Einstein–Cartan theory

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Einstein–Cartan theory in theoretical physics extends general relativity, to handle spin angular momentum correctly. Basically, it is just a fancy name to describe general relativity with a nonzero torsion.

As the master theory of classical physics general relativity has one known flaw: it cannot describe "spin-orbit coupling", that is, exchange of intrinsic angular momentum (spin) and orbital angular momentum. There is a qualitative theoretical proof showing that general relativity must be extended to Einstein-Cartan theory when matter with spin is present.

Experimental effects are too small to be observed at the present time because the spin tensor of typical macroscopic objects is often small and torsion is nonpropagating which means that torsion will only appear within a massive body. In addition, only spinning objects couple to torsion.

1 Motivation
2 Geometry and formulation
3 Derivation of field equations of Einstein–Cartan theory
4 Geometric insights from Einstein–Cartan theory
5 General relativity plus matter with spin implies Einstein–Cartan theory
6 See also
7 References

[edit] Motivation

The reason that general relativity cannot describe spin-orbit coupling is rooted in the Riemannian geometry, on which general relativity is based. In Riemannian geometry, the Ricci curvature tensor

$R_{ab} \,$

must be symmetric in a and b (that is, R_ab = R_ba). Therefore the Einstein curvature tensor G_ab defined as

$G_{ab} \equiv R_{ab} - {1\over 2} R g_{ab}$

must be symmetric. 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 (also called the energy-momentum-stress tensor since it contains all such)

$P_{ab} \,$

(We denote the stress-energy tensor by $P \,$ because the customary symbol $T \,$ in general relativity is used in Einstein-Cartan theory to denote affine torsion.) The symmetry of the Einstein curvature tensor forces the momentum portion of the tensor to be symmetric. However, when spin and orbital angular momentum are being exchanged, the momentum tensor is known to be nonsymmetric according to the general equation of conservation of angular momentum

(divergence of spin current) $\propto$ ( $P_{ab} \,$ – $P_{ba} \,$ ) $\neq 0$ .

(See spin tensor for more details.)

Therefore general relativity cannot properly model spin-orbit coupling.

In 1922 Elie 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 now known as Riemann–Cartan geometry.

[edit] Geometry and formulation

The basic mathematics underlying spacetime physics are the ideas of affine connections and differential geometry, in which we endow an n dimensional differentiable manifold M with a law of parallel translation of vectors along paths in M. (At each point of a differentiable manifold, we have a linear space of tangent vectors, but we have no way to transport vectors to another point, or to compare vectors at two points in M.) The parallel translation preserves linear relationships between vectors; that is, if two vectors u and v at the same point of M parallel translate along a curve to vectors u' and v', then

a u + b v

parallel translates to a

a u' + b v'.

Parallelism is path-dependent; that is, if you parallel translate a vector along two different paths with the same starting and ending points, the resulting vectors at the end point in general differ. The difference between parallel-translating a vector along different curves is the essential meaning of curvature, which is the central concept in differential geometry.

In (pseudo) Riemannian geometry, an n dimensional differential manifold M is endowed with a Riemannian metric g, which is a nondegenerate linear map that maps two tangent vectors to a real number. The metric uniquely determines a law of parallel translation that preserves inner products between vectors and has zero torsion. This law of parallel translation is called the Levi-Civita connection.

(In the more abstract language of fiber bundles, if the metric g is preserved by the connection, then the structure group of the principal bundle is reducible to the orthogonal group O(p,q), where the metric g has p principal directions with positive length and q principal directions with negative length.)

A Riemann–Cartan geometry is uniquely determined by

a metric tensor field g that specifies all lengths of vectors and angles between vectors.
the requirement that lengths and angles are preserved by parallel transport. This is expressed by the condition that the covariant derivative of the metric tensor vanishes:

$\nabla \bold{g}=0$

where ∇ is the covariant derivative determined by the affine connection.

an affine torsion field Θ

$\nabla_{\bold{u}}\bold{v}-\nabla_{\bold{v}}\bold{u}-[\bold{u},\bold{v}]=\Theta(\bold{u},\bold{v})$

where u and v are vector fields and [,] is the Lie bracket. (See Lie algebra for the definition of Lie bracket.)

In Riemann–Cartan geometry, the curvature tensor has a rotational part

R_k,l,jⁱ,

analogous to the curvature in Riemannian geometry, and a translational part, the affine torsion

T_i,j^k.

The rotational curvature R_k,l,jⁱ describes the rotation in the i,j plane experienced by a vector that is parallel translated around a small loop in the k,l plane in the base manifold. The translational curvature T_i,j^k describes the translation in the i direction resulting from 'developing' a small loop in the base manifold M into a flat manifold X that has the same dimension as M. (Developing a curve

C_m: [0,1] → M

into a curve

C_x : [0,1] → X

means defining a curve Cx in X that has the same pattern of accelerations as the curve Cm in M. The motivation for development of a curve Cm is to create a curve Cx whose shape is determined by the same pattern of accelerations as Cm, but without the impact on shape of curve Cx from the curvature of the ambient space M.)

