Exact charged black-hole solutions in D-dimensional f(T) gravity: torsion vs curvature analysis
Salvatore Capozziello, P. A. Gonzalez, Emmanuel N. Saridakis, Yerko Vasquez
aa r X i v : . [ h e p - t h ] O c t Exact charged black-hole solutions in D-dimensional f ( T ) gravity: torsion vs curvature analysis S. Capozziello a,b
P. A. Gonz´alez c,d
E. N. Saridakis e,f,g
Y. V´asquez h a Dipartimento di Scienze Fisiche, Universit‘a di Napoli “Federico II”, Napoli, Italy b INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126,Napoli, Italy c Escuela de Ingenier´ıa Civil en Obras Civiles. Facultad de Ciencias F´ısicas y Matem´aticas, Uni-versidad Central de Chile, Avenida Santa Isabel 1186, Santiago, Chile. d Universidad Diego Portales, Casilla 298-V, Santiago, Chile. e Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens,Greece f CASPER, Physics Department, Baylor University, Waco, TX 76798-7310, USA g Institut d’Astrophysique de Paris, UMR 7095-CNRS, Universit´e Pierre & Marie Curie, 98bisboulevard Arago, 75014 Paris, France h Departamento de Ciencias F´ısicas, Facultad de Ingenier´ıa, Ciencias y Administraci´on, Universi-dad de La Frontera, Avenida Francisco Salazar 01145, Casilla 54-D, Temuco, Chile.
E-mail: [email protected] , [email protected] , Emmanuel [email protected] , [email protected] Abstract:
We extract exact charged black-hole solutions with flat transverse sections inthe framework of D-dimensional Maxwell- f ( T ) gravity, and we analyze the singularitiesand horizons based on both torsion and curvature invariants. Interestingly enough, we findthat in some particular solution subclasses there appear more singularities in the curvaturescalars than in the torsion ones. This difference disappears in the uncharged case, or in thecase where f ( T ) gravity becomes the usual linear-in- T teleparallel gravity, that is GeneralRelativity. Curvature and torsion invariants behave very differently when matter fields arepresent, and thus f ( R ) gravity and f ( T ) gravity exhibit different features and cannot bedirectly re-casted each other. Keywords:
Modified gravity; f(T) gravity; teleparallel gravity; black holes; singularities ontents f ( T ) extension 23 D-dimensional Teleparallel Gravity and its Maxwell- f ( T ) extension 34 Exact charged solutions 5 Teleparallel equivalent of General Relativity (TEGR) [1, 2] is an equivalent formulation ofgravity, but, instead of using curvature invariants defined by the Levi-Civita connection,the Weitzenb¨ock connection is adopted. Therefore TEGR exhibits no curvature but onlytorsion. The dynamical objects in such a framework are the four linearly independent vierbeins and the advantage of this framework is that the torsion tensor is formed solely byproducts of first derivatives of these vierbeins. In such a formulation, as described in [2],the Lagrangian density, T , can then be constructed from this torsion tensor assuming theinvariance under general coordinate transformations, global Lorentz transformations, andthe parity operation, along with demanding the Lagrangian density to be second order inthe torsion tensor. In [3–6] an extension of the above idea was constructed, making theLagrangian density a function of T , similar to the f ( R ) extension of the Hilbert-Einsteinaction. f ( T ) gravity has gained a significant attention in the literature, and proves toexhibit interesting cosmological implications [3–51].Such an approach can be framed within the class of new gravity theories aimed toextend General Relativity in order to solve its shortcomings at Infra-Red and Ultra-Violetscales [52]. Clearly, in extending the geometry sector, one of the goals is to solve thepuzzle of dark energy and dark matter