# Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory

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In Einstein-Cartan gravity, the action is the usual Einstein-Hilbert action but now the Torsion tensor is allowed to vary as well (in usual GR, it is just set to zero).

Variation with respect to the metric gives:

$$R_{ab}-\frac{1}{2}R g_{ab}=\kappa P_{ab} \quad (1)$$

where $P_{ab}$ is the canonical stress energy tensor. Variation with respect to the torsion tensor ${T^{ab}}_c$ gives:

$${T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = \kappa {\sigma_{ab}}^c \quad (2)$$

where ${\sigma_{ab}}^c$ is the Spin Tensor.

By contracting that equation, I can see that if the spin tensor is zero, the Torsion tensor is identically zero:

$${T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = \kappa {\sigma_{ab}}^c = 0$$ $${g^b}_c({T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d) = 0$$ $${T_{ab}}^b + {g_a}^b{T_{bd}}^d - {g_b}^b {T_{ad}}^d = 0$$ $${T_{ab}}^b + {T_{ad}}^d - 4 {T_{ad}}^d = 0$$ $${T_{ad}}^d = 0$$ $${T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = 0 \quad \Rightarrow \quad {T_{ab}}^c =0$$

My understanding is that:

The spin tensor and the stress energy tensor, are defined entirely in terms of whatever matter Lagrangian we add to the theory. Therefore, from above, the equations in vacuum are exactly the same as normal GR (so solving for outside matter, only the boundary conditions with the matter could be different).

Assuming my understanding up to here is correct, my line of questioning is:

1. How is the Spin tensor (and hence Torsion) related to the concept of material with intrinsic spin?

2. Hopefully answered by 1, but if the matter has zero intrinsic spin, yet we have an extended "spinning" body, is the spin tensor still zero (as I'd consider that orbital angular momentum then)?

3. Does this mean Einstein-Cartan predictions are identical to normal Einstein GR if (and only if) the instrinsic spin of any particles and fields in the theory are zero?

This post imported from StackExchange Physics at 2014-10-02 10:37 (UTC), posted by SE-user CuriousKev

edited Oct 2, 2014
If torsion is going to be interesting, then you have to have something that acts as a source of torsion. The idea of letting the spin-1/2 of fermions act as its source is only an assumption. Good question, though, as to why orbital angular momentum can't be a source. I'd like to see an explanation of that. It might simply be that orbital angular momentum is easily ruled out empirically.

This post imported from StackExchange Physics at 2014-10-02 10:37 (UTC), posted by SE-user Ben Crowell
If torsion is going to be interesting, then you have to have something that acts as a source of torsion. The idea of letting the spin-1/2 of fermions act as its source is only an assumption. Good question, though, as to why orbital angular momentum can't be a source. I'd like to see an explanation of that. It might simply be that orbital angular momentum is easily ruled out empirically.

This post imported from StackExchange Physics at 2014-10-03 02:53 (UTC), posted by SE-user Ben Crowell
@BenCrowell What do you mean the source of torsion is only an assumption? Are you saying we need to specify something outside of the Lagrangian? The spin tensor shows up from varying the action with respect to the torsion, so I assumed this must already be defined in terms of the matter Lagrangian. If not, where does it come from? (sorry if these are all stupid questions, I don't know how to obtain that equation of motion myself)

This post imported from StackExchange Physics at 2014-10-02 10:37 (UTC), posted by SE-user CuriousKev
@BenCrowell What do you mean the source of torsion is only an assumption? Are you saying we need to specify something outside of the Lagrangian? The spin tensor shows up from varying the action with respect to the torsion, so I assumed this must already be defined in terms of the matter Lagrangian. If not, where does it come from? (sorry if these are all stupid questions, I don't know how to obtain that equation of motion myself)

This post imported from StackExchange Physics at 2014-10-03 02:53 (UTC), posted by SE-user CuriousKev

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Please let me first refer you to the following review by I. L. Shapiro, which contains a lot of theoretical and phenomenological information on spacetime torsion. The answer will be mainly based on this review.

In the basic Einstein-Cartan theory, in which the antisymmetric part of the connection is taken as independent additional degrees of freedom, the torsion is non-dynamical: (apart from a surface term which does not contribute to the equations of motion).

