In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory or the Einstein–Sciama–Kibble theory is a classical theory of gravitation similar to general relativity but relaxing the assumption that the connection has vanishing torsion, so that the torsion can be coupled to the spin angular momentum of matter, much in the same way in which the curvature is coupled to the energy momentum of matter. The theory was first proposed by Élie Cartan in 1922 and expounded in the following few years. Dennis Sciama and Tom Kibble independently revisited the theory in the 1950s, and an important review was published in 1976. Albert Einstein became affiliated with the theory in 1928 during his unsuccessful attempt to match torsion to the electromagnetic field tensor as part of a unified field theory. This line of thought led him to the related but different theory of teleparallelism.
Einstein–Cartan theory has been historically overshadowed by its torsion-free counterpart and other alternatives like Brans–Dicke theory because torsion seemed to add little predictive benefit at the expense of the tractability of its equations. In particular, many effects of torsion are hypothesized to be non-propagating so that its direct observation is expected to be difficult; however, the fact that torsion does not propagate ensures that in vacuum there is no torsion, which implies that the Einstein-Sciama-Kibble theory is indistinguishable from the Einstein theory in all experiments performed so far, and therefore the non-propagation of torsion is also the main reason for which the Einstein-Sciama-Kibble theory is still a viable completion of Einstein gravity. Experiments apt to detect torsion, or torsional effects, must thus be done in presence of matter with spin, but even then one of the problems torsion had was due to the fact that its effects were expected to be relevant only at the Planck scale; this was due to the fact that the Einstein–Cartan theory is only the most straightforward but not the most general torsional completion of Einstein gravity, which was instead obtained in recent years  and where torsional effects manifested as spin-spin contact interactions in the Dirac equation can be relevant at much larger scales. Since the Einstein–Cartan theory is purely classical, it also does not fully address the issue of quantum gravity; this too is a delicate issue, since in the Einstein–Cartan theory the Dirac equation becomes non-linear, and therefore the superposition principle used in usual quantization techniques would not work. Recently, interest in Einstein–Cartan theory has also been driven toward cosmological implications. The theory is still considered viable and remains an active topic in the physics community.
|This section does not cite any references or sources. (August 2011)|
Although general relativity can accommodate particles with spin, including spin-1/2, by using the tetrad formalism, it cannot couple the spin to orbital angular momentum. Spin-orbit coupling is a well documented phenomenon in quantum mechanics, so a theory of gravitation that hopes to be fit eventually into a quantum theory of gravity should be expected to incorporate spin directly into its field equations.
must be symmetric in a and b (that is, Rab = Rba). Therefore the Einstein curvature tensor Gab defined as
must be symmetric. (gab is the metric tensor that defines lengths of vectors and inner products of pairs of vectors). In general relativity, the Einstein curvature tensor models local gravitational forces, and it is equal (up to a gravitational constant) to the stress-energy tensor
(We denote the stress-energy tensor by because the customary symbol in general relativity is used in Einstein–Cartan theory to denote affine torsion. The stress-energy tensor is also sometimes called the momentum tensor, the energy-momentum tensor, or the energy-momentum-stress tensor.)
The symmetry of the Einstein curvature tensor forces the momentum tensor to be symmetric. However, when spin and orbital angular momentum are being exchanged, the momentum tensor is known to be nonsymmetric, whose antisymmetric part being proportional to the divergence of the spin current) (See spin tensor for more details.)
Therefore general relativity cannot properly model spin-orbit coupling.
In 1922 Élie Cartan conjectured that general relativity should be extended by including affine torsion, which allows the Ricci tensor to be non-symmetric. Although spin-orbit coupling is a relatively minor phenomenon in gravitational physics, Einstein–Cartan theory is quite important because
- (1) it makes clear that an affine theory, not a metric theory, provides a better description of gravitation;
- (2) it explains the meaning of affine torsion, which appears naturally in some theories of quantum gravity; and
- (3) it interprets spin as affine torsion, which geometrically is a continuum approximation to a field of dislocations in the spacetime medium.
The extension of Riemannian geometry to include affine torsion is known as Riemann–Cartan geometry.
ω-consistency and derivation from general relativity
General relativity is acknowledged to be ω-inconsistent in the sense that there is a limit of the theory that is not in the theory. Consider a fluid of rotating black holes. These black holes possess orbital angular momentum only and can be seated in a torsion-free spacetime. In the continuum limit, however, the angular momentum turns into spin angular momentum, and the metric becomes contorted. As the limit has torsion, it is not allowed in general relativity by assumption. Einstein–Cartan theory can thus be viewed as the minimal ω-consistent extension of general relativity.
