I Is acceleration absolute or relative - revisited

[PeterDonis]

We can just look up the Christoffel symbols

This is not a good way of looking for "influences", since those are supposed to be invariant but the Christoffel symbols are not. "Influences" should be described by invariants. The relevant invariants to look at would be the proper acceleration and vorticity of the congruence of worldlines describing the bucket.

In the first case, "rotating bucket in static universe", we describe the bucket using the Langevin congruence in flat Minkowski spacetime. The proper acceleration of this congruence is $- \omega^2 r / \left( 1 - \omega^2 r^2 \right)$ and the vorticity is $\omega / \left( 1 - \omega^2 r^2 \right)$. The proper acceleration is what accounts for the curved shape of the water surface inside the bucket, and the vorticity is what tells us the bucket is rotating.

In the second case, "static bucket in rotating universe", now that that OP has clarified that he intends this to mean an actual change in spacetime geometry, we describe the bucket using a Fermi-Walker transported congruence centered on a comoving worldline in the Godel spacetime. The proper acceleration and vorticity of this congruence are both zero. The zero proper acceleration tells us that the surface of the water in the bucket is flat, and the zero vorticity tells us that the bucket is not rotating.

 

[Dale]

This is not a good way of looking for "influences", since those are supposed to be invariant but the Christoffel symbols are not. "Influences" should be described by invariants. The relevant invariants to look at would be the proper acceleration and vorticity of the congruence of worldlines describing the bucket.

I don’t disagree in principle, but in practice I think that it is sufficient for this thread. As the OP was interested in “gravitational fields” (ala Einstein’s lecture) $\Gamma_{tt}^r$ is the “gravitational field” associated with a spinning frame, such as a traditional rotating space station’s artificial gravity.

It is clear that such a “gravitational field” leads to a flattened blob. Of course the details can be computed in a fully invariant manner, and that would be more satisfying. But I am lazy and only wanted to point out that there is a difference in the easiest way possible.

 

[Peter Leeves]

You haven't just been asking questions, though. You've been making definitive statements, many of which are wrong.

My post started by describing a scenario and suggesting a possible answer. Quote "I haven't come down on one side or the other yet, but I do at least see the argument that acceleration is relative."

My understanding of acceleration types was completely wrong at that time (probably still is) as I didn't have a clear understanding of what relative meant and what absolute (invariant) meant. I definitely used relative incorrectly there. I was trying to say "acceleration is invariant" (and might therefore apply equally in both scenarios - pehaps).

Throughout the entire thread, I don't think it was necessary to punctuate every sentence with a "?" to establish my contributions as a "question". It must be obvious to all but the blind, if I start off by saying "I don't know which is true, can you please help me understand ?" that all following discussion is trying to dig further to (hopefully) arrive at a conclusion. Yes, some of that dialogue is made in the form of statements without question marks. You can have my apology for not stating the "bleeding obvious".

 

[PeterDonis]

As the OP was interested in “gravitational fields” (ala Einstein’s lecture) $\Gamma_{tt}^r$ is the “gravitational field” associated with a spinning frame, such as a traditional rotating space station’s artificial gravity.

It is clear that such a “gravitational field” leads to a flattened blob.

Only with the appropriate caveats about the choice of coordinates. But one of the key points of discussion during this thread has been that all choices of coordinates are equally valid, and that actual physical observables must be described by invariants. That point is all the more important now that the OP has clarified that he intends the two scenarios to be different in an invariant sense, not just as a matter of choice of coordinates.

To put it another way, the concept of "gravitational field" as embodied in the Christoffel symbols is a coordinate-dependent concept; but the OP has said he's not interested in coordinate-dependent concepts, but in two scenarios that have an invariant difference between them. So the difference should be described in terms of invariants.

 

[PeterDonis]

Throughout the entire thread, I don't think it was necessary to punctuate every sentence with a "?" to establish my contributions as a "question". It must be obvious to all but the blind, if I start off by saying "I don't know which is true, can you please help me understand ?" that all following discussion is trying to dig further to (hopefully) arrive at a conclusion.

Yes, you said you wanted to improve your understanding, but if your method of doing that is to state what your current understanding is in definitive statements--and that is the method you adopted in this thread--then the only way for us to help you improve your understanding is for us to tell you when your statements are wrong, in the same definitive manner that you used to make the statements. Which is what I did. And when I make such a statement about something you've said, and then you make a subsequent post that says the same wrong thing, definitively, that I've already told you is wrong, you can expect me to remind you, definitively, that I've already told you it's wrong. Which is what I did.

 

[Peter Leeves]

And when I make such a statement about something you've said, and then you make a subsequent post that says the same wrong thing, definitively, that I've already told you is wrong, you can expect me to remind you, definitively, that I've already told you it's wrong. Which is what I did.

What's the purpose of this forum if not to willingly share knowledge and increase understanding - both of the learner and perhaps also the teacher, through his efforts to educate ?

I've done my best to keep up and have re-read each and every post SEVERAL times to try and understand better (and follow suggested links !). If I made you repeat yourself, that can be frustrating and I apologise (sincerely this time). I've done nothing but try to grasp a LOT of new terminology and take on a HEAP of new concepts (Christoffel symbols anyone ?) in a very short space of time. All of this may well be "baby talk" to you. I assure you it's not to me. As mentioned earlier, a lot of people contributing to this thread have helped enormously - but also with patience and good grace. Strikes me as a much nicer way of doing business for all.

This has gone way too far off physics now. I'm not going to respond further to anything other than the OP and physics.

 

[PeterDonis]

What's the purpose of this forum if not to willingly share knowledge and increase understanding

Yes, that is the purpose, and that's what we've been doing. If the discussion has helped you, that's good.

 

[Peter Leeves]

To all contributors, I made the following summary for my own (educational) benefit. Since it's merely a quick "copy and paste", I thought it might be of value to anyone coming across the thread in the future.

There's no need to read it, and it might be better if there are no further responses / clarifications / corrections / arguments etc. I'm definitely not expecting anything else. I thank you ALL for your time and patience sharing your knowledge.

