Einstein–Cartan theory

From Wikipedia, the free encyclopedia

Jump to: navigation, search

Einstein–Cartan theory in theoretical physics extends general relativity to account for spin angular momentum. The theory is named after Albert Einstein and Élie Cartan.

1 Introduction
2 Motivation
3 Geometric Foundations
4 Derivation of field equations of Einstein–Cartan theory
5 Geometric insights from Einstein–Cartan theory
6 General relativity plus matter with spin implies Einstein–Cartan theory
7 Significance of Einstein-Cartan theory
8 Basis for loop quantum gravity
9 Other basic physical theories that employ affine torsion
10 See also
11 References
12 Further reading

[edit] Introduction

General relativity is (almost) the master theory of classical (non-quantum) physics. It unifies classical mechanics, special relativity, gravitation, electrodynamics, phenomena at near-light speeds, and cosmic scales of time and length. It provides the basic framework for cosmology; it even includes a rudimentary classical model of dark energy in the form of the cosmological constant that can make cosmological models expand exponentially after early history.

Within the scope of classical physics, general relativity has one flaw that stands out: it cannot model spin-orbit coupling (conversions between intrinsic angular momentum (spin) and orbital angular momentum). This flaw stands out because angular momentum is one of the two main conserved currents (along with momentum) arising from the basic symmetries of classical spacetime, so any master theory of classical physics should handle angular momentum well. While spin-orbit coupling is usually studied in quantum systems, there are at least two cases where it can occur in classical physics: in very dense matter with spin (as in neutron stars or possibly the early universe), and in dust models of galaxies with correlated rotation (as in cosmological models with spin).

Einstein-Cartan theory resolves the problems with spin angular momentum. It extends general relativity by extending Riemannian geometry (the mathematical foundation of general relativity) to includes affine torsion, yielding what is known as Riemann-Cartan geometry. In Einstein-Cartan theory, torsion appears only within the spinning matter that is its source; that is, torsion is a nonpropagating field.

It has been shown that general relativity and spin angular momentum together generate affine torsion that enters the equations as in Einstein-Cartan theory, with no additional assumptions.

The effects of torsion are negligible for any celestial body with current technology, due to the smallness of the gravitational coupling and the non-propagation of torsion.

[edit] Motivation

The reason that general relativity cannot describe spin-orbit coupling is rooted in 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. (g_ab 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 momentum tensor

$P_{ab} \,$

(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 momentum tensor is also called the stress-energy tensor, the energy-momentum tensor, and the energy-momentum-stress tensor. Special relativity shows that energy, momentum, momentum flux, and stress are different spacetime components of the same covariant object, known most compactly as the momentum 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 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 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;^{[citation needed]}

(2) it explains the meaning of affine torsion, which appears naturally in some theories of quantum gravity;^{[citation needed]} 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.

[edit] Geometric Foundations

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 u' + b v'.

Parallelism is path-dependent; that is, if you parallel translate a vector along two different paths that both start at point x and end at point y in M, the resulting vectors at the end point y in general differ. Similarly, if we parallel translate a frame (a linear basis) of vectors at x along two different paths to y, then the two resulting frames at y differ. The difference (more precisely the linear transformation) between these two frames is the essential meaning of curvature, which is the central concept in differential geometry (and general relativity).

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

${R_{klj}}^i$

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

${T_{kj}}^i$ .

The rotational curvature ${R_{klj}}^i$ 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 ${T_{kj}}^i$ describes the translation in the k direction resulting from 'developing' a small loop in the i,j plane in the base manifold M into a flat manifold X that has the same dimension as M. (Developing a curve

$C_m: [0,1]\rightarrow M$

into a curve

$C_x: [0,1] \rightarrow X$

means defining a curve $C x$ in X that has the same pattern of accelerations as the curve $C m$ in M. The motivation for development of a curve $C m$ is to create a curve $C x$ whose shape is determined by the same pattern of accelerations as $C m$ , but without the impact on shape of curve $C x$ 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 outlined 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

${\sigma_{ab}}^k$

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 rotations (generalized to include boosts, that is 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 "spiral 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 sets the affine torsion to zero, because including torsion 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,

$\mathrm{div} \, P = P^{ab}{}_{;b}=0$

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:

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

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

where

$σ a b k$ is the spin current density 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 Palatini 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 (usually very small) terms that are products of curvature and torsion (Petti 1976), specifically

$\nabla_{k}$

G d k

= -

T a b c

R c d a b

–

G a c

S c d a

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. (In language adapted from metallurgy, this is a screw dislocation with timelike Burghers vector.)

