Algebraic Rainich conditions for the tensor V

TL;DR

This paper explores algebraic Rainich conditions for the tensor V, showing it shares properties with the Bel-Robinson tensor and identifying conditions under which these tensors form a basis for energy expressions in gravitational theories.

Contribution

The paper introduces the tensor V as a new candidate satisfying Rainich-like conditions, expanding the understanding of algebraic properties of fourth rank tensors in gravitational energy expressions.

Findings

Tensor V satisfies Rainich conditions like the Bel-Robinson tensor.

Only two tensors, B and V, form a basis for good energy expressions in the small sphere limit.

Algebraic Rainich conditions require tensors to be totally trace free or causal.

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 $B_{\alpha\beta\mu\nu}$, 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 $B_{\alpha\beta\mu\nu}$ fulfill these conditions, but so also does our recently proposed tensor $V_{\alpha\beta\mu\nu}$, which has many of the desirable properties of $B_{\alpha\beta\mu\nu}$. For the quasilocal small sphere limit restriction, we found that there are only two fourth rank tensors $B_{\alpha\beta\mu\nu}$ and $V_{\alpha\beta\mu\nu}$ 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.