Magnetic solutions in Einstein-massive gravity with linear and nonlinear fields
Seyed Hossein Hendi, Behzad Eslam Panah, Shahram Panahiyan, Mehrab Momennia
aa r X i v : . [ g r- q c ] J un Magnetic solutions in Einstein-massive gravity with linear and nonlinear fields
Seyed Hossein Hendi , ∗ , Behzad Eslam Panah , † Shahram Panahiyan , , ‡ , and Mehrab Momennia § , Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran Helmholtz-Institut Jena, Fr¨obelstieg 3, D-07743 Jena, Germany Physics Department, Shahid Beheshti University, Tehran 19839, Iran
The solutions of U (1) gauge-gravity coupling is one of the interesting models for analyzing thesemi-classical nature of spacetime. In this regard, different well-known singular and nonsingularsolutions have been taken into account. The paper at hand investigates the geometrical propertiesof the magnetic solutions by considering Maxwell and power Maxwell invariant (PMI) nonlinearelectromagnetic fields in the context of massive gravity. These solutions are free of curvature sin-gularity, but have a conic one which leads to presence of deficit/surplus angle. The emphasizeis on modifications that these generalizations impose on deficit angle which determines the totalgeometrical structure of the solutions, hence, physical/gravitational properties. It will be shownthat depending on the background spacetime (being anti de Sitter (AdS) or de Sitter (dS)), thesegeneralizations present different effects and modify the total structure of the solutions differently. I. INTRODUCTION
Existence of topological defects have been reported in various aspects of the physics and their important roles inphysical properties of the systems have been highlighted. From gravitational/cosmological point of view, the effectsand importance of the topological defects could be related to their role as a possible dark matter source [1, 2], theirrole in large scale structure of the universe [3–5] anisotropy in the Cosmic Microwave Background (CMB) [6, 7]and their lensing properties [8] (which are due to existence of deficit angle). Essentially, the topological defects incosmology are produced due to symmetries that are broken in phase transition that has taken place in the earlyuniverse [3–5]. Depending on the number and type of the symmetries that are broken, these topological defectsare categorized into domain walls (a discrete symmetry is broken and it divides the universe into blocks), cosmicstrings (axial or cylindrical symmetry is broken and have applications in regard to grand unified particle physicsmodels/electroweak scale), monopoles (a spherical symmetry is broken) and textures (several symmetries are broken).Since these topological defects may be formed during the early universe, they may also carry valuable informationof this era which highlights yet another importance of studying them. The topological defects are located at theboundaries of regions which have chosen different minima during the early universe phase transition. So far, thesetopological defects have inspired large number of publications which among them one can point out; cosmic strings inthe presence of Maxwell theory [9, 10], their superconducting property in the presence of different models of gravity(such as Einstein [11], Brans-Dicke [12] and dilaton gravity [13]), the QCD [14] and quantum [15] applications of themagnetic strings, limits on the cosmic string tension using CMB temperature anisotropy maps [16], gravitational wavesproduced by cosmic strings [17] and decaying domain walls [18] and localization of fields and chiral spinor on domainwalls [19] (For further studies regarding topological defects, we refer the reader to an incomplete list of Refs. [20–25]).Motivated by these studies and their interesting results, here we investigate a type of topological defects which areknown as magnetic branes (generalization of magnetic string), in the presence of two generalizations; massive gravityand nonlinear electromagnetic field which are generalizations in gravitational and matter field sectors, respectively.Although the Maxwell electrodynamics (linear electrodynamics) is one of the most successful theories in the historyof physical science, it does not provide very precise results in some scales. On the other hand, due to the fact that themost physical systems are nonlinear in the nature, the generalization of linear electrodynamics to nonlinear ones seemsto be logical. In addition, owing to specific properties of nonlinear electrodynamics in the gauge/gravity