Einstein–Cartan theory

In 1922 Élie Cartan conjectured that general relativity must be extended to include affine torsion, which allows for an asymmetric Ricci tensor. The extension of Riemannian geometry to include affine torsion is now known as Riemann-Cartan geometry. A Riemann-Cartan geometry is uniquely determined by:

1. a choice of metric tensorial field (which specifies all lengths of vectors and angles between vectors),
2. an aphin torsion field, and
3. the requirement that the lengths and angles are preserved by parallel translation (as in Riemann's geometry where the torsion is zero).

A Riemann geometry is a Riemann-Cartan geometry with zero torsion, so it is uniquely determined by a metric tensor.

As the main theory of classical physics, general relativity has a known flaw: it cannot adequately describe the exchange between intrinsic (spin) angular momentum and orbital angular momentum. The problem is rooted in the fundamentals of general relativity. General relativity is based on Riemannian geometry, in which the Ricci curvature tensor R_ij must be symmetric in i and < i>j (ie, R_ij = R_ji). In general relativity, R_ij models the local gravitational forces, and its symmetry forces the momentum tensor (we use P and leave T for torsion): P_ij to be symmetric, so general relativity cannot accommodate the general conservation equation of angular momentum: divergence of spin current ½(P_ij - P_ji) = 0.

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

• edge dislocations (formed by adding an additional semi-plane of atoms to a perfect crystal, so you get a defect in the regular crystalline structure along the line where the additional semi-plane ends), and
• the dislocations of "screw" (formed by inserting "a parking garage ramp" that extends the edges of the garage in a structure, which otherwise would be perfectly stacked).

A Riemann-Cartan geometry can be thought of as uniquely determined by the lengths and angles of vectors and the density of dislocations in the affine structure of space.

General relativity fixed affine torsion at zero, because it did not seem necessary to provide a model of gravitation (with a consistent set of equations leading to a well-defined initial value problem).

The derivation of the field equations from the Einstein-Cartan theory

Einstein-Cartan's general relativity and theory both use the climbing curvature as a lagrangian. The general relativity obtains its equations of the field varying the integral action (integral of the lagrangian on space-time) with respect to the metric tensor ${displaystyle {g_{ij}}{{{}}{{}}{displaystyle {g_{ij}}}}{{}}{{displaystyle}}$. The result is Einstein's famous equations:

${displaystyle R_{ij}-{frac {1}{2}}}g_{ij}R={frac {8pi G}{c^{4}}{c}}}}$

where

• ${displaystyle {R_{ij}}{{displaystyle}}{displaystyle {R_{ij}}}{{}}}{{}}{}{displaystyle}}$ is Ricci's curvature tensor (a contraction of Riemann's full curvature tensor that has four indices: ${displaystyle {R^{k}}_{ikj}}}$) (Einstein's convention is followed: a higher repeated index (contravariant indicator) and a lower one (covariant indicator) involve a summation on that index.)
• ${displaystyle {g_{ij}}{{{}}{{}}{displaystyle {g_{ij}}}}{{}}{{displaystyle}}$ is the metric tensor (not degenerated, symmetrical),
• ${displaystyle {R_{}{}{}{}{}}{displaystyle {R_{}{}{}{}{}}{}{}{}{}{}}{}}}}}{$ is the climbing curvature: ${displaystyle {R_{}{}{}{}{R_{ij}g^{ij}}}}}$
• ${displaystyle {P_{ij}}{{displaystyle}}{$ is the energy-moment tensor
• ${displaystyle {G_{}{}{}{}{}}{displaystyle} {$ is the Universal Gravitational Constant of Newton and ${displaystyle {c_{}{}{}{}{}}{displaystyle {c_{}{}{}{}{}}{}{}{}{}}{}{}}}}}}{}}{displaystyle {c}{}{}{}{}{}{}{}{displaystyle {c}}}}}}{c}}{c}}}{c}}}}{c}}{c}}}}}{$ It's the speed of light.

Bianchi's "Second Identity Contracted" of Riemannian geometry becomes, in general relativity, div(P)=0, which makes conservation of energy and momentum equivalent to an identity of Riemannian geometry.

A basic question in formulating Einstein-Cartan's theory is what variables in action should vary to get the equations in the field. Can vary the metric tensor ${displaystyle {g_{ij}}{{{}}{{}}{displaystyle {g_{ij}}}}{{}}{{displaystyle}}$ and torsion tensor, ${displaystyle {T_{ij}}{k}}}$. However, this makes the equations of Einstein-Cartan's theory more dirty than necessary and disguises the geometric content of theory. The key intuition is to let the symmetry group of Einstein-Cartan theory be the non-homogeneous rotation group (which includes space and time translations). Non-homogeneous rotary symmetry is broken by the fact that the zero point in each tangent fiber remains a privileged point, as it is in the ordinary geometry of Riemann based on the group homogeneous of rotation. We vary the action with respect to the aphin connection coefficients associated with translating and rotation symmetries. A similar approach in general relativity is called "Palatini variation", in which the action with respect to the revolving coefficients of the connection instead of the metric; the general relativity has no translating coefficient in the connection.

The field equations resulting from the Einstein-Cartan theory are:

${displaystyle R_{ak}-{frac {1}{2}}}g_{ak}R={frac {8pi G}{c^{4}}}{p_{ak}}}}}$

${displaystyle {S_{ab}}{k}={frac {8pi G}{c^{4}}}{sigma _{ab}}}{{k}}}}$

where

• ${displaystyle {sigma}{ab}}{k}}$ is the thorn of all matter and radiation
• ${displaystyle {S_{ab}}{k}}}$ is the modified torsion tensor: ${displaystyle {S_{ab}}}{k}={T_{ab}}}^{k+{g_{a}{k}{T_{bm}}{m}}{m}-{g_{b}}}{k}{T_{am}}{m}}}}{$

• ${displaystyle {T_{ab}}{k}}$ It's the affinity torsion tensor.

