Does this contradict or require GR?

[Buckethead]

This is a rather offbeat question, but I was wondering just the same. Imagine a spiral galaxy. Imagine a space surrounding that galaxy, somewhat larger. Imagine that this space surrounding the galaxy were to rotate somewhat with the galaxy, as a whole, stiff like a bike wheel. (Side note: by rotating space, I mean if a person were to be standing at the axis of this rotation and rotating with this space, they would feel no rotational forces). Here are my questions:

1. I know space can expand, and that spacetime can be distorted such that inertial paths appear to be curved when viewed from outside. Based on this, is it completely unreasonable to imagine that space can do as I describe above. In other words, is is utterly completely impossible.

2. If this is not utterly impossible, is this something that might be considered a possible "solution" to GR. In other words, if someone wanted to investigate if this were possible would they start by finding a new solution to Einstein's field equations or is this under the realm of another law?

3. If it is not utterly impossible, but a solution to GR is not where one would start, where would one start?

4. If it is utterly impossible, is that because it contradicts an already known solution to GR? Or even that it contradicts SR? And if so, can you elaborate? I can't imagine that it contradicts SR in any way as it would have no effect on the speed of light when measured locally (just like expanding space), but still trying to cover all the bases here. Thanks.

6. One more question. If this is a possible anomaly would this be considered a modification of gravity? ( I don't think so but that's why I'm asking)

 

[kimbyd]

Space-time definitely rotates around rotating objects. You can start with the metric for a rotating black hole, the Kerr metric (https://en.wikipedia.org/wiki/Kerr_metric). This is General Relativity, though, so nothing about it is easy or simple.

 

[Buckethead]

Isn't the effect that results from the Kerr metric (The Lense-Thirring effect) a non linear effect, i.e. the degree of spacetime rotation depends on the distance from the rotating body? I think my question is more about the Schwarzschild metric as I'm asking more about the galaxy as a whole and even though it is rotating, certainly not as fast as the black hole in the center. So the question really comes down to, can the spacetime around the entire galaxy rotate/distort as a whole, due to the rotation of the entire galaxy, and more specifically could the result of this be that the space (not spacetime) rotates as a stiff single entity when viewed from outside this system?

 

[Bandersnatch]

You do know the entire galaxy doesn't rotate as a whole at the same angular velocity?

 

[Buckethead]

You do know the entire galaxy doesn't rotate as a whole at the same angular velocity?

Yes. I was asking about the space (not spacetime) around the galaxy. Could it rotate at the same angular velocity as a result of the Kerr metric (or something else) for example.

 

[kimbyd]

Isn't the effect that results from the Kerr metric (The Lense-Thirring effect) a non linear effect, i.e. the degree of spacetime rotation depends on the distance from the rotating body? I think my question is more about the Schwarzschild metric as I'm asking more about the galaxy as a whole and even though it is rotating, certainly not as fast as the black hole in the center. So the question really comes down to, can the spacetime around the entire galaxy rotate/distort as a whole, due to the rotation of the entire galaxy, and more specifically could the result of this be that the space (not spacetime) rotates as a stiff single entity when viewed from outside this system?

Yes, the degree of spacetime rotation depends strongly on the distance from the rotating body. Frame dragging is a local effect (and has been measured around the Earth).

In practice, the amount of frame dragging around the galaxy is likely to be so tiny as to be unmeasurable. You could probably get a rough estimate of how much frame dragging there should be using the Kerr metric and using the total mass within a radius $R$ to estimate how much there would be. My guess is not much.

As for the Schwarzschild metric, that metric isn't a rotating metric, so you can't apply it to this question.

 

[kimbyd]

Yes. I was asking about the space (not spacetime) around the galaxy. Could it rotate at the same angular velocity as a result of the Kerr metric (or something else) for example.

The difference between space and space-time is a choice of coordinates. To talk about one is to talk about the other.

 

[Buckethead]

Yes, the degree of spacetime rotation depends strongly on the distance from the rotating body. Frame dragging is a local effect (and has been measured around the Earth).

In practice, the amount of frame dragging around the galaxy is likely to be so tiny as to be unmeasurable. You could probably get a rough estimate of how much frame dragging there should be using the Kerr metric and using the total mass within a radius $R$ to estimate how much there would be. My guess is not much.