I fully accept that the two scenarios are NOT equivalent and the observations would NOT be identical. Further, within the reference frame (excuse my little play on words) of my limited abilities, I have understood both the reasoning and the proof.

My take from this thread (what I now understand - or think I do, lol):

Proper "anything" means it is independent of coordinate system / reference frame, it is not relative and it can't be transformed away. eg proper acceleration or proper rotation.

Invariant means the same / unchanging in all reference frames, and is not limited to inertial coordinate systems. eg. Proper acceleration or charge on an object.

Relative means changing depending on the reference frame. eg. Velocity

A coordinate system typically consists of one coordinate for time and three mutually perpendicular coordinates for space.

Proper acceleration (aka absolute acceleration, aka invarient acceleration) is invariant (unchanging in all coordinate systems / reference frames). It is physically measured by an accelerometer. The physics (the outcome of any experiment) depends on the proper acceleration, not the coordinate acceleration. All observers agree on proper acceleration.

Relative acceleration (aka coordinate acceleration) is dependent on a coordinate system/reference frame and is the second derivative of the space coordinates with respect to time.

In an inertial frame, the coordinate acceleration and the proper acceleration are the same (at non-relativistic velocities). They can be very different in non-inertial frames.

An object falling to Earth has coordinate acceleration (reference frame is Earth) but no proper acceleration (subject to no forces). An object on the Earth's surface has proper acceleration (upwards force from the ground).

With respect to equivalence of rotating bucket / rotating universe scenarios - this is only true as a statement about choices of coordinates for the same spacetime geometry; it is not true as a statement about invariants. As far as invariants are concerned, "rotating bucket in non-rotating universe" is not the same as "rotating universe with non-rotating bucket"; the latter would be a different spacetime geometry from the former. [This concludes the two scenarios are NOT equivalent and the observations would NOT be identical, for the stated reason.]

The equivalence principle is local, not the entire universe. [Thought only - if the entire universe was proper rotating / proper angular accelerating, would this make the equivalence principle non-local ?]

The global equivalence of different choices of coordinates on the same spacetime geometry is called "general covariance", not the equivalence principle. So the bucket thought experiment is an illustration of general covariance, not the Equivalance Principle.

Einstein said that when the traveling twin fires his rocket to turn around, this can be viewed as creating a gravitational field in which the stay-at-home twin is at a much higher altitude than the traveling twin, and this accounts for the stay-at-home twin's much greater elapsed time during the period when the field is present (i.e., when the traveling twin is firing his rocket).

In the spacetime geometry in question, there is no frame dragging. Frame dragging is not a coordinate effect; it is an effect of the spacetime geometry, and only certain kinds of spacetime geometries have it.

The way that the rest of the matter in the universe influences the bucket is by determining the spacetime geometry.

It really is the bucket that is rotating.

Einstein’s 1918 lecture was based on coordinate acceleration, not proper acceleration.

For equivalence it must not merely have some influence, it must have the exact same influence. It does not. [Conclusion that the two scenarios are NOT equivalent and the observations would NOT be identical].

It is actually fairly simple to show that it does not exert the same influence. We can just look up the Christoffel symbols in the Catalog of Spacetimes.
Where equation 2.1.30 (rotating bucket in the bucket's frame) [missing equation] has but equation 2.10.2 (rotating universe in the bucket's frame) has [missing equation]. [Proof the two scenarios will NOT yield identical observations - but see later comment that "this is not a good way of looking for influences"].

The shape of the water in the bucket will be very different in the two cases. So if you meant (2), your original claim in the OP of this thread that "the gravitational field of the rotating universe" will make the water in the bucket climb up the sides of the bucket is wrong. [Conclusion the two scenarios will NOT yield identical observations].

*** This is the proof that the two scenarios are NOT equivalent ***

In the first case, "rotating bucket in static universe", we describe the bucket using the Langevin congruence in flat Minkowski spacetime. The proper acceleration of this congruence is and the vorticity is . The proper acceleration is what accounts for the curved shape of the water surface inside the bucket, and the vorticity is what tells us the bucket is rotating.

In the second case, "static bucket in rotating universe", now that that OP has clarified that he intends this to mean an actual change in spacetime geometry, we describe the bucket using a Fermi-Walker transported congruence centered on a comoving worldline in the Godel spacetime. The proper acceleration and vorticity of this congruence are both zero. The zero proper acceleration tells us that the surface of the water in the bucket is flat, and the zero vorticity tells us that the bucket is not rotating.
*************************************************************

But one of the key points of discussion during this thread has been that all choices of coordinates are equally valid, and that actual physical observables must be described by invariants.

To put it another way, the concept of "gravitational field" as embodied in the Christoffel symbols is a coordinate-dependent concept; but the OP has said he's not interested in coordinate-dependent concepts, but in two scenarios that have an invariant difference between them. So the difference should be described in terms of invariants.

I should avoid using words like “ACTUALLY” “truth” “truly” and “reality” as they can be misconstrued.

PeroK considers the notion of a proper rotating universe being equivalent to a proper stationary universe (now properly discounted) to be mysticysm / metaphysics / philosophy.

And my final take is ... "There is no spoon"

Things I didn't understand and need to further research:

Proper acceleration can be written mathematically as a covariant derivative (?).

What Einstein was calling a gravitational field is technically the Christoffel symbols. Those are indeed relative to the individual frame. The Christoffel symbols do not affect the proper acceleration (?).

What you are describing here is not frame dragging. It is those Christoffel symbols (?).

Acceleration type is related to Mach's Principle (?).

The shape of water's surface would be subject to Lorentz contraction the same as any other shape. Doesn’t change the measured proper acceleration though... we’re looking at the geodesic deviation between adjacent volumes of water in the bucket (?).

In an empty universe a particle can still undergo inertial forces because Minkowski spactime solves the field equations of an empty universe (?).

This is all about Mach's principle. The question is: does the (inertial) mass m of the water depend on all the other mass M of the universe? Mach believed so; he believed that, whatever m(M) is, the inertial property of it should vanish if M vanishes. It's not clear if and how the (inertial) mass of the water is fully determined by all the other (inertial) mass in the universe (?).

