Hair or a Boundary Source? Covariant Phase Space of a Static Type-I Black Hole

Abstract

We study the two-parameter static type-I vacuum black hole recently obtained from an electromagnetic seed by a nonlinear solution-generating map. Its parameter $B$ changes dimensionless horizon geometry, yet the Komar mass variation at fixed asymptotic time is incompatible with the entropy variation. Using the Iyer--Wald, Barnich--Brandt, and Lee--Wald constructions, we derive the surface-charge one-form throughout the regular solution space and show that it has a nonzero curl. The same obstruction is measured by symplectic flux through the limiting Weyl boundary. A field-dependent normalization of the stationary Killing field supplies an integrating factor; the resulting energy $H=m/(1+B^2m^2)^{3/2}$ is uniquely selected by Schwarzschild mass normalization within this class. Homothetic scaling leaves the fixed-$G$ phase space two-dimensional because its tangent carries nonzero exact-symmetry charges. In Weyl coordinates, spatial infinity at finite mass is a closed surface of revolution. Its Brown--York canonical response reproduces the Lee--Wald flux pointwise. The semi-infinite annular source of the massless background emerges as a nonuniform limit of this surface. Finally, with $y=Bm$, the mechanics takes the global form $\text{d} M=T\text{d} S+Ψ_y\text{d} y$. Thus $B$ labels a genuine geometric deformation while its variation acts as external-source work in the Hamiltonian associated with fixed boundary time.