Curvature tensors in a 4D Riemann–Cartan space: Irreducible decompositions and superenergy Jens Boos and Friedrich W. Hehl [email protected] [email protected]"oeln.de University of Alberta University of Cologne & University of Missouri (uesday, %ugust 29, 17:0. Geometric Foundations of /ravity in (artu Institute of 0hysics, University of (artu) Estonia Geometric Foundations of /ravity Geometric Foundations of /auge Theory Geometric Foundations of /auge Theory ↔ Gravity The ingredients o$ gauge theory: the e2ample o$ electrodynamics ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: redundancy conserved e2ternal current 5 ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: Complex spinor 6eld: redundancy invariance conserved e2ternal current 5 conserved #7,8 current ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: Complex spinor 6eld: redundancy invariance conserved e2ternal current 5 conserved #7,8 current Complete, gauge-theoretical description: 9 local #7,) invariance ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: iers Complex spinor 6eld: rce carr ry of fo mic theo rrent rosco rnal cu m pic en exte att desc gredundancyiv er; N ript oet ion o conserved e2ternal current 5 invariance her f curr conserved #7,8 current e n t s Complete, gauge-theoretical description: gauge theory = complete description of matter and 9 local #7,) invariance how it interacts via gauge bosons ,3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincaré gauge theory *3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincaré gauge theory *3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincar; gauge theory *3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincar; gauge theory *3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincaré gauge theory *3,. Curvature tensors Riemann curvature tensor Cartan<s torsion tensor electrodynamics :ang–Mills theory /eneral Relativity 0oincaré gauge theory *3,. Curvature tensors rotational curvature translational curvature electrodynamics :ang–Mills theory /eneral Relativity 0oincaré gauge theory *3,. Curvature tensors rotational curvature translational curvature % Riemann–Cartan geometry ! is a $our!dimensional mani$old) 4hose torsion tensor and curvature tensor satis$y (he 6rst =ianchi identity links dynamical properties o$ torsion to algebraic properties o$ curvature. → In the presence of non!vanishing torsion) the curvature tensor has diferent algebraic properties. %nalyze this. *3,. :oung decomposition o$ a general ran"!p tensor =ased on Achur–Weyl duality that lin"s representations o$ An and /B(4) R8) see literature. Here) is the C!th allo4ed :oung diagram) and is the "!th :oung tableau2 of the :oung diagram . Bastly) denotes the :oung symmetrizer associated 4ith a certain :oung tableau2. (his decomposition is block diagonal in the sense that . → Bet us apply this to the Riemann tensor o$ a #4 geometry 4ith curvature and torsionD E3,. :oung decomposition o$ the Riemann curvature tensor 7,3*8 Aymmetries o$ the Riemann tensor: F 9 double *!$orm: (algebraic curvature tensor8 EG 9 =ianchi identity (if torsion vanishes8 ,G 9 implications: (if torsion vanishes8 ,H I , :oung decomposition of the Riemann tensor ( ): Weyl tensor “paircom” tensor pseudoscalar symmetric tracefree Ricci tensor antisymmetric Ricci tensor Ricci scalar 4/10 :oung decomposition o$ the Riemann curvature tensor 7*3*8 /B74) R) AL7,)E8 J:oung decompositionK JIrreducible decompositionK H3,. %n algebraic superenergy tensor in 0oincaré gauge theory of gravity 7,3E8 (he Bel tensor can be defned in terms o$ the duals o$ the Riemann tensor: (he :oung decomposition is In /eneral Relativity) the Bel–Robinson tensor is constructed analogously $rom the Weyl tensor. It is also related to superenergy: a positive de6nite Muantity $or a timelike observer. Ho4 to generalize to 0oincaré gauge theoryN → Introduce =el trace tensor and subtract traces to de6ne an algebraic Bel–Robinson tensor. G3,. %n algebraic superenergy tensor in 0oincaré gauge theory of gravity 7*3E8 12plicit $orm o$ the decomposition o$ the =el tensor: -3,. %n algebraic superenergy tensor in 0oincaré gauge theory of gravity 7E3E8 (he $ollo4ing is our "nal result: O3,. Some remarks on the =el trace tensor (he =el trace tensor lists ho4 diferent curvature ingredients contribute to traces: 9 In /eneral Relativity) implies . 9 In other theories (diferent Bagrangian) diferent geometry 4ith torsion) ...8) the vacuum 6eld equations may impose other constraints on the curvature. 9 Lnly the Weyl tensor does not appear in the =el trace tensor. (his is because it is traceless) ) and it also satis6es . Conclusions (he =el trace tensor allo4s us to defne a tensor that has the same algebraic properties as the =el–Robinson tensor. Further 4ork needs to be done: 9 Would a spinorial treatment give rise to a deeper algebraic understandingN 9 What about diferential properties o$ the algebraic =el–Robinson tensorN #han$ you for your attention% +3,. Lutloo": $urther decompositions in $our dimensions AL7,)E8 /B(4,R8 R ( /B(4,R8 AL7,)E8 X Z Y ZR V W S4 geometry with vanishing torsion. #4 geometry with non-vanishing torsion. ,.3,..