A Riemann–Cartan geometry with zero torsion is a Riemannian geometry.

How to include spacetime translations in fiber bundle gauge theories has been a subject of controversy for 50 years, because spacetime symmetries are not internal symmetries of the bundle structure group. A consistent approach to including translations in spacetime theories is outline in (Petti 2006).

The best way to formulate Einstein–Cartan theory is to distinguish between tangents to the spacetime M and tangents to an associated flat affine fiber space, X. X is a (pseudo-) Euclidean space (a Minkowski space) with metric g and no origin, so you cannot add two points in X or multiply a point in X by a scalar. The affine connection tells us how to parallel translate points in X and tangents to X along curves in M, not how to parallel translate tangents to M. The translational part of the affine connection acts like an (inverse) frame field that enables us to identify tangents to M with tangents to X, and pulls back the metric g on X to a metric on M. While the distinction between tangents to M and tangents to X at first may seem artificial, the equations of Einstein–Cartan theory become conceptually and computationally simpler when conserved currents (like momentum and spin) are represented by tangents to X, which are parallel translated by the connection, and directions in M (and flux boxes through which conserved currents flow) are represented by tangents to M, which never need to be parallel translated along a curve in M. In this article, we use Roman indices i,j,k,... to denote tangent vectors to M and Roman indices a,b,c,... to denote tangents to the fiber space X. For example the momentum tensor

P_a^k

represents the flux normal to the spacetime k-direction of momentum in the a-direction, and the spin tensor

Spin_a,b^k

represents the flux normal to the spacetime k-direction of spin in the a,b plane.

(Advanced point: In order to accommodate spinor fields, all of the constructions of Riemannian and Riemann–Cartan geometry can be generalized from orthogonal groups, principal orthogonal frame bundles and associated tangent bundles to spin groups, principal spin bundles and associated spinor bundles. A spacetime manifold admits a spin bundle over its principal frame bundle only if the second Stiefel-Whitney class of M is zero. The Riemann tensor is the curvature form for (generalized to include boosts) rotations (i.e. the spin(p,q) part) while torsion is the curvature form for translations (R⁴.)

A geometric interpretation of affine torsion comes from continuum mechanics of solid materials. Affine torsion is the continuum approximation to the distribution of dislocations that are studied in metallurgy and crystallography. The simplest kinds of dislocations in real crystals are

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).

We can think of a Riemann–Cartan geometry as uniquely determined by the lengths and angles of vectors and the density of dislocations in the affine structure of the space.

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).

[edit] Derivation of field equations of Einstein–Cartan theory

General relativity and Einstein–Cartan theory both use the scalar curvature as Lagrangian. General relativity obtains its field equations by varying the Einstein-Hilbert action (integral of the Lagrangian over spacetime) with respect to the metric tensor. The result is the famous Einstein equations:

$R_{ab}-\frac{1}{2}R g_{ab}=\frac{8 \pi G}{c^4}P_{ab}$

where

$R a b$ are the Ricci tensor components (a contraction of the full Riemann curvature tensor that has four indices).

$g a b$ are the (symmetric nondegenerate) metric tensor components.

$R$ is the scalar curvature (Ricci scalar).

$P a b$ are the energy-momentum tensor components. (We reserve the symbol $T$ , which is the usual symbol for the energy-momentum tensor in general relativity, for the affine torsion.)

$G$ is the Newtonian gravitational constant.

$c$ is the speed of light.

The contracted second Bianchi identity of Riemannian geometry becomes, in general relativity,

d i v P = P a b; b = 0

which makes conservation of energy and momentum equivalent to an identity of Riemannian geometry.

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), that is, the analogue of the Euclidean group. (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:

$R_{a}{}^{k} - \frac{1}{2} g_{a}{}^{k} R = \frac{8 \pi G}{c^4} P_{a}{}^{k}$

$S_{ab}{}^{k} = \frac{8 \pi G}{c^4} Spin_{ab}{}^{k}$

where

$S p i n a b k$ is the spin tensor of all matter and radiation

$S a b k = T a b k + g a k T b m m$ – $g b k T a m m$ is the modified torsion tensor

$T a b k$ is the affine torsion tensor.

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 contracted second Bianchi identity of Riemann–Cartan geometry becomes, in Einstein–Cartan theory,

div(P) = some very small terms that are products of curvature and torsion,

div(Spin) = – antisymmetric part of $P a j$ .

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.

[edit] Geometric insights from Einstein–Cartan theory

[edit] First geometric insight

Spin (intrinsic angular momentum) consists of a (continuous or discrete) distribution of dislocations in the fabric of spacetime. For ordinary fermions (particles with half-integer 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.

