Spin and torsion in general relativity: I. Foundations

Abstract

From physical arguments space-time is assumed to possess a connection % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFepeea0de9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa83Kdm% aaDaaajeaObaGaa8xAaiaa-PgaaeaacaWFRbaaaOGaa8xpamaacmaa% baqbaeqabiqaaaqaaiaa-TgaaeaacaWFPbGaa8NAaaaaaiaawUhaca% GL9baacaWFRaGaam4uamaaDaaaleaacaWFPbGaa8NAaaqaaiaa-bca% caWFRbaaaOGaa8xlaiaadofadaqhaaWcbaGaa8NAaiaa-bcacaWFGa% Gaa8hiaiaa-LgaaeaacaWFGaGaa83Aaaaakiaa-TcacaWGtbWaa0ba% aSqaaiaa-bcacaWFGaGaa8hiaiaa-LgacaWFQbaabaGaa83Aaaaaki% aa-1dadaGadaqaauaabeqaceaaaeaacaWFRbaabaGaa8xAaiaa-Pga% aaaacaGL7bGaayzFaaGaa8xlaiaadUeadaqhaaWcbaGaa8xAaiaa-P% gaaeaacaWFGaGaa8hiaiaa-Tgaaaaaaa!5E41! $$\Gamma _{ij}^k = \left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\} + S_{ij}^{ k} - S_{j i}^{ k} + S_{ ij}^k = \left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\} - K_{ij}^{ k} $$ . % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFepeea0de9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaafa% qabeGabaaabaacdaGaa83Aaaqaaiaa-LgacaWFQbaaaaGaay5Eaiaa% w2haaaaa!3AEE! $$\left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\}$$ is Christoffel's symbol built up from the metric g ij and already appearing in General Relativity (GR). Cartan's torsion tensor % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFepeea0de9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacdiGaa83uam% aaBaaajeaObaacdaGaa4xAaiaa+PgaaeqaaOWaaWbaaKqaGgqabaGa% a43Aaaaakiabg2da9maaleaaleaacaaIXaaabaGaaGOmaaaakiaacI% cajaaqcqqHtoWrkmaaDaaajeaObaGaa4xAaiaa+PgaaeaacaGFRbaa% aOGaeyOeI0scaaKaeu4KdCKcdaqhaaqcbaAaaiaa+PgacaGFPbaaba% Gaa43AaaaakiaacMcaaaa!4AC0! $$S_{ij} ^k = \tfrac{1}{2}(\Gamma _{ij}^k - \Gamma _{ji}^k )$$ and the contortion tensor K ij k , in contrast to the theory presented here, both vanish identically in conventional GR. Using the connection introduced above in this series of articles, we will discuss the consequences for GR in the framework of a consistent formalism. There emerges a theory describing, in a unified way, gravitation and a very weak spin-spin contact interaction. In section 1 we start with the well-known dynamical definition of the energy-momentum tensor σ ij ∼ δℒ/δg ij , where ℒ represents the Lagrangian density of matter (section1.1). In sections1.2,3 we will show that due to geometrical reasons, the connection assumed above leads to a dynamical definition of the spin-angular momentum tensor according to τk ji ∼ δℒ/δK ij k . In section1.4, by an ideal experiment, it will become clear that spin prohibits the introduction of an instantaneous rest system and thereby of a geodesic coordinate system. Among other things in section1.5 there are some remarks about the role torsion played in former physical theories. In section 2 we sketch the content of the theory. As in GR, the action function is the sum of the material and the field action function (sections2.1,2). The extension of GR consists in the introduction of torsion S ij k as a new field. By variation of the action function with respect to metric and torsion we obtain the field equations in a general form (section2.3). They are also valid for matter described by spinors; in this case, however, one has to introduce tetrads as anholonomic coordinates and slightly to generalize the dynamical definition of energy-momentum (sections2.4,5).