A covariant framework reveals non-closed scalar charges with bulk contributions in ESGB black holes that become closed under shift symmetry and interpret spontaneous scalarization via the Smarr formula.
Bekenstein,Exact solutions of Einstein conformal scalar equations,Annals Phys.82(1974) 535
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Non-closed scalar charge in four-dimensional Einstein-scalar-Gauss-Bonnet black hole thermodynamics
A covariant framework reveals non-closed scalar charges with bulk contributions in ESGB black holes that become closed under shift symmetry and interpret spontaneous scalarization via the Smarr formula.