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I'm a physicist and the mathematics around the Hodge Decomposition is way formal than I can currently follow (I'm trying to better myself but it'll take a while).

Specifically what I'm (pragmatically) interested in is the existence of some decomposition of a (co)vector field in a $3+1$ Lorentzian manifold into (gradient of a) scalar and non-scalar parts. I understand that this is not generically possible but I was reading this answer

and a few Einstein-Maxwell papers which imply that in some (not really very physically restrictive) restricted circumstances it is possible.

So my question is what fails in attempting direct generalisation to non-compact Lorentzian manifolds? With the follow up question: is global hyperbolicity enough of a restriction to allow such a generalisation?

Isaac Browne
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user223345
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  • Sorry, can't seem to make new lines work.... I think I've found some answer: in the introduction here http://www.ams.org/journals/distribution/mmj/vol9-4-2009/troyanov.pdf

    Theorem 14.3 of http://link.springer.com/article/10.1007%2FBF02922133

    They state (and prove in the second case) that the hodge decomposition holds for any complete (Riemannian) manifold

     – user223345 Mar 16 '15 at 11:56
  • (1) The Lorentzian and Riemannian cases are vastly different. (2) Igor's answer on the linked MO post gives a pretty good discussion. What else exactly are you looking for? – Willie Wong Mar 18 '15 at 11:21

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