We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel-Robinson-type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors.
We present a study of Rainich-like conditions for symmetric and trace-free tensors T. For arbitrary even rank we find a necessary and sufficient differential condition for a tensor to satisfy the source-free field equation. For rank 4, in a generic case, we combine these conditions with previously obtained algebraic conditions to gain a complete set of algebraic and differential conditions on T for it to be a superenergy tensor of a Weyl candidate tensor, satisfying the Bianchi vacuum equations. By a result of Bell and Szekeres, this implies that in vacuum, generically, T must be the Bel-Robinson tensor of the spacetime. For the rank 3 case, we derive a complete set of necessary algebraic and differential conditions for T to be the superenergy tensor of a massless spin-3/2 field, satisfying the source-free field equation.
We study the problem of constructing tensors satisfying the dominant property, a generalization of the dominant energy condition T ab u a v b ≥ 0 for all future directed causal vectors u, v. The construction is done on the paravector subspace of the r-fold Euclidean Clifford algebra ⊗rCℓp and is a generalization of the representation of superenergy tensors with complex 2-spinors. Especially, as with 2-spinors, we are able to construct causal tensors of arbitrary rank, contrary to earlier constructions using tensors or the r-fold Lorentzian Clifford algebra ⊗rCℓp,1 that only produce causal tensors of even rank. An advantage of the construction in ⊗rCℓp is that several algebraic properties become trivial due to the Euclidean norm on it.