Eigenvalues
The characteristic polynomial of the Einstein tensor in a minimally coupled massless scalar field solution must have the form
In other words, we have a simple eigvalue and a triple eigenvalue, each being the negative of the other. Multiply out and using Gröbner basis methods, we find that the following three invariants must vanish identically:
Using Newton's identities, we can rewrite these in terms of the traces of the powers. We find that
We can rewrite this in terms of index gymanastics as the manifestly invariant criteria:
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