Einstein-Cartan theory
In 1922 Elie Cartan conjectured that general relativity should be extended by including affine torsion, which allows the Ricci tensor to be non-symmetric. The extension of Riemannian geometry to include affine torsion is now known as Riemann-Cartan geometry. A Riemann-Cartan geometry is uniquely determined by
- a choice of metric tensor field (which specifies all lengths of vectors and angles between vectors),
- an affine torsion field, and
- the requirement that lengths and angles are preserved by parallel translation (as in Riemannian geometry where the torsion is zero).
As the master theory of classical physics, general relativity has one known flaw: it cannot adequately describe exchange of intrinsic angular momentum (spin) and orbital angular momentum. The problem is rooted in the foundations of general relativity. General relativity is based on Riemannian geometry, in which the Ricci curvature tensor
- Rij
- Pij
- divergence of spin current ½(Pij − Pji) = 0.
- edge dislocations (formed by adding an extra half-plane of atoms to a perfect crystal, so you get a defect in the regular crystal structure along the line where the extra half-plane ends), and
- screw dislocations (formed by inserting a "parking garage ramp" that extends to the edges of the garage into an otherwise perfectly layered structure).
General relativity set the affine torsion to zero, because it did not appear necessary to provide a model of gravitation (with a consistent set of equations that led to a well-defined initial value problem).
General relativity and Einstein-Cartan theory both use the scalar curvature as Lagrangian. General relativity obtains its field equations by varying the action integral (integral of the Lagrangian over spacetime) with respect to the metric tensor g{i,j}. The result is the famous Einstein equations:
A basic question in formulating Einstein-Cartan theory is which variables in the action to vary to get the field equations. You can vary the metric tensor g[i,j] and the torsion tensor T[i,j,^k]. However, this makes the equations of Einstein-Cartan theory messier than necessary and disguises the geometric content of the theory. The key insight is to let the symmetry group of Einstein-Cartan theory be the inhomogeneous rotation group (which includes translations in space and time). (The inhomogeneous rotational symmetry is broken by the fact that the zero point in each tangent fiber is still a preferred point, as in ordinary Riemannian geometry based on the homogeneous rotation group.) We vary the action with respect to the affine connection coefficients associated with translational and rotational symmetries. (A similar approach in general relativity is called "Palatini variation," in which the action is varied with respect to the rotational connection coefficients instead of the metric; general relativity has no translational connection coefficients.)
The resulting field equations of Einstein-Cartan theory are:
The contracted second Bianchi identity of Riemann-Cartan geometry becomes, in Einstein-Cartan theory,
Derivation of field equations of Einstein-Cartan theory
where
The "contracted second Bianchi identity" of Riemannian geometry becomes,
in general relativity, div(P)=0, which makes conservation of energy and momentum equivalent to an identify of Riemannian geometry.
where
T{a,b,^k} + g{a,^k} T{b,m,^m} ÃÂ g{b,^k} T{a,m,^m}.
The first equation is the same as in general relativity, except that the affine torsion is included in all the curvature terms, so P{a,j} need not be symmetric.
The conservation of momentum is altered by products of gravitational field strength and spin density. These terms are exceedingly small under normal conditions, and they seem reasonable in that the gravitational field itself carries energy. The second equation is conservation of angular momentum, in a form that accommodates spin-orbit coupling.
Spin (intrinsic angular momentum) consists of dislocations in the fabric of spacetime. For ordinary fermions (particles with spin such as protons, neutrons and electrons), these are screw dislocations (parking garage ramps) with timelike direction of the screw. That is, for a particle with spin in the +z direction, traversing a space-like loop in the x-y plane around the particle parallel translates you into the past or the future by a small amount.
