Conservation laws on riemann-cartan and lorentzian spacetimes

Abstract

In this chapter we examine in details the conditions for existence of conservation laws of energy-momentum and angular momentum for the matter fields of on a general Riemann-Cartan spacetime (M, η, ∇, τη, ↑) and also in the particular case of Lorentzian spacetimes M = (M, η, D, τη, ↑) which as we already know model gravitational fields in the GRT [3]. Riemann-Cartan spacetimes are supposed to model generalized gravitational fields in so called Riemann-Cartan theories. In what follows, we suppose that in (M, η, ∇, τη, ↑) (or M) a set of dynamic fields live and interact. Of course, we want that the Riemann-Cartan spacetime admits spinor fields, and from what we learned in Chap. 7, this implies that the orthonormal frame bundle must be trivial. This permits a great simplification in our calculations. Moreover, we will suppose, for simplicity that the dynamic fields ϕA, A = 1, 2,…, n, are in general distinct r-forms,1 i.e., each ϕA ∈ sec Λr T*M → Cℓ(M, g), for some r = 0,1,…4. Before we start our enterprise we think it is useful to recall some results which serve also the purpose to fix the notation for this chapter.