A mathematical proof has been published that general relativity plus a fluid of many tiny rotating black holes generate affine torsion that enters the field equations exactly as in the equations of Einstein–Cartan theory (Petti, 1986, 2013). If we introduce a classical spin fluid with spin-orbit coupling, torsion is necessary to describe the spin-orbit coupling. (Example of a classical spin fluid: Approximate a distribution of galaxies with correlated rotations as a classical fluid with spin. In this approximation, the rotational angular momentum of the galaxies becomes intrinsic angular momentum, that is, spin.) The mathematical proof starts with a standard Kerr-Newman rotating black hole solution of general relativity, and it computes the non-zero time-like translation that occurs when you parallel-translate an affine frame (keeping track of translation as well as rotation) around an equatorial loop near the black hole. The main conclusion (that general relativity plus spin-orbit coupling implies nonzero torsion and Einstein–Cartan theory) is derived from classical general relativity and classical differential geometry without additional assumptions for parameters and without recourse to quantum mechanical spin or spinor fields.
The proof is similar to derivations of continuum equations of motion from equations for discrete particles in fluid mechanics and electrodynamics. The proof consists of four steps.
- Start with a Kerr-Newman rotating black hole. Compute the translational holonomy around an equatorial spacelike loop of constant radius.
- Construct an ensemble of very many very small rotating black holes with correlated rotations and take the continuum limit.
- In the continuum limit, the translational holonomy becomes affine torsion, the distribution of rotating black holes generates spin density, and the torsion and spin density are related exactly as in Einstein–Cartan theory.
- For completeness, compute the translational holonomy for loops in other hyperplanes, to prove that the other loops generate no affine torsion in the continuum limit.
Adamowicz showed that general relativity plus a linearized classical model of matter with spin yields the same linearized equations for the time-time and space-space components of the metric as linearized Einstein–Cartan theory (Adamowicz 1975). Adamowicz does not treat the time-space components of the metric, the spin-torsion field equation, spin-orbit coupling and the non-symmetric momentum tensor, the geometry of torsion, or quantum mechanical spin. Also, Adamowicz does not show that Einstein–Cartan theory follows from general relativity plus spin. He says, “It is possible a priori to solve this problem [of dust with intrinsic angular momentum] exactly in the formalism of general relativity but in the general situation we have no practical approach because of mathematical difficulties.” Adamowicz’s conclusion is at best incomplete: it is not possible to solve the full problem exactly in general relativity, including spin-orbit coupling, without adopting the larger framework of Einstein–Cartan theory.
Normally a physical theory is accepted if it is the simplest candidate that explains previously unexplained observations or generates new predictions that are verified by empirical results. Einstein–Cartan theory currently satisfies a substantial array of validations: (a) it can be derived from general relativity with no added assumptions or parameters; (b) it extends general relativity to describe spin-orbit coupling, which fixes the most outstanding problem in general relativity as the master theory of classical physics; (c) it is the minimal extension of general relativity that can fix this problem, because it arises merely by relaxing the ad-hoc assumption that torsion is zero, and because Einstein–Cartan theory is identical with general relativity where spin density is zero; (d) cosmological models based in Einstein–Cartan theory provide an explanation for key cosmological observations based on classical geometry, without relying on speculations about quantum fields as does inflation theory; and (e) it generates new predictions that can in principle validate or falsify the theory, but it cannot be validated by empirical results due to current limitations in technology.
Any successful quantum theory of gravitation must include spin, and therefore affine torsion or a precursor, because its classical limit must include affine torsion.
In order to understand why torsion should be related to spin, we can consider how the presence of torsion may affect observers living in contorted spacetime. The first equation to consider is the geodesic equation, which is unchanged. We may alternatively rewrite the geodesic equation in terms of the contortion tensor defined above:
where is the proper time. This shows that the connection can be split into pure metric and pure torsion contributions. In the limit of Minkowski spacetime and a completely antisymmetric torsion tensor , a vector parallelly transported along a geodesic (i.e. straight line for Minkowski spacetime) described by the reduced equation
rotates about the translation axis.
