Meissner effect in Kerr–Bertotti–Robinson spacetime

Abstract 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 $$\Phi _N=-2\pi Ba^2\sqrt{1-B^2a^2}$$ Φ N = - 2 π B a 2 1 - B 2 a 2 , which is nonzero and maximal in magnitude at $$Ba=1/\sqrt{2}$$ B a = 1 / 2 . Complete expulsion occurs only in the static near-horizon boundary limit $$Ba\rightarrow 1^-$$ B a → 1 - , equivalently $$k\rightarrow 0$$ 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, $$\Omega _x|_{r_e}=0$$ Ω x | r e = 0 and $$\Omega _r|_{r_e}=B^2a$$ Ω r | r e = B 2 a , both consequences of the double-root structure of the horizon function $$\Delta $$ Δ . Together they force the azimuthal potential $$A_\phi |_{r_e}$$ 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 $$F_{x\phi }|_{r_e}$$