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Algebraic and differential Rainich conditions for symmetric trace-free tensors of higher rank. (English) Zbl 1206.83109
Summary: 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.

MSC:
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
83C40 Gravitational energy and conservation laws; groups of motions
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