coupling, therelations between the general relativity (GR) and nonlinear electrodynamics attract significant attention. Nonlinearelectrodynamic theories have some interesting results and predictions, and therefore, various nonlinear models ofelectrodynamics have been introduced by many authors. For typical examples, one may look at the Born-Infeldtheory [26], logarithmic form [27], exponential Lagrangian [28], arcsin nonlinear electrodynamics [29, 30] and etc.Black hole and magnetic solutions by considering these nonlinear electrodynamics have been investigated in Refs ∗ email address: [email protected] † email address: [email protected] ‡ email address: [email protected] § email address: [email protected] [31–52]. Also, other aspects of these nonlinear models have been studied in the context of quantum level [53–56]and astrophysical area [57–59] as well. The extensive usages of these nonlinear theories provide the validity of theirauthenticity.Taking into account the conformal invariance, one may find that it plays an important role in the structure of thesome interesting models of string theory. In other words, conformal invariance is a kind of criterion for obtainingcovariant equations of motion for the on-shell classical background in the low energy effective of string theory [60, 61].Regarding the equations of motion in the classical Einstein gravity, one finds the conformal invariance is equivalent tothe existence of a traceless stress-energy tensor. It is evident that the Maxwell theory enjoys the conformal invarianceonly in 4 − dimensions. But in three or higher dimensional spacetime, the conformal symmetry will be broken. Inother words, the stress-energy tensor of Maxwell theory is traceless only for 4 − dimensions. In order to keep theconformal invariance symmetry in arbitrary dimensions, one should generalize the Maxwell field to the so-calledpower Maxwell invariant (PMI) theory. PMI theory is one of the interesting branches of nonlinear electrodynamicsin which its Lagrangian is an arbitrary power of Maxwell Lagrangian ( L P MI ( F ) ∝ F s , where F is the Maxwellinvariant) [62–65]. This theory of nonlinear electrodynamics has more interesting properties with regard to linearelectrodynamics (Maxwell case), and for the case of s = 1, it reduces to the Maxwell theory. Another attractiveproperty is related to conformal invariance property. In other words, when the power of the Maxwell invariant isa quarter of spacetime dimensions ( s = d/
4, where d is dimensions), this theory enjoys conformal invariance, andtherefore, its energy-momentum tensor will be traceless. In this case, one may obtain the Reissner-Nordstr¨om likesolutions in higher dimensions [62]. The effects of considering PMI source for the classical black hole solutions invarious gravities have been studied in literature, for example; Lovelock and Lifshitz black holes with PMI field havebeen investigated in [65–68], BTZ black holes in the Einstein and F ( R ) gravities with this nonlinear electrodynamicmodel have been studied in Refs. [69–71], the effects of PMI for BTZ black hole with a scalar hair in the Einsteingravity are reported in [72]. Thermodynamics of topological black holes in the Brans-Dicke theory in the presencePMI field has been studied before [73]. Geometrical thermodynamics and the van der Waals like phase transition ofblack holes in higher dimensional spacetimes with PMI theory have been evaluated in Einstein and dilaton gravity[74–77]. Moreover, holographic superconductors and magnetic branes (string) supported by PMI source have beeninvestigated in Refs. [78–80].Although most of physicists believe that we should respect to conformal invariance symmetry, they believe thatLorentz invariance symmetry should be broken in high energy regimes. Considering a nonzero mass for the gravitonsmay leads to such breaking symmetry. Recent observations of gravitational waves from a binary black hole mergerprovided a firmly evidence of Einstein theory [81]. However, graviton in Einstein gravity is a massless particle, whereasthere are several arguments that state graviton may be a massive object [81]. Therefore, GR can be generalized toinclude massive gravitons. The first attempt for such generalization was done by Fierz and Pauli [82] by using a lineartheory. However, propagators of this theory do not reduce to those of GR in limit of vanishing graviton mass, m = 0(van Dam, Veltman and Zakharov discontinuity). In order to remove this substantial problem, Vainshtein introduceda mechanism which