Meissner effect in Kerr–Bertotti–Robinson spacetime

We study the black-hole Meissner effect for extremal Kerr–Bertotti–Robinson (Kerr–BR) black holes, exact Einstein–Maxwell solutions describing a rotating black hole immersed in a uniform Bertotti–Robinson electromagnetic universe. We show that extremality alone is not sufficient for complete magnetic-flux expulsion in this family. For a purely magnetic external BR field, finite extremal holes with $$0<Ba<1$$ 0 < B a < 1 carry the exact northern-hemisphere flux Φ N=-2π Ba²√1-B²a² Φ N = - 2 π B a 2 1 - B 2 a 2 , which is nonzero and maximal in magnitude at Ba=1/√2 B a = 1 / 2 . Complete expulsion occurs only in the static near-horizon boundary limit Ba→ 1⁻ B a → 1 - , equivalently k→ 0 k → 0 , where the U (1) fibration of the near-horizon metric trivialises. The proof relies on two exact identities that hold at extremality for all values of the external field, Ω ₓ|rₑ=0 Ω x | r e = 0 and Ω ᵣ|rₑ=B²a Ω r | r e = B 2 a , both consequences of the double-root structure of the horizon function Δ Δ . Together they force the azimuthal potential A_φ |rₑ A ϕ | r e in the purely magnetic sector to become independent of the polar angle in the static boundary limit, so that the gauge-invariant horizon flux density Fxφ|rₑ