The family of worldlines describing the motion of objects "at rest relative to the universe" will be integral curves of a timelike Killing vector field that is hypersurface orthogonal (which is what "static" translates to in more technical GR language); whereas the family of worldlines describing the motion of the bucket will be integral curves of a timelike Killing vector field (assuming the bucket's angular velocity of rotation relative to the universe is constant) that is not hypersurface orthogonal (in more technical jargon, the bucket's motion will be stationary but not static) (?).

Links followed (or yet to be followed):
https://en.wikipedia.org/wiki/Proper_acceleration#In_curved_spacetime
Mach's principle.
https://en.wikipedia.org/wiki/Brans–Dicke_theory
https://arxiv.org/abs/0904.4184

P.S. I knew you couldn't resist reading it

 

[PeterDonis]

A coordinate system typically consists of one coordinate for time and three mutually perpendicular coordinates for space.

This is not correct.

First, there are many spacetimes in GR for which it is impossible to find such a coordinate chart globally at all.

Second, there are many spacetimes for which, even if such a coordinate chart is possible, there will be scenarios where a different coordinate chart is more useful.

The only actual requirements on a coordinate chart are that all four coordinates are real functions on spacetime, that the functions are continuous, and that the mapping of coordinate 4-tuples to points in spacetime is one-to-one. There is no requirement that one coordinate be timelike and the other three spacelike, and there is no requirement that the coordinate "grid lines" be orthogonal. (In fact, as noted, there are cases where these last two things are not even possible.)

if the entire universe was proper rotating / proper angular accelerating, would this make the equivalence principle non-local

First, the concepts of "proper acceleration" and "proper rotation" make no sense with respect to the universe. They only make sense with respect to individual objects in the universe.

That said, the EP is always local. No choice of spacetime geometry can change that.

It is actually fairly simple to show that it does not exert the same influence. We can just look up the Christoffel symbols in the Catalog of Spacetimes.

I think the caveats that I stated in my earlier exchange with @Dale about this are important. The fact that this heuristic happens to work in the particular cases discussed in this thread is a matter of being fortunate in the choice of coordinates that the Catalog of Spacetimes happens to have made for those cases. It cannot be extended to a general rule that always works. The general rule that always works is to look at the invariants I specified, proper acceleration and vorticity, as described later on in your post.

 

[Peter Leeves]

This is not correct.

First, there are many spacetimes in GR for which it is impossible to find such a coordinate chart globally at all.

All your input relating to multiple alternatives duly noted and agreed, other than my statement being incorrect. I didn't say "only", but "typical". The most familiar coordinate system to all humans is the one in which we live, which is orthagonal and consists of the 4 coordinates stated - hence it is typical. "Typical" also specifically requires there to be other types. It cannot imply "only", by definition. It would have been more accurate if you'd begun, "Yes, that's one familiar coordinate system, and here are some others ..." Or even better, just don't set out to make me appear incorrect when I'm not, in the first place.

First, the concepts of "proper acceleration" and "proper rotation" make no sense with respect to the universe.

Your comment suggests you're failing to grasp this thread is about a thought experiment, not reality. My idea (now answered) was essentially - Is it possible to devise a thought experiment in which a proper rotating universe might be considered eqivalent and produce identical observations to reality, and therefore be considered equally valid ? The fact you proved it can't isn't the point.

My idea was never about literally proper accelerating the universe to observe the influence on a proper stationary bucket. A spaceship firing it's rocket does not in reality remain proper stationary against a proper accelerating linear gravitational field magically generated by the rocket firing, and thus accelerate the entire universe away from the spaceship. Einstein merely proposed that it was equivant. Your comment is the same as telling Einstein, "don't be an idiot - firing a spaceship rocket cannot cause the entire universe to accelerate". Whether he was right or wrong doesn't matter. He was entitled to devise his ridiculous concept and ask - would this be equivalent ? I was just as entitled to ask if my thought experiment might work, particularly as they share some similar (though not identical) concepts.

Yes, you have disproved my idea for which I will be eternally grateful. But in the context of my thought experiment, a true rotating universe does make sense. After all, how did you manage to disprove something that "makes no sense" to you ? The answer is, it did make sense to you. Once again, it appears the real purpose of your comment was to make me appear wrong, not to make a valid contribution.

I think the caveats that I stated in my earlier exchange with @Dale about this are important.

I agree. An answer providing a general rule for all cases is preferable.

 

[Dale]

This is not correct.

I disagree. I think “typical” is an accurate description. After all, such a coordinate system is in fact the prototypical coordinate system, and how can the prototypical system not be considered typical. Your point regarding global coordinates is correct, but even in such spacetimes you can and often do construct local coordinates. When you do, they are typically as described.

First, the concepts of "proper acceleration" and "proper rotation" make no sense with respect to the universe. They only make sense with respect to individual objects in the universe.

This is terminology that I introduced, and I think your assertion goes too far. There are some spacetimes where the concepts may make no sense, but many where it makes perfect sense.

When I say that I am undergoing proper acceleration or rotation I mean that an accelerometer at rest with respect to my body and located in or on my body will measure said acceleration or rotation. Similarly, if an accelerometer is at rest with respect to the matter of the universe then it can be said to measure the proper acceleration or rotation of the universe. In the Goedel spacetime the universe has rotation in this sense.

Obviously, some universes have material that is not at rest with respect to other material, in which case you can specify the local distribution of matter and see if that rotation or acceleration is homogenous throughout the universe. Other universes are vacuum solutions, where there isn’t any matter to be at rest with respect to. For those I agree that it wouldn’t make sense.

 

[weirdoguy]

Your comment suggests you're failing to grasp this thread is about a thought experiment, not reality.

Thought experiments can't be in contradiction with known physics. If they are, we have no basis to analyse them.

 

[Peter Leeves]

Thought experiments can't be in contradiction with known physics.