requires the system to be considered in a nonlinear regime [83]. Nonetheless, we encounter withBoulware-Deser ghost in the generalization of Fierz and Pauli massive theory to the nonlinear regime [84]. To solvesuch problem, another class of massive gravity was proposed by de Rham, Gabadadze and Tolley (dRGT) [85, 86].dRGT massive theory is free of Boulware-Deser ghost and it can be used in higher dimensions with admissible validity[87, 88]. It is noteworthy that, in order to obtain exact solutions with massive terms, an additional metric (calledthe reference metric) is invariably needed. The reference metric is required due to the fact that the interactionterms that can be formed from the metric alone, cannot be used to construct a mass term. In addition, this is anunphysical metric that does not have a direct influence on the geometrical nature of spacetime and it just helps usto find exact solutions and get rid of Boulware-Deser ghost instability. Considering the suitable reference metric,one finds various interesting publications in the context of dRGT massive gravity. Relativistic stars and black objectsolutions in dRGT massive gravity with interesting results have been investigated in [89–93]. On the other hand, itwas shown that massive gravity can be expressible on an arbitrary reference metric [88]. Therefore, a modificationin the reference metric could lead to another dRGT like massive theory. In this respect, Vegh has introduced a newreference metric with broken translational symmetry property [94]. In this massive gravity, similar to dRGT theory,massive terms are built by using this kind of reference metric which has an additional property. Also, in this theory,graviton may behave like a lattice and exhibits a Drude peak [94], and it is stable and free of ghost [95]. Neutron starshave been studied in this theory and it was found that the maximum mass of the neutron stars can be more than 3 M ⊙ ( M ⊙ is mass of the Sun) [96]. Black hole solutions and its thermodynamic properties have been investigated in Refs.[97–99]. Besides, the generalizations of this massive theory to include higher derivative gravity [100], and gravity’srainbow extension [101] have been studied as well. In addition, black hole and magnetic solutions with (non)linearelectrodynamics have been explored in the context of massive gravity [102, 103].In this paper, we want to study the magnetic solutions of Einstein-massive gravity with linear and nonlinearelectrodynamics in four and higher dimensions. This paper is one of interesting papers for considering the effects ofmassive gravitons on the horizonless solutions of nonsingular spacetime. Before proceeding, we provide some briefmotivations for considering arbitrary higher dimensional spacetimes. In the 20th century, Kaluza and Klein introduceda new theory of gravity in five dimensions which unified gravitation and electromagnetism [104, 105]. In addition,development of string and M-theories led to further progresses in higher dimensional gravity. Another motivationoriginates from the anti de Sitter/conformal field theory (AdS/CFT) correspondence which relates the properties of d -dimensional black holes with quantum field theory in ( d − II. BASIC FIELD EQUATIONS
The d -dimensional action in Einstein-massive gravity coupled to electromagnetic field is given by I G = − π Z M d d x √− g " R −
2Λ + L ( F ) + m X i =1 c i U i ( g, f ) , (1)where R is the scalar curvature, Λ = ± ( d − d − / l is the negative/positive cosmological constant for asymptot-ically AdS/dS solutions, and L ( F ) is an arbitrary Lagrangian of electrodynamics. In addition, f is a fixed symmetrictensor, c i ’s are massive coefficients, and U i ’s are symmetric polynomials of the eigenvalues of matrix K µν = √ g µα f αν U = [ K ] , U = [ K ] − (cid:2) K (cid:3) , U = [ K ] − K ] (cid:2) K (cid:3) + 2 (cid:2) K (cid:3) , U = [ K ] − (cid:2) K (cid:3) [ K ] + 8 (cid:2) K (cid:3) [ K ] + 3 (cid:2) K (cid:3) − (cid:2) K (cid:3) . Now, we can obtain the field equations by using variational principle. Varying the action (1) with respect to bothmetric tensor and gauge potential, one can obtain the following field equations R µν − g µν ( R − m χ µν = T µν , (2) ∂ µ (cid:0) √− g L F F µν (cid:1) = 0 , (3)where L F = d L ( F ) /d F and F = F µν F µν is the Maxwell invariant in which F µν = ∂ µ A ν − ∂ ν A µ is the Faraday tensorand A µ is the gauge potential. In addition, χ µν is the massive term with the following form χ µν = − c U g µν − K µν ) − c (cid:0) U g µν − U K µν + 2 K µν (cid:1) − c U g µν − U K µν +6 U K µν − K µν ) − c U g µν − U K µν + 12 U K µν − U K µν + 24 K µν ) . (4)and the energy-momentum tensor of electromagnetic source in Eq. (2) can be introduced as T µν = 12 g µν L ( F ) − L F F µλ F λν . (5) III. MAGNETIC SOLUTIONS IN EINSTEIN-MASSIVE GRAVITY WITH MAXWELL FIELD
