Abstract
We consider a Dirac field coupled minimally to the Mielke–Baekler model of gravity and investigate cosmological solutions in three dimensions. We arrive at a family of solutions which exists even in the limit of vanishing cosmological constant.
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Appendix: Irreducible decompositions
Appendix: Irreducible decompositions
In this section we give briefly the irreducible pieces of torsion, contortion and curvature in three dimensions. Firstly torsion which has nine components can be decomposed
where \({}^{(2)}T^a = - \frac{1}{2} (\iota _b T^b) \wedge e^a, {}^{(3)}T^a = \frac{1}{3} \iota ^a (e_b \wedge T^b)\) and \({}^{(1)}T^a = T^a - {}^{(2)}T^a - {}^{(3)}T^a\). In this section the notation with the number under a brace is for the number of components of that part. They have the properties, \({}^{(1)}T^a \wedge e_a = {}^{(2)}T^a \wedge e_a =0\) and \(\iota _a {}^{(1)}T^a = \iota _a {}^{(3)}T^a = 0\). Our choice (25) corresponds to
which means \(4=3\oplus 1\). After the solution (32) we are left only with \({}^{(3)}T^{a}\).
Secondly one can decompose the contortion having nine components
where \({}^{(2)}K_{ab}=\frac{1}{2}[e_a \wedge (\iota ^cK_{cb}) - e_b \wedge (\iota ^cK_{ca})], {}^{(3)}K_{ab}=-\frac{1}{6}\iota _{ab} (K_{cd}\wedge e^{cd})\) and \({}^{(1)}K_{ab} = K_{ab} - {}^{(2)}K_{ab} - {}^{(3)}K_{ab}\). They have the properties \(\iota _a{}^{(1)}K^{ab}=\iota _a{}^{(3)}K^{ab}=0\) and \( {}^{(1)}K_{ab} \wedge e^{ab} = {}^{(2)}K_{ab} \wedge e^{ab}=0\). For our case (26) we possess again \({}^{(1)}K_{ab}=0\), but nonzero \({}^{(2)}K_{ab}\) and \({}^{(3)}K_{ab}\), i.e. \(4=3\oplus 1\). Besides after the solution (32) only \({}^{(3)}K_{ab}\) survives.
Finally one can split the curvature with nine components
where \({}^{(2)}R_{ab}=\frac{1}{2}(e_a \wedge \iota _b - e_b \wedge \iota _a)(e^c \wedge R_c), {}^{(3)}R_{ab}=\frac{1}{6} R e_{ab}\) and \({}^{(1)}R_{ab} = R_{ab} - {}^{(2)}R_{ab} - {}^{(3)}R_{ab}\) with \(R_a = \iota ^bR_{ba}\) and \(R=\iota ^aR_a\). They have the properties \(\iota _{ab}{}^{(1)}R^{ab}=\iota _{ab}{}^{(2)}R^{ab}=0, {}^{(1)}R_{ab} \wedge e^b = {}^{(3)}R_{ab} \wedge e^b=0\) and \(e_b \wedge \iota _a {}^{(1)}R^{ab}=0\). For the solution (32) although all three pieces of curvature and of torsion are nonzero, since \(DT^a =0\) the Ricci tensor is symmetric.
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Sert, Ö., Adak, M. Dirac field in topologically massive gravity. Gen Relativ Gravit 45, 69–78 (2013). https://doi.org/10.1007/s10714-012-1460-2
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DOI: https://doi.org/10.1007/s10714-012-1460-2