# Who’s spin in the Einstein-Cartan theory? Source’s or test particle's?

1. Aug 2, 2011

### mings6

Who’s “spin” is in the Einstein-Cartan theory? Source’s spin or the test particle’s spin?

I quantum mechanics, when we say spin orbit interaction, such as in hydrogen atom, it’s about the test particle's, the electron’s spin and the electron’s orbit interact to each other.

In the Einstein’s general relativity, with weak field condition, it can be described with Gravitoelectromagnetism, in which the source’s spin interacts with the test particles, that leads to the frame dragging and the Lense-Thirring effect.

Its said that “general relativity has one flaw that it cannot model spin-orbit coupling, so we need the Einstein-Cartan theory”. But who’s “spin” is here in the Einstein-Cartan theory? Source’s spin or the test particle’s spin?

From the relation,

(divergence of spin current) = P_{ab} – P_{ba} <> 0

it seems the spin is the source’s spin. Than what the difference between the Einstein-Cartan theory and Gravitoelectromagnetism? What happens if the test particle has spin?

2. Aug 2, 2011

### tom.stoer

I haven't seen any Einstein-Cartan equations for motion of spinning test particles; this would be something a "geodesic equation plus spin of the test particle".

In Einstein-Cartan theory the l.h.s. of the equations is always "geometry", the r.h.s. is "matter", just like ordinary GR. Therefore the spin current is always the source term for torsion.

In general the equation for matter would of course not be the equation for a test particle i.e. a "geodesic equation plus spin of the test particle", but a field equation for the matter field coupled to spacetime = curvature + torsion, for example the Dirac equation.

3. Aug 3, 2011

### mings6

Thanks for reply. I still do not understand, if general relativity can not model spin-orbit coupling, then how can we have the Kerr metric that describes the geometry of spacetime around a rotating massive body? And also Gravitoelectromagnetism can do some jobs for frame dragging and the Lense-Thirring effect?

4. Aug 3, 2011

### tom.stoer

GR in standard metric formulation can describe orbital angular momentum, but not intrinsic spin. In order to do that you need at least the tetrad formalism. Doing that one observes that it seems to unnatural to restrict the geometry to vanishing torsion. Allowing torsion automatically means introducing spin currents.