Here, we are going to study the magnetic solutions of Eqs. (2) and (3) by considering the Maxwell electromagneticfield, namely L ( F ) = −F . To do so, we consider the metric of d − dimensional spacetime in the following explicit form ds = − ρ l dt + dρ g ( ρ ) + l g ( ρ ) dϕ + ρ l h ij dx i dx j , i, j = 1 , , , ..., n, (6)where g ( ρ ) is an arbitrary function of radial coordinate ρ which should be determined, h ij dx i dx j is the Euclideanmetric on the ( d − l is related to the cosmological constantΛ. In addition, the angular coordinate ϕ is dimensionless and ranges in 0 ≤ ϕ ≤ π while x i ’s range is ( −∞ , + ∞ ).The motivation of considering the metric gauge [ g tt ∝ − ρ and ( g ρρ ) − ∝ g ϕϕ ] instead of the usual Schwarzschildlike gauge [( g ρρ ) − ∝ g tt and g ϕϕ ∝ ρ ] comes from the fact that we are looking for the magnetic solutions instead ofelectric ones. In addition, one can obtain such magnetic metric with local transformations t → ilϕ and ϕ → it/l inthe horizon flat Schwarzschild like metric, ds = − g ( ρ ) dt + dρ g ( ρ ) + ρ dϕ + ρ l h ij dx i dx j . In other words, using suchtransformation, the metric (6) can be mapped to d -dimensional Schwarzschild like spacetime locally, but not globally,and therefore, both spacetimes are distinct.In order to obtain exact solutions, we should make a suitable choice for the reference metric. Regarding thementioned local transformation, we consider the following ansatz for the reference metric f µν = diag ( − c l , , , c l h ij ) , (7)where c in the above equation is a positive constant. Before we go on, we discuss the reason for considering such areference metric (7). In case of d dimensional black holes, the metric with ( − , + , ..., +) signature is given by ds = − g ( ρ ) dt + dρ g ( ρ ) + ρ dϕ + ρ l h ij dx i dx j , i, j = 1 , , , ..., n. (8)The black hole solutions in massive gravity are obtained using the ansatz metric f µν = diag (0 , , c , c l h ij ) forreference metric. In electrical black hole solutions, the metric function, g ( ρ ), is coupled with radial and tem-poral coordinates whereas in magnetic spacetime metric (Eq. 6), it is coupled with radial and spatial coordi-nates. Therefore, to obtain exact solutions in an axially symmetric spacetime with the form (6), reference metric f µν = diag (0 , , c , c l h ij ) should be modified into f µν = diag ( − c l , , , c l h ij ). It should be noted that using thereference metric ( f µν = diag ( − c l , , , c l h ij )), results into a new class of nontrivial solutions. In addition, it is worthmentioning that this choice for reference metric, first, cannot produce any infinite value for the bulk action, since thebulk action contains non-negative powers of f µν , and second, it does not preserve general covariance in the transversecoordinates t , x , x , .... .Using the metric ansatz (7), U i ’s can be calculated in the following forms U = d cρ , U = d d c ρ , U = d d d c ρ , U = d d d d c ρ , (9)where d i = d − i . Due to our interest to investigate the magnetic solutions, we should assume a suitable gaugepotential which leads to consistent field equations A µ = h ( ρ ) δ ϕµ . (10)Using the Maxwell equation (3) with L ( F ) = −F , and the metric (6), one finds the following differential equation d F ϕρ + ρF ′ ϕρ = 0 , (11)where F ϕρ = h ′ ( ρ ) and ”prime” denotes differentiation with respect to ρ . The solution of Eq. (11) is F ϕρ = qρ d , (12)where q is an integration constant which may be related to electric charge. Substituting Eqs. (6) and (10) in the fieldequation (2), one can obtain l g ′ ( ρ ) + ρd F ϕρ − d l m (cid:18) cc d + c c ρ + d c c ρ + d d c c ρ (cid:19) − (cid:16) g ( ρ ) + ρ d d (cid:17) ρ = 0 , (13) l g ′′ ( ρ ) + d l ρ g ′ ( ρ ) − F ϕρ − d l (cid:26) m (cid:18) cc ρ + d c c ρ + d d c c ρ + d d d c c ρ (cid:19) − d g ( ρ ) ρ − d (cid:27) = 0 . (14)Using the above equations, one can calculate the metric function g ( ρ ) as g ( ρ ) = m ρ d − ρ d d + 2 d q d ρ d + m (cid:18) c c + cc ρd + d c c ρ + d d c c ρ (cid:19) , (15)which m is an integration constant related to the mass. It is worthwhile to mention that in the absence of massiveparameter ( m = 0), the metric function (15) reduces to the Einstein-standard Maxwell [10].