From a logical standpoint, I find it curious that frame dragging doesn't extend far beyond a galaxy as what is holding it back? As an example, if you stirr the water in a cup, the water as a whole rotates with the exception of the areas bound by friction. In a similar way, what is holding back spacetime from rotating to much greater radii? Not the stars in the system as they are orbiting as well, so the spacetime should be going at least as fast as the stars, then beyond the galaxy there is even less resistance to the phenomenon until you reach other galaxies.

As for the Schwarzschild metric, that metric isn't a rotating metric, so you can't apply it to this question.

But that metric specifies the geometry of the spacetime and thought it might fit here when describing the system on a larger scale that is not depending on the Kerr metric. In other words, if you have a black hole that is rotating with the same angular velocity as the average angular velocity of the stars in the system so that more or less everything is has some overall angular velocity, then I would think that the spacetime surrounding that system would rotate with a constant angular velocity due to something like the Lense-Thirring effect but which wouldn't be so localized.

My question is not exactly clear, but from a logical standpoint, why would the Kerr metric act as if there was a lot of "friction" beyond the rotating body causing it to drop off so rapidly? What is fighting against it having full freedom of rotation?

 

[kimbyd]

From a logical standpoint, I find it curious that frame dragging doesn't extend far beyond a galaxy as what is holding it back? As an example, if you stirr the water in a cup, the water as a whole rotates with the exception of the areas bound by friction. In a similar way, what is holding back spacetime from rotating to much greater radii? Not the stars in the system as they are orbiting as well, so the spacetime should be going at least as fast as the stars, then beyond the galaxy there is even less resistance to the phenomenon until you reach other galaxies.

Nothing "holds it back". It's just that gravity is local. The motion of any single object only impacts the space-time near it. Frame dragging is a local effect for the same basic reason that in the classical approximation the gravitational force falls off as $1/r^2$ with distance.

But that metric specifies the geometry of the spacetime and thought it might fit here when describing the system on a larger scale that is not depending on the Kerr metric. In other words, if you have a black hole that is rotating with the same angular velocity as the average angular velocity of the stars in the system so that more or less everything is has some overall angular velocity, then I would think that the spacetime surrounding that system would rotate with a constant angular velocity due to something like the Lense-Thirring effect but which wouldn't be so localized.

It's not the geometry. It's a way of describing the geometry of space-time outside a neutrally-charged, non-rotating, spherically-symmetric object. If your object doesn't follow those very strict rules, then the Schwarzschild metric will not describe the system: it will only be an approximate description.

The Kerr metric adds rotation to the mix, so that it can describe such objects. It still assumes neutral electric charge and spherical symmetry, however.

My question is not exactly clear, but from a logical standpoint, why would the Kerr metric act as if there was a lot of "friction" beyond the rotating body causing it to drop off so rapidly? What is fighting against it having full freedom of rotation?

Because the universe as a whole isn't rotating.

 

[kimbyd]

@Bruce:

Your question would be better-asked in its own thread, but a reasonably-clear explanation for the distance measures and where they come from is in Hogg's 1999 paper linked here.

I'm not going to go through the time to parse and understand that VB code, sorry.

 

[Buckethead]

Nothing "holds it back". It's just that gravity is local. The motion of any single object only impacts the space-time near it. Frame dragging is a local effect for the same basic reason that in the classical approximation the gravitational force falls off as 1/r21/r^2 with distance.

I appreciate the drop off rate, but it can't be the same thing. For example, let me offer this (really terrific) analogy. Imagine concentric balloons, one inside the other, with a gap in between the balloons. There is air between each concentric layer of balloon. The first layer has air at 1 atmosphere, the second is at 1/4 atmosphere, the 3rd at 1/9th and so on, so the air pressure falls at the same 1/r^2 rate. If the outer balloon is in a vacuum and the inner balloon starts to spin, the outer balloon will also spin at the same angular velocity because there is nothing preventing it from doing so. On the other hand if someone were to hold the outer balloon from spinning, then we would see a rapid drop in the angular rotations of each balloon. Why is this analogy not applicable to this situation?

