Einstein-Cartan theory

The Einstein-Cartan theory  (ECT, also Einstein-Cartan-Sciama-Kibble theory , ECSK theory) is a generalization of the general relativity theory to the Riemann-Cartan geometry. In the Cartan geometry, the torsion appears as an additional degree of freedom, which results in an additional field equation in the ECT. This second field equation couples the torsion with the spin density tensor.

The ECT replicates all the results of general relativity, but predicts additional effects in the case of very high spin densities. However, the required spin densities are so high that the deviations are only relevant when looking at the Big Bang . Accordingly, the deviations are not yet measurable. However, the ECT is also exciting from a theoretical point of view, as it represents a calibration theory formulation of gravity.

history

Elie Cartan dealt with Riemannian manifolds in the early 1920s . Cartan expanded the concept of the connection to include so-called torsion, which gives the connection additional degrees of freedom. Cartan then tried to generalize the general theory of relativity to this Cartan geometry. He coupled the additional degrees of freedom to the spin density tensor. Cartan soon gave up these attempts because the deviations were too small and the concept of spin (in the sense of quantum field theory ) had not yet been developed. Cartan's spin density tensor was the tensor from the theory of elasticity developed by the brothers Francois and Eugène Cosserat .

When considering the unification of gravitation and electromagnetism, Hermann Weyl and others try to take up the concept again and to associate torsion with electromagnetic potential. That failed, however, and Weyl instead developed the concept of gauge theory of electromagnetism, which was later generalized by Yang, Mills, and Utiyama.

In the following years, in which the concept of gauge theories proved to be successful and quantum field theory achieved more and more successes, the ECT lost some of its attention. That changed in the early 1960s when Dennis W. Sciama and Tom Kibble used the theory again, but for a different motivation: In his article on the Yang Mills theory , Utiyama treated gravity as the gauge theory of the Lorentz group. In doing so, Utiyama had to make two ad hoc assumptions, in that he had postulated the relationship as symmetrical (or had said that the antisymmetrical components would disappear) and had also assumed the tetrads to be symmetrical. Sciama first published a connection between spin and gravity, Kibble then developed the gauge theory of the Poincaré group (called the complete Lorentz group by Kibble). This gauge theory is the most widely recognized form of ECT to this day. The translative part creates the metric (or the tetrad fields) and the rotating part creates the torsion.

In the following years the ECT was able to gain attention because it enables a theoretical consideration of gravitation. A review article by Hehl, von der Heyde and Kerlick appeared in 1976

Based on the ECT, further gauge theories in the Riemann-Cartan geometry have emerged, which predict other effects, such as a gauge theory with a quadratic Lagrangian (meaning the Ricci scalar).

torsion

Torsion as a concept can be understood from two standpoints: If one considers a (pseudo-) Riemannian manifold with a Levi-Cevita connection (shown as a Christoffel symbol ), the connection can be expanded by an antisymmetric part , where the torsion is. The transformation rule for the context must be retained. Now one demands that the metric condition still be met . This denotes the covariant derivative. The torsion is now always a tensor and is therefore also called a torsion tensor. Alternatively, one can proceed in exactly the opposite direction from a metric-affine manifold, i.e. a manifold on which metric and torsion are defined, but independently of each other, and then demand the metric condition. The difference between the connection and the Christoffel symbol is then the torsion. ${\displaystyle \{_{\mu \nu }^{\lambda }\}}$ ${\displaystyle \Gamma _{\mu \nu }^{\lambda }=\{_{\mu \nu }^{\lambda }\}+S_{\mu \nu }^{\lambda }}$ ${\displaystyle S_{\mu \nu }^{\lambda }}$ ${\displaystyle \Gamma }$ ${\displaystyle \nabla _{\lambda }g_{\mu \nu }=\partial _{\lambda }g_{\mu \nu }-g_{\sigma \nu }\Gamma _{\lambda \mu }^{\sigma }-g_{\mu \sigma }\Gamma _{\lambda \nu }^{\sigma }=0}$ ${\displaystyle \nabla }$ ${\displaystyle S}$

The torsion can be imagined as a closing error of an infinitesimal parallelogram: If you take two infinitesimal vectors and move them parallel to one another, the result is not exactly the same point. This can be represented with the following calculation: Let be a tensor scalar, then . This closing error is also referred to as Cartan displacement . ${\displaystyle dx}$ ${\displaystyle dy}$ ${\displaystyle A}$ ${\displaystyle \nabla _{\mu }\nabla _{\nu }A-\nabla _{\nu }\nabla _{\mu }A=-2(\partial _{\sigma }A)S_{\mu \nu }^{\sigma }}$

Field equations

The presentation here follows Hehl, von der Heyde and Kerlick. Let be the Lagrangian of any matter field. Then we define with the coupling constant ${\displaystyle {\mathfrak {L}}_{m}={\sqrt {-g}}L_{m}}$ ${\displaystyle k}$

${\displaystyle {\sqrt {-g}}k\sigma _{\mu \nu }={\frac {\delta {\mathfrak {L}}_{m}}{\delta g^{\mu \nu }}}}$ the symmetric energy-momentum tensor and

${\displaystyle {\sqrt {-g}}k\mu _{\lambda }^{\mu \nu }={\frac {\delta {\mathfrak {L}}_{m}}{\delta S_{\mu \nu }^{\lambda }}}}$ the spin energy potential.

