# The Vanishing of Torsion in Relativity?

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I was reading the replies to a thread on another site:

One posters states that torsion vanishes from Riemann geometry and general relativity ''by construction.'' This is still a common myth, even by experienced posters online. Torsion does not ''vanish'' in spacetime as some intrinsic part of relativity, in fact, such a statement has no empirical support.

It has been noted by some authors, that general relativity without torsion is actually, the most boring kind of model you can work with.The torsion is of course part of the Poincare spacetime symmetries from properties found in rotation.Torsion, if we would accept for a moment could have solid foundation in reality, could offer solutions to standing problems in QM and even Cosmology.

''The Einstein–Cartan theory eliminates the general-relativistic problem of the unphysical singularity at the Big Bang.[9] The minimal coupling between torsion and Dirac spinors generates an effective nonlinear spin-spin self-interaction, which becomes significant inside fermionic matter at extremely high densities. Such an interaction replaces the singular Big Bang with a cusp-like Big Bounce at a minimum but finite scale factor, before which the observable universe was contracting. This scenario also explains why the present Universe at largest scales appears spatially flat, homogeneous and isotropic, providing a physical alternative to cosmic inflation.[8]

Torsion allows fermions to be spatially extended[11] instead of "pointlike", which helps to avoid the formation of singularities such as black holes and removes the ultraviolet divergence in quantum field theory. According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular black hole. In the Einstein–Cartan theory, instead, the collapse reaches a bounce and forms a regular Einstein–Rosen bridge (wormhole) to a new, growing universe on the other side of the event horizon.''

I've also shown, through investigations into the work of Venzo de Sabbata and C Sivaram, (who have both done extensive work in the past on torsion theories) that the torsion should arise in a Friedmann equaion in the following way, by entering the effective density ~

$(\frac{\ddot{R}}{R})^2 = \frac{8 \pi G}{3}(\rho - k\sigma^2)$

In which the Torsion is related to the Poisson equation

$\nabla^2 \phi = 4 \pi G(\rho - \mathbf{k}\sigma^2)$

You can calculate the torsional part by the following the dimensional arguments:

$\mathbf{k}\sigma^2 = \frac{Gm^2v^2R^2}{c^4L^6} = \frac{Gm^2}{c^2L^4}$

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Yes I agree on the misconceptions which arise from not fully examining the field equations to higher orders as mentioned and described in those papers. It is an excellent point to make others aware of

However Gravity probe B does tighten the constraints on torsion with the final results found here. This is the 2011 paper on the G probe

I don't have any doubt that you and I will agree on further tests are required. While the gravity Probe data currently favors GW as opposed to Cartan the results isn't conclusive enough.

Just a side note BCrowell I highly respect on his knowledge on GR.  He has several books on the topic with a Ph.D.

PS all symmetric solutions are boring in any metric the fun is the assymetry.

Edited by Mordred
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9 hours ago, Mordred said:

I don't have any doubt that you and I will agree on further tests are required.

Indeed, more stringent tests need to be done. Perhaps torsion coupling can only be seen with more significant contribution of mass, like I expect around black holes.

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It would certainly be a good arena to measure it. For other readers torsion applies to fluid hydrodynamics involved in the other entries of the stress tensor. The components involving stress flux and vorticity all involve torsion equations. This is true even in GR.

density doesn't induce stress however pressure does, but there is further stress terms beyond stress pressure. These additional stress terms involve torsion as the vector directions require multiple rotations.

Now in GR for free fall motion we initially set the stress tensor to zero to comply with the two conservation of energy/momentum laws. (linear and angular). This does not mean torsion does not apply, where it applies is when the stress tensor cannot be set to zero.

In Cartan where you always modelling rotating fields the stress tensor is always involved ie will affect the pressure term of the previous

The fundamental difference between the two metrics is in essence how you examine the system.

This breaks down to the question of the differences between frame dragging and torsion.

In frame dragging the stress tensor itself is unchanging and it is the metric tensor that undergoes change.( oversimplification to above) so don't take too literally.

The difference can be understood by comparing the ds^2 seperation distance between the two line elements.

[mathd{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2][/math]

In Cartan theory the line element contains 1 additional term $\omega$

So your tensor organizations will reflect the additional term

The line element identifies to the null geodesic equations involved and more complexely the Bianchi identities that apply in the geodesic equations. (the line element itself is not specifically the geodesic equation)

Edited by Mordred
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The Gravity probe B results are very close to the expected results predicted by GR, Mordred.

GR has frame dragging and a vanishing torsion, while Einstein-Cartan has torsion.
Doesn't the GPB result for frame dragging ( from your post ) put a nail in the coffin of Cartan torsion ?

Or can a third model provide both ?

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This is actually tricky to answer, as it really depends largely on the system being measured as to whether or not a local system probe is a sufficient test. Its not unknown that localized dynamics can wash out global dynamics.

Personally The upper bounds and studies of the CMB provided a far stronger test between Cartan and GR. This is something I tried without much success in pointing out to the OP in his speculation thread.

The gravity probe B test itself was by design a low orbit test, so it was viable a global intrinsic torsion as per say a rotating universe could be washed out.

PS the CMB studies also favor GR.

Edited by Mordred

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