The first equation is as in general relativity, except that the affin torsion is included in all the terms of the curvature, so ${displaystyle {P_{ak}}{}{displaystyle}}{$ It doesn't need to be symmetrical.

The second Bianchi identity contracted from Riemann-Cartan geometry becomes, in Einstein-Cartan theory,

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

• div(σ) = - antisymmetric part of PThe conservation of the moment is altered by the products of the force of the field and the gravitational density of the spine. These terms are extremely small under normal conditions, and seem reasonable since the gravitational field itself carries energy. The second equation is the preservation of the angular moment, in a way that accommodates the spinal-orbit coupling.

The geometric insights of the Einstein-Cartan theory

First geometric intuition

Spin (intrinsic angular momentum) consists of dislocations in the fabric of space-time. For ordinary fermions (particles with half-integer spin such as protons, neutrons, and electrons), these are screw dislocations (parking garage ramps) with the time-type screw direction. That is, for a particle with spin in the +z direction, traversing a space-like loop in the x-y plane around the particle moves it parallel to the past or future by a small amount.

Third geometric intuition

In Einstein-Cartan theory, one must distinguish between tensor indices that represent conserved currents (such as momentum and spin) and indices that represent spacetime boxes (through which energy flows). currents are measured). This is similar to other gauge theories, such as electromagnetism and the Yang-Mills theory, where you would never confuse spacetime indices representing flux boxes with fiber indices representing conserved currents.

Writing the Einstein-Cartan theory in its simplest form requires distinguishing two classes of tensor indices:

1. addresses in Minkowski's "fibre space" idealized at every point of space-time (the space of tangent vectors).
2. tangents to the variety of space-time that describe the flow boxes, and

These two types of indices have two roles in the theory.

1. Conserved currents are represented by fiber indices.
2. All derivative indices in Einstein-Cartan theory are space-time indices. In addition, derivatives are all external derivatives, which measure current flows through space-time boxes (or divergences, which are external derivatives disguised as "Hodge's chart"). Derivative indices are time-space indices, as are all indices with which external derivatives are anti-symtrized (or indices with which derivative indices are contracted in the case of divergences).

For example, in the field equations of the Einstein-Cartan theory given above, one should interpret the indices a, b as fiber indices and the indices i, j as basis space indices. The momentum tensor P_a^k describes the flow of a-momentum through the flow box normal to the k-direction in spacetime, and the tensor of spin σ_ab^k describes the flow of angular momentum in the axb-plane through the flow box normal to the k-direction in spacetime.

Before the distinction between these types of indices became clear, researchers varied the action with respect to the metric to get what they called the "momentum tensor" (the 'wrong' one) and also sometimes they varied with respect to the translation coefficients of the connection and got a different moment tensor (the 'correct' one) and didn't know what the one was. real moment tensor. The theory equations had many unnecessary terms because no distinction was made between base space and fiber space indices.

Fourth geometric intuition

The Einstein-Cartan theory is about defects in the affine structure of space-time (Euclidean type but curved); it is not a metric theory of gravitation.

Affine torsion is a continuous model of dislocation density. The full rotary (or Riemannian) tensor,

R^a_bij of curvature also has an interpretation as defect density in continuum mechanics. It is the continuous model of a density of "disclination defects." An offset results when a cut is made in a continuum (a radial cut is made from the edge to the center of a rubber disk) and an angular wedge of material is inserted (or removed) so that the sum of the angles surrounding to the end point of the cut is more (or less) than 2π radians. In fact, this procedure can turn a flat disk into a bowl by making many small radial rim cuts with varying lengths all the way to the center, excising wedges of material of the appropriate angular width, and sewing up the cuts.

The central role of affine defects explains why the clean way to do the Einstein-Cartan theory is to vary the translational and rotational coefficients of the connection (not the metric) and to distinguish between the base space and base space indices. the fiber. The connection coefficients are following the dislocation and disclination defects in the affine structure of space-time. It is as if space-time is made up of many microcrystals of perfectly flat Minkowski space, and these perfect micro-chunks are bound together with defects like dislocations and disclinations. The central role of translational and rotational connection coefficients as field variables is recognized in modern efforts to quantify general relativity under the name of "Ashtekar variables". The Ashtekar variables are essentially the translational and rotational coefficients of the connection, conveniently worked into a Hamiltonian formulation of general relativity.

General relativity plus matter with spin implies the Einstein-Cartan theory

For decades, the Einstein-Cartan theory was thought to be based on an independent assumption to include affine torsion. Since the effect of torsion is too small to measure empirically until now, the Einstein-Cartan theory was considered (and largely ignored) one of many speculative extensions of general relativity. General relativity plus a fluid of many tiny rotating black holes has been shown to generate affine torsion and essentially the equations of the Einstein-Cartan theory. The "test" uses a standard rotating Kerr-Newman black hole solution of general relativity. Computes the time-like non-zero translation that occurs when an affine frame is parallel-translated (following the translation as well as the rotation) around an equatorial loop near the black hole. The word "test" it appears in quotes because, while it is intuitively mandatory that this implies the Einstein-Cartan theory, the proof of the convergence of the equations of the Einstein-Cartan theory has not yet been made.