In the following, only minimal coupling to gravity will be considered (in which the flat metric tensor is replaced by the full metric tensor and the derivatives are replaced by covariant derivatives). There is a huge number of suggestions for nonminimal couplings in most of which the torsion becomes dynamical.

The torsion contribution to the gravitational part of the Lagrangian is quadratic in the torsion components, please see Shapiro equation: (2.15), where the additional terms to the torsion can be more economically expressed using the contorsion tensor whose components are linear combinations of the torsion tensor:

$$K_{\alpha\beta\gamma} = T_{\alpha\beta\gamma} -T_{\beta\alpha\gamma}-T_{\gamma\alpha\beta}$$

A scalar field minimal coupling to gravity, does not require covariant derivatives because the covariant derivative of a scalar is identical to its ordinary derivative, thus the scalar field Lagrangian does not depend on the torsion, therefore in the case of a scalar field coupled to gravity, the torsion remains sourceless, and the its equations of motion imply its vanishing.

When the Dirac field coupled to gravity with torsion, the Lagrangian can be written in the following form

$$\mathcal{L} = e^{\mu}_a\bar{\psi}\gamma^a (\partial_{\mu} -\frac{i}{2}\omega_{\mu}^{cd}\sigma_{cd} )\psi + e_{\mu a} K_{\alpha\beta\gamma} \epsilon^{\mu\alpha\beta\gamma} \bar{\psi}\gamma^a \gamma_5 \psi$$

($e$ are the vielbeins and $\omega$, the torsionless part of the spin connection, they both do not depend on the torsion).

Taking the variation with respect to the contorsion components, we obtain an algebric equation of motion for the contorsion tensor:

$$K^{\alpha\beta\gamma} = \frac{\kappa}{4}e_{\mu a} \epsilon^{\mu\alpha\beta\gamma} \bar{\psi}\gamma^a \gamma_5 \psi$$

($\kappa = 8 \pi G$). The last term in the Lagrangian is just of the form:

$$\mathcal{L_K} = K_{\alpha\beta\gamma} \sigma^{\alpha\beta\gamma}$$

Where $\sigma^{\alpha\beta\gamma}$ is the intrinsic spin part of the Noether current corresponding to the local Lorentz symmetry:

$$M^{\alpha\beta\gamma} = x^{\alpha}\Theta^{\beta\gamma}-x^{\beta}\Theta^{\alpha\gamma}+\sigma^{\alpha\beta\gamma}$$

$\Theta$ is the stress energy tensor. This example shows that for the Dirac field, the torsion source is the spin tensor.

As can be observed, the torsion couples axially to the Dirac field. This type of coupling is known to produce anomalies. A careful analysis shows that in the baryonic sector, the same anomaly cancellation criteria of the standard model lead to the cancellation of the axial anomalies due to torsion as well but not in the leptonic sector. This is one of the difficulties of this theory. One possible solution is to absorb the torsion contribution into the definition of the axial current. In contrary to the gauge and photon fields, where this contribution is not gauge invariant, since the torsion field is not a gauge field, this redefinition seems possible. This seems also consistent with the Atiyah-Singer index theorem which states that the anomaly density must be equal the Pontryagin class which is a topological invariant, while torsion can be introduced without altering the topology.

There is another difficulty related to the torsion coupling coming from the fact that the torsion couples only to the intrinsic part of the fields:

In the case of gauge fields such as the Maxwell field. The intrinsic spin is not gauge invariant and only the sum of the spin and the orbital angular momentum is gauge invariant. Thus, although the minimal coupling leads to a coupling of the torsion to the intrinsic spin, the gauge invariance is lost. The following recent article by Fresneda, Baldiotti and Pereira reviews some of the suggestions to overcome this problem.

This post imported from StackExchange Physics at 2014-10-03 02:53 (UTC), posted by SE-user David Bar Moshe
answered Oct 2, 2014 by (4,355 points)

This is interesting and all, but it does not answer the question asked. The question is about non-propagating torsion for classical spin-densities in standard GR, i.e. Einstein-Cartan theory, not for how to couple dynamical torsion to other theories. In cases of interest, when you are doing quantum field theory, you don't need to specifically couple the torsion to anything, it emerges naturally when you minimally couple spinning fields, and look at configurations with a net density of spin. It's unavoidable in standard GR, and it doesn't need to be added in by hand.

