Algebraic and differential Rainich conditions for symmetric trace-free tensors of higher rank

TL;DR

This paper develops algebraic and differential Rainich-like conditions for symmetric trace-free tensors of higher rank, characterizing when they represent superenergy tensors of specific field solutions in vacuum spacetimes.

Contribution

It introduces necessary and sufficient differential conditions for higher-rank tensors to satisfy source-free field equations, extending Rainich theory.

Findings

For even rank tensors, a differential condition characterizes source-free fields.

For rank 4, combined algebraic and differential conditions identify superenergy tensors of Weyl candidates.

For rank 3, conditions characterize superenergy tensors of massless spin 3/2 fields.

Abstract

We study 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 obtain 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.