[edit] Second geometric insight

It has long been known that the spin angular momentum tensor

${\sigma_{ab}}^k$

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 (Petti 1976):

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 ...)
indices that represent conserved currents like momentum and spin. Geometrically, these indices represent directions in the idealized Minkowski "fiber space" X over each point of spacetime. (Notation used here: a, b, c...)

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 (Petti 1976, Petti 2001).

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.

Interpretation of Einstein-Cartan theory in terms of disclinations (for rotational curvature) and dislocations (for affine torsion) fits into a broader geometric interpretation of gauge theories. A third kind of line defect (so-called because the disruption of lattice order occurs along a one-dimensional line through the medium) in a regular lattice is a "dispiration", which is the discrete analog of internal symmetries that are the foundation of SU(n) unitary symmetries that form the foundation of high energy physics. For explanantions with pictures, see (Petti 2001).

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

For decades, Einstein–Cartan theory was considered one of many speculative (and largely ignored) extensions of general relativity, for two reasons.

It was thought that Einstein–Cartan theory is based on an independent assumption to include affine torsion.
The effects of torsion are too small to measure empirically at this time. For example, in the best case scenario in which the earth would consist completely of equally spin-oriented constitiuents, the angular momentum coming from all spins would be smaller by a factor 10^-15 than that from the earth's rotation^[1].

Binary pulsar PSR J0737-3039A/B may be a test of relativistic spin-orbit coupling (Breton, et al., 2008).

[edit] Mathematical proof

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). 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 recourse to quantum mechanical spin or spinor fields.

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 Eeinstin-Cartan theory.

[edit] Two types of spin

When we say the general relativity plus spin implies torsion, it is important to distinguish two types of spin:

Spin tensors that have fiber indices (a,b,c here), such as the spin density $J a b k$ . The spinor indices of Dirac fields are fiber indices(Petti 1976) that couple to a spin connection (which is the affine connection raised from the rotation group to the spin group). In principle, this can be extended to higher order spin that couples to the connection. In the language of tensor indices, coupling to the affine conection (or spin connection) means that a tensor has fiber indices.
Spin tensors that have only base space indices (i,j,k here) and no fiber indices. For example, the Maxwell vector potential is a connection coefficient on a U(1) bundle, so its index structure is $A k$ . This is why Maxwell fields do not contribute to the spin density J, and they do not imply the presence of affine torsion.

The pedestrian distinction between base space and fiber indices is a shorthand for a deeper distinction between what each tensor field is in the fiber bundle structure of a theory. The distinction between a vierbein and an ordinary frame field is another attempt to describe the difference between a base space and fiber tensor, without reference to the underlying fiber bundle structure. Below is how this distinction is expressed with vierbeins.

If we introduce spinorial fields, then 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 as 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.

[edit] The meanings of proof

The concept of proof has different meannigs in different contexts and communities. For example,

Physicists generally require a proof to include experimental evidence that validates a proposition, and that could in principle falsify the proposition.

Example: Physicists did not regard black holes to be proven to exist until they were observed, and numerous specific predictions of black hole models were verified experimentally.

Negative example: String theory has not produced any verifiable and falsifiable propositions, hence the oft-heard charge that "string theory is not even wrong."

Applied mathematicians and physicists generally regard a mathematical proof to mean a derivation in symbolic mathematics.
Pure mathematicians regard a proof to be a deduction at the level of sets and mappings that establishes existence, uniqueness, or convergence in ways that other scientists usually take for granted. (Physicists usually do not insist on this level of proof, except in field theory where existence, uniqueness and convergence of solutions are central issues.)

The mathematical proof that general relativity plus spin imply torsion currently meets only the second standard. There is no experimental evidence of the type required in the first type of proof, and proofs that general relativity converges to an Einstein-Cartan solution, even in specific cases, have not been made in a way that would satisfy someone wearing his pure mathematician's hat.

