Horizon fractalization in black strings ungravity

In this paper, we study the scalar (tensor) and vector unparticle corrections for cosmic and black strings. Initially, we consider a static cosmic string ansatz from which we obtain the solution in terms of first- and second-kind Bessel functions. We also obtain the solution for a black string in the unparticle scenario. We identify two regimes, namely, a gravity-dominated regime and an ungravity-dominated regime. In the gravity-dominated regime, the black string solution recovers the usual solution for black strings. The Hawking temperature is also studied in both regimes. As in the static and rotating black hole, we find a fractalization of the event horizon. This points to the fact that fractalization is a natural consequence of unparticles. Finally, we study the thermodynamics of the black string in the ungravity scenario by computing the entropy, heat capacity, and free energy. For both cases, we find that, depending on the region of the parameter dU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_U$$\end{document}, phase transitions are possible.


I. INTRODUCTION
The unusual properties of non-trivial scale invariance of matter in the IR regime was first studied by Banks and Zaks [1].Georgi suggested [2,3] a coupling between the standard model (SM) fields and this scale invariant sector of a high energy physics described by the Banks-Zaks (BZ) fields.The interactions between the standard model fields and the BZ fields are mediated by the exchange of highly massive particles M U , obeying the scale-supressed non-renormalizable Lagrangian density where O stands for the field operators of dimension d S M and d BZ , while k = d S M + d BZ − 4 in order to guarantee a dimensionless action.It was argued by Georgi [3] that below some scale energy limit (Λ U ) the interactions between the standard model and the hidden sector become stronger.As the scale-invariance is manifested, massive particles are not allowed and this sector was dubbed "unparticles", since the notion of mass in this sector is meaningless.
Since the proposal of the unparticle physics several papers were published in particle physics [4][5][6], condensed matter analogue models [7,8], extra dimensions [9] among others.A very interesting work by Gaete and Spallucci [10] study the unparticle dynamics in a standard quantum field theory scenario, and they construct effective actions for scalar, gravitation and vector gauge unparticles.Also in cosmological and astrophysical scenarios, several were carried out including studies with black holes [11][12][13][14] and white dwarfs [15,16].One of these black holes works draws the attention to the fact that the exterior event horizon presents a surface with fractal dimension equals to d U [11].Recently, some of the present authors generalized the work of [11] to rotating black holes and found that the the fractalization is also present [14].
It is also noticeable that a recent paper on Casimir effect with unparticles shows a fractalization in the parallel plates dimension [17].The dimensional reduction seems to be a natural behavior in high energy and its comprehension in fundamental at this regime scales [18,20].However, as far as we know, no studies were carried out in order to investigate the influence of the unparticle physics in cosmic strings and black strings.
Objects such as cosmic strings may have been formed due to phase transitions in the early universe [21].
The spacetime geometry related to an infinitely long and straight cosmic string is characterized by a planar angle deficit on the two-surface orthogonal to the string.In addition to that, the spacetime is flat, locally, except on the top of the string.Cosmic strings can also be described in the realm of a classical field theory when the energy-momentum tensor related to the vortex configuration of the Higgs-Maxwell system [22] couples to the Einstein equations.
On the other hand, black strings are higher dimensional solutions of the Einstein equations in Ddimensional spacetime [23].These vacuum objects can be understood as symmetric generalizations of black holes under translations being the event horizon topologically equivalent to S 2 × R (or S 2 × S 1 in the case of black rings) and spacetime is asymptotically M D−1 × S 1 for a zero cosmological constant or AdS D−1 × S 1 for a negative one [24,25].A black string is a particular case of a black p-brane solution (with p = 1), where p is the number of extra spatial dimensions to ordinary space.Hence a p-brane gives rise to a (p + 1)-dimensional world-volume in spacetime.In string theory, a black string is described by a D1-brane surrounded by a horizon.It is widely known that since an open string propagates through spacetime, it is necessary that its endpoints to lie on a D2-brane on which it satisfies Dirichlet boundary condition, and the dynamics on the D-brane worldvolume is described by a gauge field theory [26,27].In some models our universe is understood as a higher dimensional world-brane, thus the gravitational collapse of galactic matter would produce black holes lying on the brane.Black string solutions are nevertheless unstable to long wavelength perturbations (at least in asymptotically flat spaces) since the localized black hole is entropically preferred to a long segment of string.The string's horizon therefore has a tendency to form a line of black holes.However, in AdS the space acts like a confining box which prevents fluctuations of long wavelengths from developing.For instance, in a Randall Sundrum domain wall in a five dimensional AdS spacetime the black string is stable at least far from the AdS horizon.Near the horizon, the string is likely to pinch off to become a stable shorter cigar-like singularity [28].
In this paper we study the unparticle corrections for cosmic and black strings.This paper is organized as follows: In the next section we present a brief review of the unparticle theory.In section III we study the unparticle corrections for cosmic strings and in section IV we obtain the black string solution in the unparticle scenario.In section V we study the thermodynamic properties of black strings in the unparticle physics domain and in section VI we draw our conclusions.

