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In an axisymmetric stationary spacetime, the horizon angular velocity is the constant coefficient in the null horizon generator $\chi =k+{\Omega }_{H}m$, where $k$ is the stationary Killing vector normalized at infinity and $m$ generates axial orbits with period $2\pi$. It measures rotation relative to the stationary frame at infinity. For a Kerr black hole, ${\Omega }_H=a/(r_{+}^2+a^2)$. It enters the rotating physical-process first law through the horizon flux of $E-{\Omega }_{H}J$.

A regular hypersurface $f=0$, with $df\ne 0$, is a null hypersurface if its normal covector has zero squared norm:Its normal vector is then also tangent to the hypersurface; the induced metric tensor is degenerate.

For the Kerr metric the normal to a constant-$r$ surface has squared normFor $0<a<M$ the two roots are positive and distinct. To verify that they are genuine regular null hypersurfaces, rather than trusting singular Boyer-Lindquist coordinates, introduce ingoing Kerr coordinatesSubstitution givesThere are no ${\Delta }^{-1}$ terms. At either root $\Sigma >0$, and the determinant is $-{\Sigma }^2{\sin }^2\theta$ away from the angular coordinate poles. Thus the metric tensor extends smoothly across each root. The raised normal on either surface isThis is a nonzero null Killing vector field normal to the surface. Both surfaces are Killing horizons, with horizon angular velocities ${\Omega }_{\pm }$.

The event-horizon terminology in the paper needs qualification. In the usual asymptotically flat Kerr black hole extension, relative to a chosen exterior's future null infinity, the future branch of $r=r_{+}$ is the event horizon, the boundary $\partial J^{-}({\mathcal{I}}^{+})$. In the intervening black-hole region $\Delta <0$, the coordinate $r$ has timelike gradient and future-directed curves move toward smaller $r$; they cannot return across the outer boundary to that exterior. The inner surface $r=r_{-}$ is a Cauchy horizon, not a second event horizon for the same asymptotic end. Nullness is a local property; identifying an event horizon additionally requires this global causal structure.^[ref]

The allowed endpoint $a=0$ provides an especially concrete counterexample to the unrestricted description of both roots. The Kerr metric then becomes Schwarzschild spacetime, $r_{+}=2M$ is its regular event horizon, but $r_{-}=0$ is a curvature singularity:It is not a regular inner null hypersurface. Thus the two-horizon calculation above assumes $0<a<M$.

Use geometrized units $G=c=1$. The Physical-process first law of black-hole mechanics concerns a small perturbation of a rotating black hole in an initially stationary spacetime by infalling matter, followed by relaxation to another stationary spacetime. For an uncharged black hole, or neutral accretion with charge held fixed, its first-order statement isHere $\kappa >0$ is the background surface gravity, ${\Omega }_H$ is the background horizon angular velocity, and $\delta A$ is the change in event horizon area. The energy $\delta E$ and angular momentum $\delta J$ are the fluxes carried into the event horizon; when these account for the changes of the black hole, they equal $\delta M$ and its change of angular momentum. The result is a linearized law, not an exact identity for a finite violent accretion event. The perturbation must be small enough that focusing does not produce new caustics or invalidate the expansion about the existing generators.

Let $k^a$ be the stationary Killing vector field normalized at infinity, $m^a$ the axial Killing vector field with $2\pi$-periodic orbits, andOn the background Killing horizon, ${\chi }^a$ is its future-directed null generator and satisfies ${\chi }^b{\nabla }_b{\chi }^a=\kappa {\chi }^a$. Choose a parameter $v$ with $\chi ={\partial }_v$. The null expansion in this parameter is $\theta ={\partial }_v\log dA$. The generator is not affinely parametrized; if $\ell ={\partial }_{\lambda }$ is affine, then $\chi =\kappa \lambda \ell$ after an appropriate choice of affine origin, with $\lambda \propto e^{\kappa v}$. Rescaling the Null Raychaudhuri equation givesThe null twist vanishes because the generators are normal to the event horizon. By the Einstein field equations, $R_{ab}{\chi }^a{\chi }^b=8\pi T_{ab}{\chi }^a{\chi }^b$. Background null expansion and null shear vanish on a Killing horizon of the background stationary spacetime. Thus $\theta$ and $\hat{\sigma }$ are first order, their squares are second order, and the linearized equation isThe final stationary spacetime supplies the boundary condition $\theta \to 0$ in the future. Explicitly,This future boundary condition reflects the global definition of the event horizon. For localized infall, $\theta$ also tends to zero in the remote past. Integrating the linearized equation along the generators, and then over a background cross-section with area element $d{A}_0$, givesUsing the background area element introduces only second-order corrections.

With orientations chosen so that future infalling positive energy has positive flux, the conserved Killing energy and axial angular momentum giveConsequently $\delta E-{\Omega }_H\delta J={\int }_{\mathcal{H}}\delta T_{ab}{\chi }^a{\chi }^{b}dvd{A}_0$, proving the displayed Physical-process first law of black-hole mechanics. Restoring $G$ replaces $8\pi$ by $8\pi G$. The null energy condition makes the integral nonnegative and therefore implies area increase in this regime; that condition is needed for the sign conclusion, not for the linearized flux identity itself. Charged infall would require the additional electrostatic work term, and an extremal background needs separate treatment because the proof uses $\kappa >0$.