Positive smooth horizon area and the exact degenerate angular-momentum test

Work in ZFC with The Axiom of Choice. In particular The Axiom of Countable Choice (AC_ω) holds: apply choice to the family of nonempty sets in a sequence and compose the choice function with that sequence. Countable choice is the inherited hypothesis of the chart-partition integration in [F2] and [F3].

Horizon, stationarity, axisymmetry and degeneracy terminology is that of Staticity, axisymmetry and black-hole regularity hypotheses. Smoothness, compactness, nonemptiness and spacelikeness of the two-dimensional cross-section $S$ are explicit hypotheses here. Its smooth induced positive-definite tensor $h$ is a Riemannian metric in Riemannian metric and riemannian manifold.

The density Riemannian volume density is ${\mu }_h=\sqrt{\det (h_{ab})}\mathrm{\mid }d{x}^{1}d{x}^2\mathrm{\mid }$ in a surface chart. By The riemannian volume density is coordinate independent these formulas glue to a positive smooth density independent of the chart.

Use Riemannian volume of a compactly supported smooth density with $f\equiv 1$ on the compact surface to define $A={\int }_S{\mu }_h$. Under [A1], Orientation-free density integration and its properties makes this integral independent of charts and partition, nonnegative, and strictly positive for every nonzero nonnegative smooth density. The compact-support construction is a finite sum of finite chart integrals and requires no orientation.

For the second assertion, the extra data are a specified real scalar $J$ attached to this component and the assumed equality $A=8\pi \mathrm{\mid }J\mathrm{\mid }$, with $A$ defined in [F3]. In the intended degenerate stationary axisymmetric vacuum application, take the charge $J_S$ defined in [F5] for this same section, with the equality in that normalization supplied independently. The lemma uses only this scalar-data contract and the equality; the area–angular-momentum relation remains a premise.

For the application terminology, vacuum is the geometric predicate $\mathrm{Ric}(g)=0$ near $S$, with the Ricci tensor of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. Use the chosen $2\pi$ axial generator $Z$ of Staticity, axisymmetry and black-hole regularity hypotheses. Specify an orientation of its Lorentzian neighborhood and an orientation of $S$ for this charge. Metric duality The inverse metric, musical isomorphisms and index raising gives $Z^{\mathrm{♭}}=g(Z,\cdot )$, and Lorentzian exterior calculus and Hodge star gives the smooth two-form ${\star }_{g}d{Z}^{\mathrm{♭}}$. With inclusion $j:S\hookrightarrow M$, define $J_S:=(16\pi {)}^{-1}{\int }_S{j}^{\ast }({\star }_{g}d{Z}^{\mathrm{♭}})$ using Integration on an oriented embedded submanifold. Reall (8.25) fixes the integrand and factor in units $G=c=1$; the finite-section functional is defined here. Its orientations and axial scale are supplied data. No equality to a charge at infinity, independence of cross-section, or derivation of the area relation is asserted by this definition.