Nonetheless, it remains valid to ask "Is this thought experiment in line with known physics ?". It's equally valid to receive the answer "No, and here's why not". And the person asking the question has increased their knowledge in a meaningful way.

Perhaps you meant to say, "Successful thought experiements won't be in contradiction with known physics." ?

 

[vanhees71]

Concerning the question whether the universe rotates or not, afaik, what's meant is the question whether the large-scale averaged spacetime is described by a standard FLRW solution of Einstein's equations or whether it's described by a solultion "with spin" (like Gödel's solution). Also I think today there's no hint of such a rotating universe from observations, but it's a possible cosmological model within standard GR, and it's also empirically investigated (e.g., by evaluating CMBR observables). Here is a not too old paper about this issue. I don't know, whether there are newer ones with newer measurements.

https://arxiv.org/abs/0902.4575
https://doi.org/10.1088/0004-637X/703/1/354

 

[Peter Leeves]

Dale, may I pose a question to you directly, as it relates to your Post #87 ? (all welcome to comment of course).

Prior to Post #87, I asked if the mass of the entire visible univese (proper rotating) would have any influence at all on the water ? You replied "Certainly, it does have some influence. But for equivalence it must not merely have some influence, it must have the exact same influence. It does not."

I fully take onboard that statement. Particularly the "It does not." bit. And yet confirming there would be some influence is like a red rag to a bull.

Given there is some influence on the water (though not identical), implies we may simply not be considering the correct magnitude of gravitational field or the correct magnitude of proper angular acceleration or some combination of both. Maybe if the gravitational field were stronger or we proper rotated it faster (or both) it might have identical influence on the water after all. I merely seek to close out this possibility.

Can you please clarify that no magnitude of gravitational field and no magnitude of proper rotational acceleration (or combination of both) could ever duplicate the obervation ?

 

[Dale]

Given there is some influence on the water (though not identical), implies we may simply not be applying the correct magnitude of gravitational field or the correct magnitude of proper angular acceleration or some combination of both.

How would it imply that? In science the “correct” quantity is the one that matches experiment.

 

[Peter Leeves]

How would it imply that?

How one thing implies the other:

We agreed just before post #87 all matter in the visible universe communicates gravitationally with all other matter at the speed of light c in a vacuum, and this must have some influence on the water. In a proper stationary universe this would likely have little (or no) net influence on the water, especially if you choose to assume matter is more or less evenly distributed. Perhaps just a tiny, quite possibly imperceptible, bulge in all directions would be my guess.

It then occurred that if the entire visible universe was proper rotationally accelerated, the effect on the water might be far from net zero. Now the entire universe's mass would exert a circumferential torque on the water. (I'm not certain, but that seems to be a description of rotational frame dragging, by-the-by, forget it). So when you confirmed, yes, there would be some influence - but not the same, it occurred upon subsequent re-reading that perhaps we were just not considering a large enough gravitational field or a large enough proper rotational acceleration (or both). The question then becomes, is there any magnitude that would apply an indentical torque to the water and so produce an identical observation ?

That's my best shot at explaining the implication.

As ever, this is a question and not a statement. I also thought clarifying there was no magnitude that could ever duplicate the obervation would close the issue and add even more value to this thread.

(I also considered a slim chance that on reflection you might agree. That if there was some influence, then adding more might well produce an indentical influence - but shhhh, don't tell anyone. For me, I can't shake that proper acceleration (and the resulting observation) is invarient and MUST be present in all reference frames. If the proper rotating universe is a legitimate reference frame then, by definition, it MUST contain the same proper acceleration and identical observation. That's just simple logic. If that's correct then it's possible I've borrowed from Einstein and described the mechanism by which it works - as an equivalent, not as a reality. Nonetheless, both would be equivalent and therefore equally valid viewpoints).

 

[Dale]

That's my best shot at explaining the implication.

That is pretty far from an implication. X implies Y means that if X is true then logically Y must also be true.

As I mentioned above, the "correct" value for a quantity in physics is the value that matches experiments. So the desire to produce something that matches some philosophical desideratum has nothing whatsoever to do with the correctness of a physical quantity. In this case, you have very little room to change anything. GR has exactly 1 tuneable parameter, and it cannot take any value other than what it is without departing from experimental observations.

 

[PeterDonis]

GR has exactly 1 tuneable parameter

Are you referring to the gravitational constant $G$?

 

[Peter Leeves]

That is pretty far from an implication. X implies Y means that if X is true then logically Y must also be true.

I gave it my best shot. My reasoning was logical, even if it didn't amount to what you consider to be a legitimate implication. Your definition is incorrect/incomplete. X implies Y, could equally mean that if X is true then logically Y must be untrue. It's a completely muddled and frankly pointless definition. Can we please stick to physics rather than definitions of words which is adding little ?

Is a rotating universe a legitimate frame of reference ? If the answer is yes, then given proper acceleration (and resulting observation) is invarient, then it (plus the corresponding observation) must by definition also exist in the rotating universe reference frame. Therefore your answer can only be no, a rotating universe is not a legitimate reference frame ?

 

[PeterDonis]

I can't shake that proper acceleration (and the resulting observation) is invarient and MUST be present in all reference frames. If the proper rotating universe is a legitimate reference frame

You're mixing up the concept of "reference frame" with the concept of "spacetime geometry".

General covariance means that all reference frames are equivalent for describing the same spacetime geometry. So if a given object in a given spacetime geometry has a given proper acceleration in one frame, it will have the same proper acceleration in all frames.

However, when you talk about a proper rotating universe, as compared to a proper non-rotating universe, you are talking about two different spacetime geometries. It doesn't even make sense to compare reference frames in these two different spacetime geometries (except locally). Reference frames that describe things in a proper rotating universe have nothing whatever to do with reference frames that describe things in a proper non-rotating universe.

 

[Dale]

Are you referring to the gravitational constant $G$?

No, I was referring to the cosmological constant.

I guess you could talk about tuning G, but that is basically an artifact of the units, or at least it is fixed so that we get Newtonian gravity as a limit of GR. So I would tend not to list G as a tuneable parameter of GR.

