Is frame dragging the same as torsion?

[friend]

Is frame dragging in GR the same as torsion in curved spacetime?

 

[tom.stoer]

No.

The mathematical setup for GR is Riemann geometry with Levi-Cevita connection and vanishing torsion. Frame-dragging does exist in GR.

Einstein-Cartan gravity is an extension of GR with non-vanishing torsion, especially relevant when coupling gravity to fermions.

 

[friend]

No.

The mathematical setup for GR is Riemann geometry with Levi-Cevita connection and vanishing torsion. Frame-dragging does exist in GR.

Einstein-Cartan gravity is an extension of GR with non-vanishing torsion, especially relevant when coupling gravity to fermions.

So does that mean the frame dragging IS torsion even though torsion in not in Eistein's GR?

 

[nicksauce]

GR has frame dragging. GR does not have torsion. Therefore frame dragging cannot be the same as torsion.

 

[tom.stoer]

To be more specific: the torsion tensor T is defined in terms of the connection Gamma as

T^a_{bc} = \Gamma^a_{bc} - \Gamma^a_{cb}

and determines the antisymmetric part of the connection.

As I said, GR uses Riemann geometry with the rather special Levi-Cevita connection which is constructed from the metric

\tilde{\Gamma}^a_{bc} = \frac{1}{2}g^{ad}(\partial_{c}g_{db}+\partial_{d}g_{dc} - \partial_{d}g_{bc})

in such a way that this Gamma is symmetric in the lower indices

\tilde{\Gamma}^a_{bc} = \tilde{\Gamma}^a_{cb}

and therefore torsion vanishes in Riemann geometry and GR by construction.

Another remark: frame dragging is an effect which follows from the specific dynamics of GR (i.e. from the coupling of matter with its energy-momentum-tensor) to the geometry, whereas torsion is already introduced (or absent) on the purely geometric level. It's like asking whether the centrifugal force has something to do with Euclidean geometry: not really b/c Euclidean geometry is pure geometry whereas the centrifugal force has to be explained with Newtonian dynamics which is formulated in terms of the geometry, but which is not identical with geometry.

 

[bcrowell]

Experimentally:

Searches for torsion have given negative results:
http://www.npl.washington.edu/eotwash/sites/www.npl.washington.edu.eotwash/files/webfiles/publications/pdfs/lowfrontier2.pdf

Frame dragging has been confirmed by Gravity Probe B: http://en.wikipedia.org/wiki/Gravity_Probe_B

Sometimes if you're trying to distinguish two concepts it helps to consider how they're actually measured.

 

[tom.stoer]

Experimentally:

Searches for torsion have given negative results

I doubt that this can be a general result. Reading the paper I can't see any hint how they rule out torsion.

As we know Einstein-Cartan theory - as an extension to Einstein's General Relativity - is formulated in terms of Riemann-Cartan geometry with torsion. But here torsion is not a dynamical i.e. propagating d.o.f, therefore
a) the torsion tensor can always be expressed algebraically in terms of spin-densities of matter
b) torsion must be absent in vacuum i.e. in spacetime with vanishing matter distribution

But when fermionic matter is present we may have a spin current which acts as a source for non-vanishing torsion. So
c) torsion can exist inside matter.
But inside matter you can't test this b/c matter effects like (orbital) angular momentum dominate torsion.

I strongly believe that Einstein-Cartan theory - which is equivalent to GR experimentally, except for the tiny torsion effects - is a much more natural mathematical setup for dynamical spacetime, especially when coupled to fermions. And I haven't ever seen an experiment that is able to measure macroscopic spin densities and torsion inside matter.

 

[TrickyDicky]

Frame dragging has been confirmed by Gravity Probe B: http://en.wikipedia.org/wiki/Gravity_Probe_B

No, the geodetic effect was confirmed, the result was not conclusive for frame dragging.

 

[A.T.]

I think the idea that frame dragging means "torsion" comes from visualizations like these:

http://einstein.stanford.edu/MISSION/mission1.html

http://imagine.gsfc.nasa.gov/docs/features/news/20apr04.html

http://cr4.globalspec.com/blogentry/1670/Extreme-Frame-Dragging

Are they inaccurate? Misleading? Is there a better way to visualize it?

 

[tom.stoer]

Are they inaccurate? Misleading?

No; I think it's fine.

But geometrically this is not torsion.

 

[A.T.]

No; I think it's fine.

But how should they be interpreted? What does it mean physically, when the radial lines and the circumferences are not orthogonal? Would radially falling photons be diverted tangentially, and "spiral down" instead of going straight down?

 

[tom.stoer]

What does it mean physically, when the radial lines and the circumferences are not orthogonal?

Nothing. At least not directly b/c what is shown are coordinates = reference frames. And they are unphysical i.e. cannot be observed.

Would radially falling moving photons be diverted tangentially, and "spiral down" instead of going straight down?

Yes! But this does not follow from the reference frames but from the geodesics.

No; I think it's fine.

Hm, perhaps I was wrong ;-(