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. 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 F-KN(r, theta; M, a, Q) which characterizes the Kerr-Newman black-hole spacetime [here {r, theta} are respectively the radial and polar coordinates of the curved spacetime and {M, J = Ma, 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 (a) over cap equivalent to a/r(+) [here r(+)(M, a, Q) is the black-hole horizon radius], (a) over cap (-)(crit) = root 3 - 2 root 2 and (a) over cap (+)(crit) = root 5 - 2 root 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 (a) over cap < <(a)over cap>(-)(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 (a) over cap (-)(crit) <= (a) over cap <= (a) over cap (+)(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 (a) over cap > (a) over cap (+)(crit) are characterized by a non-monotonic spatial behavior of the Maxwell electromagnetic invariant F-KN(r = r(+), theta; 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 (a) over cap (2) . (cos(2)theta)(max) = 5 - 2 root 5.