I’d like to pose a sort of brain-teaser about Relativity and Mach’s Principle, to see if I understand them correctly. I’ll post my answer in rot13.
Here goes: Assume the universe has the same rules it currently does, but instead consists of just you and two planets, which emit visible light. You are standing on one of them and looking at the other, and can see the surface features. It stays at the same position in the sky.
As time goes by, you gradually get a rotationally-shifted view of the features. That is, the longitudinal centerline of the side you see gradually shifts. This change in view could result from the other planet rotating, or from your planet revolving around it while facing it. (Remember, both planets emit light, so you don’t see a different portion being in a shadow like the moon’s phases.)
Question: What experiment could you do to determine whether the other planet is spinning, or your planet is revolving around it while facing it?
My answer (rot13): Gurer vf ab jnl gb qb fb, orpnhfr gurer vf ab snpg bs gur znggre nf gb juvpu bar vf ernyyl unccravat, naq vg vf yvgreny abafrafr gb rira guvax gung gurer vf n qvssrerapr. Gur bayl ernfba bar zvtug guvax gurer’f n qvssrerapr vf sebz orvat npphfgbzrq gb n havirefr jvgu zber guna whfg gurfr gjb cynargf, juvpu sbez n onpxtebhaq senzr ntnvafg juvpu bar bs gurz pbhyq or pbafvqrerq fcvaavat be eribyivat.
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. If your planet is heavy and dense like hell, you could see the difference between a “rotating” Kerr metric and a “static” Schwarzschild metric.
Of course, general relativity is generally covariant, and any motion can be interpreted as a free fall in some gravitational field, and more, there is no absolute background spacetime with respect to which to measure acceleration. So you can likely find coordinates in which the planet is static and the pendulum movement explain by changing gravitational field. The price paid is that it will be necessary to postulate weird boundary conditions in the infinity. 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.
Carlo Rovelli in his Quantum Gravity (once I downloaded it from arXiv, now it seems unavailable, but probably it could still be found somewhere on the net) considers eight versions of Mach principle (MP). This is what he says (he has discussed the parabolic water surface of a rotating bucket before instead of two planets or Foucault pendula):
MP1:Distant stars can affect local inertial frame. True. Because matter affects the gravitational field.
MP2:The local inertial frame is completely determined by the matter content of the universe. False. The gravitational field has independent degrees of freedom.
MP3:The rotation of the inertial frame inside the bucket is in fact dragged by the bucket, and this effect increases with the mass of the bucket. True. This is the Lense-Thirring effect: a rotating mass drags the inertial frames in the vicinity.
MP4:In the limit in which the mass is large, the internal inertial reference frame rotates with the bucket. Depends on the details of the way the limit is taken.
MP5:There can be no global rotation of the universe. False. Einstein believed this to be true in GR, but Goedel’s solution is a counter-example.
MP6:In the absence of matter, there would be no inertia. False. There are vacuum solutions of the Einstein equations.
MP7:There is no absolute motion, only motion relative to something else, therefore the water in the bucket does not rotate in absolute terms, it rotates with respect to some dynamical physical entity. True. This is the basic physical idea of GR.
MP8:The local inertial frame is completely determined by the dynamical fields of the universe. True. In fact, this is precisely Einstein key idea.
I think number 4 is especially relevant here. The boundary conditions or the global topology of the universe have to be taken into account, else the two-planet scenario is not entirely defined.
Edit: The last remark doesn’t make much sense after all. The planets aren’t thought to be too heavy and the dragging effect shouldn’t be too big, and its relation to boundary conditions isn’t straightforward. Nevertheless, the boundary conditions still play an important role (see my subcomment here).
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.
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.
Couldn’t you tell whether your planet is revolving or rotating using a Foucault’s pendulum? I’m not sure whether you can get all the information about the planets’ relations with a complex set of Foucault’s pendula or not, but you could get some.
Also, I think your answer is a map-territory confusion. While GR does not distinguish certain types of motion from each other, and while GR seems to be the best model of macroscopic behavior we have, to claim that this means that there is really no fact of the matter seems a little overconfident.
Couldn’t you tell whether your planet is revolving or rotating using a Foucault’s pendulum? I’m not sure whether you can get all the information about the planets’ relations with a complex set of Foucault’s pendula or not, but you could get some.
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?
Also, I think your answer is a map-territory confusion. While GR does not distinguish certain types of motion from each other, and while GR seems to be the best model of macroscopic behavior we have, to claim that this means that there is really no fact of the matter seems a little overconfident.
It’s the other way around. The fact that there is no test that would distinguish your location along a dimension means that no such dimension exists, and any model requiring such a distinction is deviating from the territory.
Yes, GR could be wrong, but for it to be wrong in a way such that e.g. you actually can distinguish acceleration from gravity would require more than just a refinement of our models; it would mean the universe up to this point was a lie.