that, up to now, seems to have no counterpart atfundamental level. In other words, both f ( T ) gravity and f ( R ) gravity could be reliableapproaches to address the problems of missing matter and accelerated expansion withoutasking for new material ingredients that have not been detected yet by the experiments[53]. – 1 –n this work we investigate D-dimensional f ( T ) gravity, considering additionally theelectromagnetic sector. Exact black-hole solutions with flat transverse section (Banados-Teitelboim-Zanelli (BTZ)-like solutions [54]) can be derived for a given range of parameterspace. Then we analyze the singularities of these solutions based on the torsion scalar andthe curvature scalar, pointing out differences with respect to f ( R ) gravity. It is importantto stress that searching for exact solutions is a fundamental step to set a new field theory.Exact solutions allow a full control of the systems and can contribute to the well-formulationand well-position of the Cauchy problem (for a discussion on this point see [55]).The paper is organized as follows. In Sec. 2, we present a brief review of TEGR infour dimensions, as well as of its f ( T ) extension. In Sec. 3 the D-dimensional teleparallelgravity is formulated and the analysis is extended to D-dimensional Maxwell- f ( T ) gravity.In Sec. 4 we derive exact charged static solutions and Sec. 5 is devoted to the investigationof singularities and horizons. Finally, in Sec. 6 we discuss some physical implications ofthe results. f ( T ) extension In this section we briefly review Teleparallel Equivalent to General Relativity (TEGR) infour dimensions and its f ( T ) extension. Throughout the manuscript we use the followingnotation: Greek indices µ, ν, ... run over all space-time coordinates 0, 1, 2, 3; lower caseLatin indices (from the middle of the alphabet) i, j, ... run over spatial coordinates 1, 2,3; capital Latin indices A, B, ... run over the tangent space-time 0, 1, 2, 3, and lower caseLatin indices (from the beginning of the alphabet) a, b, ... run over the tangent space spatialcoordinates 1, 2, 3.As we mentioned above, the dynamical variable of teleparallel gravity is the vierbeinfield e A ( x µ ), which forms an orthonormal basis for the tangent space at each point x µ of the manifold, that is e A · e B = η AB , with η AB = diag (1 , − , − , − e A can be expressed in terms of its components e µA in a coordinate basis, namely e A = e µA ∂ µ . In such a formulation the metric tensor is acquired from the dual vierbein as g µν ( x ) = η AB e Aµ ( x ) e Bν ( x ) . (2.1)Although in General Relativity one uses the torsionless Levi-Civita connection, in thepresent construction one uses the curvatureless Weitzenb¨ock connection [56], whose torsiontensor reads T λµν = w Γ λνµ − w Γ λµν = e λA ( ∂ µ e Aν − ∂ ν e Aµ ) . (2.2)Moreover, the contorsion tensor, which gives the difference between Weitzenb¨ock and Levi-Civita connections, is given by K µνρ = − ( T µνρ − T νµρ − T µνρ ), while it proves convenientto define S µνρ = (cid:0) K µνρ + δ µρ T ανα − δ νρ T αµα (cid:1) . For a detailed exposition of torsion tensorproperties see [57].In conclusion, in the present formulation the torsion tensor T λµν includes all the infor-mation concerning the gravitational field. Using the above definitions one can construct the– 2 –implest form of the “teleparallel” Lagrangian, which is the torsion scalar, that is [58, 59] L = T ≡ T ρµν T ρµν + 12 T ρµν T νµρ − T ρρµ T νµν . (2.3)Thus, the simplest action of teleparallel gravity reads: S = 12 κ Z d xe ( T + L m ) , (2.4)where κ = 8 πG , e = det( e Aµ ) = √− g and L m accounts for the matter Lagrangian. It isworth noticing here that the Ricci scalar R and the