[edit] Second geometric insight

It has long been known 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. General relativity without translational connection coefficients (which would introduce affine torsion into the theory) does not provide a clean derivation of the momentum as the Noether current of translational symmetry.

[edit] Third geometric insight

Expressing Einstein–Cartan theory in the simplest form requires distinguishing two kinds of tensor indices:

indices that represent conserved currents like momentum and spin. Geometrically, these indices represent directions in the idealized Minkowski "fiber space" at each point of spacetime. (Notation used here: a, b, c...)
indices that represent spacetime boxes through which fluxes of the currents are measured. Geometrically, these indices represent tangents to the spacetime base manifold that describe boxes through which the fluxes of conserved currents are measured. (Notation used here: i, j, k ...)

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.

All the derivative indices in Einstein–Cartan theory are spacetime (base space) indices. Furthermore, the derivatives are all 'exterior derivatives,' which measure fluxes of currents through spacetime boxes (or divergences, which are Hodge duals of exterior derivatives). All the indices that are antisymmetrized with derivative indices in exterior derivatives (or the indices with which the derivative indices are contracted in divergences) are also spacetime indices. All these indices are part of the calculus of flux boxes in spacetime, and do not represent the conserved currents themselves.

The statement that all derivatives are covariant exterior derivatives boils down to the fact that the affine connection is a law of parallel translation for points in the affine fiber space X, and not a law of parallel translation for tangent vectors to the base manifold M. In fact, we don't have ANY connection on the tangent bundle TM of M. The Levi–Civita metric connection that is often highlighted in treatments of general relativity is merely a computational convenience for writing exterior derivatives and divergences involving spacetime indices, which has nothing to do with parallel translation.

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

describes 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 a × b plane through a flux box normal to the k-direction in spacetime.

NB: 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.)

[edit] Fourth geometric insight

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

$R_{klb}{}^{a} \,\!$

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 (imagine 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 than (or less than) 2π 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.)

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.

[edit] General relativity plus matter with spin implies Einstein–Cartan theory

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 (Petti, 1986). 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.

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. 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 EC theory.

A GR theory with only scalar and Maxwell (i.e. massless vector) fields need not have a nonzero torsion. However, if we introduce spinorial fields, when we also have to introduce a spin connection. The Euler-Lagrange equation for pure GR or GR with scalar and Maxwell fields does not involve the connection except in the Einstein-Hilbert action and states that the theory is torsionless. However, once we introduce spinor fields which have to couple to the spin connection, the Euler-Lagrange equation now equates the torsion with the result of varying the matter action with the spin connection. In fact, the standard definition of the stress-energy tensor as the result of varying the matter action with respect to the metric tensor can no longer apply because spinors couple to vierbeins and the spin connection instead. Instead, we now have to define the stress-energy tensor has the result of varying the matter action with respect to the vierbein. This stress-energy tensor is now no longer symmetric and if we define the spin tensor as the result of varying the matter action with respect to the spin connection, we find that the antisymmetric part of the stress-energy tensor is equal to the divergence of the spin tensor. Of course, it is possible to get rid of the torsion by fiat if we simply introduce a Lagrange multiplier term, but even that would not be stable under the renormalization group once quantum effects are taken into account.

[edit] See also

Classical theories of gravitation

[edit] References

Adamowicz, W. (1975). Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), no. 11, 1203-1205.
Cartan, E. (1922). Comptes Rendus 174, 437-439, 593-595, 734-737, 857-860, 1104-1107.
Gronwald, F. and Hehl, F. W. (1996). On the Gauge Aspects of Gravity gr-qc/9602013
Hehl, F. W. (1973). Gen. Rel. Grav., 4, 333.
Hehl, F. W. (1974). 5, 491.
Kerlick, G. D. (1975). thesis, Department of Physics, Princeton U.
Kibble, T. W. B. (1961). J. Math. Phys., 2, 212.
Kleinert, H. (1987). In "Gauge Fields in Condensed Matter" (World Scientific Publishing). See especially “Part IV: Differential Geometry of Defects and Gravity with Torsion.”
Kleinert, H., (2000). Gen. Rel. Grav. 32, 769.
Petti, R.J. (1976). Gen. Rel. Grav. 7, 869-883.
Petti, R. J. (1986). Gen. Rel. Grav. 18, 441-460.
Petti, R. J. (2001). Gen. Rel. Grav. 33, 209-217.
Petti, R. J. (2006). Translational spacetime symmetries in gravitational theories Class. Quantum Grav. 23, 737-751.
Saa, Alberto (1993). Einstein–Cartan theory of gravity revisited, gr-qc/9309027
Venzo de Sabbata, and C. Sivaram, "Spin and Torsion in Gravitation", World Scientific 1994.