Yes, GR could be wrong, but for it to be wrong in a way such that e.g. you actually can distinguish acceleration from gravity would require more than just a refinement of our models; it would mean the universe up to this point was a lie.
This isn’t really true. In GR, you can in principle always distinguish acceleration from gravity over finite stretches of spacetime by measuring the tidal forces. There is no distribution of mass that would produce an ideally homogeneous gravitational field free of tidal forces whose effect would perfectly mimic uniform acceleration in flat spacetime. The equivalence principle holds only across infinitesimal regions of spacetime.
This isn’t really true. In GR, you can in principle always distinguish acceleration from gravity over finite stretches of spacetime by measuring the tidal forces. …
Yes, I was just listing an offhand example of an implication of GR and I didn’t bother to specify it to full precision. My point was just that in order for a certain implication to be falsified (specifically, that there is no fact of the matter as to e.g. what the velocity of the universe is), you would need the laws of the universe to change, not just a refinement in the GR model.
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.
If the two planets aren’t revolving around each other, wouldn’t gravity pull them together? But maybe space is expanding at precisely the rate necessary to keep them at the same distance despite gravity? To test that, build a rocket on your planet and push it (the planet) slightly, either toward the other planet or away from it. If the planets are revolving around each other, you’ve just changed a circular orbit into an elliptical one, so you should see an oscillation in the distance between the two planets. If they are not revolving around each other, then they’ll either keep getting closer together or further apart, depending on which direction you made the push.
(This is all based on my physics intuition. Somebody who knows the math should write down the two equations and check if they’re isomorphic. :)
If the two planets aren’t revolving around each other, wouldn’t gravity pull them together?
Gravity would pull, yes, but the rotation of a body also distorts space in such a way to produce another effect you have to consider.
ETA: Look at a similar scenario. Same as the one I proposed, but you always see the same portion of the other planet. How do you know how fast the two planets are revolving around each other? Isn’t this the same as asking how fast the entire universe is rotating?
Exactly as fast as needed to keep them on cyclical orbit (assuming you don’t experience change of the distance to the second planet). For this, you can quite safely use the Newton laws.
In general-relativistic language, what exactly do you mean by “how fast the entire universe is rotating”?
In general-relativistic language, what exactly do you mean by “how fast the entire universe is rotating”?
I mean nothing. In GR, the very question is nonsense. The universe does not have a position, just relative positions of objects.
The universe does not have a velocity, just relative velocities of various objects. The universe does not have an acceleration, just relative accelerations of various objects. The universe does not have a rotational orientation, just relative rotational orientations of various objects. The universe does not have a rotational velocity, just relative rotational velocities of various objects.
There is no way in this universe to distinguish between a bucket rotating vs. the rest of the universe rotating around the bucket. There is also no such thing as how fast the universe “as a whole” is rotating.
I’m not sure if what you write makes sense. Take one simple example: a flat Minkowski spacetime, empty except for a few light particles (so that their influence on the metric is negligible). This means that special relativity applies, and it’s clearly consistent with GR.
Accelerated motions are not going to be relative in this universe, just like they aren’t in Newton’s theory. You can of course observe an accelerating particle and insist on using coordinates in which it remains in the origin (which is sometimes useful, as in e.g. the Rindler coordinates), but in this coordinate system, the universe will not have the above listed properties in any meaningful sense.
You write “In GR, the very question is nonsense. [0] The universe does not have a position, just relative positions of objects. [1] The universe does not have a velocity, just relative velocities of various objects. [2] The universe does not have an acceleration, just relative accelerations of various objects.” This passage incorrectly appeals to GR to lump together three statements that GR doesn’t lump together.
See http://en.wikipedia.org/wiki/Inertial_frames_of_reference and note the distinction there between “constant, uniform motion” and various aspects of acceleratedness. Your [0] and [1] describe changes within an inertial frame of reference, while [2] gets you to a non-inertial frame. Not coincidentally, your [0] and [1] are predicted by GR and are consistent with centuries of careful experiment, while [2] is not predicted by GR and is inconsistent with everyday observation with Mark I eyeballs. (With modern vehicles it’s common to experience enough acceleration in the vicinity of some low-friction system to notice that acceleration causes conservation of momentum to break down in ways that a constant displacement and/or uniform motion doesn’t.)
[2] is not predicted by GR and is inconsistent with everyday observation with Mark I eyeballs.
I ask, in return, that you read this. Eliezer Yudkowsky had argued that GR implies it’s impossible to measure the acceleration of the universe, and no one had objected. Now, EY is not the pope of rationality, but I suggest things aren’t as simple as you’re making them.
(With modern vehicles it’s common to experience enough acceleration in the vicinity of some low-friction system to notice that acceleration causes conservation of momentum to break down in ways that a constant displacement and/or uniform motion doesn’t.)