It has long been know that the spin angular momentum tensor Spin(a,b,^k} is the "Noether current" of rotational symmetry of spacetime, and the momentum tensor P{a,k} is the Noether current of translational symmetry. (The Noether theorem states that, for every symmetry of a physical system, there is a corresponding conserved current derived by performing the symmetry transformation on the Lagrangian.) Einstein-Cartan theory provides a clean derivation of momentum as the Noether current of translational symmetry. It may be that general relativity without rotational connection coefficients (which would have introduced affine torsion to the theory) cannot provide a clean derivation of the momentum as the Noether current of translational symmetry.
In Einstein-Cartan theory, we should distinguish between tensor indices that represent conserved currents (like momentum and spin) and indices that represent spacetime boxes (through which fluxes of the currents are measured). (This is similar to other gauge theories, like electromagnetism and Yang-Mills theory, where we would never confuse spacetime indices that represent flux boxes with the fiber indices that represent the conserved currents.)
Writing Einstein-Cartan theory in the simplest form requires distingushing two kinds of tensor indices:
(Before the distinction between these types of indices became clear, researchers would vary the action with respect to the metric to get what they called the "momentum tensor" (the 'wrong' one) and also sometimes vary with respect to the translational connection coefficients and get a different momentum tensor (the 'right' one) and they did not know which one was the real momentum tensor. The equations of the theory had many unnecessary terms because they did not distinguish between the base space and fiber space tensor indices.)
Einstein-Cartan theory is about defects in the affine (Euclidean-like but curved) structure of spacetime; it is not a metric theory of gravitation.
We have seen above that the affine torsion is a continuum model of dislocation density. The full rotational (or Riemannian) curvature tensor
The central role of affine defects explains why the clean way to do Einstein-Cartan theory is to vary the translational and rotational connection coefficients (not the metric) and to distinguish between the base space and fiber indices. The connection coefficients are keeping track of the dislocation and disclination defects in the affine structure of spacetime. It is as if spacetime were composed of many microcrystals of perfectly flat Minkowski space, and these perfect micro-pieces are fit together with defects like dislocations and disclinations.
The central role of the translational and rotational connection coefficients as field variables is recognized in modern efforts to quantize general relativity under the name "Ashtekar variables." The Ashtekar variables are essentially the translational and rotational connection coefficients, suitably worked into a Hamiltonian formulation of general relativity.
For decades, it was thought that Einstein-Cartan theory is based on an independent assumption to include affine torsion. Since the effect of torsion is too small to measure empirically so far, Einstein-Cartan theory was considered one of many speculative (and largely ignored) extensions of general relativity.
It has been shown that general relativity plus a fluid of many tiny rotating black holes generate affine torsion and essentially the equations of Einstein-Cartan theory. The "proof" uses a standard Kerr-Newman rotating black hole solution of general relativity. 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 word "proof" appears in quotes because, while it is intuitively compelling that this implies Einstein-Cartan theory, the proof of convergence to the equations of Einstein-Cartan theory has not been done.Geometric insights from Einstein-Cartan theory
First Geometric Insight
Second Geometric Insight
Third Geometric Insight
These two types of indices have two roles in the theory.
For example, in the field equations of Einstein-Cartan theory stated above, we should interpret the indices a,b as fiber indices and the indices i,j as base space indices. The momentum tensor P{a,^k} describe the flux of a-momentum through a flux box normal to the k-direction in spacetime, and the spin tensor Spin{a,b,^k} describes the flux of angular momentum in the axb plane through a flux box normal to the k-direction in spacetime.Fourth Geometric Insight
also has an interpretation as a density of defects in continuum mechanics. It is the continuum model of a density of "disclination defects." A disclination results when you make a cut into a continuum (imaging making a radial cut from the edge to the center of a disk of rubber) and insert (or excise) an angular wedge of material, so that the sum of the angles surrounding the endpoint of the cut is more (or less) than 2 pi radians. (Indeed, this procedure can convert a flat disk into a bowl: make many small radial cuts from the edge with varying lengths part-way to the center, excise wedges of material of the appropriate angular width, and sew up the cuts.)General relativity plus matter with spin implies Einstein-Cartan theory