The geodesic deviation equation is modified in the presence of torsion. Let be a deviation vector between two infinitesimally neighboring geodesics, and let be a tangent to the geodesics along which is transported. The modified geodesic deviation equation that expresses the acceleration between neighboring geodesics is
Whereas the Riemann curvature applies a tidal acceleration, the torsion causes the geodesics to twist around each other. The infinitesimal picture must be emphasized. Imagine a rod made up of a lattice of particles. The relative positions of the particles are fixed to be at the spacing of the lattice, but the particles are otherwise non-interacting. As the rod moves along, the rod as a whole does not rotate but rather each of the infinitesimally small particles making up the rod will want to spin about its own axes.
The fundamental equations of any field theory are declared as brute facts, for example D'Alembert's principle in Newtonian mechanics. It is epistemically useful, however, to consider the more restricted class of field equations that can be derived by a variational principle applied to an action because this eases the implementation of Noether's theorem or one of its analogs. For example, the Einstein field equations of general relativity can be derived, among other ways, by postulating the Einstein-Hilbert action to be the true action of spacetime and then varying that action with respect to the metric tensor. The field equations of Einstein–Cartan theory come from exactly the same approach. Let represent the Lagrangian density of matter and represent the Lagrangian density of gravitation. We use the convention that Lagrangian densities are tensor densities. First, the Einstein-Hilbert action is postulated, i.e. the Lagrangian density of gravitation is assumed to be the Ricci scalar (up to an overall constant),
Variation with respect to the metric yields equations reminiscent of general relativity:
where is the Ricci tensor and is the canonical energy-momentum tensor. The reader must be cautioned that, despite similarity of form, this is not the same as the Einstein field equation because the Ricci tensor is no longer symmetric but instead contains information about the nonzero torsion tensor as well. The right hand side of the equation cannot be symmetric either, so must also contain information about the nonzero spin tensor. This canonical energy-momentum tensor is related to the more familiar symmetric energy-momentum tensor by the Belinfante–Rosenfeld procedure.
We may also consider the variation of with respect to the torsion tensor . This yields a new equations
where is the spin tensor. Note that in the first field equation, the trace of the Ricci tensor has been modified. Compare to the second field equation, where the trace of the torsion tensor has been altered. The steps between the variation of the action to the final field equations are given in the appendix of the Hehl et al. review.
Avoidance of singularities
In 2011, Nikodem Poplawski  and independently Luca Fabbri  have shown employing two different methods that the presence of torsion requires fermions to be spatially extended: the proof is the mathematical implementation of the idea that because the presence of torsion introduces rotational degrees of freedom, then the spinorial matter distribution must have an angular momentum, and thus it cannot be point-like but instead it must display a spreading; in turn this implies that the ultraviolet divergence occurring in quantum field theory because of singular matter distributions is avoided. This is true in particular for the formation of singularities in black holes: according to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular black hole. In the Einstein–Cartan theory, however, it forms a regular Einstein-Rosen bridge (wormhole) with a new, growing universe on the other side of the event horizon.
Einstein–Cartan theory eliminates the general-relativistic problem of the unphysical singularity at the Big Bang. The minimal coupling between torsion and Dirac spinors generates a spin-spin interaction which is significant in 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.
Other basic physical theories that employ affine torsion
|This section relies on references to primary sources. (August 2011)|
For completeness, below are references to some speculative physical theories that employ torsion in ways that are different from Einstein–Cartan theory.
There are proposals to describe what is called propagating torsion. This is done by expressing the torsion as a gradient of some other field which propagates. However, since the torsion couples directly to spin, the propagating field couples to the gradient of the spin current density, so that the interaction is again completely local (it leads to a four-Fermi interaction), and the effect of torsion cannot propagate away from matter after all. Moreover, the photon has spin 1, but the torsion must be forbidden to couple to it, since the photon is so far the lowest-mass particle in the universe. But if torsion coupled to the spin-1/2 electrons and protons, this will make the photons massive by vacuum polarization.
To avoid these problems, the only way out so far is to assume for the gauge field description of the Einstein-Cartan Theory the Einstein action in the teleparallel form, where torsion is just an equivalent alternative to curvature.
Moreover, it can be shown that there exists an infinity of equivalent descriptions where any amount of torsion can be moved into Cartan curvature. All these theories are connected by a new type of multivalued gauge transformations.
- Classical theories of gravitation
- Gauge gravitation theory
- Gauge theory gravity
- Loop quantum gravity
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