1. Geometric Properties
In order to discuss the geometric properties of spacetime, we should focus on special points of spacetime (such asroots of the metric function) and boundary of radial coordinate (both ρ → ρ → ∞ ) as well.Since the second term (Λ term) of the metric function is dominant for large values of ρ , the asymptotical behaviorof the solution (15) is adS or dS provided Λ < > R µνλκ R µνλκ = (cid:18) d g ( ρ ) dρ (cid:19) + 2 d (cid:18) ρ dg ( ρ ) dρ (cid:19) + 2 d d (cid:18) g ( ρ ) ρ (cid:19) . (16)Using the metric function (15), it is easy to show that the Kretschmann scalar (16) diverges at ρ = 0, and therefore,one may guess that there is a curvature singularity located at ρ = 0, but as we will show, the spacetime will neverachieve ρ = 0. There are two possible cases for the metric function: first, the metric function has no root which isinterpreted as naked singularity, and second, the metric function has one or more roots. We assume that r + is thelargest real positive root of the metric function g ( ρ ). Therefore, the metric function g ( ρ ) will be negative for ρ < r + and positive for ρ > r + . This indicates that signature of the metric at this root changes from ( − , + , + , + , ..., +) changeto ( − , − , − , + , ..., +). In general relativity and gravity, although the field equations are metric dependent, they mustnot depend on the signature of metric [107–110]. The mentioned change in the signature of metric indicates that fieldequations for ρ > r + and ρ < r + are different resulting into two sets of different metric functions. To avoid suchinconsistency, the possibility of extending the spacetime to ρ < r + must be removed. To do so, we introduce a newradial coordinate r as r = ρ − r = ⇒ dρ = r r + r dr , (17)where ρ ≥ r + leads to r ≥
0. Applying this coordinate transformation, the metric (6) should be written as ds = − r + r l dt + r (cid:0) r + r (cid:1) g ( r ) dr + l g ( r ) dϕ + r + r l dX , (18)in which the coordinate ϕ assumes the value 0 ≤ ϕ < π , as usual. The metric function g ( r ) (Eq. (15)) is now givenby g ( r ) = m r d − (cid:0) r + r (cid:1) d d + 2 d q d r d + m c c + cc q r + r d + d c c q r + r + d d c c (cid:0) r + r (cid:1) , (19)The nonzero component of the electromagnetic field in the new coordinates can be given by F ϕr = q (cid:0) r + r (cid:1) d / . (20)One can show that all curvature invariants are functions of g ′′ , g ′ /r , and g/r . Since these terms do not diverge inthe range 0 ≤ r < ∞ , one finds that all curvature invariants are finite. Therefore, this spacetime has no curvaturesingularity and no horizon. However, the spacetime (18) has a conic geometry and has a conical singularity at r = 0,because the limit of the ratio ”circumference/radius” is not 2 π ,lim r −→ r r g ϕϕ g rr = 1 . (21)The conical singularity can be removed if one exchanges the coordinate ϕ with the following period P eriod ϕ = 2 π (cid:18) lim r −→ r r g ϕϕ g rr (cid:19) − = 2 π (1 − µ ) , (22)where µ is given by µ = 14 (cid:20) − lr g ′′ ( r ) | r =0 (cid:21) , (23)in which g ( r ) | r =0 = g ′ ( r ) | r =0 = 0, and g ′′ ( r ) | r =0 is g ′′ ( r ) | r =0 = − d − d ) q d r d + + m (cid:18) cc r + + d c c r + d d c c r + d d d c c r (cid:19) , (24)where shows that the metric (18) describes a spacetime which is locally flat, but has a conical singularity at r = 0with a deficit angle as δφ = 8 πµ. (25)Here, we skip investigation of physical properties of the obtained results. After obtaining the consequences ofnonlinear case, we give a detailed discussion with comparison. IV. MAGNETIC SOLUTIONS IN THE EINSTEIN-MASSIVE GRAVITY WITH PMI FIELD
In this section, we are going to obtain d -dimensional magnetic brane solutions in the presence of PMI field. There-fore, we consider the PMI Lagrangian with the following form L P MI ( F ) = ( − κ F ) s , (26)where κ and s are coupling and positive arbitrary constants, respectively. Since the Maxwell invariant is negative instatic spacetimes, hereafter, we set κ = 1 without loss of generality to obtain real solutions. Also, it is easy to showthat when s goes to 1, the PMI Lagrangian (26) reduces to the standard Maxwell Lagrangian ( L Maxwell ( F ) = −F )which we have investigated in the previous section. It is easy to show that for the case of power = dimension/
