# What fields have known Rainich conditions?

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As it is known, in 1925 Rainich published necessary and sufficient conditions to be satisfied by the Ricci tensor, in order to correspond via Einstein's equation to the stress-energy tensor of a source-free electromagnetic field. These conditions were rediscovered by Misner and Wheeler, in the so called "already unified theory". Also, there are known similar conditions for the Weyl neutrino field. Some results were obtained for Yang-Mills.

Are there results for other fields, containing similar conditions like those of Rainich, that is, necessary and sufficient conditions that ensure that the Ricci tensor comes from the stress-energy tensor of that field?

asked Jan 18, 2016

Could you provide a reference for the Rainich conditions you allude to in your last sentence (ie the one for Yang-Mills fields). I figure you meant more general non-abelian type yang mills rather than the "easy" abelian example such as EM.

@R.Rankin there's a link in that very sentence, the one that I knew about when I wrote it

## 1 Answer

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Yes, there are generalizations of Rainich conditions to many others fields, including scalar fields, perfect fluids and null electromagnetic fields. (Indeed! The original form of the Rainich conditions were not conveniently formulated for electromagnetic fields $F\in \bigwedge ^2M$ whose both Poincaré invariant vanishes:  $F \wedge \star F = 0$ and $F \wedge F = 0$. This even lead some very good relativists even to doubt that null-electromagnetic fields could be present in electrovacuum solutions of Einstein-Maxwell equations. Cf. for instance this paper by Louis Witten: "Geometry of Gravitation and Electromagnetism" here. Then Peres and Bonnor found some plane-fronted wave solutions to Einstein-Maxwell and showed that they were perfectly consistent.)

A complete review for all these types of Rainich conditions can be found is:

https://arxiv.org/abs/1308.2323

https://arxiv.org/abs/1503.06311

PS: Peres and Bonnor solutions (which describes some interesting coupled system of electromagnetic-gravitational waves) are here:

http://journals.aps.org/pr/abstract/10.1103/PhysRev.118.1105

http://projecteuclid.org/euclid.cmp/1103841572 (free access)

answered Dec 8, 2016 by (540 points)

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