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In this note we analyze the BPS black hole equations in extendedsupergravities and we find two interesting relations involving first and secondderivatives of combinations of the central charges. One relation is a newidentity that solely relies on the geometric properties of the scalar manifoldsof extended supergravity theories. The other relation is a generalization of ascalar weak gravity conjecture relation conjectured by Palti and usesproperties of the underlying black hole solution. We also provide for the firsttime an explicit covariant construction of the BPS squared action for suchsolutions.

Covariant formulation of BPS black holes and the scalar weak gravity conjecture