The torsion for macroscopic spinning matter cannot be inconsistent or anomalous, or anything else, because it is simply what happens to GR when there is a spin density around. The absence of usual gravitational anomalies must be sufficient to rule out specific new torsion anomalies, because there is nothing new in this type of torsion beyond consistent coupling to the spinning field.

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The spin tensor is used to give a continuum macroscopic model of a material with lots of teeny-tiny spinning sources inside, but which does not have any large-scale gross material motion, so no ordinary visible mechanical angular momentum. You should think of a ferromagnet, where the electrons in the ferromagnet are aligned, so there is a net macroscopic amount of angular momentum around the z-axis hidden inside the system, and it's hidden because the ferromagnet as a whole isn't spinning at all, it's just staying put.

In the famous Einstein de-Haas experiment, reversing the magnetic field on such a ferromagnet leads the magnetic electron spins to flip, and the entire ferromagnet to twist slightly on a torsion pendulum so as to conserve the total mechanical plus spin angular momentum. This verifies both that the spinning electrons carry the same kind of angular momentum as rotating mechanical bodies, and also from the magnitude of the effect that it is the electron spin alignment that is responsible for the magnetism. This was the definitive experiment that established that electron spin is really spinning electrons, not just a two-quantum-state something or other.

So whenever you think of a macroscopic spin tensor, you should keep the Einstein deHaas ferromagnet in your head. If the aligned spin is in the z-direction, the angular momentum density in the stationary ferromagnet is all spin, there is no net macroscopic momentum density, and in some ordering of the indices, S^t_{xy} is the nonzero spin-tensor component of interest--- that's the density (t component) of the spin angular momentum in the x-y plane (xy components).

Einstein-Cartan shows how this macroscopic configuration gravitates. Inside the ferromagnet, macroscopically, there is torsion, but this torsion is zero everywhere outside the ferromagnet. You could today understand this completely intuitively by modelling the spin-density using a continuous fluid of infinitesimal Kerr black holes (see below). Considering Einstein's work on frame-dragging preceded this, I am not sure this isn't exactly how he thought of it, although the Kerr solution was unavailable then of course.

If you do this, you see that the Einstein Cartan theory is simply the consistent extension of GR so that a spinning continuum gravitates the same way as a continuum made of infinitesimal Kerr sources, and using Einstein-Cartan, you don't need to take the limit by hand, you can just formulate the equations with a spinning source. It is such a conservative extension of GR, that it's really what people mean by GR nowadays.

In short, eveything you said in the question is right. For the specific questions:

1. The spin tensor models materials with intrinsic spin, like a ferromagnet. The intrinsic spin is stored by spinning particle sources internally, and is not seen in the macroscopic mechanical stress-energy tensor, so that the total angular momentum includes a continuous density of contributions beyond r cross p (as in the Einstein de-Haas case, where p=0).

2. Yes, there is no spin tensor when there are no spinning particles (or fields with indices, vector, spinor, or tensor fields), the spin angular momentum is the difference of the total angular momentum $L^{\mu\nu\sigma}$ in the theory, and the mechanical angular momentum $x^\mu T^{\nu\sigma} - x^\nu T^{\mu\sigma}$, where T is the naive mechanical stress tensor. If there is no spinning matter, the difference is zero.

I should add that the case of spinning fields, i.e. fields with indices, is subtler, because with the appropriate field theory definition worked out by Belinfante, this tensor difference is zero again, the whole field angular momentum is again the naive r cross p angular momentum determined from the Poynting stresses. But this is best understood in quantum field theory, because there are actual spinning particles in the real world. All the Belinfante is showing is that if you choose the T right in the field theory, this angular momentum shows up as a Poynting style rotating field momentum for the case of continuous fields, although this doesn't work for describing spinning electrons by a classical field with a Poynting-like spinning field momentum because the Dirac equation (which can be modified for GR) doesn't have a classical field limit.

3. Yes, the two theories are identical when there are no spinning matter sources. They are also have identical vacuum equations outside spinning matter, where the torsion vanishes. I would go so far as to say that they are really the same theory, Einstein-Cartan is just showing what spinning matter does in standard GR. Einstein knew about ferromagnets.