 

[Dale]

Is a rotating universe a legitimate frame of reference ? If the answer is yes, then given proper acceleration (and resulting observation) is invarient, then it (plus the corresponding observation) must by definition also exist in the rotating universe reference frame. Therefore your answer can only be no, a rotating universe is not a legitimate reference frame ?

Yes, it is legitimate. In technical language, you can make a tetrad that is at every event comoving with the local matter.

Yes, the proper rotation of the universe exists in every frame, meaning that an accelerometer at rest with respect to the universe will measure rotation. Regardless of whether you are describing things in the frame described above or in some other frame.

I am unclear why you think my answer must be “no”.

 

[Peter Leeves]

You're mixing up the concept of "reference frame" with the concept of "spacetime geometry".

I've learned in this thread that proper (true ?) acceleration is invarient and thus applies in all reference frames, including inertial ones. It's detected by an accelerometer, and all physics and corresponding results would be the same to all observers no matter what their reference frame. Was that wrong ? If it's correct, then I have at least a rudimentary grasp on "reference frame".

Spacetime geometry I understand as a fabric comprising one dimension of time and three dimensions of space. Within this fabric, mass determines the geometry of the three spatial dimensions, and the geometry of the three spatial dimentions determines how mass moves within it. Properties of mass, energy and light within spacetime are determined by Special Relativity (constant velocity) and General Relativity (acceleration, or via equivalence, gravity). Is that wrong or incomplete ? If so, I'm always happy to learn.

General covariance means that all reference frames are equivalent for describing the same spacetime geometry. So if a given object in a given spacetime geometry has a given proper acceleration in one frame, it will have the same proper acceleration in all frames.

Took several re-reads, but I now understand General Covariance. Thanks.

I may understand better now. Can you please confirm you mean "stationary universe" is one spacetime geometry, and all reference frames WITHIN THAT GEOMETRY must show the same proper acceleration (as measured by an accelerometer). But if you jumped into a DIFFERENT spacetime geometry "rotating universe" it wouldn't show the same (original) proper acceleration ?

However, when you talk about a proper rotating universe, as compared to a proper non-rotating universe, you are talking about two different spacetime geometries. It doesn't even make sense to compare reference frames in these two different spacetime geometries (except locally). Reference frames that describe things in a proper rotating universe have nothing whatever to do with reference frames that describe things in a proper non-rotating universe.

If my understanding of different spacetime geometries is now correct, then yes, I can now follow each and every point in this paragraph.

However, I'm uncertain as to whether "rotating universe" is a different spacetime to "stationary universe". This may be blindingly obvious to you. Sorry, I need to think about it. My immediate thought is it's actually the same spacetime geometry, simply rotating (as a thought experiment) and hence invariant acceleration would still be applicable and result in indentical observation.

Could you please explain why it's a different spacetime to the original one ? I have an argument for it being the original one - but rather think about it a bit.

 

[PeterDonis]

I was referring to the cosmological constant.

I would view this as a solution-specific parameter, and if we're considering those, we have to consider the entire effective stress-energy tensor (which the cosmological constant would be part of--it's just a piece of the SET of the form $\Lambda g_{\mu \nu}$), which gives a total of ten tuneable parameters.

 

[Dale]

I would view this as a solution-specific parameter, and if we're considering those, we have to consider the entire effective stress-energy tensor (which the cosmological constant would be part of--it's just a piece of the SET of the form $\Lambda g_{\mu \nu}$), which gives a total of ten tuneable parameters.

I would not consider the sources to be tuneable parameters of the theory. I can see your point that the cosmological constant could be considered part of the sources rather than a parameter of the theory. I know many people do that so you are at least in good company, but I have always felt that it deserves its own spot outside of the SET. I don't have a particularly good reason for that, other than the way that I was first exposed to GR.

In any case, since I would not consider the sources to be parameters of the theory then I would say that if you want to include the cosmological constant in the sources then GR would have no tuneable parameters at all.

 

[PeterDonis]

I've learned in this thread that proper (true ?) acceleration is invarient and thus applies in all reference frames, including inertial ones. It's detected by an accelerometer, and all physics and corresponding results would be the same to all observers no matter what their reference frame. Was that wrong ?

No. But "all reference frames" here really means "all reference frames that describe the same spacetime geometry".

Spacetime geometry I understand as a fabric comprising one dimension of time and three dimensions of space.

More precisely, one of an infinite number of possible such "fabrics", described by different physically distinct solutions of the Einstein Field Equation. The flat Minkowski spacetime of special relativity, which is what we have implicitly been using the term "non-rotating universe" to mean, is one such solution. The curved spacetime of the Godel universe, which is what we have implicitly been using the term "rotating universe" to mean (once you clarified that that's what you intended, in a post I'll quote below), is another, different such solution. So they are two different spacetime geometries.

Within this fabric, mass determines the geometry of the three spatial dimensions

No. Mass (more precisely, stress-energy) determines the geometry of all four dimensions. The three spatial dimensions by themselves do not have a well-defined geometry. Only the spacetime, consisting of all four dimensions, does.

Properties of mass, energy and light within spacetime are determined by Special Relativity

Only if the spacetime is flat Minkowski spacetime. See below.

(constant velocity) and General Relativity (acceleration, or via equivalence, gravity).

No. You have to use GR whenever spacetime is curved. But you can have acceleration (proper acceleration) in flat spacetime, and SR can handle that just fine. And you can have geodesic motion (free fall, zero proper acceleration) in curved spacetime, and you need GR to handle that.

Can you please confirm you mean "stationary universe" is one spacetime geometry, and all reference frames WITHIN THAT GEOMETRY must show the same proper acceleration (as measured by an accelerometer).

Yes, this is correct. More precisely, once you've specified a particular state of motion for an object in a given spacetime geometry, that fixes its proper acceleration, which will be the same in all reference frames that describe that spacetime geometry.

if you jumped into a DIFFERENT spacetime geometry "rotating universe" it wouldn't show the same (original) proper acceleration ?

