Einstein–Cartan theory - AbsoluteAstronomy.com

Theoretical physics

Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory or the Cartan–Sciama–Kibble theory is a classical theory of gravitation similar to general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

but relaxing the assumption that the metric

Metric

Metric may refer to:* the metric system of measurement** International System of Units, or Système International , the modern form of the metric system** Metric ton, a measurement of mass equal to 1,000 kg...

be torsion

Torsion tensor

In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet-Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves In the...

-free. Introducing torsion allows greater freedom to couple classical spin angular momentum to the metric. The theory was first proposed by Élie Cartan

Élie Cartan

Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...

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

Albert Einstein

Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

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

Teleparallelism

Teleparallelism , was an attempt by Einstein to unify electromagnetism and gravity...

.

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 nonzero torsion are hypothesized to be non-propagating so that its direct observation is expected to be difficult. Since Einstein-Cartan theory is purely classical, it also does not fully address the issue of quantum gravity

Quantum gravity

Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

. Interest in Einstein-Cartan theory has been renewed in recent years, however, as theorists try to incorporate torsion into novel quantum theories or as they explore its cosmological

Cosmology

Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...

implications. The theory is still considered viable and remains an active topic in the physics community.

Spin-orbit coupling

Although general relativity can accommodate particles with spin, including spin-1/2, by using the tetrad formalism

Tetrad formalism

The tetrad formalism is a non-coordinate approach to General Relativity.In the tetrad formalism all tensors are represented in terms of a chosen basis, a set of four vectors called the tetrad....

, 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.

The reason that general relativity cannot describe spin-orbit coupling is rooted in Riemannian geometry

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

, on which general relativity is based. In Riemannian geometry, the Ricci curvature tensor

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

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

(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 momentum tensor is also called the stress-energy tensor

Stress-energy tensor

The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...

, 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)

(

–

)

(See spin tensor for more details.)

Therefore general relativity cannot properly model spin-orbit coupling.

In 1922 Élie Cartan

Élie Cartan

Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...

conjectured that general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

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

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 hole

Rotating black hole

A rotating black hole is a black hole that possesses spin angular momentum.-Types of black holes:There are four known, exact, black hole solutions to Einstein's equations, which describe gravity in General Relativity. Two of these rotate...

s. 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). 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.

The proof consists of these 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.

Mathematical background

The two most important concepts in Einstein-Cartan theory that may be new to the general relativist are the understanding of the torsion tensor

Torsion tensor

and how it relates to the spin tensor. Since torsion is a more mathematical than phenomenological concept, we will discuss it first. The reader should be familiar with the mathematical foundation of general relativity at the level of Wald

General Relativity (book)

In physics and especially relativity, General Relativity is a popular textbook on Einstein's theory of general relativity written by Robert Wald....

, whose notation and axiomatic approach have been borrowed. An appropriate accompanying text in the same notation is onlinehttp://www.slimy.com/~steuard/teaching/tutorials/GRtorsion.pdf and may be useful to the beginner. The equations below use abstract index notation

Abstract index notation

Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical...

with implied summation over repeated indices unless otherwise noted. Brackets denote the antisymmetric part of a tensor.

A covariant derivative

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

is an operator satisfying four defining properties.

Linearity:
Product rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...

:
Commutativity
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

with tensor contraction
Tensor contraction
In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...

(summation emphasized):
A tangent vector is a directional derivative
Directional derivative
In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...

of a scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...

:

Given a symmetric (0,2) tensor

called the metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

and an antisymmetric (1,2) tensor

known as the torsion tensor

Torsion tensor

, there exists a unique covariant derivative denoted by a semicolon with two extra properties.

Compatibility of the metric:
Commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...

of the covariant derivative:

In the case

(torsion-free), this theorem is known as the fundamental theorem of Riemannian geometry

Fundamental theorem of Riemannian geometry

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric...

. It is this torsion-free assumption that Einstein-Cartan theory relaxes. The uniqueness of the covariant derivative nevertheless remains assured for nonzero torsion. The torsion tensor may also be interpreted as an antisymmetric part of the Christoffel symbols

Christoffel symbols

In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...

but such a relation is not manifestly covariant. If we define the general relativistic connection coefficient as

where we have used comma derivative notation, then we may define an auxiliary (1,2) tensor

called the contortion tensor defined by

where we note again that the last relation is not manifestly covariant. While the loss of symmetry is more general, it complicates many of the equations relevant to understanding curvature. For example, the Riemann curvature tensor

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...

now behaves nontrivially under certain permutations of its indices:

The Bianchi identity must also be adjusted:

Likewise, the Ricci tensor is no longer symmetric but instead has the property

Geometric intuition

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

Proper time

In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

. 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

Parallel transport

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...

along a geodesic (i.e. straight line for Minkowski spacetime) described by the reduced equation

rotates about the translation axis.

The geodesic deviation equation

Geodesic deviation equation

In general relativity, the geodesic deviation equation is an equation involving the Riemann curvature tensor, which measures the change in separation of neighbouring geodesics or, equivalently, the tidal force experienced by a rigid body moving along a geodesic...

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.

Field equations

The fundamental equations of any field theory are declared as brute facts, for example D'Alembert's principle

D'Alembert's principle

D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert...

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

Variational principle

A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions...

applied to an action

Action

Action may refer to:* Action theory , concerning the processes causing intentional human bodily movements* Social actions, person taking account of others* Action , a characteristic of a stringed instrument...

because this eases the implementation of Noether's theorem

Noether's theorem

Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...

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

Einstein-Hilbert action

The Einstein–Hilbert action in general relativity is the action that yields the Einstein's field equations through the principle of least 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 the exact 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

Tensor density

In differential geometry, a tensor density or relative tensor is a generalization of the tensor concept. A tensor density transforms as a tensor when passing from one coordinate system to another , except that it is additionally multiplied or weighted by a power of the Jacobian determinant of the...

. 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),

where

is the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

of the metric, and

is a physical constant

involving the gravitational constant

Gravitational constant

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

and the speed of light

Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

. By Hamilton's principle

Hamilton's principle

In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...

, the total variation of

should vanish:

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

Belinfante–Rosenfeld stress-energy tensor

In mathematical physics, the Belinfante–Rosenfeld tensor is a modification of the energy-momentum tensor that is constructed from the canonical energy-momentum tensor and the spin current so as to be symmetric yet still conserved....

.

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.

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 variables 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 it possible to pursue quantization of it using well-known techniques from quantum gauge field theory.

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 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.

Spin-orbit coupling

ω-consistency

Mathematical background

Geometric intuition

Field equations

Basis for loop quantum gravity

Other basic physical theories that employ affine torsion

See also

Further reading