The easiest way to get comfortable with the emergence of torsion in spinning solutions today, after Kerr, is to consider parallel transport of vectors around an infinitesimal square containing a Kerr black hole at the center of even more infinitesimal mass, and then adding up lots of spinning infinitesimal sources to a density. If the black hole is spinning along the z-axis, and you take an x-pointing vector along the y axis, so that the square surroundes the origin, where the black hole is, there is a clear failure of the symmetry relation for connection coefficients, i.e. the transport of the x-component in the y-direction is not equal to the transport of the y-component in the x-direction, and the failure is in the time direction.

To picture the torsion condition, consider a vector pointing in the x-direction being dragged by parallel transport an infinitesimal distance along a vector pointing along the y direction, so as to fill out a square as the dragging path. The classical interpretation of this dragging in the original Einstein-era variational-geometric point of view is that you have a bunch of parallel geodesics, starting at each separate point along the infinitesimal y-vector, which get no closer and no further away to leading order.  The "No torsion" condition means geometrically that you fill out the same infinitesimal square (to leading order) by dragging the y-axis vector by parallel transport along the x axis vector, i.e. you fill out the same 2-d patch of points in spacetime no matter which way you drag the vector, x along y or y along x.

This relation fails in spacetimes with an infinitesimal spin at the center of the square because of the frame-dragging around the spin. The x-vector you are dragging in the y-direction becomes an infinitesimal geodesic of constant length. At the beginning of the transport, the x-geodesic is (say) going along the direction of the spin so that it is temporally helped along by the frame dragging in the spinning spacetime. After it crosses the Kerr-center, above the complicated central region, the geodesic in the same direction is going against the direction of the spin, so it ends up with time component deflected. By contrast, the relevant y-geodesic, the other side of the square, is crossing the spin in the opposite sense, so that at first it is going against the spin, and after crossing the Kerr-center it is going along with the spin, so that it gets a time-component deflected the other way. This means that the square fails to close into a proper 2-d sheet, the two transport paths are torn apart in the time direction, and this is the Einstein-Cartan torsion showing up as a global mismatch in the Kerr solution. The failure to close up a 2d sheet is clearly proportional to the spin-density, which in this case is a delta function at the origin with magnitude equal to the angular momentum L of the black hole. $S^t_{xy}$ is nonzero, and that's an asymmetric time parallel transport component when comparing transport of x and y vectors around a spinning center.

The general case can be understood from this by limits and superposition, just from the slowest falloff components of the Kerr metric. The difference is extremely intuitive, it is the difference in time between an observer that flies by against the rotation and one that flies by with the rotation, and it is due to the tilt of the time axis away from the spinning center just by the Machian frame-dragging of the spin.

To comment on the manipulations in the question, you should complete the solution of the EC equation of motion explicitly for the torsion. In d dimensions:

$${1\over \kappa} T_{ab}^c = \sigma_{ab}^c + {\delta^c_a \sigma_{br}^r - \delta^c_b \sigma_{ar}^r \over (2-d)}$$

The first term is what you get from the Kerr business above, it gives that the torsion is proportional to the spin-density. There are no traces for Kerr angular momentum, as the density is a t-component of x-y angular momentum. This generalizes by covariance to:

$${1\over \kappa} T_{ab}^c = \sigma_{ab}^c$$

Supplemented with the additional covariant condition that all the spin angular momentum is due to spinning particles $\sigma_{ab}^b = 0$, that it is Kerr-like.

This is the right physical condition for everything in the universe, I can't imagine the situation with a nonzero trace of $\sigma$, as this is not a superposition of spinning particles (which are traceless sigma). Moving to the rest frame of the density, where the top component is pure time, such a thing would be a time-space angular momentum, which by Noether's theorem is not an intrinsic angular momentum at all, but an intrinsic center of mass location. Unlike hidden intrinsic spin, there is no intrinsic hidden center of mass, but I suppose Einstein and Cartan gave the right equation to deal with such a situation, although how it would ever arise is beyond me, it wouldn't arise from spinning matter, that's for sure. I would just reduce the equation to the physical traceless case, as the case with nonzero trace for $\sigma$ looks purely mathematical, although if someone has an example where there is such a thing as "intrinsic center of mass location" I would be happy to hear it.

answered Oct 3, 2014 by (7,720 points)

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