Rotation and Revolution in relativity

[arindamsinha]

I have a question about the concepts of rotation and revolution - on how they are treated in relativity.

Since all motion is relative, a revolution of a planetary body around a central body could also be seen instead as a rotation of the central body w.r.t. a fixed (non-revolving) planetary body. Both points of view should be equally valid.

Now, why should properties of any object then depend on its state of rotation or revolution?

For example, I am thinking about the concept of static and rotating black holes having different properties. When we identify a rotating black hole somewhere, we could also equally consider it to be static, with us revolving around it, surely?

So why should there be any observable difference in properties of such an object, which appears to be the case, when we can flip the notions of static and rotating by using appropriate reference frames?

Or, is there some sort of preferred reference frame in the Universe which allows us to determine whether something is rotating or not (e.g. ECIF)? If so, how is that taken into account in relativity?

 

[Dale]

Since all motion is relative

This is incorrect. Only inertial motion is relative. Non-inertial motion, such as rotation, can be measured without reference to any external object so it is not relative.

Both points of view should be equally valid.

Both points of view are equally valid and will give you the same experimental outcomes in all cases. However, the metric has a different form in the two points of view, so even though they are both valid, they are distinguishable.

 

[Bill_K]

Both points of view are equally valid and will give you the same experimental outcomes in all cases. However, the metric has a different form in the two points of view, so even though they are both valid, they are distinguishable.

In this connection one should always mention that, although special relativity handles acceleration just fine, it does not handle just fine accelerating coordinate systems, which if used at all can be used only in a local region.

Either for rotation or linear acceleration, eventually at large distances the relative velocity exceeds c, leading to a paradox. The proper description of acceleration uses a different concept, a set of basis vectors at each point, variously called a tetrad, a vierbein, or frame.

 

[arindamsinha]

This is incorrect. Only inertial motion is relative. Non-inertial motion, such as rotation, can be measured without reference to any external object so it is not relative.

How is that possible? What is rotation without reference to an external frame?

In other words, how can we tell if a body is static or rotating in an otherwise empty Universe (or very far away from all other matter)?

 

[Dale]

How is that possible? What is rotation without reference to an external frame?

In other words, how can we tell if a body is static or rotating in an otherwise empty Universe (or very far away from all other matter)?

Use a gyroscope or a ring interferometer attached to the body. This is very similar to how you tell if a body is accelerating or inertial using an accelerometer.

 

[arindamsinha]

Use a gyroscope or a ring interferometer attached to the body. This is very similar to how you tell if a body is accelerating or inertial using an accelerometer.

We can't reasonably do that to a black hole. How can we determine if a black hole is rotating or not?

 

[Bill_K]

For a real black hole, this is a question in observational astronomy. But in principle, a rotating black hole would exhibit frame dragging, that might be seen for example as an asymmetry in the deflection of grazing light rays.

 

[Dale]

We can't reasonably do that to a black hole. How can we determine if a black hole is rotating or not?

Measure the Lense-Thiring effect near the black hole.

 

[Mentz114]

We can't reasonably do that to a black hole. How can we determine if a black hole is rotating or not?

Measure the tidal forces around an observer. In the absence of rotation the $\phi$-component is equal to the $\theta$-component. They are not equal if rotation is present. I've done this calculation in Doran coords, but in the comoving frame basis. The extra contribution to the $\theta$-component is $3a^2m/r^5$ where a is the angular momentum parameter.

 

[jtbell]

In other words, how can we tell if a body is static or rotating in an otherwise empty Universe (or very far away from all other matter)?

You can tell if you yourself are rotating, even if your eyes are closed, can't you? (At least if you're rotating fast enough...)

 

[arindamsinha]

You can tell if you yourself are rotating, even if your eyes are closed, can't you? (At least if you're rotating fast enough...)

This essentially goes back to my original question. Yes, we are able to tell. Why are we able to tell?

Does this go back to Mach's principle, that it is dependent on the rest of the matter in the Universe (which in some sense creates an Universal frame)? The example above is very similar to what Mach had in fact talked about.

I know Einstein took some of Mach's principle as inputs, but it is not clear that he maintained all of that through development of GR. There does not seem to be any reference to an Universal frame, and all phenomena seem to be defined locally. Did he in fact abandon Mach's idea, or does it still underly GR?

 

[George Jones]

We can't reasonably do that to a black hole. How can we determine if a black hole is rotating or not?

For a real black hole, this is a question in observational astronomy. But in principle, a rotating black hole would exhibit frame dragging, that might be seen for example as an asymmetry in the deflection of grazing light rays.

We should be able to "see" this within a few years. A nice non-technical article on this is "Portrait of a Black Hole",

https://www.cfa.harvard.edu/~loeb/sciam2.pdf,

from the December 2009 issue of Scientific American. The February 2012 issue of Sky & Telescope has a more recent but less detailed article on this, "Einstein's Shadow".

This is very exciting, because it will give observational tests of images predicted by strong-field general relativity near event horizons.

[edit]Fixed broken link. Thanks, Mentz114[/edit]

 

[Mentz114]

George, the link seems to be broken.

Found this https://www.cfa.harvard.edu/~loeb/sciam2.pdf

 

[arindamsinha]

We should be able to "see" this within a few years. A nice non-technical article on this is "Portrait of a Black Hole" ...

... The February 2012 issue of Sky & Telescope has a more recent but less detailed article ...

This is very exciting, because it will give observational tests of images predicted by strong-field general relativity near event horizons.

Yes, most interesting. Thanks for the link [and Mentz for the working one]. I did see the Sky & Telescope article.

Still interested in any views on the questions in my previous post.

 

[Bill_K]

Still interested in any views on the questions in my previous post.

I thought we had answered it? Well then to repeat, in general relativity acceleration and rotation are absolute. A nonaccelerating, nonrotating inertial frame can be defined locally without reference to the distant stars. Mach's Principle may have played a role in the development of Einstein's ideas, but does not hold true in the final theory.

There are examples in which the rotation of one body influences nearby inertial frames. But at the same time there are "anti-Machian" cosmologies, in which the motion of distant matter is in complete disagreement with local inertial frames. So the general principle does not hold, and is not really useful as a guide.

 

[Dale]

Did he in fact abandon Mach's idea, or does it still underly GR?

Unfortunately, Mach's principle is pretty vague so it is quite open to interpretation as to how to check if a given theory is Machian.

There is a generalization of GR called Brans-Dicke gravity which has a free parameter that, to my understanding, represents the Machian-ness of the universe with 0 being a perfectly Machian universe and infinity corresponding to GR. So Brans and Dicke, at least, think that GR is not Machian.

Unfortunately, according to that criterion the universe appears to be non-Machian since the current lower bound on the parameter is something like 40000. So my interpretation is that Mach's principle is either too vague to be tested or it has been tested and found contrary to experiment. Either way it is not useful.

 

[arindamsinha]

...Mach's Principle may have played a role in the development of Einstein's ideas, but does not hold true in the final theory.

...there are "anti-Machian" cosmologies...

...it is quite open to interpretation as to how to check if a given theory is Machian.

...So Brans and Dicke, at least, think that GR is not Machian.

...So my interpretation is that Mach's principle is either too vague to be tested or it has been tested and found contrary to experiment. Either way it is not useful.

OK. That's to the point and helpful.