4, onecan obtain T µµ = 0 in PMI theory, which is confirmation of its conformal invariance properties in this case.Considering Eq. (26), the electromagnetic field equation (3) reduces to(2 s − ρh ′′ ( ρ ) + d h ′ ( ρ ) = 0 , (27)with the following solutions F ϕρ = h ′ ( ρ ) = ( qρ , s = d / (2 s − d ) q (2 s − ρ d / (2 s − , otherwise . (28)where q is an integration constant. Using Eqs. (6) and (28), one can show that the gravitational field equation (2)reduces to d d g ( ρ ) ρ + 2 d g ′ ( ρ ) ρ + g ′′ ( ρ ) + 2Λ + (cid:0) F ϕρ (cid:1) s − d m ρ (cid:0) c cρ + d c c ρ + d d c c ρ + d d d c c (cid:1) = 0 , (29) d d g ( ρ ) ρ + d g ′ ( ρ ) ρ + 2Λ + (1 − s ) (cid:0) F ϕρ (cid:1) s − d m ρ (cid:0) c cρ + d c c ρ + d d c c ρ + d d d c c (cid:1) = 0 . (30)Substituting Eq. (28) in the above equations, it is straightforward to show that the metric function g ( ρ ) has thefollowing form g ( ρ ) = m ρ d − ρ d d + m (cid:18) c c + cc ρd + d c c ρ + d d c c ρ (cid:19) + A , (31)in which A = ( √ q ) d ρ d ln (cid:0) ρl (cid:1) , s = d / s ρ (2 s − d (2 s − d ) (cid:16) (2 s − d ) q (2 s − ρ d / (2 s − (cid:17) s , otherwise . It is worthwhile to mention that in the absence of massive parameter ( m = 0), the metric function (31) is just likethe metric function which was obtained before in Ref. [111].Considering the Kretschmann scalar (16), one can show that the metric (6) with the metric function (31), like theMaxwell case, has a singularity at ρ = 0. However, as we mentioned before, it is not possible to extend the spacetime FIG. 1:
PMI solutions: δ ( φ ) versus r + for q = 0 . c = c = c = c = c = l = 1, d = 5, s = 0 . m = 0 (continuous line), m = 0 . m = 0 . m = 1 (dashed-dotted line). Left diagram:
Λ = − Right diagram:
Λ = 1. to ρ < r + because of signature changing (see Ref. [80], for more details). Also, one can apply the coordinatetransformation (17) to the metric (6) and find the metric function as g ( r ) = m r d − (cid:0) r + r (cid:1) d d + m c c + cc q r + r d + d c c q r + r + d d c c (cid:0) r + r (cid:1) + ( √ q ) d ( √ r + r ) d ln (cid:18) √ r + r l (cid:19) , s = d / s ( r + r ) (2 s − d (2 s − d ) (cid:18) (2 s − d ) q (2 s − ( √ r + r ) d / (2 s − (cid:19) s , otherwise , (32)and the electromagnetic field in the new coordinate is F ϕr = q √ r + r , s = ( d − / (2 s − d ) q (2 s − ( √ r + r ) d / (2 s − , otherwise . (33)In this case, like the Maxwell case, this spacetime has a conical singularity at r = 0 with the deficit angle δ ( φ ) = 8 πµ where µ is modified due to the nonlinear electrodynamics with the following form g ′′ ( r ) | r =0 = − d + m (cid:18) cc r + + d c c r + d d c c r + d d d c c r (cid:19) + d / q d r d , s = d / s (2 s − d (cid:18) q ( d − s ) (2 s − r d / (2 s − (cid:19) s , otherwise . (34)Due to the complexity of obtained relation in Eq. (34), it is not possible to calculate the root and divergence pointsof deficit angle analytically, therefore, we study them in some graphs.Before starting, we should point it out that we have an upper limit of −∞ < δφ ≤ π on the values that deficitangle can acquire. This limit is marked with a horizontal dotted line in plotted diagrams. The value of deficit angledetermines the geometrical structure of solutions. Depending on geometrical properties, gravitational effects andlensing properties of the magnetic solutions, hence topological defects will be different. Here, we see that dependingon choices of different parameters, deficit angle could be positive/negative and it may have roots and divergencepoints. In order to highlight the effects of background spacetime, we have plotted two series of diagrams for AdS (leftpanels of Figs. 1-5) and dS (right panels of Figs. 1-5). FIG. 2:
PMI solutions: δ ( φ ) versus r + for c = c = c = c = c = m = l = 1, d = 5, s = 0 . q = 0 (continuous line), q = 0 .
