Elsevier

Annals of Physics

Volume 431, August 2021, 168557
Annals of Physics

Geodesic motion around hairy black holes

https://doi.org/10.1016/j.aop.2021.168557Get rights and content

Abstract

In a recent paper Ovalle et al. (2021) it has been obtained new static black hole solutions with primary hairs by the Gravitational Decoupling. In this work we either study the geodesic motion of massive and massless particles around those solutions and restrict the values of the primary hairs by observational data. In particular, we obtain the effective potential, the innermost stable circular orbits, the marginally bounded orbit, and the periastron advance for time-like geodesics. In order to restrict the values taken by the primary hairs we explore their relationship with the rotation parameter of the Kerr black hole giving the same innermost stable circular orbit radius and give the numerical values for the supermassive black holes at Ark 564 and NGC 1365. The photon sphere and the impact parameter associated to null geodesics are also discussed.

Introduction

Through the developments of general relativity black holes (BHs) solutions have been a subject of study, discussions and analysis. From being considered as simply mathematical constructions lacking for physical reality to be one of the central topics in recent researches, BHs are undoubtedly one of the most known and intriguing objects in literature. Indeed, the most recent Nobel Prize in Physics has been given not only for theoretical developments on BHs [1] but for the studies of orbits followed by massive objects (stars) around the presumably super massive BH in the center of the Milky Way, Sagittarius A* [2], [3], [4], [5]. In this sense, it is clear that the study of the motion around BHs by solving the geodesic equations of test particles is important not only to show that they indeed are real objects but with the aim to understand some of their main features given that study of geodesics exposes geometric properties of space–time. More precisely, the geodesic equations can be applied to calculate observable quantities like the shadow of a black hole, the periastron shift of a bound orbit and many other quantities which allows to characterize the BH geometry [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53].

The non-hair conjecture states that, independent of the manner in which a BH is formed, it can be only characterized by three parameters namely, the mass, charge and angular momentum [54], [55]. However, it has been demonstrated that under certain circumstances the existence of a BH characterized with more than these three parameters is a possibility [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66]. In this sense, taken hairy BH as real possibility it could be interesting to explore how the geodesic motion of particles around it deviates from their behavior when the hairs are ignored.

Among all the possible hairy BH in literature we could consider in this paper, we concentrate on one recent solution reported in [67]. This hairy BH was obtained by the now well-known Gravitational Decoupling (GD) [68] through the Minimal Geometric Deformation (MGD) [69], [70] (for implementation in 3+1 and 2+1 dimensional spacetimes see [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92], [93], [94], [95], [96], [97], [98], [99], [100], [101], [102], [103], [104], [105], [106], [107], [108], [109], [110], [111], [112], [113], [114], [115], [116], [117], [118], [119], [120], [121], [122] and references there in) in its extended version [91]. As it is reported in [67], these black holes were obtained by demanding they should fulfill the strong (SEC) or the dominant (DEC) between the horizon of the BH to infinity. As demonstrated in [67], all the new hairy BHs solutions correspond to deformations of the Schwarzschild vacuum. It is our main goal here to explore how the presence of such primary hairs which deform the Schwarzschild geometry affects the geodesics of massive and massless test particles around. In particular we study the innermost stable circular orbits (ISCO), the marginally bound orbit (MBO) for massive test particles. Additionally we obtain the periastron advance by numerical computations and analyze how much the orbits are affected in comparison to the Schwarzschild case. Besides, we explore the values that should be taken by the primary hairs in order to mimic the same effect that Kerr solution has on test particles at the ISCO radius. This analysis allows to restrict the values of such primary hairs using observed data. Finally, we also consider the behavior of null geodesics to see how the hairs affect the photon sphere.

This work is organized as follows. In the next section we review the main aspects on geodesic motion around a central object. Next, in Section 3 we introduce the hairy BHs reported in [67]. Section 4 is devoted to the analysis of the impact parameters, ISCO and MBO for each of the BH models introduced. In Section 5 the bounded orbits are studied and next, in Section 6, we obtain the numerical values of the primary hairs that mimic the spin parameter of the Kerr solution. In Section 7 an analysis on null geodesics is performed and we conclude our work in the last section.

Section snippets

Geodesic equations

In this section we review the basic concepts related to the geodesic motion around black holes. The geodesic motion of test particles in a spherically symmetric space–time with metric ds2=fdt2f1dr2r2dΩ2,is described by the geodesic equations ṙ2=E2fr2Q+L2+εr2θ̇2=1r4QL2cot2θϕ̇=Lr2csc2θṫ=Ef, where ε=0 for null geodesics, ε=1 for timelike geodesics, Q is the Carter constant, and L and E are the specific angular momentum and energy, respectively. The orbits of particles can be described by

Hairy BH solutions

In this section we briefly review the hairy BH solutions obtained in Ref. [67]. It is worth mentioning that all the models in which we will base our geodesic analysis were previously obtained by gravitational decoupling (GD) which has become a powerful tool in the study of BH [77], interior solutions [68], f(R) [109], f(R,T) [110] and f(G) [113] gravity, Brans–Dicke [114] and cosmology [104]. In the particular case [67], GD was used to obtain hairy BH by the extended minimal geometric

Timelike geodesics

In this section, we study the impact parameters, ISCO, MBO and bounded orbits for each of the BH models, presented in the previous section, and compared with the Schwarzschild BH.

Bounded orbits

In this section we analyze the bounded orbits for timelike geodesics around each hairy BH under consideration including the Schwarzschild solution. All the numerics have been performed by considering L=4 and E0.95.

As shown in Fig. 5, all the orbits have an epicyclic motion with variation in the average size of the orbit in dependence of the parameters α and 0. Model 1, in comparison with the Schwarzschild BH, shows an extension of the bound epicyclic motion to larger distances, which means

Hairy black hole as a rotating solution mimicker

The ISCO radius of test particles following corotating orbits around Kerr BH is given by risco=3+Z2(3Z1)(3+Z1+2Z2),where Z1=1+(1a3+1+a3)1+a23Z2=3a2+Z12 Now, it is possible to find a relationship between the rotation parameter and the primary hairs {α,l0} such that the hairy BH found here serves as a mimicker of the rotating solution. In Fig. 6 we show the primary hairs for each model as a function of a. In all the cases it is observed that the hairy BHs can mimic the rotation parameter in an

Null geodesics

In this section we study the null geodesics in order to get the photon radius rph for the different models here considered. The null geodesic equation is obtained when vanishing ε, Q and ṙ in Eq. (2), which becomes r=fLE.In Fig. 7 it is shown the radius of the photon sphere corresponding to each of the four models considered. We have as reference the well known radius of the Schwarzschild photon sphere which corresponds to rph=3. On one hand, in the left panel the rph for model 1 and model 2

Conclusions

It is well-known that the apparition of stable matter/field configurations around the black hole geometry is possible, leading thus to the so-called hairy black holes solution. This was precisely the topic under study in this paper. In this work we considered hairy black holes which correspond to deviation from the Schwarzschild geometry by a generic fluid which can be thought of as a kind of non-linear electrodynamics. Besides, we performed a complete analysis regarding the behavior of both

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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