Hi, I have a few questions, maybe someone can answer some of them?
1. Why is it not possible to classify Einstein spaces (vacuum solutions to the Einstein equation) with the eigenvalues of the metric tensor? Why was it done with the eigenvalues of the Weyl tensor (as a linear operator on a bivector space) instead? I think perhaps the answer is that the eigenvalues of the metric tensor (and their multiplicity) are not invariant under a general coordinate transformation, whereas the ones of the Weyl tensor are? Or at least their multiplicity is invariant (similar to how the number of positive and
negative eigenvalues is an invariant)? But has anyone actually shown this? Couldn't see it from a cursory reading of Petrov's paper.
2. Can the classification be generalized to when (non-gravitational) matter/energy is present?
3. Is there a more general version of the Poincare-Hopf theorem to n-tensor fields, not just vector fields? I saw something in a hydrodynamics journal but it was only for Euclidian 2D spaces and I'm not sure it's correct for the reason in (1) above.

On Friday, August 13, 2021 at 11:45:08 AM UTC-7, iuval wrote:
> Hi, I have a few questions, maybe someone can answer some of them?
> 1. Why is it not possible to classify Einstein spaces (vacuum solutions to the Einstein equation) with the eigenvalues of the metric tensor? Why was it done with the eigenvalues of the Weyl tensor (as a linear operator on a bivector space) instead? I think perhaps the answer is that the eigenvalues of the metric tensor (and their multiplicity) are not invariant under a general coordinate transformation, whereas the ones of the Weyl tensor are? Or at least their multiplicity is invariant (similar to how the number of positive and
> negative eigenvalues is an invariant)? But has anyone actually shown this? Couldn't see it from a cursory reading of Petrov's paper.
> 2. Can the classification be generalized to when (non-gravitational) matter/energy is present?
> 3. Is there a more general version of the Poincare-Hopf theorem to n-tensor fields, not just vector fields? I saw something in a hydrodynamics journal but it was only for Euclidian 2D spaces and I'm not sure it's correct for the reason in (1) above.

Why you no Ricci? There is Kerr.

On Friday, August 13, 2021 at 11:45:08 AM UTC-7, iuval wrote:
> Hi, I have a few questions, maybe someone can answer some of them?
> 1. Why is it not possible to classify Einstein spaces (vacuum solutions to the Einstein equation) with the eigenvalues of the metric tensor?

That's because the metric tensor, being a bilinear map (rank (2,0)), does not have eigenvalues. They are defined for maps of
linear spaces (rank (1,1)). What persists for different (orthogonal) base choices is the number of + and - signs.

> Why was it done with the eigenvalues of the Weyl tensor (as a linear operator on a bivector space) instead?

Because it's a linear operator from a certain (complex 3D) space V to itself.

> I think perhaps the answer is that the eigenvalues of the metric tensor (and their multiplicity) are not invariant under a general coordinate transformation, whereas the ones of the Weyl tensor are?

Yes.

> Or at least their multiplicity is invariant (similar to how the number of positive and
> negative eigenvalues is an invariant)?

Yes.

> But has anyone actually shown this? Couldn't see it from a cursory reading of Petrov's paper.

Has anyone shown what? The constant number of +/- on the diagonal (they are not eigenvalues) is
an old theorem.

> 2. Can the classification be generalized to when (non-gravitational) matter/energy is present?

Not sure what you mean by non-gravitational matter/energy. Can you give an example of such thing?

> 3. Is there a more general version of the Poincare-Hopf theorem to n-tensor fields, not just vector fields? I saw something in a hydrodynamics journal but it was only for Euclidian 2D spaces and I'm not sure it's correct for the reason in (1) above.

I don't know.

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Jan