In this thread on only two grains of sand in the universe,
a discussion came up about rotating frames of reference. In Newtonian mechanics, space is absolute so it’s simple to define a rotating frame of reference. But how exactly do you do it in General Relativity. If there really were only solid thing (and this is relativity not quantum mechanics remember), how can I tell if it’s spinning or not? What would it be spinning relative to?
If there are only two solid things, why can’t I say the geodesic line connecting them is not rotating. Again what would it be rotating with respect to. But it would seem to me in this latter case, we must be able to tell. If they are not orbiting around their center of mass, then they’ll head right at each other and crash.
If they are orbiting, I assume they’ll radiate energy in gravitational waves and eventually crash, but it will take much longer.
To expand on this, the classical laws of motion only hold in inertial reference frames, and something in a rotating frame isn’t in an inertial reference frame, so a rotating frame would observe laws of motion which aren’t the same as a non-rotating one. You can trace this variance to the fact linear momentum is conserved; something feeling what’s observed as a centrifugal force in the rotating frame is being blocked from following an inertial straight-line path in its own reference frame.
I don’t know how much this has to do with GR, however.
I understand that. In Newtonian mechanics space is considered an absolute and I can define rotation with respect to that, and a rotating frame in not inertial.
However, in General Relativity, it is my understanding that space is not absolute. So if there is only one thing in space what can it be rotating with respect to? If there are only two things in space, what can the line between them be said to be rotating with respect to? There isn’t anything else unless I’m wrong and the “fabric” of space is an absolute just as in classical mechanics.
It’s rotating with respect to any frame that does not have centrifugal forces. Space is not absolute, but that doesn’t mean that all frames of reference are equivalent, just the inertial ones.
I have wondered about this too. All I can glean from the answers is that, while there is no fixed reference frame, there is a fixed rotation frame, one that has no centrifugal force.
Mach’s principle proposes that the distinction between rotating and non-rotating frames is determined by the overall distribution of matter in the universe. I.e. we can determine which frames of reference are inertial through local experiments, and we can observe that, in these frames, the distant galaxies are not rotating around us, and this is not just one huge coincidence.
This is exactly the type of thing I was looking for. It seems that others (way back) were worried about this question as well. If Mach’s principle is true, then I’d think in a universe with just one particle, there would be no way to tell if it were spinning. The same might be true with just two particles.
All this may not be correct, of course, but it is nice to know that others have worried about this question.
I would break your question down into 3 different questions:
What is the basic difference between a rotating and non-rotating frame of reference and does that basic difference still manifest itself in empty space?
Can we tell if a an object with mass-energy is rotating in otherwise empty space?
My answers to those questions would be as follows:
In general relativity the frame of an observer can be represented by a local coordinate system. If the observer is not inertial then these coordinates will not be normal coordinates. We can go further and look at different types of non-inertial frames: pure rotation and pure linear acceleration in general relativity can be differentiated by precisely which Christoffel symbols fail to vanish in the local coordinate system. None of this depends on whether space is empty or not.
If we restrict ourselves to highly symmetric objects and make the (big) assumption that the space outside such an object is represented by the equally symmetric Ernst vacuums, we can clearly see the difference between rotating and non-rotating objects as the rotation manifests itself in the parameters of that family of vacuums. So the answer is that there is a difference between mass-energy that rotates and mass-energy that doesn’t rotate in otherwise empty space in general relativity.
If that’s true, then there’s no way to tell if the universe, as a whole, is spinning. Only objects within it, relative to the entirety. I’m not sure whether that makes any sense. Wouldn’t a spinning universe exhibit properties that a stationary one wouldn’t, even if it’s not spinning relative to anything?
We anxiously await your third question and more importantly, your third answer.
A thought to perhaps muddy the waters of the last dozen-ish posts.
If the “object” we’re observing in the otherwise empty universe is macroscopic, then parts of it at different radii will experience either zero centripetal force or some differing centripetal forces. So we’re only ignorant of its rotation if we consider it an observationally opaque object.
This tell also vanishes in the case of point objects of zero physical size. But what does it even mean to say an idealized point is rotating? That seems to my barely educated eye to be a non sequitur.
Our thought experiments already assume a magical observer in this otherwise empty universe. My (perhaps woefully clueless) bottom line is this:
Either the object is a point, and as a logical necessity rotation is an undefined concept, or the object is a non-point and our magical observer can detect rotation by the forces acting on the various constituent regions of the non-point.