Your point just seems to be a version of the bucket argument: “acceleration must be real, because it has real, detectable, frame-independent consequences like breakage and pain and ficticious forces”. I think I posed the same challenge in an open thread a month or two ago. And as the link you gave says,
In Newton’s time the fixed stars were invoked as a reference frame, supposedly at rest relative to absolute space. In reference frames that were either at rest with respect to the fixed stars or in uniform translation relative to these stars, Newton’s laws of motion were supposed to hold. In contrast, in frames accelerating with respect to the fixed stars, an important case being frames rotating relative to the fixed stars, the laws of motion did not hold in their simplest form, but had to be supplemented by the addition of fictitious forces, for example, the Coriolis force and the centrifugal force. Two interesting experiments were devised by Newton to demonstrate how these forces could be discovered, thereby revealing to an observer that they were not in an inertial frame: the example of the tension in the cord linking two spheres rotating about their center of gravity, and the example of the curvature of the surface of water in a rotating bucket.
But under Mach’s principle (the version that says only relative motion is meaningful, and which GR agrees with), these consequences of acceleration you describe only exist because of the frame against which to describe the acceleration, which is formed by the (relatively!) non-accelerating the rest of the universe. Therefore, if all of the universe were to accelerate uniformly, there would be no relative motion and therefore no experimental consequences, and we should regard the very idea as nonsense.
So if the universe were only you and your vehicle, you would not be able notice joint accelerations of you and the vehicle, only acceleration of yourself relative to the vehicle.
Now, you can disagree with this application of Mach’s principle, but the observations you describe do not contradict it.
I should also add one of the great insights I got out of Barbour’s book The End of Time (from which EY got his love of Mach’s principle and timelessness). The insight is that the laws of physics do not change in a rotating reference frame. Rather, there is a way you can determine if any given object is not in uniform motion relative to the rest of the universe, and this method also allows you to define an “inertial clock” which gives you an appropriate measure of time.
Most importantly, if you are spinning around, and there’s some other object accelerating relative to the rest of the universe, this method allows you to detect its acceleration, no matter how much or in what way your own frame is moving!
Perhaps the root of our disagreement is that you think (?) that the GR field equations constrain their solutions to conform to Mach’s principle, while I think they admit many solutions which don’t conform to Mach’s principle, and that furthermore that Vladimir_M is probably correct in his sketch of a family of non-Mach-principle solutions.
EY’s article seems pretty clear about claiming not that Mach’s principle follows deductively from the equations of GR, but that there’s a sufficiently natural fit that we might make an inductive leap from observed regularity in simple cases to an expected identical regularity in all cases. In particular EY writes “I do not think this has been verified exactly, in terms of how much matter is out there, what kind of gravitational wave it would generate by rotating around us, et cetera. Einstein did verify that a shell of matter, spinning around a central point, ought to generate a gravitational equivalent of the Coriolis force that would e.g. cause a pendulum to precess.” I think EY is probably correct that this hasn’t been verified exactly—more on that below. I also note that from the numbers given in Gravitation, if you hope to fake up a reasonably fast rotating frame by surrounding the experimenter with a rotating shell too arbitrarily distant to notice, you may need a very steep quantity discount at your nonlocal Black-Holes-R-Us (Free Installation At Any Velocity), and more generally that apparently solutions which locally hide GR’s preferred rotational frame seem to be associated with very extreme boundary conditions.
You write “under Mach’s principle (the version that says only relative motion is meaningful, and which GR agrees with), these consequences of acceleration you describe only exist because of the frame against which to describe the acceleration, which is formed by the (relatively!) non-accelerating the rest of the universe.” I think it would be more precise to say not “which GR agrees with” but “which some solutions to the GR field equations agree with.” Similarly, if I were pushing a Newman principle which requires that the number of particles in the universe be divisible by 2, I would not say “which GR agrees with” if there were any chance that this might be interpreted as a claim that “the equations of GR require an even number of particles.” Solutions to the GR field equations can be consistent with Mach’s principle, but I’m pretty sure that they don’t need to be consistent with it. The old Misner et al. Gravitation text remarks on how a point of agreement with Mach’s principle “is a characteristic feature of the Friedman model and other simple models of a closed universe.” So it seems pretty clear that as of 1971, there was no known requirement that every possible solution must be consistent with Mach’s principle. And (Bayes FTW!) if no such requirement was known in 1971, but such a requirement was rigorously proved later, then it’s very strange that no one has brought up in this discussion the name of the mathematical physicist(s) who is justly famous for the proof.
(I’m unlikely to look at The End of Time ’til the next time I’m at UTDallas library, i.e., a week or so.)
Refer to the Rovelli paper mentioned in this discussion:
MP7: There is no absolute motion, only motion relative to something else, therefore the water in the bucket does not rotate in absolute terms, it rotates with respect to some dynamical physical entity. True. This is the basic physical idea of GR.
