Second order symmetry operators

Abstract

Using systematic calculations in spinor language, we obtain simple descriptions of the second order symmetry operators for the conformal wave equation, the Dirac-Weyl equation and the Maxwell equation on a curved four-dimensional Lorentzian manifold. The conditions for existence of symmetry operators for the different equations are seen to be related. Computer algebra tools have been developed and used to systematically reduce the equations to a form which allows geometrical interpretation.

Invariant prolongation of the Killing tensor equation

“…where the last equation again follows from (61). Now we look at the second term on the right-hand-side of (59): using (61) we get that Then putting (62) and (63) together, for S BCDE ∈ V * ⊗ V (2,1) , finishes the proof. Now assume that D C is the Thomas D-operator and K DE is symmetric such that (64)…”

“…The following lemma will give a formula for the projection P , when restricted to T ⊗ T (2,1) , i.e., applied to S BCDC ∈ T * ⊗ T (2,1) . Lemma 21.…”

“…Lemma 21. Let P := P (2,2) be the projection of ⊗ 4 T * onto T (2,2) defined above and S BCDC ∈ T * ⊗ T (2,1) . Then For the the third term on the right-hand-side in (59) we compute…”

“…Proposition 22. Let D be the Thomas D-operator for a projective structure with curvature W AB C D and let P be the projection from T * ⊗ T (2,1) to T (2,2) . Then K ∈ Γ(T * (2) ) satisfies D (A K BC) = 0, i.e., D A K BC ∈ T (2,1) , if and only if (2,2) .…”

“…On a Riemannian manifold (M, g) a tangent vector field k ∈ X(M ) is an infinitesimal automorphism (or symmetry) if the Lie derivative of the metric g in direction of k vanishes. In terms of the Levi-Civita connection ∇ = ∇ g , this may be written as (1) ∇ (a k b) = 0…”