II. UNPARTICLE THEORY
The un-gravity action is [11] where κ 2 = 8πG, κ 2 * is represents the un-gravitational Newton constant, and −1 [11], and D 2 is the generally covariant D'alambertian.The Einstein Field Equations (EFE) is given by The second term of right side of EFE is the unparticle correction of energy-momentum tensor.In Ref. [12] it was found an exact solution of EFE for vector particles, and this solution is Reissner-Nordstrom like.In this paper, we'll consider an axially symmetric ansatz for the metric, (g µν ) = diag g tt (r), g rr (r), g φφ (r), g zz (r) , since we are interested in studying solutions of Cosmic String and Black String with unparticle corrections.
Let us consider a static string which has a density of energy ρ(r) = µ(2πr) −1 δ(r) = µδ(x)δ(y), where µ is the linear density of mass.We can compute the contribution of unparticle in equation ( 3) by calculating the term T U .We achieve that by switching to isotropic (free-falling), Cartesian-like coordinate for computational convenience, where X is a vector in a direction perpendicular to the string.Hence, by applying the Schwinger represen- Yielding for the equation ( 5) Finally, when we compute this integration, we find which allow us to write a proper expression for the energy density of a static string with unparticle as When we replace in equation ( 9), we find where Also it is important to highlight that the R parameter sets a proper unparticle length scale so that, similarly to the authors in [11,14], it is possible to distinguish between two regimes.The gravity dominated (GD) regime is defined by R << r and the ungravity dominated (UGD) regime by R >> r.Note that in the GD regime the second term of ( 10) vanishes and we recover the usual cosmic string energy density.
In the next section, we will present the solution of static cosmic string with unparticle corrections.