This is a much stronger claim than the one you pretended I was making, that GR agrees my selected Mach’s principle—rather, the pure relativity of universe is the basic idea of GR, not something simply shared between Mach’s principle and GR (like with your modulo 2 example).
And (Bayes FTW!) if no such requirement was known in 1971, but such a requirement was rigorously proved later, then it’s very strange that no one has brought up in this discussion the name of the mathematical physicist(s) who is justly famous for the proof.
Here’s another possible experiment. Send a robot to the other planet, cut it in half, and then build a beam to push the two halves apart. If that planet is rotating, then due to conservation of angular momentum, this should cause its rotation to slow down, and you’d see that. If the two planets are just revolving around each other, then you won’t observe such a slowdown in the apparent rotation of the other planet.
ETA: I’m pretty curious what the math actually says. Do we have any GR experts here?
What experiment could you do to determine whether the other planet is spinning, or your planet is revolving around it while facing it?
check whether you are experiencing a centrifugal force.
Regarding your answer, standard physics seems to indicate that you can tell the difference, unless the laws of physics change to violate newton’s laws when there are fewer than 3 bodies. Mach proposed this (I think) but people seem to doubt him.
Relativity says that as motion becomes very much slower than the speed of light, behavior becomes very similar to Newton’s laws. Everyday materials (and planetary systems) and energies give rise to motions very very much slower than the speed of light, so it tends to be very very difficult to tell the difference. For a mechanical experimental design that can accurately described in a nontechnical blog post and that you could reasonably imagine building for yourself (e.g., a Foucault-style pendulum), the relativistic predictions are very likely to be indistinguishable from Newton’s predictions.
(This is very much like the “Bohr correspondence principle” in QM, but AFAIK this relativistic correspondence principle doesn’t have a special name. It’s just obvious from Einstein’s equations, and those equations have been known for as long as ordinary scientists have been thinking about (speed-of-light, as opposed to Galilean) relativity.)
Examples of “see, relativity isn’t purely academic” tend to involve motion near the speed of light (e.g., in particle accelerators, cosmic rays, or inner-sphere electrons in heavy atoms), superextreme conditions plus sensitive instruments (e.g., timing neutron stars or black holes in close orbit around each other), or extreme conditions plus supersensitive instruments (e.g., timing GPS satellites, or measuring subtle splittings in atomic spectroscopy).
And the example I posited is a superextreme condition: the two bodies in question make up the entire universe, which amplifies the effects that are normally only observable with sensitive instruments. See frame-dragging.
Amplifies? The Schwarzschild spacetime (which behaves like Newtonian gravitational field in large distance limit) needs only one point-like massive object. What do you expect as a non-negligible difference made by (non-)existence of distant objects?
What do you expect as a non-negligible difference made by (non-)existence of distant objects?
The fact that there’s no longer a frame against which to measure local rotation in any sense other than its rotation relative to the frame of the other body. So it makes a big difference what counts as “the rest of the universe”.
People believed for a quite long period of time that the distant stars don’t provide a stable reference frame. That it is the Earth which rotates was shown by Foucault pendulum or similar experiments, without refering to outer stellar frame.
(two points, one about your invocation of frame-dragging upstream, one elaborating on prase’s question...)
point 1: I’ve never studied the kinds of tensor math that I’d need to use the usual relativistic equations; I only know the special relativistic equations and the symmetry considerations which constrain the general relativistic equations. But it seems to me that special relativity plus symmetry suffice to justify my claim that any reasonable mechanical apparatus you can build for reasonable-sized planets in your example will be practically indistinguishable from Newtonian predictions.
It also seems to me that your cited reference to wikipedia “frame-dragging” supports my claim. E.g., I quote: “Lense and Thirring predicted that the rotation of an object would alter space and time, dragging a nearby object out of position compared with the predictions of Newtonian physics. The predicted effect is small—about one part in a few trillion. To detect it, it is necessary to examine a very massive object, or build an instrument that is very sensitive.”
You seem to be invoking the authority of standard GR to justify an informal paraphrase of version of Mach’s principle (which has its own wikipedia article). I don’t know GR well enough to be absolutely sure, but I’m about 90% sure that by doing so you misrepresent GR as badly as one misrepresents thermodynamics by invoking its authority to justify the informal entropy/order/whatever paraphrases in Rifkin’s Entropy or in various creationists’ arguments of the form “evolution is impossible because the second law of thermo prevents order from increase spontaneously.”
point 2: I’ll elaborate on prase’s “What do you expect as a non-negligible difference made by (non-)existence of distant objects?” IIRC there was an old (monastic?) thought experiment critique of Aristotelian “heavy bodies fall faster:” what happens when you attach an exceedingly thin thread between two cannonballs before dropping them? Similarly, what happens to rotational physics of two bodies alone in the universe when you add a single neutrino very far away? Does the tiny perturbation cause the two cannonballs discontinously to have doubly-heavy-object falling dynamics, or the rotation of the system to discontinously become detectable?
