Dynamical Horizons and their Properties

Core claim

Fix two cross-sections $S_1$ and $S_2$ of a dynamical horizon $H$ and let $R$ be the area radius, $a_S = 4πR^2$. Combining the scalar and vector constraints on $H$ and integrating over the portion $ΔH$ bounded by the two cross-sections gives the area balance law $$(R_2 - R_1)/2G = ∫_{ΔH} T_{ab}τ̂^a ξ^b_{(R)} $d^{3}$V + (1/16πG)∫_{ΔH} N_R(|σ|^2 + 2|ζ|^2) $d^{3}$V,$$ where $σ_{ab}$ is the shear of the outgoing null normal and $ζ^a = q̃^{ab} r̂^c ∇_c ℓ_b$ measures how the null normal twists as one moves off the cross-section. The first integral is matter energy flux; the second is proposed as the flux of gravitational energy, and it is manifestly non-negative. The authors argue this term meets the criteria normally used to certify the Bondi flux at null infinity: gauge invariance, positivity, locality, vanishing in spherical symmetry, agreement with perturbation theory off Kerr, a balance-law structure, and a Hamiltonian interpretation. They then introduce angular momentum $J^φ_S = -(1/8πG)∮_S K_{ab}φ^a r̂^b d^2V$ for any vector field $φ^a$ tangent to the cross-sections, obtain a matching balance equation for it, and combine the two into an integral generalization of the first law, $dE_t = (κ̄_r/8πG)da + Ω dJ$. For axi-symmetric horizons they single out a preferred time field whose effective surface gravity and angular velocity are exactly those of the Kerr solution with the same area and angular momentum, so that each cross-section is assigned the mass it would have if it were instantaneously Kerr. A byproduct is a topology result: cross-sections are two-spheres when $Λ>0$ and two-spheres or tori when $Λ=0$, the torus case being degenerate with all fluxes vanishing.