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 FKNrθMaQ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{F}}_{\ extrm{KN}}\\left(r,\ heta; M,a,Q\\right) $$\\end{document} 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 â≡a/r+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{a}\\equiv a/{r}_{+} $$\\end{document} [here r+ (M, a, Q) is the black-hole horizon radius], âcrit−=3−22\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\hat{a}}_{\ extrm{crit}}^{-}=\\sqrt{3-2\\sqrt{2}} $$\\end{document} and âcrit+=5−25\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\hat{a}}_{\ extrm{crit}}^{+}=\\sqrt{5-2\\sqrt{5}} $$\\end{document}, 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 â<âcrit−\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{a}<{\\hat{a}}_{\ extrm{crit}}^{-} $$\\end{document} regime are characterized by negative definite Maxwell electromagnetic invariants that increase monotonically towards spatial infinity, (ii) for black holes in the intermediate spin regime âcrit−≤â≤âcrit+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\hat{a}}_{\ extrm{crit}}^{-}\\le \\hat{a}\\le {\\hat{a}}_{\ extrm{crit}}^{+} $$\\end{document}, 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 â<âcrit+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{a}<{\\hat{a}}_{\ extrm{crit}}^{+} $$\\end{document} are characterized by a non-monotonic spatial behavior of the Maxwell electromagnetic invariant FKNr=r+θMaQ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{F}}_{\ extrm{KN}}\\left(r={r}_{+},\ heta; M,a,Q\\right) $$\\end{document} along the black-hole horizon with a spin and charge dependent global maximum whose polar angular location is characterized by the dimensionless functional relation â2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\hat{a}}^2 $$\\end{document} · (cos2θ)max = 5 – 25\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 2\\sqrt{5} $$\\end{document}.