Generalizations of the Kerr-Newman solution Contents 1 Topics 663 1.1 ICRANetParticipants. 663 1.2 Ongoingcollaborations. 663 1.3 Students ............................... 663 2 Brief description 665 3 Introduction 667 4 Thegeneralstaticvacuumsolution 669 4.1 Line element and field equations . 669 4.2 Staticsolution ............................ 671 5 Stationary generalization 673 5.1 Ernst representation . 673 5.2 Representation as a nonlinear sigma model . 674 5.3 Representation as a generalized harmonic map . 676 5.4 Dimensional extension . 680 5.5 Thegeneralsolution ........................ 683 6 Static and slowly rotating stars in the weak-field approximation 687 6.1 Introduction ............................. 687 6.2 Slowly rotating stars in Newtonian gravity . 689 6.2.1 Coordinates ......................... 690 6.2.2 Spherical harmonics . 692 6.3 Physical properties of the model . 694 6.3.1 Mass and Central Density . 695 6.3.2 The Shape of the Star and Numerical Integration . 697 6.3.3 Ellipticity .......................... 699 6.3.4 Quadrupole Moment . 700 6.3.5 MomentofInertia . 700 6.4 Summary............................... 701 6.4.1 Thestaticcase........................ 702 6.4.2 The rotating case: l = 0Equations ............ 702 6.4.3 The rotating case: l = 2Equations ............ 703 6.5 Anexample:Whitedwarfs. 704 6.6 Conclusions ............................. 708 659 Contents 7 PropertiesoftheergoregionintheKerrspacetime 711 7.1 Introduction ............................. 711 7.2 Generalproperties ......................... 712 7.2.1 The black hole case (0 < a < M) ............. 713 7.2.2 The extreme black hole case (a = M) .......... 714 7.2.3 The naked singularity case (a > M) ........... 714 7.2.4 The equatorial plane . 714 7.2.5 Symmetries and Killing vectors . 715 7.2.6 The energetic inside the Kerr ergoregion . 716 7.2.7 The particle’s energy and effective potential . 717 7.2.8 Stability of circular orbits and notable radii . 718 7.2.9 Zero-energy particles . 722 7.3 Blackholes.............................. 723 7.3.1 The set BH I : a [0, a˜ ] ................. 723 − ∈ 1 7.3.2 The set BH II : a ]a˜ , a˜ ] ................ 727 − ∈ 1 2 7.3.3 The set BH III : a ]a˜ , M] ............... 729 − ∈ 2 7.3.4 Some final notes on the BH case ............ 729 − 7.4 Nakedsingularities . 730 7.4.1 The set NS I : a ]1, a˜ ] ................. 730 − ∈ 3 7.4.2 The set NS II : a ]a˜ , a˜ ] ................ 734 − ∈ 3 4 7.4.3 The set NS III : a ]a˜ , a˜ ] ............... 737 − ∈ 4 5 7.4.4 The set NS IV : a ]a˜ , +∞] .............. 737 − ∈ 5 7.5 Theextremeblackhole . 737 7.6 Thestaticlimit............................ 740 7.7 Summary of black holes and naked singularities . 742 7.8 Conclusions ............................. 745 8 Charged boson stars 749 8.1 Introduction ............................. 749 8.2 The Einstein-Maxwell-Klein-Gordon equations . 751 8.3 Charge, radius, mass and particle number . 754 8.4 Numericalintegration . 756 8.4.1 Klein-Gordon field and metric functions . 757 8.4.2 Mass, charge, radius and particle number . 760 8.5 Neutralbosonstars ......................... 768 8.6 Conclusions ............................. 772 9 A stationary q metric 781 − 9.1 Introduction ............................. 781 9.2 The q metricanditsproperties. 782 − 9.3 The rotating q metric ....................... 783 − 9.4 Conclusions ............................. 785 660 Contents 10 RepulsivegravityintheKerr-Newmanspacetime 787 10.1 Introduction ............................. 787 10.2 Theeffectivemass. 788 10.3 Curvature invariants and eigenvalues . 790 10.4 Naked singularities with black hole counterparts . 794 10.5 Final outlooks and perspectives . 800 Bibliography 803 661 1 Topics Generalizations of the Kerr-Newman solution • Properties of Kerr-Newman spacetimes • 1.1 ICRANet Participants Donato Bini • Andrea Geralico • Roy P. Kerr • Hernando Quevedo • Jorge A. Rueda • Remo Ruffini • 1.2 Ongoing collaborations Medeu Abishev (Kazakh National University - KazNU, Kazakhstan) • Kuantay Boshkayev (Kazakh National University - KazNU, Kazakhstan) • Calixto Gutierrez (University of Bolivar, Colombia) • Cesar´ S. Lopez-Monsalvo´ (UNAM, Mexico) • Orlando Luongo (University of Naples, Italy) • Daniela Pugliese (Silesian University in Opava, Czech Republic) • 1.3 Students Saken Toktarbay (KazNU PhD, Kazakhstan) • Viridiana Pineda (UNAM PhD, Mexico) • Pedro Sanchez´ (UNAM PhD, Mexico) • 663 2 Brief description One of the most important metrics in general relativity is the Kerr-Newman solution that describes the gravitational and electromagnetic fields of a rotat- ing charged mass. For astrophysical purposes, however, it is necessary to take into account the effects due to the moment of inertia of the object. To attack this problem we have derived