I have only a superficial understanding of GR, but nevertheless, your question seems a bit unclear and/or confused. A few important points:
Whether GR is actually a Machian theory is a moot point, because it turns out that Mach’s principle is hard to formulate precisely enough to tackle that question. See e.g. here for an overview of this problem: http://arxiv.org/abs/gr-qc/9607009
According to the Mach’s original idea—whose relation with GR is still not entirely clear, and which is certainly not necessarily implied by GR—a necessary assumption for the “normal” behavior of rotational and other non-inertial motions is the large-scale isotropy of the universe, and the fact that enormous distant masses exist in every direction. If the only other mass in the universe is concentrated nearby, you’d see only weak inertial forces, and they would behave differently in different directions.
The geometry of spacetime in GR is not uniquely determined by the distribution of matter. You can have various crazy spacetime geometries for any distribution of matter. (As a trivial example, imagine you’re living in the usual Minkowski or Schwarzschild metric, and then a powerful gravitational wave passes by.) In this sense, GR is deeply anti-Machian.
That said, assuming nothing funny’s going on, in the scenario you describe, the classical limit applies, and the planets would move pretty much according to Newton’s laws. This means they’d both be orbiting around their common center of mass, so it’s not clear to me that the observations you listed would be possible. [ETA: please ignore this last point, my typing was faster than my thinking here. See the replies below.]
Therefore, the only way I can make sense of your example would be to assume that the other planet is much heavier than yours, and that the Schwarzschild metric applies and gives approximately Newtonian results, so we get something similar to the Moon’s rotation around the Earth. Is that what you had in mind?
it’s not clear to me that the observations you listed would be possible. … the only way I can make sense of your example would be to assume that the other planet is much heavier than yours
I don’t understand. The listed observations are in accordance with Newton, whatever the masses of the planets.
Yes, you’re right. It was my failure of imagination. I thought about it again, and yes, even with similar or identical masses, the rotations of individual planets around their own axes could be set so as to provide the described view.
I have only a superficial understanding of GR, but nevertheless, your question seems a bit unclear and/or confused. A few important points:
Whether GR is actually a Machian theory is a moot point, because it turns out that Mach’s principle is hard to formulate precisely enough to tackle that question. See e.g. here for an overview of this problem: http://arxiv.org/abs/gr-qc/9607009
According to the Mach’s original idea—whose relation with GR is still not entirely clear, and which is certainly not necessarily implied by GR—a necessary assumption for the “normal” behavior of rotational and other non-inertial motions is the large-scale isotropy of the universe, and the fact that enormous distant masses exist in every direction. If the only other mass in the universe is concentrated nearby, you’d see only weak inertial forces, and they would behave differently in different directions.
The geometry of spacetime in GR is not uniquely determined by the distribution of matter. You can have various crazy spacetime geometries for any distribution of matter. (As a trivial example, imagine you’re living in the usual Minkowski or Schwarzschild metric, and then a powerful gravitational wave passes by.) In this sense, GR is deeply anti-Machian.
That said, assuming nothing funny’s going on, in the scenario you describe, the classical limit applies, and the planets would move pretty much according to Newton’s laws. This means they’d both be orbiting around their common center of mass, so it’s not clear to me that the observations you listed would be possible. [ETA: please ignore this last point, my typing was faster than my thinking here. See the replies below.]
Therefore, the only way I can make sense of your example would be to assume that the other planet is much heavier than yours, and that the Schwarzschild metric applies and gives approximately Newtonian results, so we get something similar to the Moon’s rotation around the Earth. Is that what you had in mind?
I don’t understand. The listed observations are in accordance with Newton, whatever the masses of the planets.
Yes, you’re right. It was my failure of imagination. I thought about it again, and yes, even with similar or identical masses, the rotations of individual planets around their own axes could be set so as to provide the described view.