ETA: I’m not asking because I don’t know the standard ways to measure cetrifugal force, I’m asking because the standard measurement methods don’t work when the universe is just two planets.
The equipment is already calibrated. You have said that everything works in the same way as today, except the universe consists of two planets. Which I have interpreted like that the observer already knows the value of the gravitational constant in units he can use. If the gravitational constant has to be independently measured first, then it is more complicated, of course.
The equipment is already calibrated. You have said that everything works in the same way as today, except the universe consists of two planets.
Right: you know the laws of physics. You don’t know your mass though, and you don’t know any object that has a known mass. I posit this because, in the history of science, they made certain measurements that aren’t possible in a two-planet universe, and to assume you can calibrate to those measurements would assume away the problem.
But still, in the rotating scenario the attractive force wouldn’t be perpendicular to the planet’s surface, and this can be established without knowing the gravitational constant. If the planet is spherical and you already know what is perpendicular, of course.
If you’re revolving about the other planet, the direction of tidal forces on your planet should rotate as well. If both planets are fixed, the gradient on your planet should be constant.
edit: Nevermind, after seeing that you specified that the orbit is synchronous.
I’d like to pose a sort of brain-teaser about Relativity and Mach’s Principle, to see if I understand them correctly. I’ll post my answer in rot13.
Here goes: Assume the universe has the same rules it currently does, but instead consists of just you and two planets, which emit visible light. You are standing on one of them and looking at the other, and can see the surface features. It stays at the same position in the sky.
As time goes by, you gradually get a rotationally-shifted view of the features. That is, the longitudinal centerline of the side you see gradually shifts. This change in view could result from the other planet rotating, or from your planet revolving around it while facing it. (Remember, both planets emit light, so you don’t see a different portion being in a shadow like the moon’s phases.)
Question: What experiment could you do to determine whether the other planet is spinning, or your planet is revolving around it while facing it?
My answer (rot13): Gurer vf ab jnl gb qb fb, orpnhfr gurer vf ab snpg bs gur znggre nf gb juvpu bar vf ernyyl unccravat, naq vg vf yvgreny abafrafr gb rira guvax gung gurer vf n qvssrerapr. Gur bayl ernfba bar zvtug guvax gurer’f n qvssrerapr vf sebz orvat npphfgbzrq gb n havirefr jvgu zber guna whfg gurfr gjb cynargf, juvpu sbez n onpxtebhaq senzr ntnvafg juvpu bar bs gurz pbhyq or pbafvqrerq fcvaavat be eribyivat.
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. If your planet is heavy and dense like hell, you could see the difference between a “rotating” Kerr metric and a “static” Schwarzschild metric.
Of course, general relativity is generally covariant, and any motion can be interpreted as a free fall in some gravitational field, and more, there is no absolute background spacetime with respect to which to measure acceleration. So you can likely find coordinates in which the planet is static and the pendulum movement explain by changing gravitational field. The price paid is that it will be necessary to postulate weird boundary conditions in the infinity. 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.
Carlo Rovelli in his Quantum Gravity (once I downloaded it from arXiv, now it seems unavailable, but probably it could still be found somewhere on the net) considers eight versions of Mach principle (MP). This is what he says (he has discussed the parabolic water surface of a rotating bucket before instead of two planets or Foucault pendula):
I think number 4 is especially relevant here. The boundary conditions or the global topology of the universe have to be taken into account, else the two-planet scenario is not entirely defined.
Edit: The last remark doesn’t make much sense after all. The planets aren’t thought to be too heavy and the dragging effect shouldn’t be too big, and its relation to boundary conditions isn’t straightforward. Nevertheless, the boundary conditions still play an important role (see my subcomment here).
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.
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.
Couldn’t you tell whether your planet is revolving or rotating using a Foucault’s pendulum? I’m not sure whether you can get all the information about the planets’ relations with a complex set of Foucault’s pendula or not, but you could get some.
Also, I think your answer is a map-territory confusion. While GR does not distinguish certain types of motion from each other, and while GR seems to be the best model of macroscopic behavior we have, to claim that this means that there is really no fact of the matter seems a little overconfident.
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?
It’s the other way around. The fact that there is no test that would distinguish your location along a dimension means that no such dimension exists, and any model requiring such a distinction is deviating from the territory.
Yes, GR could be wrong, but for it to be wrong in a way such that e.g. you actually can distinguish acceleration from gravity would require more than just a refinement of our models; it would mean the universe up to this point was a lie.
