Non-closed scalar charge in 4-dimensional Einstein-scalar-Gauss-Bonnet black hole thermodynamics

TL;DR

This work develops a covariant, differential-form framework to define scalar charges for stationary black holes in $4$-dimensional EsGB gravity with a general coupling $f(\phi)$, identifying a bulk obstruction $\mathcal{W}_k$ that renders the contracted scalar charge non-closed in general. In the shift-symmetric case, $\mathcal{W}_k=0$ and the scalar charge obeys a Gauss law, linking it to boundary data and topological results; for general couplings a bulk term enters the Smarr formula and the generalized Komar charge, yielding a richer thermodynamic structure. The analysis provides a covariant interpretation of spontaneous scalarization as the dynamical generation of scalar charge via the obstruction term, connecting horizon data, asymptotic charges, and bulk contributions. Special couplings (linear and dilatonic) illustrate when bulk terms vanish and how the Smarr relation reduces to a surface-term description, while general couplings reveal a controlled breakdown of purely boundary-determined charges. The framework offers a unified geometric understanding of scalar charges, black-hole thermodynamics, and scalarization in EsGB gravity with potential extensions to rotation and holography.

Abstract

We develop a covariant differential-form framework to define scalar charges for stationary, asymptotically flat black holes in $4$--dimensional Einstein-scalar-Gauss-Bonnet gravity with a general scalar coupling function. Contracting the scalar field equation of motion with the horizon generator $k$ yields a non-closed-form scalar charge, revealing a bulk contribution encoded in a $3$--form, which measures the obstruction to its closedness. In the presence of shift-symmetry, this obstruction vanishes and the $2$--form scalar charge satisfies a Gauss law, depending solely on boundary data. Geometrically, this reproduces known topological results in the shift-symmetric limit. This framework allows us to analyze the role of the non-closed scalar charges in black hole thermodynamics through the Smarr formula for more general couplings and provide a covariant, charge-based interpretation of the spontaneous scalarization mechanism, showing how the behavior of the scalar charge and the bulk term capture the instability of scalar-free black holes and the emergence of scalar hair. Our results offer a unified geometric understanding of the role of scalar charges and the mechanism of spontaneous scalarization in Einstein-scalar-Gauss-Bonnet gravity.