Analytic study of the Maxwell electromagnetic invariant in spinning and charged Kerr-Newman black-hole spacetimes

Abstract

The Maxwell invariant plays a fundamental role in the mathematical description of electromagnetic fields in charged spacetimes. In particular, it has recently been proved that spatially regular scalar fields which are non-minimally coupled to the Maxwell electromagnetic invariant can be supported by spinning and charged Kerr-Newman black holes. Motivated by this physically intriguing property of asymptotically flat black holes in composed Einstein-Maxwell-scalar field theories, we present a detailed analytical study of the physical and mathematical properties of the Maxwell electromagnetic invariant $$ {\mathcal{F}}_{\textrm{KN}}\left(r,\theta; M,a,Q\right) $$ F KN r θ M a Q which characterizes the Kerr-Newman black-hole spacetime [here {r, θ} are respectively the radial and polar coordinates of the curved spacetime and {M, J = M a, Q} are respectively the mass, angular momentum, and electric charge parameters of the black hole]. It is proved that, for all Kerr-Newman black-hole spacetimes, the spin and charge dependent minimum value of the Maxwell electromagnetic invariant is attained on the equator of the black-hole surface. Interestingly, we reveal the physically important fact that Kerr-Newman spacetimes are characterized by two critical values of the dimensionless rotation parameter $$ \hat{a}\equiv a/{r}_{+} $$ a ̂ ≡ a / r + [here r+ (M, a, Q) is the black-hole horizon radius], $$ {\hat{a}}_{\textrm{crit}}^{-}=\sqrt{3-2\sqrt{2}} $$ a ̂ crit − = 3 − 2 2 and $$ {\hat{a}}_{\textrm{crit}}^{+}=\sqrt{5-2\sqrt{5}} $$ a ̂ crit + = 5 − 2 5 , which mark the boundaries between three qualitatively different spatial functional behaviors of the Maxwell electromagnetic invariant: (i) Kerr-Newman black holes in the slow-rotation $$ \hat{a}<{\hat{a}}_{\textrm{crit}}^{-} $$ a ̂ < a ̂ crit − regime are characterized by negative definite Maxwell electromagnetic invariants that increase monotonically towards spatial infinity, (ii) for black holes in the intermediate spin regime $$ {\hat{a}}_{\textrm{crit}}^{-}\le \hat{a}\le {\hat{a}}_{\textrm{crit}}^{+} $$ a ̂ crit − ≤ a ̂ ≤ a ̂ crit + , the positive global maximum of the Kerr-Newman Maxwell electromagnetic invariant is located at the black-hole poles, and (iii) Kerr-Newman black holes in the super-critical regime $$ \hat{a}<{\hat{a}}_{\textrm{crit}}^{+} $$ a ̂ < a ̂ crit + are characterized by a non-monotonic spatial behavior of the Maxwell electromagnetic invariant $$ {\mathcal{F}}_{\textrm{KN}}\left(r={r}_{+},\theta; M,a,Q\right) $$ F KN r = r + θ M a Q along the black-hole horizon with a spin and charge dependent global maximum whose polar angular location is characterized by the dimensionless functional relation $$ {\hat{a}}^2 $$ a ̂ 2 · (cos2θ)max = 5 – $$ 2\sqrt{5} $$ 2 5 .