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

The Maxwell invariant plays a fundamental role in the mathematical description of electromagnetic fields in charged spacetimes. We present a detailed {\it analytical} study of the physical and mathematical properties of the Maxwell electromagnetic invariant ${\mathcal{F}}_{\text{KN}}(r,\theta ;M,a,Q)$ which characterizes the Kerr-Newman black-hole spacetime. 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 $\widehat{a}\equiv a\mathrm{/}{r}_{+}$, ${\widehat{a}}_{\text{crit}}^{-}=\sqrt{3-2\sqrt{2}}$ and ${\widehat{a}}_{\text{crit}}^{+}=\sqrt{5-2\sqrt{5}}$, 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 $\widehat{a}<{\widehat{a}}_{\text{crit}}^{-}$ regime are characterized by negative definite Maxwell electromagnetic invariants that increase monotonically towards spatial infinity, (ii) for black holes in the intermediate spin regime ${\widehat{a}}_{\text{crit}}^{-}\le \widehat{a}\le {\widehat{a}}_{\text{crit}}^{+}$, 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 $\widehat{a}>{\widehat{a}}_{\text{crit}}^{+}$ are characterized by a non-monotonic spatial behavior of the Maxwell electromagnetic invariant along the black-hole horizon with a spin and charge dependent global maximum whose polar angular location is characterized by the dimensionless functional relation ${\widehat{a}}^2\cdot ({\cos }^2\theta {)}_{\text{max}}=5-2\sqrt{5}$.