We choose as the Lagrangian of the gravitational field . This corresponds formally to the Lagrangian of general relativity, but this Lagrangian is defined by the Ricci scalar of the Riemann-Cartan manifold and accordingly contains a portion of the torsion. As in the case of the general theory of relativity, the variation takes place with the help of the Palatine identity, which takes the form in the case of a Cartan connection . The following applies . The result is ${\displaystyle {\mathfrak {L}}_{g}={\sqrt {-g}}R}$ ${\displaystyle g^{\mu \nu }\delta R_{\mu \nu }=-2{\overline {S}}_{\lambda }^{\mu \nu }\delta \Gamma _{\mu \nu }^{\lambda }}$ ${\displaystyle {\overline {S}}_{\lambda }^{\mu \nu }=S_{\lambda }^{\mu \nu }+g^{\lambda \mu }S_{\nu \sigma }^{\sigma }-g^{\lambda \nu }S_{\mu \sigma }^{\sigma }}$

${\displaystyle k\sigma ^{\mu \nu }={\frac {1}{\sqrt {-g}}}{\frac {\delta {\mathfrak {L}}_{g}}{\delta g_{\mu \nu }}}=G^{\mu \nu }-{\overset {*}{\nabla }}_{\lambda }({\overline {S}}^{\lambda \mu \nu }+{\overline {S}}^{\mu \nu \lambda }+{\overline {S}}^{\nu \mu \lambda })}$

The following applies . The field equation for the torsion is given by ${\displaystyle {\overset {*}{\nabla }}_{\lambda }=\nabla _{\lambda }-2S_{\lambda \sigma }^{\sigma }}$

${\displaystyle k\mu ^{\lambda \mu \nu }={\frac {1}{\sqrt {-g}}}{\frac {\delta {\mathfrak {L}}_{g}}{\delta S_{\lambda \mu \nu }}}=-2({\overline {S}}^{\nu \mu \lambda }+{\overline {S}}^{\mu \lambda \nu }+{\overline {S}}^{\lambda \mu \nu })}$

These field equations are usually reduced to a simpler form by means of the canonical energy-momentum tensor and the spin density tensor ${\displaystyle T^{\mu \nu }=\sigma ^{\mu \nu }-{\overset {*}{\nabla }}_{\lambda }\mu ^{\mu \nu \lambda }}$ ${\displaystyle \tau ^{\lambda \mu \nu }={\frac {1}{4}}(\mu ^{\mu \nu \lambda }-\mu ^{\nu \mu \lambda })}$

${\displaystyle kT^{\mu \nu }=G^{\mu \nu }}$

${\displaystyle k\tau ^{\lambda \mu \nu }={\overline {S}}^{\lambda \mu \nu }}$

Web links

• Andrzej Trautman: Einstein-Cartan Theory. In: Encyclopedia of Mathematical Physics, edited by J.-P. Fran ̧coise, GL Naber and Tsou ST Oxford: Elsevier, 2006, vol. 2, pp. 189-195. (PDF file; 156 kB) arxiv : gr-qc / 0606062
• Heinicke, Christian: Exact solutions in Einstein's theory and beyond. Dissertation, University of Cologne, 2005. urn : nbn: de: hbz: 38-14637

literature

Miletun Blagojević, Friedrich W. Hehl: Gauge Theories of Gravitation, A Reader with Commentaries . Imperial College Press, 2013.

Individual evidence

1. ^ Cartan, Élie: On a generalization of the notion of Riemann curvature and spaces with torsion . In: Comptes rendus de l'Académie des Sciences . 1922, p. 593–595 (French: Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion . Translated by DG Kerlick (English)).
2. ^ Cartan, Élie: Space with a Euclidian connection . In: Riemannian Geometry in an Orthogonal Frame . S. 121-144 .
3. ^ Hermann Weyl: Electron and Gravitation . In: Journal of Physics . tape 56 , 1929, pp. 330-352 , doi : 10.1007 / BF01339504 .
4. ↑ Ryoyu Utiyama: Invariant Theoretical Interpretation of Interaction . In: Physical Review . tape 101 , 1955, pp. 1597-1607 .
5. ^ DW Sciama: The Analogy between charge and spin in general relativity . In: Recent Developments in General Relativity, Feldschrift for Infeld . Pergamon Press, Oxford, 1962.
6. ↑ TWB Kibble: Lorentz Invariance and the Gravitational Field . In: Journal of Mathematical Physics . tape 2 , 1961, p. 212-221 .
7. ↑ ^{a b} Friedrich W. Hehl, Paul van der Heyde, G. David Kerlick: General relativity with spin and torsion: Foundations and prospects . In: Review of Modern Physics . tape 48 , 1976, p. 393-416 .
8. ^ Friedrich W. Hehl, Jürgen Nitsch, Paul von der Heyde: Gravitation and the Poincaré Gauge Field Theory with Quadratic Lagrangian . 1980.