III. COSMIC STRING WITH UNPARTICLE CORRECTIONS
In this section, we will study the spacetime solution which is static and has cylindrical symmetry.Consider the ansatz of the line-element as follows The components of the Ricci tensor are We know that the T 0 0 = −ρ(r), so we can see by EFE G This equation can be rewritten as Now, we need to properly solve the equation (15).Let us perform the following variable transformation b(r) = √ r f (r), such that the equation ( 15) becomes First, we need to solve this equation for r > 0. After we impose that when R/r → 0 we must find cosmic string spacetime.When we consider r > 0 the equation ( 16) simplifies to Now, we can perform a transformation of variable x = µκ 2 2π , allowing the equation ( 17) to be written as This is the Bessel differential equation.The solution of equation ( 18) depends on 1 2du−2 .When we define ξ ≡ 1/(2du − 2) as a positive real number the equation ( 18) becomes whose solution is Here J ξ (x) and N ξ (x) are the first and second kind Bessel function of order ξ, respectively.Let us present some considerations regarding the boundary conditions.Initially, we want that when R/r → 0, b(r) = (1 − 8Gµ) 1/2 r, that is, the spacetime becomes the cosmic string spacetime.Since x is proportional to (R/r) d U −1 and b(r) = √ r f (x(r)), we can study the equation ( 20) when x 1.In regime x 1 and ξ > 0, equation ( 20) can be written as 21), when we impose the limit R/r 1 we shall find f (r) = (1 − 8Gµ) 1/2 √ r.Hence C 1 = 0 and Consequently, the component g φφ of the metric is The metric component g φφ = b 2 (r), the behavior of this component is given at Figure 2. In order to understand the structure of this spacetime, that is, how the unparticles modify the cosmic string spacetime, We must analyze the curvature scalars.Ricci's scalar for the ansatz dr 2 .Then Ricci's scalar for our solution is given by where we used equation ( 16).
For the black string in the un-gravity scenario, the Einstein-Hilbert effective action is composed by (2) added by the cosmological constant contribution, i.e., The Einstein equation in (3) remains valid, however the left-hand-side will be modified due to the different ansatz for the metric in (24).The EFE for this ansatz are We can see that the energy-momentum tensor for the ansatz of equation ( 24) is T µ ν = −ρ(r) diag(1, 1, 0, 0) + p l (r) diag(0, 0, 1, 1), where p l = p φ = p z .This way we can find f (r) by solving G t t − 3α 2 = −κ 2 ρ(r), we find with R given by equation (11).Note that the unparticle black string solution is divergent for d U = 3/2.The plus signal is taken for the scalar unparticle case, while the minus one is for the vector case.
The above solution for the black string in the ungravity scenario recovers the usual black string solution [29] in the GD regime, i.e., Also, it is important to highlight here that the unparticle black string may exhibit naked singularity, violating therefore the cosmic censorship, for the scalar case when 3/2 < d U < 2.

V. THERMODYNAMIC PROPERTIES OF THE BLACK STRING WITH UNPARTICLE CORRECTIONS
To make easy the calculations, we can rewrite the equation ( 30) as f (r) = h(r) + µs(r), where Our black string solution in the un-gravity scenario has the horizon curves defined by f (r h ) = 0, so µ = − h(r h ) s(r h ) , which gives us In the GD regime, the mass parameter µ reduces to µ = α 3 4 r 3 h , which is the usual mass parameter for the black-string in General Relativity.In [14] the authors have found that for an unparticle black hole the mass parameter becomes negative for the vector unparticle case.For the unparticle black string, as we can see in fig.(4), both vector and scalar cases present the possibility of negative mass parameter.For the scalar unparticle case we have a negative mass parameter for values of 3/2 < d U < 2. For the vector unparticle case the mass parameter becomes negative for 1 The authors in [12] does not take into consideration the regions where the mass parameter is negative.
However it is known that a negative mass parameter in a black hole solution suggests that it has a nontrivial topology [31].Therefore we can expect that these negative mass regions are also related to nontrivial topologies of the the black string.In possession of the solution for the static black string in the ungravity scenario given by (30), we are able to study the thermodynamics of the black string by computing the Hawking's temperature by means of , where r h is the radius of the horizon.Note that in terms of h(r) and s(r) we find where we used µ = − h(r h ) s(r h ) .When we substitute the expressions for h(r h ) and s(r h ), we find The usual black string Hawking temperature, i.e.T H = 3α 2 4π r h , is recovered in the GD regime and in UGD regime, we find for both GD and UGD regimes.The Hawking temperature for the unparticle black string presents a similar behaviour for both scalar and vector unparticle cases.The behaviour of the Hawking temperature as a function of r for the scalar unparticle case for some values of d U is presented in figure (5).We can see from Note that the temperature is aproximally proportional to D − 1, where D is the dimension of the spacetime.So, when we confront the equation (37) with this fact, we see the fractalization of the event horizon with where d H = D − 2. To reinforce the fractalization of the event horizon, we can compute the density of entropy by length S althrough dS = dµ/T H , which in terms of h(r) and s(r), where we used equation (35) and again µ = − h(r h ) s(r h ) .So we find Now, we want to find analytically the entropy by length of the black string with unparticle.It can be done by integration of equation ( 40), but it is necessary to analyze the regions due to the possible singularity of this equation.Up to now we have discussed the scalar and vector cases simultaneously, but now, for the sake of simplicity we will consider them separately.
A. scalar case: Let us begin by considering the case when d U < 3/2.In this case we can rewrite the equation (40) as So, in case d U < 3/2 there is not singularity.By using the identity where 2 F 1 (a , b ; c ; x) is the hypergeometric function and γ > 0, the integration of equation ( 41) can be written as Note that, GD regime (R → 0) the argument of hypergeometric funcion is proportional to R 2d U −2 , which approaches to 0 when R → 0. It is, the entropy in GD recovers S = 2παr H 2 4 as shall be.
The heat capacity and the free energy of the system are defined by The heat capacity can be obtained directly from the equations ( 34) and (36), which yields The free energy can also be obtained from (34), (36) and (43), yielding The behaviour of the Hawking temperature, entropy, heat capacity and free energy for d U < 3/2 is depicted in figure (6).Now, we need to investigate the case d U > 3/2.In this case, there is a singularity in equation (40) when It is easy to see that we have two disjunte regions and when r h > r 0 (r h < r 0 ) the differential of entropy by length, equation (40), approaches infinity (minus infinity).First, let us compute the entropy for r h > r 0 .We can rewrite equation (40) as The heat capacity can be obtained straightforwardly from the equations (34) and (36), yielding The free energy can also be obtained from (34), ( 36) and (43), from which follows that where B z (a, b) is the incomplete Euler Beta function.
As it is widely known, the thermodynamical stability of black holes (black strings for our case) is directly related to the sign of the heat capacity.A positive heat capacity indicates that the system is thermodynamically stable, while its negativity imply a thermodynamical instability.As we can see in figure (6), the value of the d U controls the black string stability, being the black string stable in the ungravity regime for d U < 3/2 and unstable for d U > 3/2.

