# Radiating Kerr-like regular black hole

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## Abstract

We derive a radiating regular rotating black hole solution, radiating Kerr-like regular black hole solution. We achieve this by starting from the Hayward regular black hole solution via a complex transformation suggested by Newman–Janis. The radiating Kerr metric, the Kerr-like regular black hole and the standard Kerr metric are regained in the appropriate limits. The structure of the horizon-like surfaces are also determined.

## 1 Introduction

The formation of spacetime singularities is a quite common phenomenon in general relativity and, indeed, celebrated theorems, proved by Penrose and Hawking [1], state that under some circumstances singularities are inevitable in general relativity. As these theorems use only the laws of general relativity and some properties of matter, they are valid generally. It is widely accepted that spacetime singularities do not exist in Nature; they are a limitation or creation of the classical theory. The existence of a singularity implies that there exists a point in spacetime where the laws of physics break down or signal a failure of the physical laws. It turns out that what amounts to a singularity in general relativity could be adequately explained by some other theory. If physical laws do exist at those extreme situations, then we should turn ourselves to a theory of quantum gravity. However, we are yet a long distance away from a definite theory of quantum gravity. So a line of action is to understand the inside of a black hole and resolve its singularity by carrying out research of classical or semi-classical black holes, with regular, i.e., nonsingular, properties. This can be motivated by quantum arguments. Sakharov [2] and Gliner [3] proposed that spacetime in the highly dense central region of a black hole should be de Sitter-like for \(r \simeq 0\) (see also, [4, 5, 6, 7]). This indicates that an unlimited increase of spacetime curvature during a collapse process can stop the collapse if quantum fluctuations dominate the process. This places an upper bound on the value of the curvature and necessitates the formation of a central core.

Bardeen [8] realized concretely the idea of a central matter core, by proposing the first regular black hole solution of the Einstein equations. Bardeen’s regular metric is a solution of the Einstein equations in the presence of an electromagnetic field, yielding an alteration of the Reissner–Nordström metric. But near the center the solution tended to a de Sitter core solution. Subsequently, there has been enormous development in investigating the properties of regular black hole solutions [9, 10, 11, 12, 13, 14, 15, 16, 17], but most of these regular black hole solutions are more or less based on Bardeen’s proposal. In particular, an interesting proposal is made by Hayward [17] for the formation and evaporation of regular black holes, in which the static region is the Bardeen-like black hole. The dynamic regions are Vaidya-like black hole regions, with negative energy flux during evaporation and ingoing radiation of positive energy flux during collapse. The latter is balanced by outgoing radiation of positive energy flux and a surface pressure at a pair creation surface. This is the only non-stationary or dynamical regular black hole. However, these non-rotating metrics cannot be tested by astrophysical observations, as the black hole spin plays an important and fundamental role in any astrophysical process.

The generalization of these stationary regular black holes to the axially symmetric case, the Kerr-like regular black hole, was addressed recently [18, 19, 20]. In particular, it was established [18, 19] that the rotating regular black hole solutions can be obtained starting from regular black hole solutions by a complex coordinate transformation previously suggested by Newman and Janis [21]. However, this is obviously not the most physical scenario and we would like to consider dynamical black hole solutions, i.e., black holes with non-trivial time dependence. Further, the axially symmetric counterpart of the regular Vaidya-like black hole is still unexplored, e.g., the radiating generalization of the regular Kerr-like black hole is still unknown. It is the purpose of this paper to obtain this metric. Thus we extend a recent work [17] on radiating regular black holes to include rotation, and it is also a non-static generalization of the Kerr-like regular black hole solution [18]. We also carry out a detailed analysis of the horizon structure of radiating Kerr-like regular black holes, which also is valid for a static Kerr-like regular black hole and which has not been done earlier. It should be pointed out that the Kerr metric [22] is undoubtedly the single most significant exact solution in the Einstein theory of general relativity, which represents the prototypical black hole that can arise from gravitational collapse. The radiating or non-static counterpart of the Kerr black hole was obtained by Carmeli [23]. We also show that the Kerr-like regular black hole, the Kerr black hole, and the radiating Kerr-like black hole arise as special cases of the radiating Kerr-like regular black hole.