Because the universe as a whole isn't rotating.

This could be an answer to the above question but wouldn't it depend of the strength of the gravitational field from the universe at the outer edge of the galaxy? And wouldn't this external field be very very weak?

 

[Ibix]

Why is this analogy not applicable to this situation?

Because gravity isn't electromagnetism and spacetime isn't matter. Analogies can be illustrative, but attempting to reason from an analogy won't work. If it did, we wouldn't bother with the maths.

 

[Buckethead]

Because gravity isn't electromagnetism and spacetime isn't matter. Analogies can be illustrative, but attempting to reason from an analogy won't work. If it did, we wouldn't bother with the maths.

I agree, but I'm not trying to reason from an analogy, I'm using the analogy to clarify my questions. My reasons for wanting to know how far solutions to GR can go with regard to frame dragging or rotating spacetimes is because there is an interesting relationship between these things and the dark matter problem. For example, in this paper:

https://arxiv.org/pdf/0807.0679.pdf

Kohkichi Konno makes the opening statement:

"The long range feature of frame-dragging effect under the Chern-Simon gravity well explains the flat rotation curves of galaxies...".

This is a provocative statement because I have known for some time now that there seems to be a correlation between the rotational curve of common spiral galaxies and an imaginary and simple concept of a "rotating space" (whether or not such a thing is possible) around the galaxy. To illustrate what I mean take a look at these two graphs of galaxies NGC3198 and M33. The blue lines are the observed rotational curves of the galaxies, the green lines are the expected curves if no dark matter existed in these galaxies, and the red curves, are the differences between the green and blue curves. The interesting feature of the red curves is that they approximate a simple y=kx line.

These graphs do not represent anything in Konno's paper, I put them here to illustrate the reasons for my original questions and why they are not idle questions. If the space surrounding a galaxy as a whole rotated with the galaxy (represented by a line y=kx) then from the perspective of an observer outside a galaxy devoid of DM, it would appear to rotate as represented by the blue line.

So from my limited understanding it seems that Konno is using the Chern-Simon gravity well effect to illustrate how such an effect can cause a distortion in our observations of a rotational curve.

The difference of course between my illustrations is that I am just taking a foundationless idea of a rotating space around a galaxy to show myself what would happen if such an idea was actually based on real science such as the Lense-Thirring effect or perhaps a new solution to GR that allows my foundationless idea to actually have some foundation.

I hope I have clarified why I am asking the questions I have in my first post.

 

[PeterDonis]

Imagine that this space surrounding the galaxy were to rotate somewhat with the galaxy, as a whole, stiff like a bike wheel.

You can't just "imagine" this. You have to look at what GR actually says. And GR says that spacetime (which is what you should be looking at, not space) around a galaxy won't rotate this way.

Space-time definitely rotates around rotating objects.

But not "stiff like a bike wheel".

Isn't the effect that results from the Kerr metric (The Lense-Thirring effect) a non linear effect, i.e. the degree of spacetime rotation depends on the distance from the rotating body?

To the extent that "the degree of spacetime rotation" is a meaningful concept, yes, it depends on the distance from the body. It also depends on the "latitude" relative to the axis of rotation.

I think my question is more about the Schwarzschild metric

The Schwarzschild metric describes a non-rotating spacetime around a non-rotating body, so it's not relevant to this discussion.

 

[PeterDonis]

My reasons for wanting to know how far solutions to GR can go with regard to frame dragging or rotating spacetimes is because there is an interesting relationship between these things and the dark matter problem. For example, in this paper

For future reference: you should have said this and given the link to the paper in the OP of this thread. That would have made it clear that you are asking a question that really requires an "A" level discussion, not a "B" level one. And for an "A" level discussion you would first need to understand the much simpler concepts relating to the Kerr spacetime and the metric around a rotating object, which could be a whole separate discussion, at least "I" level but quite possibly "A" level itself.

For the present I am closing this thread. @Buckethead, if you want an "A" level discussion of the paper you linked to, let me know by PM and I will spin it off into its own thread at that level.

 

[PeterDonis]

Moderator's note: I have changed the level of this thread to "I" since the OP topic can't be usefully discussed at the "B" level.