B. vector case:
Let us consider the case when d U > 3/2, which for the vector case avoid the singularity, so that the equation (40) can be written as We can clearly see that, in virtue of the similarity between the expressions (58) and (41), the same analysis performed for the scalar case when d U < 3/2 can be done for the vector case when d U > 3/2.Also the results for the scalar case when d U > 3/2 are identical to those of the vector case when d U < 3/2.
Therefore, the thermodynamic quantities depicted in figure (6a) for the scalar case when d U < 3/2 also describe the thermodynamic quantities for the vector case when d U > 3/2.Likewise, the thermodynamic quantities depicted in figure (6b) for the scalar case when d U > 3/2 are also describing the thermodynamic quantities for the vector case when d U < 3/2.

VI. CONCLUSION
In this paper we study the unparticle corrections for cosmic and black strings.Initially we have discussed some general features about the unparticle physics from which we obtained a general expression for the free energy for the scalar case and plots are depicted in figure (6) for a particular choice of parameters.We could see that the value of the d U controls the black string stability in the ungravity scenario, being the black string stable for d U < 3/2 and unstable for d U > 3/2.
Finally we point that the source 59 will give modified light deflection and could be found in the present era of high precision astrophysical measurements.It would also be interesting to study if the rotating case give the same fractal dimension, as in the black hole case.In fact, the study of fractal dimension in unparticle can be done to many other objects.A more ambitious project would be to show that horizon fractalization is always present in unparticle gravity.

FIG. 2 :
FIG. 2: Metric component g φφ of the unparticle static cosmic string as a function of the radial coordinate r for some values of d U , with Λ U = 10, µ = 0.7, κ * = 10 3

figure ( 5 )FIG. 5 :
figure(5) that when 1 < d U < 3/2 the temperature increases slowly with the radius in comparison with the usual black string temperature.When 3/2 < d U < 2 the unparticle correction promotes an increasing in the temperature in comparison with the black string usual case.