# Cloud of strings as source in \(2+1\)-dimensional \(f\left( R\right) =R^{n}\) gravity

## Abstract

We present three parameters exact solutions with possible black holes in \( 2+1\)-dimensional \(f\left( R\right) =R^{n}\) modified gravity coupled minimally to a cloud of strings. These three parameters are *n*, the coupling constant of the cloud of strings \(\xi \), and an integration constant *C*. Although in general one has to consider each set of parameters separately, for *n* an even integer greater than one we give a unified picture providing black holes. For \(n\ge 1\) we analyze a null/timelike geodesic within the context of particle confinement.

## Keywords

Black Hole Event Horizon Black Hole Solution Null Geodesic Null Energy Condition## 1 Introduction

The advantages of working in lower-dimensional gravity, specifically in \( 2+1\) dimensions has been highlighted extensively during the recent decades. The interest started all with the discovery of a \(2+1\)-dimensional black hole solution by Banados, Teitelboin and Zanelli (BTZ) [1, 2, 3, 4]. The physical source of the BTZ black hole was a cosmological constant, which was subsequently extended with the presence of different sources [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. How significant are these sources physically? The addition of scalar [18, 19, 20, 21, 22, 23] and electromagnetic fields, both linear and non-linear, has almost become a routine, while exotic and phantom fields also found room of applications in the problem. A source that is less familiar is a cloud of strings [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], which was considered in Einstein’s general relativity. Within this context in \(3+1\) dimensions the importance of a string cloud has been attributed to the action-at-a distance interaction between particles. For a detailed geometrical description of a string cloud we refer to [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. In this study we extend such a source to the \(f\left( R\right) =R^{n}\) gravity which is a modified version of general relativity [35, 36, 37, 38, 39]. In *D*-dimensional spacetime the energy-momentum tensor for a string cloud is represented by \(T_{\mu }^{\nu }=\frac{\xi }{r^{D-2}}\mathrm{diag}\left( 1,1,0,0,\ldots ,0\right) \) where \(\xi \) is a positive constant. In 3 dimensions, which will be our concern here, this amounts to \( T_{0}^{0}=T_{1}^{1}=\frac{\xi }{r},\) with \(T_{2}^{2}=0\) where our labeling of coordinates is \(x^{\mu }=\left( t,r,\theta \right) .\) Compared with the energy-momentum of the scalar and electromagnetic sources, which are of the order \(\frac{1}{r^{2}}\) in \(2\,+\,1\) dimensions, the order \(\frac{1}{r}\) for a string cloud may play an important role. Briefly the singularity at \(r=0\) is weaker in comparison with different sources. This forms the motivation for us to conduct the present study. We note that our cloud of strings is reminiscent of the wormhole “fur coat” model in the 5*D* Reissner–Nordström black hole case [40].

Such a cloud of 1-dimensional strings plays the role of particles in analogy with 1-dimensional gas atoms. The spatial geometry is confined to the polar plane with the cyclic angular coordinate. With the addition of a time variable the sheet description of the string becomes more evident. The strings are open, originating at the singular origin and extending to infinity, vanishing with \(r\rightarrow \infty .\) Thus, for \(r\rightarrow \infty \) our model reduces to the source-free (vacuum) \(f\left( R\right) \) model, which derives its power from the curvature of geometry. For \(\xi =0\) in \(f\left( R\right) =R\) model the spacetime is automatically flat unless supplemented by other sources. In \(f\left( R\right) \) gravity, on the other hand, even though we can take \(\xi =0\) we have still room for a non-flat metric albeit this may not be a black hole.

We investigate the field equations of \(f\left( R\right) \) gravity in the presence of a string cloud. In general, these are highly non-linear differential equations but owing to the simplicity of our source and the \( 2+1\) dimensions we are fortunate to obtain a large class of exact solutions. For a particular choice of the parameter *n* our solution can be interpreted as black holes, while for other choices it corresponds to cosmological models.