exact solutions of Einstein-Maxwell equations which posses an infinite set of gravitational and electromagnetic multipole moments. Several analysis have been performed that investigate the physi- cal effects generated by a rotating deformed mass distribution in which the angular momentum and the quadrupole determine the dominant multipole moments. In this connection, we propose an approximate method based upon the Hartle formalism to study slowly rotating stars in hydrostatic equilibrium in the framework of Newtonian gravity. All the relevant quantities are con- sidered up to the second order in the angular velocity. It is shown that the gravitational equilibrium conditions reduce to a system of ordinary differen- tial equations which can be integrated numerically. Moreover, we find ex- plicitly the total mass of rotating configurations, the moment of inertia, the quadrupole moment, the eccentricity and the equation that relates the mass and the central density of the rotating body. We also investigate in detail the circular motion of test particles on the equatorial plane of the ergoregion in the Kerr spacetime. We find all the regions inside the ergoregion where circular motion is allowed, and ana- lyze their stability properties and the energy and angular momentum of the test particles. We show that the structure of the stability regions has def- inite features that make it possible to distinguish between black holes and naked singularities. The naked singularity case presents a very structured non-connected regions of orbital stability. The properties of the circular or- bits turn out to be so distinctive that they allow the introduction of a complete classification of Kerr spacetimes, each class of which is characterized by dif- ferent physical effects that could be of particular relevance in observational astrophysics. The presence of counterrotating particles and zero angular mo- mentum particles inside the ergoregion of a specific class of naked singulari- ties is interpreted as due to the presence of a repulsive field generated by the central source of gravity. In an attempt to include an arbitrary quadrupole into the Kerr spacetime, we derive a stationary generalization of the static q metric, the simplest gen- eralization of the Schwarzschild solution that contains− a quadrupole param- 665 2 Brief description eter. It possesses three independent parameters that are related to the mass, quadrupole moment and angular momentum. We investigate the geometric and physical properties of this exact solution of Einstein’s vacuum equations, and show that it can be used to describe the exterior gravitational field of ro- tating, axially symmetric, compact objects. To understand the physical properties of the Reissner-Nordstron spacetime in the presence of a scalar field, we study time-independent, spherically sym- metric, self-gravitating systems minimally coupled to a scalar field with U(1) gauge symmetry: charged boson stars. We find numerical solutions to the Einstein-Maxwell equations coupled to the relativistic Klein-Gordon equa- tion. It is shown that bound stable configurations exist only for values of the coupling constant less than or equal to a certain critical value. The metric coefficients and the relevant physical quantities such as the total mass and charge, turn out to be in general bound functions of the radial coordinate, reaching their maximum values at a critical value of the scalar field at the origin. We discuss the stability problem both from the quantitative and qual- itative point of view. We take into account the electromagnetic contribution to the total mass, and investigate the stability issue considering the binding energy per particle. We verify the existence of configurations with positive binding energy in which objects that are apparently bound can be unstable against small perturbations, in full analogy with the effect observed in the mass-radius relation of neutron stars. An interesting physical effect that has been found in Kerr-Newman space- times is that test particles under certain circumstance experience a repulsive force in a region close to the horizons. Repulsive gravity has been investi- gated in several scenarios near compact objects by using different intuitive approaches. Here, we propose an invariant method to characterize regions of repulsive gravity, associated to black holes and naked singularities. Our method is based upon the behavior of the curvature tensor eigenvalues, and leads to an invariant definition of a repulsion radius. The repulsion radius determines a physical region, which can be interpreted as a repulsion sphere, where the effects due to repulsive gravity naturally arise. Further, we show that the use of effective masses to characterize repulsion regions can