Correct. More precisely, you would need to specify a particular state of motion of an object in this different spacetime geometry, and once you did that, that would fix the proper acceleration of the object, which would be the same in all reference frames that describe this different spacetime geometry. But you would have no reason in general to expect that this proper acceleration would be the same as the one in the previous case above. And even if it was, that wouldn't necessarily tell you anything useful.

I'm uncertain as to whether "rotating universe" is a different spacetime to "stationary universe".

I thought the previous discussion had made that clear. It is.

More precisely, when you answered that your intent was (2) here...

I meant (2), not just coordinate change.

...you were answering the question quoted above, giving the answer "yes".

 

[PeterDonis]

I have always felt that it deserves its own spot outside of the SET.

Historically, it did have such a separate spot, yes. However, the fact that the value of the cosmological constant in our actual universe has to be determined from observation, just like the rest of the stress-energy distribution, to me means it should be considered to be part of the sources, i.e., as a solution-dependent parameter just like the rest of the SET.

If there were a way of theoretically deriving a value for the CC independently of the rest of the SET, or, to put it another way, independently of any properties of matter/energy, that would be different; but AFAIK there isn't one. The only theoretical method we have for deriving a value for the CC is from the QFT vacuum, and that method is not independent of properties of matter/energy, since it depends on which specific quantum fields are present, and that depends on what kinds of matter/energy we include in our theory. An "independent" method of deriving a value of the CC, as I am using the term, would do it purely from properties of spacetime geometry, without having to make use of any properties of other fields.

 

[Peter Leeves]

I am unclear why you think my answer must be “no”.

For a moment, put aside the question of whether we are talking about 1 spacetime covering both scenarios (rotating bucket and rotating universe) or 2 spacetimes (different spacetime required for each scenario). Up to this point I didn't realize the implications - but now I'm starting to.

So, assuming both scenarios are covered by the same spacetime geometry, we know proper acceleration (and corresponding observation) is invariant and MUST apply in all reference frames (including inertial ones). If a stationary universe and rotating universe are both legitimate reference frames, then any proper acceleration (and corresponding observation) in one MUST also be identical in the other. Ergo, if an accelerometer in stationary universe registers a proper acceleration (and the water bulges), it necessarily follows that an identical proper acceleration (and water bulge) MUST also occur in the rotating universe - because the acceleration (and observation) is invariant.

By agreeing the rotating universe is a legitimate reference frame, you have agreed that the observation (water bulge) will occur in both stationary and rotating universes (because proper acceleration is invariant and MUST occur in both). As you've been arguing up till now that the same observation wouldn't be seen in the rotating universe, of course I expected you to say it's because the rotating reference frame isn't legitimate.

 

[Dale]

However, the fact that the value of the cosmological constant in our actual universe has to be determined from observation, just like the rest of the stress-energy distribution, to me means it should be considered to be part of the sources, i.e., as a solution-dependent parameter just like the rest of the SET.

I can see your point, but on the other hand, in principle once you have determined the cosmological constant from cosmological observations then in principle a star's gravitational field should be described with a spherical distribution of matter using equations including the cosmological constant. I.e. you would take the cosmological constant (once determined) as part of the equations of the theory that you would use with any distribution of matter.

 

[Dale]

So, assuming both scenarios are covered by the same spacetime geometry,

I cannot do that. It is a false assumption and anything can be proven from a false assumption.

They are two different scenarios with different spacetime geometries represented by different metrics. This is not something that can be hypothesized away because it is central to the whole topic. The case where the universe is rotating around a non-rotating observer is physically distinct from the case where an observer is rotating within a non-rotating universe (all rotations "proper" as described above). There is no coordinate transform between the two cases because it is a completely different spacetime. Within each spacetime all frames are equally valid, but the two spacetimes do not share frames or overlap in any way, they are wholly distinct.

 

[PeterDonis]

you would take the cosmological constant (once determined) as part of the equations of the theory that you would use with any distribution of matter.

No, actually that's not what we do. We don't model the solar system (or binary pulsars, or galaxies, or even galaxy clusters) using a CC in the equations. It's true that that is because the value of the CC we determine from cosmological observations (the acceleration of the universe's expansion) is much too small to have any detectable effect on solar system observations (or even on observations of other systems like the ones I listed); but that's just another way of saying that we use observations to determine what CC parameter we use in our models, just as with the rest of the SET. If we had observations accurate enough to detect CC effects on smaller systems such as galaxies or solar systems, we would be able to test by observation whether the CC is actually a constant everywhere or whether it varies, and in our solar system models we would be using whatever CC parameter was shown by those observations, even if it was different from the one we use for our models of the universe as a whole based on cosmological observations.

 

[Peter Leeves]

I cannot do that. It is a false assumptions and anything can be proven from a false assumption.

They are two different scenarios with different spacetime geometries represented by different metrics. This is not something that can be hypothesized away because it is central to the whole topic.

Understood. That's why I stressed "assuming" - because I've just begun to understand the relevance of the two scenarios being in separate spacetimes. If it's two spacetime geometries, I now understand why you won't get the same observation in the rotating universe.

I have an argument I'd like to put that the two scenarios occur in the same spacetime geometry. I need to think on it a little and will post soon.

 

[PeterDonis]

you've been arguing up till now that the same observation wouldn't be seen in the rotating universe, of course I expected you to say it's because the rotating reference frame isn't legitimate.

No, it's because there are two different spacetime geometries.

put aside the question of whether we are talking about 1 spacetime covering both scenarios (rotating bucket and rotating universe) or 2 spacetimes (different spacetime required for each scenario).

We can't. That question must be answered before we can answer anything else.

Up to this point I didn't realize the implications - but now I'm starting to.

Yes, and that means you should take a big step back and rethink things. See below.

I have an argument I'd like to put that the two scenarios occur in the same spacetime geometry.

You can, of course, define the phrase "non-rotating bucket in a rotating universe" to mean "the same spacetime geometry as the rotating bucket in a non-rotating universe, just described in different coordinates". And if you do that, then the obvious answer to your question will be that the observed shape of the water in the bucket will be the same in both cases. But that answer has nothing to do with physics; it just has to do with how you defined the phrase "non-rotating bucket in a rotating universe".

