Extensions of the Variational Boundary Theorem: Kerr Black Holes and FRW Cosmology

We extend the Variational Boundary Theorem (VBT)—proven in its general geometric form in the companion paper 1 — from the static, spherically symmetric (Schwarzschild) case to two physically important settings: stationary axially symmetric black holes (Kerr) and Friedmann–Robertson–Walker (FRW) cosmology. For Kerr black holes with mass M and angular momentum J = aM, |a| < M, we show that the entanglement map λ : (r+,∞)×S2 → Sp(2N,R)/U(N) reaches the Satake boundary ∂Sat precisely at the outer horizon r = r+, with entropy S = A+/(4ℓ2 P). The key new ingredient is the ZAMO (Zero Angular Momentum Observer) frame, which remains timelike throughout the ergosphere and defines a local temperature diverging as (r − r+)−1/2 — the same exponent as Schwarzschild. Numerical verification confirms this universality for all spin parameters and polar angles tested. For FRW cosmology, two Satake boundaries are identified: the cosmological (de Sitter) horizon,where the Gibbons–Hawking temperature undergoes Tolman blueshift, and the Big Bang singularity at t = 0, where the cosmological temperature T(t) → ∞. The hyperbolicity of the Vacuum Time Geometry field equations, stated as a proposition in 1, is here promoted to a full theorem using the wave map theory of Shatah–Struwe and Tao for targets of nonpositive curvature, combined with the quasilinear framework of Hughes–Kato–Marsden. All proofs are complete.