Disclosing connections between black holes and naked singularities: Horizon remnants, Killing throats and bottlenecks

We study the properties of black holes and naked singularities by considering stationary observers and light surfaces in Kerr spacetimes. We reconsider the notion of Killing horizons from a special perspective by exploring the entire family of Kerr metrics. To this end, we introduce the concepts of extended plane, Killing throats and bottlenecks for weak (slowly spinning) naked singularities. Killing bottlenecks (or horizon remnants in analogy with the corresponding definition of throats in black holes) are restrictions of the Killing throats appearing in special classes of slowly spinning naked singularities. Killing bottlenecks appear in association with the concept of pre-horizon regime introduced in [1, 2]. In the extended plane of the Kerr spacetime, we introduce particular sets, metric bundles, of metric tensors which allow us to reinterpret the concept of horizon and to find connections between black holes and naked singularities throughout the horizons. To evaluate the effects of frame-dragging on the formation and structure of Killing bottlenecks and horizons in the extended plane, we consider also the Kerr-Newman and the Reissner-Norstrom spacetimes. We argue that these results might be significant for the comprehension of processes that lead to the formation and eventually destruction of Killing horizons.Comment: 33 pages, 32 multi-panels figures, 3 Table

The ergoregion in the Kerr spacetime: properties of the equatorial circular motion

We investigate in detail the circular motion of test particles on the equatorial plane of the ergoregion in the Kerr spacetime. We consider all the regions where circular motion is allowed, and we analyze the stability properties and the energy and angular momentum of the test particles. We show that the structure of the stability regions has definite features that make it possible to distinguish between black holes and naked singularities. The naked singularity case presents a very structured non-connected set of regions of orbital stability, where the presence of counterrotating particles and zero angular momentum particles for a specific class of naked singularities is interpreted as due to the presence of a repulsive field generated by the central source of gravity. In particular, we analyze the effects of the dynamical structure of the ergoregion (the union of the orbital regions for different attractor spins) on the behavior of accretion disks around the central source. The properties of the circular motion 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 different physical effects that could be of especial relevance in observational Astrophysics. We also identify some special black hole spacetimes where these effects could be relevant.Comment: 19 pages, 9 figure multi-panels; 3 Tables. This and a slightly modified version with the addition of new references and some new discussion. To appear in EPJ

Observers in Kerr spacetimes: the ergoregion on the equatorial plane

We perform a detailed analysis of the properties of stationary observers located on the equatorial plane of the ergosphere in a Kerr spacetime, including light-surfaces. This study highlights crucial differences between black hole and the super-spinner sources. In the case of Kerr naked singularities, the results allow us to distinguish between "weak" and "strong" singularities, corresponding to spin values close to or distant from the limiting case of extreme black holes, respectively. We derive important limiting angular frequencies for naked singularities. We especially study very weak singularities as resulting from the spin variation of black holes. We also explore the main properties of zero angular momentum observers for different classes of black hole and naked singularity spacetimes.Comment: 20 pages, 13 multi-panels figures, 2 table

Motion of charged test particles in Reissner--Nordstr\"om spacetime

We investigate the circular motion of charged test particles in the gravitational field of a charged mass described by the Reissner-Nordstr\"om (RN) spacetime. We study in detail all the spatial regions where circular motion is allowed around either black holes or naked singularities. The effects of repulsive gravity are discussed by finding all the circles at which a particle can have vanishing angular momentum. We show that the geometric structure of stable accretion disks, made of only test particles moving along circular orbits around the central body, allows us to clearly distinguish between black holes and naked singularities.Comment: 58 pages, 41 figures. To be published in Phys. Rev. D. This work follows the paper "Circular motion of neutral test particles in Reissner-Nordstr\'om spacetime" published in Phys.\ Rev.\ D \ Vol.83, No.2 with DOI: 10.1103/PhysRevD.83.024021 (arXiv:1012.5411v1 [astro-ph.HE]

Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization

We introduce normal coordinates on the infinite dimensional group $G$ introduced by Connes and Kreimer in their analysis of the Hopf algebra of rooted trees. We study the primitive elements of the algebra and show that they are generated by a simple application of the inverse Poincar\'e lemma, given a closed left invariant 1-form on $G$. For the special case of the ladder primitives, we find a second description that relates them to the Hopf algebra of functionals on power series with the usual product. Either approach shows that the ladder primitives are given by the Schur polynomials. The relevance of the lower central series of the dual Lie algebra in the process of renormalization is also discussed, leading to a natural concept of $k$-primitiveness, which is shown to be equivalent to the one already in the literature.Comment: Latex, 24 pages. Submitted to Commun. Math. Phy

Exterior and interior metrics with quadrupole moment

We present the Ernst potential and the line element of an exact solution of Einstein's vacuum field equations that contains as arbitrary parameters the total mass, the angular momentum, and the quadrupole moment of a rotating mass distribution. We show that in the limiting case of slowly rotating and slightly deformed configuration, there exists a coordinate transformation that relates the exact solution with the approximate Hartle solution. It is shown that this approximate solution can be smoothly matched with an interior perfect fluid solution with physically reasonable properties. This opens the possibility of considering the quadrupole moment as an additional physical degree of freedom that could be used to search for a realistic exact solution, representing both the interior and exterior gravitational field generated by a self-gravitating axisymmetric distribution of mass of perfect fluid in stationary rotation.Comment: Latex, 15 pages, 3 figures, final versio

Extracting information on black hole horizons

We present some features of Kerr black hole horizons that are replicated on orbits accessible to outside observers. We use the concepts of horizon confinement and replicas to show that outside the outer horizon there exist photon orbits whose frequencies contain information about the inner horizon and that can, in principle, be detected through the emission spectra of black holes. It is shown that such photon orbits exist close to the rotation axis of the Kerr geometry. We argue that these results could be used to recognize and further investigate black holes and their horizonsComment: 7 pages; 2 multipanels figure