 

[Dale]

I have an argument I'd like to put that the two scenarios occur in the same spacetime geometry. I need to think on it a little and will post soon.

If you put the two scenarios in the same spacetime geometry then the rotation will be coordinate rotation rather than proper rotation.

 

[Peter Leeves]

No. But "all reference frames" here really means "all reference frames that describe the same spacetime geometry".

Got it. One spacetime geometry can contain multiple reference frames and all will agree on those physics and observations. Another spacetime geometry can also contain multiple reference frames and all will agree on those physics and observations. But a reference frame in one spacetime has no (or very little, depending on the relationship between the two I guess) relevance to the reference frame in the other.

When I answered (2) earlier, it's now apparent I didn't fully understand the implications. Hopefully I have a better grasp now.

 

[Peter Leeves]

You can, of course, define the phrase "non-rotating bucket in a rotating universe" to mean "the same spacetime geometry as the rotating bucket in a non-rotating universe, just described in different coordinates". And if you do that, then the obvious answer to your question will be that the observed shape of the water in the bucket will be the same in both cases. But that answer has nothing to do with physics; it just has to do with how you defined the phrase "non-rotating bucket in a rotating universe".

I need to re-read this one several times. It's very interesting.

My immediate thought is it's similar (not identical) to what Einstein did in his linear thought experiement. I agree it's nothing to do with physics, yet it still provided a valid explanation of observations. This is all I've been trying to do since the OP. Someone was arguing if the water is stationary, then there's no reason for the it to climb the walls of the bucket. I was trying to provide the missing explanation why it would. I do think you've described the solution. Static water is just a different coorinate system in the same spacetime geometry. The acceleration is invariant in both reference frames and hence the observation has to be identical.

Einstein's thought experiment may not have been physics but it nevertheless added two things of value. Firstly he described a mechanism by which one scenario was equivalent to the other. Secondly he concluded that both viewpoints were equally valid. This answer does the same (though definitely NOT using the same or even similar mechanism).

And I've learned an awful lot in arriving at the correct answer. Thank you.

I have a sneaking suspicion you knew this all along and watched me slog my way through it - ha ha ha. Well played gents, well played

 

[PeterDonis]

One spacetime geometry can contain be described by multiple reference frames

See correction above. Reference frames aren't "contained" in spacetime geometries. They are abstract things constructed by humans to describe spacetime geometries.

 

[Peter Leeves]

Now I understand a little better, I wish to make a correction (must be hundreds required, lol). Too many times in this thread I referred to stationary or rotating "bucket", or stationary or rotating "bucket/water". I think I'm correct to say that only the water is stationary (defining it's reference frame). The bucket must rotate along with the rest of the universe.

 

[PeterDonis]

My immediate thought is it's similar (not identical) to what Einstein did in his linear thought experiement.

This was another case of defining different reference frames to describe the same spacetime geometry, yes. The spacetime geometry is flat spacetime; one reference frame is the usual global inertial frame; the other reference frame is the "accelerated" frame in which the linearly accelerating "elevator" is at rest (today we would call this frame "Rindler coordinates", which you can look up for more info). In the latter frame, a "gravitational field" exists, in the sense @Dale defined for that term--there are nonzero Christoffel symbols--and in the sense that a freely falling object will, relative to the frame, have a "downwards" coordinate acceleration, while an object at rest in the frame will have an "upwards" proper acceleration.

The part that might have confused you is that Einstein then went on to draw an analogy with a similar local situation in a different spacetime geometry, the curved spacetime geometry around a planet like the Earth. Sitting at rest inside a room on the surface of the planet "looks" locally like sitting at rest inside the linearly accelerating "elevator" in flat spacetime described above. But what "locally" means here is that we are ignoring the curvature of the spacetime geometry and restricting attention to a small patch of it, small enough that we can treat it as flat. Which means that we are really treating it, for this restricted local purpose, as the same spacetime geometry as the flat spacetime case above. So this is not actually a case of "the same" scenario in two different spacetime geometries. It's just a case of two different choices of coordinates/reference frames in the same (local) spacetime geometry.

 

[PeterDonis]

I have a sneaking suspicion you knew this all along and watched me slog my way through it

Not at all. We have been trying to figure out (or help you to figure out) what you actually mean. You appear to have changed your mind about that twice now.

 

[PeterDonis]

I think I'm correct to say that only the water is stationary (defining it's reference frame). The bucket must rotate along with the rest of the universe.

No. The bucket and the water are at rest relative to each other in every scenario we have discussed. (Note that we have been ignoring any process of "spinning up" the bucket, i.e., we have been ignoring transients and only considering steady-state situations of one sort or another. During transients the bucket and water can of course move relative to each other; but all such relative motion must disappear by the time a steady state has been reached.)

 

[Peter Leeves]

Not at all. We have been trying to figure out (or help you to figure out) what you actually mean. You appear to have changed your mind about that twice now.

I'm happy to accept that.

Ultimately I wanted to answer "Is acceleration absolute or relative". The answer is, absolute (invariant) no matter which reference frame is used or whether they are inertial or non-inertial, but only within a single spacetime geometry.

Specifically, I wanted to answer, if the water is stationary what would cause the water to climb the walls of the bucket. The answer is, the stationary water is just a different coorinate system (reference frame) in the same spacetime geometry. The acceleration is invariant in all reference frames and hence the observation has to be identical.

I bet you can't wait for my post on the delayed choice quantum eraser experiment

 

[Peter Leeves]

No. The bucket and the water are at rest relative to each other in every scenario we have discussed.

Disagree. The bucket and the water are NOT at rest relative to each other in every scenario we've discussed. The water has to spin up in every scenario and therefore the reference frame can only be with respect to the non-rotating water. If I found my opinion changed by the end of the thread, then potentially anyone else reading this thread could miss the same distinction. It's worth including for accuracy and completeness (things we all value) and costs nothing. I say this even given the immediately following proviso, which is excellent and provides the description required.

