Algebraic Rainich conditions for the fourth rank tensor V

Algebraic conditions on the Ricci tensor in the Rainich-Misner-Wheeler unified field theory are known as the Rainich conditions. Penrose and more recently Bergqvist and Lankinen made an analogy from the Ricci tensor to the Bel-Robinson tensor Bαβμν, a certain fourth rank tensor quadratic in the Weyl curvature, which also satisfies algebraic Rainich-like conditions. However, we found that not only does the tensor Bαβμν fulfill these conditions, but so also does our recently proposed tensor Vαβμν, which has many of the desirable properties of Bαβμν. For the quasilocal small sphere limit restriction, we found that there are only two fourth rank tensors, Bαβμν and Vαβμν, which form a basis for good energy expressions. Both of them have the completely trace free and causal properties, these two form necessary and sufficient conditions. Surprisingly either completely traceless or causal is enough to fulfill the algebraic Rainich conditions.