Some Remarks on Gravitational Analogs of Magnetic Charge

TL;DR

This paper explores the classification of closed p-forms derived from Lorentzian metrics on manifolds, revealing that only forms from cohomology classes or specific (n-1)-forms generalize the conservation of kink number, with implications for metric-dependent forms.

Contribution

It demonstrates that the only closed, non-exact forms are from cohomology classes or specific (n-1)-forms related to kink number conservation, and shows these forms cannot be constructed from metric and curvature derivatives.

Findings

Only cohomology class representatives generate non-exact closed forms.

Kink number is represented by (n-1)-forms depending on the metric's diffeomorphism class.

No finite-order metric, curvature, or derivative-based form can represent the kink number cohomology class.

Abstract

Existing mathematical results are applied to the problem of classifying closed $p$-forms which are locally constructed from Lorentzian metrics on an $n$-dimensional orientable manifold $M$ ($0<p<n$). We show that the only closed, non-exact forms are generated by representatives of cohomology classes of $M$ and $(n-1)$-forms representing $n$-dimensional (with $n$ even) generalizations of the conservation of ``kink number'', which was exhibited by Finkelstein and Misner for $n=4$. The cohomology class that defines the kink number depends only on the diffeomorphism equivalence class of the metric, but a result of Gilkey implies that there is no representative of this cohomology class which is built from the metric, curvature and covariant derivatives of curvature to any finite order.