(Note that we have been ignoring any process of "spinning up" the bucket, i.e., we have been ignoring transients and only considering steady-state situations of one sort or another. During transients the bucket and water can of course move relative to each other; but all such relative motion must disappear by the time a steady state has been reached.)

I understand and agree.

 

[Peter Leeves]

I haven't had time yet to do those computations in detail, but just looking at the Christoffel symbols, we have $\Gamma_{t \varphi}^{r} \neq 0$, so we would expect a nonzero proper acceleration in the $r$ direction for an object that is rotating about the origin in the $\varphi$ direction. And that means that water in a "non-rotating" bucket in this universe would not have a flat shape! Whereas water in a rotating bucket--one "rotating with the universe"--would have a flat shape!

Good old gut-instinct. I promise not to gloat if it turns out to be correct. It does seem intuitive that water in a rotating bucket in a rotating universe would stay flat. There'd be nothing to cause the water to bulge. On the other hand, if the "hand of god" (or tension in a string) grabbed hold and proper stopped the bucket, the torque I described earlier would be applied causing the water to bulge. The equivalence I've been describing between the two scenarios would be correct. Not that it would negate the coordinate solution discussed earlier. Interesting.

 

[PeterDonis]

The bucket and the water are NOT at rest relative to each other in every scenario we've discussed.

You are mistaken. See below.

The water has to spin up

But you said you understand and agree that we are ignoring the spin-up process and only considering steady state. In steady state the bucket and water are at rest relative to each other.

the reference frame can only be with respect to the non-rotating water

Whether or not the bucket and the water are at rest relative to each other is an invariant; it has nothing to do with any choice of frame and it will be equally true in every frame.

I say this even given the immediately following proviso

I have no idea what you mean by this or why you do not see that your statements are inconsistent with each other, as described above.

 

[PeterDonis]

The equivalence I've been describing between the two scenarios would be correct.

No, it wouldn't. The two spacetime geometries are still different, and that difference will still show up in the two scenarios.

I have not done the detailed computations, but here is what I think they will end up telling us. Note that I am stating everything in terms of invariants and direct observables, with no talk of "frames" at all.

First, we have to be clear about what invariants and direct observables we are talking about. There are actually four:

(1) The proper acceleration of the bucket and the water inside it. This determines the shape of the water's surface (flat for zero proper acceleration, concave for nonzero radial proper acceleration).

(2) The vorticity of the bucket and the water inside it. This determines whether the bucket is rotating (nonzero vorticity) or non-rotating (zero vorticity).

(3) The relative angular velocity of the bucket and the water inside it, with respect to the rest of the matter in the universe. Note that this is not the same as (2) above.

(4) The rotation of the rest of the matter in the universe. This is given by the vorticity of the family of worldlines that describe that matter.

Now let's look at four different scenarios and the invariants for each of them (the first two are known, the last two are what I want to verify by computation):

(NN) Non-rotating bucket in non-rotating universe (flat spacetime). The invariants are:

#1: Zero proper acceleration, flat surface.

#2: Zero vorticity, zero rotation.

#3: Zero relative angular velocity relative to rest of universe.

#4: Zero rotation of rest of universe.

(RN) Rotating bucket in non-rotating universe:

#1: Nonzero proper acceleration, concave surface.

#2: Nonzero vorticity, nonzero rotation.

#3: Nonzero angular velocity relative to rest of universe.

#4: Zero rotation of rest of universe.

(NR) Non-rotating bucket in rotating universe (Godel spacetime):

#1: Nonzero proper acceleration, concave surface.

#2: Zero vorticity, zero rotation.

#3: Nonzero angular velocity relative to rest of universe.

#4: Nonzero rotation of rest of universe.

(RR) Rotating bucket in rotating universe:

#1: Zero proper acceleration, flat surface.

#2: Nonzero vorticity, nonzero rotation.

#3: Zero angular velocity relative to rest of universe.

#4: Nonzero rotation of rest of universe.

Note how the first two invariants, which are the "local" ones, are different in each scenario; they cover all four of the possibilities, given that each of the two observables is a binary choice (zero or nonzero). That means no two of these scenarios are equivalent; each one has a different, unique set of observables.

 

[Peter Leeves]

There's no point repeating myself. If you don't wish to consider my viewpoint, that's absolutely fine. The information I felt was of genuine value and ought to be included in this thread, is now included.

But you said you understand and agree that we are ignoring the spin-up process and only considering steady state. In steady state the bucket and water are at rest relative to each other.

I do understand and agree that we are ignoring the spin-up process and only considering the steady state. At the risk of repeating myself (said I wouldn't but I'll give it one try) I feel the spin-up detail was worth including within this thread for both accuracy and completeness. Particularly for the benefit of others reading this thread at a later date. In short, it genuinely adds some value.

Whether or not the bucket and the water are at rest relative to each other is an invariant; it has nothing to do with any choice of frame and it will be equally true in every frame.

Same comment as above. Costs nothing to note the spin-up and we can all sleep better knowing we've been as accuracte and complete as possible.

I have no idea what you mean by this or why you do not see that your statements are inconsistent with each other, as described above.

My statements are consistent. You wrote a concise and excellent description covering spin up of the water and the reasons it can be ignored. It was this description I referred to as "the proviso". Very well put and definitely worth including in the thread.

 

[Dale]

The bucket and the water are NOT at rest relative to each other in every scenario we've discussed.

The traditional Newton's bucket is at rest relative to the water. I.e. the spin-up or spin-down happened a long time ago, viscous forces have caused all of the relative motion between the water and the bucket and between different parts of the water to dissipate.

"When friction between the water and the sides of the bucket has the two spinning together with no relative motion between them then the water is concave."
https://mathshistory.st-andrews.ac.uk/HistTopics/Newton_bucket/

Edit: never mind, I see that you just discussed that for completeness

 

[PeterDonis]

I do understand and agree that we are ignoring the spin-up process and only considering the steady state.

But you apparently do not agree that in the steady state, the bucket and the water are at rest relative to each other?