SilasBarta:
This isn’t really true. In GR, you can in principle always distinguish acceleration from gravity over finite stretches of spacetime by measuring the tidal forces. There is no distribution of mass that would produce an ideally homogeneous gravitational field free of tidal forces whose effect would perfectly mimic uniform acceleration in flat spacetime. The equivalence principle holds only across infinitesimal regions of spacetime.
See here for a good discussion of what the equivalence principle actually means, and the overview of various controversies it has provoked:
http://www.mathpages.com/home/kmath622/kmath622.htm
Yes, I was just listing an offhand example of an implication of GR and I didn’t bother to specify it to full precision. My point was just that in order for a certain implication to be falsified (specifically, that there is no fact of the matter as to e.g. what the velocity of the universe is), you would need the laws of the universe to change, not just a refinement in the GR model.
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.
If the two planets aren’t revolving around each other, wouldn’t gravity pull them together? But maybe space is expanding at precisely the rate necessary to keep them at the same distance despite gravity? To test that, build a rocket on your planet and push it (the planet) slightly, either toward the other planet or away from it. If the planets are revolving around each other, you’ve just changed a circular orbit into an elliptical one, so you should see an oscillation in the distance between the two planets. If they are not revolving around each other, then they’ll either keep getting closer together or further apart, depending on which direction you made the push.
(This is all based on my physics intuition. Somebody who knows the math should write down the two equations and check if they’re isomorphic. :)
Gravity would pull, yes, but the rotation of a body also distorts space in such a way to produce another effect you have to consider.
ETA: Look at a similar scenario. Same as the one I proposed, but you always see the same portion of the other planet. How do you know how fast the two planets are revolving around each other? Isn’t this the same as asking how fast the entire universe is rotating?
Exactly as fast as needed to keep them on cyclical orbit (assuming you don’t experience change of the distance to the second planet). For this, you can quite safely use the Newton laws.
In general-relativistic language, what exactly do you mean by “how fast the entire universe is rotating”?
I mean nothing. In GR, the very question is nonsense. The universe does not have a position, just relative positions of objects.
The universe does not have a velocity, just relative velocities of various objects.
The universe does not have an acceleration, just relative accelerations of various objects.
The universe does not have a rotational orientation, just relative rotational orientations of various objects.
The universe does not have a rotational velocity, just relative rotational velocities of various objects.
There is no way in this universe to distinguish between a bucket rotating vs. the rest of the universe rotating around the bucket. There is also no such thing as how fast the universe “as a whole” is rotating.
I’m not sure if what you write makes sense. Take one simple example: a flat Minkowski spacetime, empty except for a few light particles (so that their influence on the metric is negligible). This means that special relativity applies, and it’s clearly consistent with GR.
Accelerated motions are not going to be relative in this universe, just like they aren’t in Newton’s theory. You can of course observe an accelerating particle and insist on using coordinates in which it remains in the origin (which is sometimes useful, as in e.g. the Rindler coordinates), but in this coordinate system, the universe will not have the above listed properties in any meaningful sense.
You write “In GR, the very question is nonsense. [0] The universe does not have a position, just relative positions of objects. [1] The universe does not have a velocity, just relative velocities of various objects. [2] The universe does not have an acceleration, just relative accelerations of various objects.” This passage incorrectly appeals to GR to lump together three statements that GR doesn’t lump together.
See http://en.wikipedia.org/wiki/Inertial_frames_of_reference and note the distinction there between “constant, uniform motion” and various aspects of acceleratedness. Your [0] and [1] describe changes within an inertial frame of reference, while [2] gets you to a non-inertial frame. Not coincidentally, your [0] and [1] are predicted by GR and are consistent with centuries of careful experiment, while [2] is not predicted by GR and is inconsistent with everyday observation with Mark I eyeballs. (With modern vehicles it’s common to experience enough acceleration in the vicinity of some low-friction system to notice that acceleration causes conservation of momentum to break down in ways that a constant displacement and/or uniform motion doesn’t.)
I ask, in return, that you read this. Eliezer Yudkowsky had argued that GR implies it’s impossible to measure the acceleration of the universe, and no one had objected. Now, EY is not the pope of rationality, but I suggest things aren’t as simple as you’re making them.
Your point just seems to be a version of the bucket argument: “acceleration must be real, because it has real, detectable, frame-independent consequences like breakage and pain and ficticious forces”. I think I posed the same challenge in an open thread a month or two ago. And as the link you gave says,
But under Mach’s principle (the version that says only relative motion is meaningful, and which GR agrees with), these consequences of acceleration you describe only exist because of the frame against which to describe the acceleration, which is formed by the (relatively!) non-accelerating the rest of the universe. Therefore, if all of the universe were to accelerate uniformly, there would be no relative motion and therefore no experimental consequences, and we should regard the very idea as nonsense.
So if the universe were only you and your vehicle, you would not be able notice joint accelerations of you and the vehicle, only acceleration of yourself relative to the vehicle.
