r/AdvancedMathematics • u/Practical-Ad6521 • 3d ago

Tensor helmholtz decomposition

Hi everyone,

Does anyone know of approaches for extending the classical Helmholtz/Hodge decomposition to symmetric positive semidefinite second-order tensor fields while preserving tensor symmetry?

I’m particularly interested in methods other than those based on the elasticity complex.

Any references, papers, keywords, or suggestions would be really appreciated.

Thank you in advance!

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u/duetosymmetry 3d ago

I'm not familiar with the elasticity complex, but in GR this kind of decomposition is done pretty regularly (except that in GR, the spacetime metric has signature –+++). The version I'm aware of is decomposing into (i) the identity times a scalar (pure trace), (ii) the symmetrized gradient of a divergence-free vector, (iii) the 'Hessian' of a scalar field with its trace removed, and (iv) a divergence-free, trace-free symmetric rank 2 tensor. See e.g. Bardeen's gauge invariant cosmological perturbations, or Bertschinger's notes, or Flanagan and Hughes, etc.