest of the objects in the universe were rotating in unison around the earth while the earth was still, that would be observationally indistinguishable from the earth rotating. The GR equations (so I’m told[1]) account for this in that, if the rest of the universe were treated as rotating, that would send gravitaitonal waves that would jointly cause the earth to be still in that frame of reference.
Remove that external mass, and you’ve removed the gravity waves. Nothing cancels the gravity wave generated by the motion of the planets.
Yes, I think that agrees with my answer to the question.
[1] See here:
Let me write one more reply since I think my first one wasn’t entirely clear.
Let’s put all this into a thought experiment like this: Universe A contains only a light observer with a round bottle half full of water. Universe B contains all that, and moreover a lot of uniformly isotropically distributed distant massive stars. In both universes the spacetime region around the observer can be described by Minkowski metric. At the beginning, the observer sees that the water is spread near the walls of the bottle with a round vacuum bubble in the middle; this minimises the energy due to surface tension. Now, the observer gives the bottle some spin. Will the observation in universe A be different from that in universe B?
If GR is right, then no, it wouldn’t. In both, the observers will see the water concentrated in regions most distant from a specific straight line, which is reasonable to call the axis of rotation. To see that, it is enough to realise that the distant stars influence the bottle only by means of the gravitational field, and it remains almost the same in both cases—approximately Minkowskian, assumed that the bottle and the observer aren’t of black hole proportions.
Of course one can then change the coordinates to those in which the bottle is static. With respect to these coordinates, the stars in universe B would rotate, and in universe A, well, nothing much can be said. But in both universes, we will find a gravitational field which creates precisely the effects of the rotation of the now static bottle. The stars are there only to distract the attention.
We can almost do the coordinate change in the Newtonian framework: it amounts to use of centrifugal force, which can be thought of as a gravitational force (it is universal in the same way as the gravitational force; of course, this is the equivalence principle). There are only two “minor” problems in Newtonian physics: first, orthodox Newtonianism recognises only gravitational force emanating from massive objects in the way described by Newton’s gravitational law, which is why the centrifugal force has to be treated differently, and second, there is the damned velocity dependent Coriolis force.
Edit: some formulations changed
Okay, I give up. I don’t know the math well enough to speak confidently on this issue. I was just taking the Machian principles in the article I linked and extrapolating them to the scenario I envisioned, using some familiarity with frame-dragging effects.
Still, I think it’s an interesting exercise in finding the implications of a universe without the background mass, and not as easy to answer as some initially assumed.
Yes, it’s interesting, I was confused for quite a while, still the answer is simpler than what I initially assumed, which makes it a good brain teaser.
This is not so simple. The force of the gravitational waves depends on the mass of the rest of the universe. One can easily imagine the same observable rest of the universe with a very different mass (just remove all the dark matter or so). Both can’t generate the same gravitational waves, but there would be no significant observable effect on Earth. The metric around here would be still more or less Schwarzschild (or Kerr). The fact that steady state can be interpreted as rotation whose effects are cancelled by gravitational waves has not necessarily much to do with the existence of other objects in the universe. Even in empty space, the gravitational waves can come from infinity.
So, while it’s true that there is no absolute space with respect to which one measures the acceleration, there are still Foucault pendula. Because there is no absolute space, to define what constitutes rotation using any particular coordinates would be absurd. But we can still quite reasonably define rotation (extend our present definition of rotation) by use of the pendulum, or bucket, or whatever similar device. Even in single-planet universes, there can be buckets with both flat and parabolic surfaces.