Algebraic Rainich conditions for the tensor V

Abstract

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 $B_{αβμν}$, 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 $B_{αβμν}$ fulfill these conditions, but so also does our recently proposed tensor $V_{αβμν}$, which has many of the desirable properties of $B_{αβμν}$. For the quasilocal small sphere limit restriction, we found that there are only two fourth rank tensors $B_{αβμν}$ and $V_{αβμν}$ 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. Furthermore, relaxing the quasilocal restriction and considering the general fourth rank tensor, we found two remarkable results: (i) without any symmetry requirement, the algebraic Rainich conditions only require totally trace free; (ii) with a symmetry requirement, we recovered the same result as in the quasilocal small sphere limit.