The requirement for experimental verification for most extensions of physics (for example for black holes) is not analogous to a demand for experimental proof of Einstein-Cartan theory. The fundamental differences are:

General relativity (which we shall consider to be proven) fails to provide the symmetry structure that is required at the most basic level to describe spin-orbit coupling, which is a fundamental classical and quantum phenomenon. (On a more theoretical level, general relativity also fails to provide a clean derivation of momentum as the Noether current of translational spacetime symmmetry.) Einstein-Cartan theory remedies these problems in a minimally invasive way.
Ordinary models of rotating black holes converge (in the applied mathematician's sense) to spacetimes that have torsion in exactly the way it appears in Einstein-Cartan theory. If we reject Einstein-Cartan theory, then general relativity is left in a state where it is "illegal" to use the convergence arguments that every calculus student uses when defining derivatives and integrals as limits and applying l'Hopital's rule for limits of quotients, or freshman physics students use to define mass density of material (like water or sand) that may not really be a continuum; and so on in physics.

If we had experimental proof of 3-D rotational symmetry and of conservation of x and y momentum, we could mathematically deduce that z (vertical) momentum is conserved. Since it is as easy to measure conservation of z momemtum as of x and y momentum, it is reasonable to insist that we have empirical evidence before accepting that z momentum is conserved. If, however, z momentum were extremely hard to measure then, to the extent that we have confidence of our proofs of rotational symmetry and of x and y momentum conservation, it would be reasonable to accept a mathematical proof of the conwservation of z momentum as a proof of the second type, while we search for the elusive experimental evidence.

Some physicists believe that validity of a proof that general relativity and spin imply Einstein-Cartan theory is a matter of controversy. The two main objections to the claimed proof are:

Einstein-Cartan theory is well-known to be a failed attempt to extend the geometric approach of general relativity to other areas of physics. Although Cartan proposed it in 1922, Einstein spent a good deal of the later part of his career on theories that include torsion and he did not obtain a substantial result.
A proof in physics requires empirical evidence, just as physicists require empirical evidence for black holes, dark energy in the vacuum, dark matter in galaxies, and so forth.

[edit] Significance of Einstein-Cartan theory

Einstein-Cartan theory contributes to gravitation theory in several respects.

It extends general relativity to correctly describe spin and spin-orbit coupling as classical and quantum phenomena.
It provides a clean derivation of momentum as the conserved current (Noether current) associated with translational spacetime symmetry.
It provides a geometric interpretation of spin density (intrinsic angular momentum) as the continuum version of a distribution of dislocations in a spacetime lattice.
It establishes the role of affine torsion in spacetime physics, which was a topic of speculation for decades.
It is the only extension of classical (non-quantum) general relativity to be proven to be necessary since about 1920 (as of 2010). The proof consists of a mathematical derivation of affine torsion from the general relativistic description of rotating black holes, where torsion appears in the field equations as in Einstein-Cartan theory. No empirical evidence has been found; computations show that the effects are too small to measure with current technology in situations that have been investigated.

[edit] Basis for loop quantum gravity

In 1986, physicist Abhay Ashtekar completed the project which Amitabha Sen began. He clearly identified the fundamental conjugate variables of spinorial gravity: The configuration variable is as a spinoral connection - a rule for parallel transport (technically, a connection) and the conjugate momentum variable is a coordinate frame (called a vierbein) at each point. So these variable became what we know as Ashtekar variables, a particular flavor of Einstein–Cartan theory with a complex connection.General relativity theory expressed in this way, made possible to pursue quantization of it using well-known techniques from quantum gauge field theory.

[edit] Other basic physical theories that employ affine torsion

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 propagating torsion.^[2]^[3]^[4]^[5]^[6]^[7]^[8]^[9]. 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.^[10]

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.^[11]

[edit] See also

[edit] References

^ Kleinert, H., Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation. World Scientific, Singapore 2008 (online)
^ http://arxiv.org/abs/arXiv:gr-qc/9403058 Consequences of Propagating Torsion in Connection-Dynamic Theories of Gravity
^ http://prola.aps.org/abstract/PRD/v19/i2/p430_1 Propagating torsion and gravitation
^ http://prola.aps.org/abstract/PRD/v21/i4/p867_1 Gravity theories with propagating torsion
^ http://www.springerlink.com/content/u11268138235611q/ Scalar-tensor theory and propagating torsion theory
^ http://www.springerlink.com/content/4140409068712510/ Macroscopical consequences of a propagating torsion potential
^ http://www.iop.org/EJ/abstract/0264-9381/7/11/018 Propagating torsion as a massive vector field
^ http://adsabs.harvard.edu/abs/2009arXiv0905.1068H A Discussion on Massive Gravitons and Propagating Torsion in Arbitrary Dimensions
^ http://www22.pair.com/csdc/pd2/pd2fre40.htm Propagating Torsion without Spin
^ De Andrade V.C., Arcos H.I., Pereira J.G.,Torsion as Alternative to Curvature in the Description of Gravitation", Class. Quant. Grav. 21, 5193 (2004) ([1])
^ Kleinert, H., New Gauge Symmetry in Gravity and the Evanescent Role of Torsion", ([2])