In this paper, we obtain a radiating Kerr-like regular metric in Sect. 2. The Newman–Janis algorithm is applied to spherically symmetric radiating solutions, and radiating rotating solutions are obtained. We investigate the structure and locations of horizons of the radiating Kerr-like regular metric in Sect. 3. The paper ends with concluding remarks in Sect. 4.

We use units which fix the speed of light and the gravitational constant via \(G = c = 1\), and use the metric signature (\(+,\;-,\;-,\;-\)).

## 2 Radiating rotating black hole via Newman–Janis

### 2.1 Rotating radiating Hayward black hole

## 3 Physical parameters and horizons of rotating radiating Hayward black hole

If we consider radiating regular black holes, it is useful to discuss not only black hole solutions but their horizon structure. In this section, we explore horizons of the radiating regular Hayward black hole, and we discuss the effects which come from the parameter \(q\). In general, a black hole has three important surfaces [28]: the timelike limit surface (TLS), the apparent horizon (AH), and the event horizon (EH). For the non-radiating Schwarzschild black hole, the three surfaces EH, AH, and TLS coincide. For the Vaidya black hole which radiates, we have AH \(=\) TLS, but the EH is different from AH. If we break spherical symmetry, preserving stationarity, e.g., in the case of a Kerr black hole, then AH \(=\) EH but EH \( \ne \) TLS.

Here we shall focus on the investigation of these horizons for the radiating regular Kerr-like black hole. As suggested by York [28], three horizons may be obtained to \(O(L)\) by noting that (i) for a radiating black hole, we can define TLS as the locus where \(g(\partial _v, \partial _v) = g_{vv} = 0\), AHs are termed surfaces such that \(\Theta \simeq 0\), and EHs are surfaces such that \(d \Theta /dv \simeq 0\).

### 3.1 Event horizon

## 4 Conclusion

The rotating Kerr black hole relish many useful properties distinct from the non-rotating counterpart Schwarzschild black hole. However, there is a surprising connection between the two different black holes of general relativity, as analyzed by Newman and Janis [21] in their famous paper. They explicitly demonstrated that, by applying a set of complex transformations, it was possible to construct both the Kerr case starting from the Schwarzschild metric and likewise the Kerr–Newman solutions beginning with the Reissner–Nordström metric [21].

The Newman–Janis algorithm is fruitful in deriving several rotating black hole solutions starting from their non-rotating counterparts [18, 19, 21, 24], which also includes the rotating regular black hole [18, 19]. The algorithm is very useful since it directly allows us to generate rotating black holes, which otherwise could be extremely tiresome due to the nonlinearity of field equations. For a review of the Newman–Janis algorithm see, e.g., [26]. In this paper, we have generated a radiating (non-static) Kerr-like regular black hole metric, which contains the radiating Kerr metric as the special case when the deviation \(g\) vanishes, and also the standard Kerr metric when, in addition to \(g=0\), the mass function \(M(v)=M\) is constant. This metric does not arise from any particular set of field equations, but the Newman–Janis algorithm works on the spherical radiating solution to generate radiating rotating solutions. Thus, the derived radiating Kerr-like regular metric (14) bears the same relation with the rotating regular black hole as does the Vaidya metric to the Schwarzschild metric.

The structure of the three surfaces, TLSs, AHs, and EHs, of the derived radiating Kerr-like black hole were investigated by the method developed by York [28] to \(O(L)\) by a null vector decomposition of the metric. The analysis presented for determining the structure of the horizons is applicable to the stationary rotating regular case as well, but AHs coincide with EHs because stationary black holes do not accrete, i.e., \(L=0\). However, the three surfaces do not coincide with each other for radiating Kerr-like black holes. For each of TLS, AH, and EH, there exist two surfaces corresponding to the two positive roots \(r^{-}\) and \(r^{+}\), and they can be viewed, respectively, as inner and outer black hole horizons. Thus it means that as regards the presence of the term \(q\) also, we can find values of parameters so that the two inner and outer horizons still exist as in the case of radiating Kerr black hole.

To conclude, the solutions presented here provide necessary grounds to further study geometrical properties, causal structures, and thermodynamics of these black hole solutions, which will be the subject of a future project. Further generalization of such a regular black hole solution is an important direction and will be the subject of our forthcoming papers.

## Notes

### Acknowledgments

We would like to thank Pankaj Sheoran for his help in the plots. SDM also acknowledges that this work is based upon research supported by the South African Research Chair Initiative of the Department of Science and Technology and the National Research Foundation.

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