007 (dotted line), q = 0 . q = 0 . Left diagram:
Λ = − Right diagram:
Λ = 1.FIG. 3:
PMI solutions: δ ( φ ) versus r + for q = 0 . c = c = c = c = m = l = 1, d = 5, s = 0 . c = −
10 (continuous line), c = − .
87 (dotted line), c = − . c = − .
855 (dashed-dotted line) and c = − Left diagram:
Λ = − Right diagram:
Λ = 1.
Evidently, for AdS case, depending on the choices of different parameters, deficit angle could have: I) two rootsin which between roots, the deficit angle negative valued whereas before smaller and after larger roots, it is positive.II) one extreme root in which the deficit angle is always positive valued. III) two roots with one divergency wherebetween smaller/larger root and divergency the deficit angle is negative and everywhere else, it is positive valued. IV)finally, two roots with two divergencies in which the divergencies are located between the roots. In this case, betweensmaller (larger) root and smaller (larger) divergency, the deficit angle is negative valued. Between divergencies, it ispositive but its values are not in permitted area. Only before (after) smaller (larger) root, the deficit angle is positivevalued and within permitted area.On contrary, for dS case, plotted diagrams show existence of a root and a divergency for deficit angle. Before rootand after divergency, the deficit angle is positive where only before root, permitted values of the deficit angle existswhereas after divergency its values are not within permitted ones.The number of roots is a decreasing function of the mass of graviton ( m ) (left panel of Fig. 1), electric charge (leftpanel of Fig. 2), c (left panel of Fig. 3), nonlinearity parameter (left panel of Fig. 4) and dimensions (left panel ofFig. 5) for AdS case. On the other hand, for dS case, the places of root and divergency are increasing functions of m (right panel of Fig. 1), q (right panel of Fig. 2), c (right panel of Fig. 3), s (right panel of Fig. 4) and dimensions FIG. 4:
PMI solutions: δ ( φ ) versus r + for q = 0 . c = c = c = c = c = m = l = 1, d = 5, s = 0 . s = 1 (dotted line) and s = 1 . Left diagram:
Λ = − Right diagram:
Λ = 1.FIG. 5:
PMI solutions: δ ( φ ) versus r + for q = 0 . c = c = c = c = c = m = l = 1, s = 0 . d = 5 (continuous line), d = 6 (dotted line), d = 7 (dashed line) and d = 8 (dashed-dotted line). Left diagram:
Λ = − Right diagram:
Λ = 1. (right panel of Fig. 5).The existence of positive valued deficit angle results into conic like geometrical structure for our astrophysicalobjects, hence topological defects are known as horizonless magnetic solutions. On contrary, the existence of negativevalues of deficit angle leads to a saddle-like cone structure for the solutions. These two different structures for magneticsolutions could be related to different second fundamental form of spacetime. On the other hand, it was argued thatpositivty/negativity of the deficit angle results into attractive-type/repulsive-type gravitational potentials (furthersdetails could be found in Refs. [112–115]).Considering different geometrical structure depending on the sign of deficit angle, one can conclude that the root ofdeficit angle is where magnetic solutions have phase transition-like behavior. In other words, since there is a changeof sign at the root of deficit angle, magnetic solutions go under a typical topological phase transition in these points.It could be pointed out that there are cases in which roots are extreme ones. In these cases, although no change ofsign takes place, the total geometrical structure of the solutions presents diverse different comparing to the non-zerodeficit angle (absence of conic like singularity for zero deficit angle). Therefore, it could be stated that extreme rootsare also marking phase transition points. Another point which carries the properties of phase transition for magneticsolutions is divergency of the deficit angle. In other words, divergencies of the deficit angle could be interpreted as0places in which magnetic solutions go under a phase transition. This is due to the fact that deficit angle has smoothbehavior everywhere except at divergencies which are discontinuities. Usually, around these divergence points, thesign of deficit angle is changed. In other words, there is a change in the sign of deficit angle before and after divergencepoint.Although different parameters have specific contributions in existence/absence of root and divergency for deficitangle, the highest effects belong to the Λ term, hence structure of the background spacetime.For dS spacetime (positive Λ), existence of both points (root and divergence) irrespective of different parameters isevident. Before the