[edit] Further reading

Breton, R. P., et al. (2008). Science 321(5885), 104-107.
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). Gen. Rel. Grav., 5, 491.
Hehl, F. W., von der Heyde, P., Kerlick, G. D., and Nester, J. M. (1976). Rev. Mod. Phys. 48, 393.
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). See especially “Part IV: Differential Geometry of Defects and Gravity with Torsion.”
Kleinert, H. (2000). Gen. Rel. Grav. 32, 769.
Kleinert, H. (2008), Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation. (World Scientific) (online)
Petti, R.J. (1976). "Some Aspects of the Geometry of First Quantized Theories" Gen. Rel. Grav. 7, 869-883.
Petti, R. J. (1986). "On the Local Geometry of Rotating Matter" Gen. Rel. Grav. 18, 441-460.
Petti, R. J. (2001). "Affine Defects and Gravitation" Gen. Rel. Grav. 33, 209-217.
Petti, R. J. (2006). Translational spacetime symmetries in gravitational theories Class. Quantum Grav. 23, 737-751.
Poplawski, N. J. (2009). Spacetime and fields, arXiv:0911.0334
Poplawski, N. J. (2010). Cosmology with torsion - an alternative to cosmic inflation, arXiv:1007.0587
de Sabbata, V. and Sivaram, C., "Spin and Torsion in Gravitation", World Scientific 1994.
Saa, A. (1993). Einstein–Cartan theory of gravity revisited, gr-qc/9309027

[0] Kleinert, H., Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation. World Scientific, Singapore 2008 (online)

[1] ttp://arxiv.org/abs/arXiv:gr-qc/9403058 Consequences of Propagating Torsion in Connection-Dynamic Theories of Gravity

[2] ttp://prola.aps.org/abstract/PRD/v19/i2/p430_1 Propagating torsion and gravitation

[3] ttp://prola.aps.org/abstract/PRD/v21/i4/p867_1 Gravity theories with propagating torsion

[4] ttp://www.springerlink.com/content/u11268138235611q/ Scalar-tensor theory and propagating torsion theory

[5] ttp://www.springerlink.com/content/4140409068712510/ Macroscopical consequences of a propagating torsion potential

[6] ttp://www.iop.org/EJ/abstract/0264-9381/7/11/018 Propagating torsion as a massive vector field

[7] ttp://adsabs.harvard.edu/abs/2009arXiv0905.1068H A Discussion on Massive Gravitons and Propagating Torsion in Arbitrary Dimensions

[8] ttp://www22.pair.com/csdc/pd2/pd2fre40.htm Propagating Torsion without Spin

[9] De Andrade V.C., Arcos H.I., Pereira J.G.,Torsion as Alternative to Curvature in the Description of Gravitation", Class. Quant. Grav. 21, 5193 (2004) ([1])

[10] Kleinert, H., New Gauge Symmetry in Gravity and the Evanescent Role of Torsion", ([2])

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

Einstein–Cartan theory

Contents

[edit] Introduction

[edit] Motivation

[edit] Geometric Foundations

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

[edit] Geometric insights from Einstein–Cartan theory

[edit] First geometric insight

[edit] Second geometric insight

[edit] Third geometric insight

[edit] Fourth geometric insight

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

[edit] Mathematical proof

[edit] Two types of spin

[edit] The meanings of proof

[edit] Significance of Einstein-Cartan theory

[edit] Basis for loop quantum gravity

[edit] Other basic physical theories that employ affine torsion

[edit] See also

[edit] References

[edit] Further reading

Personal tools

Namespaces

Variants

Views

Actions

Search

Navigation

Interaction

Toolbox

Print/export

Languages