Advances in Mathematical Physics

Volume 2017, Article ID 1035381, 10 pages

https://doi.org/10.1155/2017/1035381

## Basic Generic Properties of Regular Rotating Black Holes and Solitons

^{1}A. F. Ioffe Physico-Technical Institute, Politekhnicheskaja 26, St. Petersburg 194021, Russia^{2}Department of Mathematics and Computer Science, University of Warmia and Mazury, Sloneczna 54, 10-710 Olsztyn, Poland

Correspondence should be addressed to Irina Dymnikova; lp.ude.mwu@aniri

Received 30 April 2017; Revised 14 June 2017; Accepted 29 August 2017; Published 16 October 2017

Academic Editor: Stephen C. Anco

Copyright © 2017 Irina Dymnikova and Evgeny Galaktionov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present a systematic description of the basic generic properties of regular rotating black holes and solitons (compact nonsingular nondissipative objects without horizons related by self-interaction and replacing naked singularities). Rotating objects are described by axially symmetric solutions typically obtained by the Gürses-Gürsey algorithm, which is based on the Trautman-Newman techniques and includes the Newman-Janis complex transformation, from spherically symmetric solutions of the Kerr-Schild class specified by . Regular spherical solutions of this class satisfying the weak energy condition have obligatory de Sitter center. Rotation transforms de Sitter center into the equatorial de Sitter vacuum disk. Regular solutions have the Kerr or Kerr-Newman asymptotics for a distant observer, at most two horizons and two ergospheres, and two different kinds of interiors. For regular rotating solutions originated from spherical solutions satisfying the dominant energy condition, there can exist the interior -surface of de Sitter vacuum which contains the de Sitter disk as a bridge. In the case when a related spherical solution violates the dominant energy condition, vacuum interior of a rotating object reduces to the de Sitter disk only.

#### 1. Introduction

Presented in the current literature, regular rotating solutions [1–10] are obtained from regular spherical solutions with using the Newman-Janis complex translation [11]. In [12], it was shown that the Newman-Janis translation works for algebraically special metrics which belong to the Kerr-Schild class [13] and can be presented as , where is the Minkowski metric and are principal null congruences. More general approach, with using basic properties of metrics from the Kerr-Schild class, was applied for obtaining the axially symmetric solutions in the noncommutative geometry [14].

Spherical metrics of the Kerr-Schild class have the form [13] where and are generated by stress-energy tensors satisfying [15–18] The energy density is , the radial pressure , and the transversal pressure . Equation (3) represents the radial part of the -dependent equation of state, . The contracted Bianchi identities () yield the second part of the -dependent equation of state, [15, 19].

All regular spherical solutions of this class, satisfying the weak energy condition (nonnegative density as measured by any local observer on a time-like curve), have obligatory de Sitter center, as with [18, 20].

Gürses and Gürsey have found that the algebraically special metrics of the Kerr-Schild class can be presented in the Lorentz covariant coordinate system, developed the general approach based on the complex Trautman-Newman techniques (which include the Newman-Janis translation), and derived the axially symmetric metric in the general, model-independent form. In the Boyer-Lindquist coordinates the metric reads [12] It involves a master function which comes from a spherically symmetric solution (2). For , it describes the Kerr-Newman geometry [21], where is the mass parameter and is the electric charge. The case corresponds to the Kerr geometry [22]. Each of these two geometries can have at most two horizons, the solutions without horizons describe naked singularities.

The parameter in (4) is defined as an affine parameter along a principal null congruence. The surfaces of constant are the confocal ellipsoids of revolution [23] which degenerate, for , to the equatorial disk centered on the symmetry axis and bounded by the ring , (in the Kerr and Kerr-Newman geometries, it comprises their ring singularities).

Regular axially symmetric solutions satisfy condition (3) in the corotating reference frame [24, 25] and describe regular rotating objects, asymptotically Kerr or Kerr-Newman for a distant observer with the mass parameter . Solutions of this class describe regular rotating black holes and spinning solitons with de Sitter vacuum interiors [25–28]. Rotation transforms de Sitter center of regular spherical solutions into de Sitter vacuum disk (8). All regular solutions presented in the literature belong to this class. Most of them are devoted to study of particular solutions specified by a choice of the mass function in (2).

In this paper, we present a systematic review of the basic generic properties of regular rotating black holes and spinning solitons replacing naked singularities and defined in the spirit of Coleman lumps [29] as nonsingular nondissipative objects without horizons keeping themselves together by their own self-interaction. For convenience and transparency in analysis of various aspects of generic behavior, we use the geometrized units . Conversion factors and instructions for transition to different nongeometrized units can be found in [30].

The paper is organized as follows. In Section 2, we present horizons, ergospheres, and ergoregions. Section 3 is devoted to basic physical properties of regular interiors. Section 4 outlines behavior of electromagnetic fields in the case of electrically charged regular black holes and electromagnetic solitons. In Section 5, we summarize and discuss the results.

#### 2. Horizons, Ergospheres, and Ergoregions

The basic features of regular rotating objects are closely related to generic properties of related regular spherical solutions (1)-(2).

Horizons are defined by . The function can be written as It follows that at zero points of the metric function , , and evolves from as to as .

A spherical metric function in (2) in the asymptotically flat case can have at most two zero points and one minimum between them [18]. Typical behavior of a spherical metric function is shown in Figure 1.