Now, you can disagree with this application of Mach’s principle, but the observations you describe do not contradict it.
I should also add one of the great insights I got out of Barbour’s book The End of Time (from which EY got his love of Mach’s principle and timelessness). The insight is that the laws of physics do not change in a rotating reference frame. Rather, there is a way you can determine if any given object is not in uniform motion relative to the rest of the universe, and this method also allows you to define an “inertial clock” which gives you an appropriate measure of time.
Most importantly, if you are spinning around, and there’s some other object accelerating relative to the rest of the universe, this method allows you to detect its acceleration, no matter how much or in what way your own frame is moving!
Perhaps the root of our disagreement is that you think (?) that the GR field equations constrain their solutions to conform to Mach’s principle, while I think they admit many solutions which don’t conform to Mach’s principle, and that furthermore that Vladimir_M is probably correct in his sketch of a family of non-Mach-principle solutions.
EY’s article seems pretty clear about claiming not that Mach’s principle follows deductively from the equations of GR, but that there’s a sufficiently natural fit that we might make an inductive leap from observed regularity in simple cases to an expected identical regularity in all cases. In particular EY writes “I do not think this has been verified exactly, in terms of how much matter is out there, what kind of gravitational wave it would generate by rotating around us, et cetera. Einstein did verify that a shell of matter, spinning around a central point, ought to generate a gravitational equivalent of the Coriolis force that would e.g. cause a pendulum to precess.” I think EY is probably correct that this hasn’t been verified exactly—more on that below. I also note that from the numbers given in Gravitation, if you hope to fake up a reasonably fast rotating frame by surrounding the experimenter with a rotating shell too arbitrarily distant to notice, you may need a very steep quantity discount at your nonlocal Black-Holes-R-Us (Free Installation At Any Velocity), and more generally that apparently solutions which locally hide GR’s preferred rotational frame seem to be associated with very extreme boundary conditions.
You write “under Mach’s principle (the version that says only relative motion is meaningful, and which GR agrees with), these consequences of acceleration you describe only exist because of the frame against which to describe the acceleration, which is formed by the (relatively!) non-accelerating the rest of the universe.” I think it would be more precise to say not “which GR agrees with” but “which some solutions to the GR field equations agree with.” Similarly, if I were pushing a Newman principle which requires that the number of particles in the universe be divisible by 2, I would not say “which GR agrees with” if there were any chance that this might be interpreted as a claim that “the equations of GR require an even number of particles.” Solutions to the GR field equations can be consistent with Mach’s principle, but I’m pretty sure that they don’t need to be consistent with it. The old Misner et al. Gravitation text remarks on how a point of agreement with Mach’s principle “is a characteristic feature of the Friedman model and other simple models of a closed universe.” So it seems pretty clear that as of 1971, there was no known requirement that every possible solution must be consistent with Mach’s principle. And (Bayes FTW!) if no such requirement was known in 1971, but such a requirement was rigorously proved later, then it’s very strange that no one has brought up in this discussion the name of the mathematical physicist(s) who is justly famous for the proof.
(I’m unlikely to look at The End of Time ’til the next time I’m at UTDallas library, i.e., a week or so.)
See also the conversational thread which runs through http://lesswrong.com/lw/qm/machs_principle_antiepiphenomenal_physics/kb3 http://lesswrong.com/lw/qm/machs_principle_antiepiphenomenal_physics/kb8 http://lesswrong.com/lw/qm/machs_principle_antiepiphenomenal_physics/kba
Refer to the Rovelli paper mentioned in this discussion:
This is a much stronger claim than the one you pretended I was making, that GR agrees my selected Mach’s principle—rather, the pure relativity of universe is the basic idea of GR, not something simply shared between Mach’s principle and GR (like with your modulo 2 example).
I did—Barbour.
Here’s another possible experiment. Send a robot to the other planet, cut it in half, and then build a beam to push the two halves apart. If that planet is rotating, then due to conservation of angular momentum, this should cause its rotation to slow down, and you’d see that. If the two planets are just revolving around each other, then you won’t observe such a slowdown in the apparent rotation of the other planet.
ETA: I’m pretty curious what the math actually says. Do we have any GR experts here?
Also, if you’ve asked the right question, would the stresses that would push the halves apart also show up as geological stresses?
check whether you are experiencing a centrifugal force.
Regarding your answer, standard physics seems to indicate that you can tell the difference, unless the laws of physics change to violate newton’s laws when there are fewer than 3 bodies. Mach proposed this (I think) but people seem to doubt him.
The universe adheres to General Relativity, not Newton’s laws. What does GR say about the effect of spinning and revolving bodies?
