Spin Density and Non-symmetric Stress-Energy Tensor

[TrickyDicky]

@TrickyDicky, regarding you statement that this is not "... defined in ... GR." and that "it will depend in the particular geometry": in GR pointlike test particles follow geodesics and b/c GR uses Riemannian geometry the geodesics are both straightest and shortest worldlines, so the two definitons coincide; it is defined in GR.

Yes, I actually didn't know exactly what Ben meant with that question because the obvious answer was: geodesics, so I figured he meant something different by "free test particle".

 

[tom.stoer]

Another intersting reference could be

http://arxiv.org/abs/1204.3672v1
Gauge Theory of Gravity and Spacetime
Authors: Friedrich W. Hehl (U Cologne and U of Missouri, Columbia)
(Submitted on 17 Apr 2012)
Abstract: The advent of general relativity settled it once and for all that a theory of spacetime is inextricably linked to the theory of gravity. From the point of view of the gauge principle of Weyl and Yang-Mills-Utiyama, it became manifest around the 1960s (Sciama--Kibble) that gravity is closely related to the Poincare group acting in Minkowski space. The gauging of this external group induces a Riemann-Cartan geometry on spacetime. If one generalizes the gauge group of gravity, one finds still more involved spacetime geometries. If one specializes it to the translation group, one finds a specific Riemann-Cartan geometry with teleparallelism (Weitzenbock geometry).

 

[KnownUniverse]

If the metric tensor is not symmetric then the energy momentum tensor is not symmetric, and so, we have the physical interpretaion that the antisymmetric part of the metric tensor is related to spin. In fact, it is the potential of the spin field (torsion). The mathematics is given in arXiv:1207.5170v1 [gr-qc].

 

[cosmik debris]

GR and EC are not mathematical identical but indistinguishable experimentally...

I thought that the torsion theories predict that matter with opposite chirality breaks the equivalence principle i.e. by following different paths through spacetime. I believe experiments were proposed using identically prepared crystals of opposite chirality and measured with an Eotvos torsion balance. I think Chinese scientists performed the experiments although the results were inconclusive. I'll try to dig up a reference.

 

[tom.stoer]

In http://arxiv.org/pdf/gr-qc/9712096.pdf Hehl puts a constraint on EC which should not produce experimentally visible effects even at densities comparable to nuclear densities

 

[cosmik debris]

In http://arxiv.org/pdf/gr-qc/9712096.pdf Hehl puts a constraint on EC which should not produce experimentally visible effects even at densities comparable to nuclear densities

Yes, interesting. I'd like to see this expanded a little.

 

[bcrowell]

As I understand it from Misner, Thorne and Wheeler's book on gravity, the fact that test particles follow geodesics is not an additional assumption in GR, but is actually provable from the field equations. I don't have it handy, so I can't actually verify that, but I seem to remember such a claim.

It's a little complicated.

Here http://arxiv.org/abs/gr-qc/0309074v1 is a paper by Ehlers and Geroch on this topic, and here http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.1 (subsection 8.1.3) is my attempt to present their argument in an accessible way. For their theorem to work, you need the dominant energy condition plus a bunch of other technical assumptions. With these assumptions, the effect of spin on the trajectory of a test particle becomes negligible in the limit where the test particle is small (in terms of radius).

 

[KnownUniverse]

... I'll try to dig up a reference.

For a review with many references, check out:
Hammond 2002 Rep. Prog. Phys. 65 599. There is a section on the experimental consequences of torsion.

 

[samalkhaiat]

As I understand it from Misner, Thorne and Wheeler's book on gravity, the fact that test particles follow geodesics is not an additional assumption in GR, but is actually provable from the field equations. ...

Yes, see post#36 in
www.physicsforums.com/showthread.php?t=547502&page=3

Sam

 

[bcrowell]

Yes, see post#36 in
www.physicsforums.com/showthread.php?t=547502&page=3

Even in the case where there's spin, the trajectory of a test particle approaches a geodesic in the appropriate limit, under the technical conditions described in the Geroch paper I linked to in #42.

 

[samalkhaiat]

In GR, two main approaches have been developed for the description of spinning particles motion in a gravitational field. The first one was initiated in 1929 when Dirac equation was generalized for curved space-time [Fock]. The second, the classical approach, was proposed by [Mathisson] in 1937 and later refined by [Papapetrou] in 1951 and [Dixon] in 1970. The equations, which are now called the MPD equations, can be written as,
$$\frac{D}{Ds}P^{a} = - \frac{1}{2}U^{b}S^{cd}R^{a}_{bcd}, \ \ (1)$$
$$\frac{D}{Ds}S^{ab}= - U^{[a}P^{b]}. \ \ (2)$$
If you want to derive these equations, go back to the thread

https://www.physicsforums.com/showthr...=547502&page=3

and instead of assumption (ii), take

$$S^{ab} \equiv \int d^{3}x \sqrt{-g} \ \delta x^{[a}T^{b]0},$$

to be the non-vanishing spin tensor. You can also derive them, as [Wong] did in 1972, by taking the semi-classical limit of the general-covariant Dirac equation; it is more fun doing it both ways.

The general form of the MPD equations indicates that:

- the dynamics of spinning particle is described by the generalized momentum
$$P^{a} = mU^{a} + U_{b}\frac{D}{Ds}S^{ab}.$$

- $P^{a}$ is not parallel to the 4-velocity $U^{a}$.

- the motion of the spin tensor $S^{ab}$ represents a precession around the 4-velocity.

- due to the coupling between the spin tensor $S^{ab}$ and the gravitational field $R_{abcd}$, the trajectory is not geodesic: for $v \ll c$, the effect of spin on the particle’s trajectory is negligible, but for particle moving with relativistic velocity, the trajectory can deviate significantly from geodesic.

To solve the MPD equations, one needs spin supplementary condition. The two well-known conditions are: the Mathisson-Pirani condition, $S^{ab}U_{b}=0$, and Dixon condition, $S^{ab}P_{b}=0$. These conditions lead to different solutions, however, all solutions coincide in the Post-Newtoian approximation.

In my opinion, the Mathisson-Pirani solution is more fundamental because it leads to helical motion with frequency which coincides exactly with the Zitterbewegung (not sure about the spelling) frequency of the Dirac equation.

Sam

[Fock] V. Fock (1929), Z. Phys. 57, 261.
[Mathisson] M. Mathisson (1937) Acta. Phys. Pol. 6, 163.
[Papapetrou] A. Papapetrou (1951) Proc. Rol. Soc. A209, 248.
[Dixon] W.G. Dixon (1970) Proc. Rol. Soc. A314, 499.
[Wong] S. Wong (1972) Int. J. Theor. Phys. 5, 221.