AI 中文总结
该研究针对双参数静态I型真空黑洞，推导其表面电荷一元形式，确定特定能量表达式，明确参数B为几何变形并给出全局力学关系。
AI 中文摘要
我们研究了一个双参数静态I型真空黑洞，该黑洞最近通过非线性解生成映射从电磁种子获得。其参数B改变无量纲视界几何，但固定渐近时间下的Komar质量变化与熵变化不兼容。利用Iyer-Wald、Barnich-Brandt和Lee-Wald构造，我们在整个正则解空间推导了表面电荷一元形式，并证明其具有非零旋度。通过极限Weyl边界的辛通量也测量到了相同的阻碍。静止Killing场的场依赖归一化提供了积分因子；由此得到的能量H=m/(1+B²m²)^(3/2)在该类中被Schwarzschild质量归一化唯一选定。同形缩放使固定G的相空间保持二维，因为其切向量携带非零的精确对称电荷。在Weyl坐标中，有限质量处的空间无穷远是闭合旋转曲面，其Brown-York正则响应逐点再现Lee-Wald通量。无质量背景的半无限环形源作为该曲面的非均匀极限出现。最后，令y=Bm，力学关系取全局形式dM=TdS+Ψ_y dy。因此B标记了真正的几何变形，而其变化在与固定边界时间相关的哈密顿量中表现为外源流功。
英文摘要
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.
