# Is there a natural relativistic invariant representing gravitational-field strength?

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To naturally measure the strength of the gravitational field, we cannot use $R$ or any Ricci $R_{\mu \nu}$ contraction as that yields zero even in very strong vacuum fields.

The second best candidate would be Kretschmann $K \equiv R^{\mu \nu \kappa \sigma} R_{\mu \nu \kappa \sigma}$ which is non-zero even in vacuum. However, the Kretschmann scalar is not positive definite. Apart from some rather pathological cases the change of sign of the Kretschmann scalar is associated with the switching of "gravitoelectric" and "gravitomagnetic" dominance e.g. in the Kerr space-time. This is true also for the other two usually considered scalars, Chern-Pontryagin and Euler. (Ref.)

We could obviously take $K^2$ or the absolute value $|K|$ but that doesn't change the fact that the value touches zero in extremely strong fields and thus misrepresents the field-strength. I am not really acquainted with the other scalars but it seems to me that at least all the Lovelock invariants will have very similar properties as they must add up to various manifold characteristics under integration over any field.

I have a feeling that the nature of GR with all the local inertial systems etc. does not allow for a strictly local characterization of field strength but instead either a higher-derivative or quasi-local notion is needed. Has anybody in the literature tried to construct such an invariant or at least discuss such a possibility?

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