Relativity says that as motion becomes very much slower than the speed of light, behavior becomes very similar to Newton’s laws. Everyday materials (and planetary systems) and energies give rise to motions very very much slower than the speed of light, so it tends to be very very difficult to tell the difference. For a mechanical experimental design that can accurately described in a nontechnical blog post and that you could reasonably imagine building for yourself (e.g., a Foucault-style pendulum), the relativistic predictions are very likely to be indistinguishable from Newton’s predictions.
(This is very much like the “Bohr correspondence principle” in QM, but AFAIK this relativistic correspondence principle doesn’t have a special name. It’s just obvious from Einstein’s equations, and those equations have been known for as long as ordinary scientists have been thinking about (speed-of-light, as opposed to Galilean) relativity.)
Examples of “see, relativity isn’t purely academic” tend to involve motion near the speed of light (e.g., in particle accelerators, cosmic rays, or inner-sphere electrons in heavy atoms), superextreme conditions plus sensitive instruments (e.g., timing neutron stars or black holes in close orbit around each other), or extreme conditions plus supersensitive instruments (e.g., timing GPS satellites, or measuring subtle splittings in atomic spectroscopy).
And the example I posited is a superextreme condition: the two bodies in question make up the entire universe, which amplifies the effects that are normally only observable with sensitive instruments. See frame-dragging.
Amplifies? The Schwarzschild spacetime (which behaves like Newtonian gravitational field in large distance limit) needs only one point-like massive object. What do you expect as a non-negligible difference made by (non-)existence of distant objects?
The fact that there’s no longer a frame against which to measure local rotation in any sense other than its rotation relative to the frame of the other body. So it makes a big difference what counts as “the rest of the universe”.
People believed for a quite long period of time that the distant stars don’t provide a stable reference frame. That it is the Earth which rotates was shown by Foucault pendulum or similar experiments, without refering to outer stellar frame.
(two points, one about your invocation of frame-dragging upstream, one elaborating on prase’s question...)
point 1: I’ve never studied the kinds of tensor math that I’d need to use the usual relativistic equations; I only know the special relativistic equations and the symmetry considerations which constrain the general relativistic equations. But it seems to me that special relativity plus symmetry suffice to justify my claim that any reasonable mechanical apparatus you can build for reasonable-sized planets in your example will be practically indistinguishable from Newtonian predictions.
It also seems to me that your cited reference to wikipedia “frame-dragging” supports my claim. E.g., I quote: “Lense and Thirring predicted that the rotation of an object would alter space and time, dragging a nearby object out of position compared with the predictions of Newtonian physics. The predicted effect is small—about one part in a few trillion. To detect it, it is necessary to examine a very massive object, or build an instrument that is very sensitive.”
You seem to be invoking the authority of standard GR to justify an informal paraphrase of version of Mach’s principle (which has its own wikipedia article). I don’t know GR well enough to be absolutely sure, but I’m about 90% sure that by doing so you misrepresent GR as badly as one misrepresents thermodynamics by invoking its authority to justify the informal entropy/order/whatever paraphrases in Rifkin’s Entropy or in various creationists’ arguments of the form “evolution is impossible because the second law of thermo prevents order from increase spontaneously.”
point 2: I’ll elaborate on prase’s “What do you expect as a non-negligible difference made by (non-)existence of distant objects?” IIRC there was an old (monastic?) thought experiment critique of Aristotelian “heavy bodies fall faster:” what happens when you attach an exceedingly thin thread between two cannonballs before dropping them? Similarly, what happens to rotational physics of two bodies alone in the universe when you add a single neutrino very far away? Does the tiny perturbation cause the two cannonballs discontinously to have doubly-heavy-object falling dynamics, or the rotation of the system to discontinously become detectable?
How would you measure the centrifugal force?
ETA: I’m not asking because I don’t know the standard ways to measure cetrifugal force, I’m asking because the standard measurement methods don’t work when the universe is just two planets.
Calculate the gravitational force on the surface of a planet of the same size and mass as yours and compare with what you actually measure.
What do you calibrate your equipment against?
The equipment is already calibrated. You have said that everything works in the same way as today, except the universe consists of two planets. Which I have interpreted like that the observer already knows the value of the gravitational constant in units he can use. If the gravitational constant has to be independently measured first, then it is more complicated, of course.
Right: you know the laws of physics. You don’t know your mass though, and you don’t know any object that has a known mass. I posit this because, in the history of science, they made certain measurements that aren’t possible in a two-planet universe, and to assume you can calibrate to those measurements would assume away the problem.
But still, in the rotating scenario the attractive force wouldn’t be perpendicular to the planet’s surface, and this can be established without knowing the gravitational constant. If the planet is spherical and you already know what is perpendicular, of course.
If you’re revolving about the other planet, the direction of tidal forces on your planet should rotate as well. If both planets are fixed, the gradient on your planet should be constant.
edit: Nevermind, after seeing that you specified that the orbit is synchronous.