divergency, the values of deficit angle are within permitted area while after it, the values are inforbidden region. The root of deficit angle in this case is located at the permitted area. Therefore, one can state thatfor dS case, the existence of deficit angle is limited to region before its divergency and in this region, deficit angleenjoys a phase transition related to the existence of root. The length of permitted region for deficit angle is a functionof massive parameters, electric charge, dimensions and nonlinearity parameter.For AdS spacetime, the situation is different. Existence of divergency depends on positivity and negativity of massivecoefficient c and it is found for sufficiently small and negative values of this parameter. Interestingly, contrary todS case, AdS spacetime could enjoy the existence of up to two divergencies in its deficit angle (for sufficiently smalland negative c ). In the case of one divergence point, the divergency exists between two roots and signature of thedeficit angle around it is the same (it is negative). In this case, the deficit angle enjoys two roots and one divergency.For the case of two divergencies, the divergence points are between two roots. Around divergencies the sign of deficitangle changes. Between the divergencies, the deficit angle is positive valued but within prohibited region. Therefore,the magnetic solutions have phase transition over a region which is marked with divergencies. This shows that inthis case, the deficit angle has two roots and one divergency with a prohibited region. The study here showed thatgeneralization to massive gravity introduces some new phase transitions into magnetic solutions. This highlights theeffects of the massive gravity in geometrical structure of the solutions, hence their physical properties. V. CONCLUSIONS
The paper was dedicated to study the nonlinearly charged magnetic brane solutions in the presence of massivegravity. The exact solutions were obtained and the absence of black hole solutions was confirmed. The existence ofconic like singularity was shown and it was pointed out that geometrical, hence, physical/gravitational properties ofthe solutions depend on a value known as deficit angle.This property of the solutions (deficit angle) determines the total structure of magnetic branes. There is a diversedifference in the geometrical properties of the solutions with positive deficit angle comparing to negative ones. Thesegeometrical properties are providing guidelines for how phenomena such as lensing property would be different. Thatbeing said, roots and divergencies could be interpreted as topological phase transition points. In roots, the transitionis being done smoothly while in the divergencies, system jumps between different deficit angles, hence geometricalstructure.In general, it was shown that existence of the divergencies for deficit angle were the background spacetime andmassive gravity dependent. If the massive coefficients are positive valued, only for dS background, deficit anglecould acquire divergency whereas, the AdS case enjoys only root in its deficit angle. On the contrary, if the massivecoefficients could be negative, for both AdS and dS backgrounds, it is possible to introduce multi geometrical phasetransition and a prohibited region. Existence of the prohibited regions indicates that our magnetic solutions arebounded by specific limits. These limiting areas and the conditions for them are rooted in massive gravity and itscoefficients. Despite the effects of other parameters on limiting areas and the conditions, in the absence of massivecoefficients, these limiting areas would rather vanish or significantly be modified. The effects of nonlinearity nature ofthe solutions in the case of AdS spacetime was in level of modifying the number of roots. Whereas for dS spacetime,it was only in level of modifying the prohibited/permitted region for deficit angle.The obtained solutions here contain magnetic brane ones. Considering the AdS nature of the solutions and theirphase transitions, it is possible to conduct studies in the context of AdS/CFT correspondence. Furthermore, onecan investigate trajectory of the particles and lensing properties of these solutions in more details to understand theeffects of massive gravity and nonlinear electromagnetic fields. In addition, it is notable that our solutions are staticand independent of time. One may modify these solutions to the case of dynamic time dependent for investigatingthe ”self-acceleration” properties [116, 117]. We leave these subjects for the future works.1
Acknowledgments
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