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Definition 1 ([Admissible purely electric exterior]). Let Ω be a connected oriented smooth n -manifold with nonempty compact smooth boundary S . The metric g , symmetric covariant two-tensor K , and vector field E are smooth up to S . The …

Definition 1 ([Admissible purely electric exterior]). Let Ω be a connected oriented smooth n -manifold with nonempty compact smooth boundary S . The metric g , symmetric covariant two-tensor K , and vector field E are smooth up to S . The metric space (Ω,g) , including S , is complete. Outside a compact set there is exactly one coordinate end Ω C≃x∈ⁿ:|x|>R_0. In particular the remaining topology is compact; there are no additional ends. Put τ=_gK and define 2μ_m, =R_g+τ²-|K|_g²-k(k-1)|E|_g²,; (J_m)_i, =∇^j(K_ij -τg_ij ). We require μ_m≥|J_m|_g, _gE=0, μ_m, |J_m|_g∈L¹(Ω, V_g). For some q>(n-2)/2 , with ρ=|x| , assume g_ij -δ_ij =O₂(ρ^-q ), K_ij =O₁(ρ^-1-q ), Eⁱ=O₁(ρ^-k ). Here O_j(ρ^-a ) bounds every coordinate derivative of order ℓ≤j by C_ℓρ^-a-ℓ ; it places no restriction on higher derivatives beyond smoothness. The following finite limits are assumed to exist: E_ADM , =1 2kω _R→∞ ∫_ρ=R (∂_{jg_} ij -∂_{ig_} jj ) (n_δ)ⁱ\, A_δ,; ( P_ADM )_i , =1 kω _R→∞ ∫_ρ=R (K_ij -τg_ij )(n_δ)^j\, A_δ, ; Q, =1ω_R→∞ ∫_ρ=R g(E,ν_R)\, A_g. The ADM fluxes use Euclidean coordinate normals and area, whereas the charge flux uses g . No timelikeness assumption is imposed on the ADM vector. On each connected component S_a choose a fixed sign ε_a∈1,-1 and require H_Sa +ε_{a_} S_a K≤0, H_Sa =_Sa ν. No condition is placed on the other expansion. The signs need not agree between components.