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

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.

New version matches published version in Physical Review D

References in corpus (47)

• Black Hole Entropy is Noether Charge
• Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy
• Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes
• Flux Compactification
• Flux compactifications in string theory: a comprehensive review
• Dilatonic Black Holes in Higher Curvature String Gravity
• A General Definition of "Conserved Quantities" in General Relativity and Other Theories of Gravity
• Spontaneous scalarization of black holes and compact stars from a Gauss-Bonnet coupling
• New Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories
• Black hole hair in generalized scalar-tensor gravity
• Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories
• Black holes in scalar-tensor gravity
• Dressing a black hole with a time-dependent Galileon
• Black hole hair in generalized scalar-tensor gravity: An explicit example
• Dynamical Horizons and their Properties
• A no-hair theorem for the galileon
• Smarr Formula and an Extended First Law for Lovelock Gravity
• Black-Hole Solutions with Scalar Hair in Einstein-Scalar-Gauss-Bonnet Theories
• Eluding the no-hair conjecture: Black holes in spontaneously broken gauge theories
• On the stability of scalarized black hole solutions in scalar-Gauss-Bonnet gravity
• The Swampland Distance Conjecture for Kahler moduli
• The first law for slowly evolving horizons
• Spontaneous scalarization
• Black hole solutions with dilatonic hair in higher curvature gravity
• On the Possibility of Large Axion Moduli Spaces
• The Refined Swampland Distance Conjecture in Calabi-Yau Moduli Spaces
• Black hole hair formation in shift-symmetric generalised scalar-tensor gravity
• Post-Newtonian dynamics and black hole thermodynamics in Einstein-scalar-Gauss-Bonnet gravity
• Black hole entropy and Lorentz-diffeomorphism Noether charge
• Black Hole Solutions in String Theory with Gauss-Bonnet Curvature Correction
• Black-hole ringdown as a probe of higher-curvature gravity theories
• Scalar charges and the first law of black hole thermodynamics
• Black hole scalar charge from a topological horizon integral in Einstein-dilaton-Gauss-Bonnet gravity
• The first law and Wald entropy formula of heterotic stringy black holes at first order in alpha prime
• Smarr Formula for Lovelock Black Holes: a Lagrangian approach
• Komar integrals for theories of higher order in the curvature and black-hole chemistry
• The first law of heterotic stringy black hole mechanics at zeroth order in alpha prime
• On scalar charges and black-hole thermodynamics
• Black hole chemistry, the cosmological constant and the embedding tensor
• Komar integral and Smarr formula for axion-dilaton black holes versus S duality
• ${\alpha }^{\prime }$-corrections and their double formulation
• Gravitational-wave constraints on scalar-tensor gravity from a neutron star and black-hole binary GW200115
• Thermodynamics sheds light on black hole dynamics
• Holographic renormalization of Horndeski gravity
• Thermodynamically Stable Phases of Asymptotically Flat Lovelock Black Holes
• No logarithmic corrections to entropy in shift-symmetric Gauss-Bonnet gravity
• Gauss-Bonnet AdS planar and spherical black hole thermodynamics and holography