## 2 String-cloud source in \(f\left( R\right) =R^{n}\) gravity

*R*,

*n*is a real constant, and

*I*with respect to \(g_{\mu \nu }\) provides the field equations (in a metric formalism),

*r*. A set of solutions to the above field equations are given by

*C*is an integration constant and

*R*, which reads

*R*.

*n*and \(\xi \), our solution is a three parameter solution. For any specific value of

*n*one has to choose an appropriate sign for \(\xi \), for which in some cases both signs may be acceptable. Setting

*n*and the sign of \(\xi \) gives us an equation for \(\alpha \) given by (17). Depending on the number of real solutions this equation may admit, we will get different metric functions. For instance let us consider \(n=2.\) In this case one finds \(\alpha ^{2}=4\left( \frac{ 12\xi }{5}\right) \), which imposes \(\xi >0\), and consequently there are two solutions for \(\alpha \), given by \(\alpha =\pm 2\sqrt{\frac{12\xi }{5}}.\) For positive/negative \(\alpha \) one finds

*C*, the solution represents a particle solution [41].

### 2.1 \(f\left( R\right) =R\) with geodesics

*R*gravity coupled to the cloud of strings minimally. Let us note that this solution was found first by Bose et al. in [42]. The solution becomes

#### 2.1.1 Geodesics confinement for \(f\left( R\right) =R\)

*R*gravity given by (24) is rather interesting if we assume \(M,\xi >0\). In this section we investigate the null and timelike geodesics of this solution. Let us start with the Lagrangian

*t*the equation of motion becomes

#### 2.1.2 Generalization to \(f\left( R\right) =R^{n}\) with *n* an even integer and \(n\ge 2\)

*n*to be even integer bigger than 1, i.e., \(n\ge 2.\) In order to keep the mathematical expression analytic we only consider the radial, null geodesics. Accordingly one finds from the Euler–Lagrange equations

*E*is the energy of the particle and \(\lambda \) is an affine parameter. Combining these two equations we find

*n*we have to appeal to the asymptotic behavior. Considering

*r*to be large i.e., \(\frac{r_{h}}{r}\ll 1\) and \(n>0\), one approximately finds

## 3 Conclusion

In search of alternative black holes to the BTZ in \(2+1\) dimensions a particular case was considered in Einstein’s theory of \(f\left( R\right) =R\) in which the source is a cloud of strings [42]. Projected in the polar plane the string can be considered as radial lines originating at the origin and extending in radial in/out directions. This excludes the possibility of a closed string in such a geometry. The advantage of such a choice of source becomes evident when substituted into the complicated \(f\left( R\right) =R^{n}\) gravity which we consider here. In other words our geometry is powered by such a cloud of strings in the \(f\left( R\right) =R^{n}\) gravity with the energy-momentum tensor \(T_{0}^{0}=T_{1}^{1}=\frac{\xi }{r}\) and \( T_{2}^{2}=0\) with \(\xi =\)constant. Obviously this satisfies the null energy conditions. Although *n* can be arbitrary in principle there are restrictions on the choice of *n* for a meaningful expression. For instance, \(n=\frac{1}{2}\) and \(n=\frac{3\pm \sqrt{5}}{4}\) are to be excluded in the class of metric solutions. Depending on the other values of *n* we obtain an infinite class of possible metrics that describe black holes/naked singularities in \(2+1\)-dimensional \(f\left( R\right) =R^{n}\) gravity theory. An interesting physical conclusion to be drawn from these solutions is the role that the power *n* plays in the confinement of (especially) null geodesics. Although the most general analytic integration of the geodesics is lacking we obtain an approximate connection between the parameter *n* and the confinement of the null geodesics for *n* an even integer greater than 1. Whether a similar relation occurs in higher dimensions \((D>3)\) for a cloud of strings as source remains to be seen.

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