torsion scalar T differ only by a totalderivative of the torsion tensor, namely [60]: R = − T − ∇ µ (cid:0) T νµν (cid:1) . (2.5)Varying the action (2.4) with respect to the vierbein we obtain the field equations e − ∂ µ ( ee ρA S ρµν ) − e λA T ρµλ S ρνµ − e νA T = 4 πGe ρA em T ρν , (2.6)where the tensor em T ρν on the right-hand side is the usual energy-momentum tensor ofmatter fields. These equations coincide with those of General Relativity for every geom-etry choice, and this is the why the theory is named “Teleparallel Equivalent to GeneralRelativity”.One can generalize the above formulation considering arbitrary functions of the torsionscalar f ( T ) in the gravitational action [3–6], although the Lorentz invariance of the lineartheory seems to be spoiled [61, 62]. Thus, the action becomes S = 12 κ Z d xe [ T + f ( T ) + L m ] . (2.7)Notice the difference in the various conventions in 4D- f ( T ) literature, since some authorsreplace T by f ( T ), while the majority replace T by T + f ( T ). In this work we follow thesecond convention, that is teleparallel gravity is acquired by setting f ( T ) = 0. Finally,variation of the action (2.7) with respect to the vierbein gives the field equations e − ∂ µ ( ee ρA S ρµν ) (cid:18) dfdT (cid:19) − e λA T ρµλ S ρνµ + e ρA S ρµν ∂ µ ( T ) d fdT − e νA [ T + f ( T )] = 4 πGe ρA em T ρν . (2.8) f ( T ) extension In this section we present teleparallel gravity in D-dimensions and its Maxwell- f ( T ) ex-tension and we explore its properties. It proves more convenient to use differential forms,where the torsion 2-form T a is simply T a = de a .We start with the gravitational teleparallel action with the most general quadraticform in the torsion tensor. Under the assumption of zero spin-connection it is given by[63, 64] S = 12 κ Z ( ρ L + ρ L + ρ L + ρ L + ρ L ) , (3.1)– 3 –here ρ i are dimensionless parameters and L = 14 e a ∧ ⋆e a , L = de a ∧ ⋆de a , L = ( de a ∧ ⋆e a ) ∧ ⋆ ( de b ∧ e b ) , L = ( de a ∧ e b ) ∧ ⋆ ( de a ∧ e b ) , L = ( de a ∧ ⋆e b ) ∧ ⋆ ( de b ∧ e a ) , (3.2)with ⋆ standing for the Hodge dual operator and ∧ for the usual wedge product. Thecoupling constant ρ = − Λ accounts for the cosmological constant term, and furthermore,since L can be completely expressed in terms of L , in the following we set ρ = 0[63]. Lastly, we mention that in the above expression κ is the D-dimensional gravitationalconstant, while the vierbeins and the metric are now D-dimensional. Therefore, in thefollowing, all the conventions adopted in Sec. 2 extend in D dimensions.Action (3.1) can be written more conveniently as S = 12 κ Z ( T − ⋆ κ Z d D x e ( T − , (3.3)where ⋆ − D − e ∧ e ∧ e ...... ∧ e D − , and the torsion scalar T is given by T = ( − D − ⋆ h ρ ( de a ∧ ⋆de a ) + ρ ( de a ∧ e a ) ∧ ⋆ ( de b ∧ e b ) + ρ ( de a ∧ e b ) ∧ ⋆ ( de b ∧ e a ) i . (3.4)Expanding this expression in its components we acquire T = 12 ( ρ + ρ + ρ ) T abc T abc + ρ T abc T bca − ρ T aca T bbc , (3.5)thus we straightforwardly see that for ρ = 0, ρ = − and ρ = 1 it coincides with (2.3)in D dimensions, namely T = 14 T abc T abc − T abc T bca − T aac T bbc . (3.6)Now, we will extend the above discussion considering arbitrary functions of the torsionscalar f ( T ) in the D-dimensional gravitational action. Thus, we consider an action of theform S = 12 κ Z d D xe [ T + f ( T ) − , (3.7)with the torsion scalar T given by (3.5), that is we keep the general coefficients ρ i . Indifferential forms the above action can be written as S = 12 κ Z { [ f ( T ) + T − ⋆ } , (3.8)where now T is given by (3.4). Finally, note that teleparallel D-dimensional gravity dis-cussed above is obtained by setting f ( T ) = 0.Lastly, we extend the discussion incorporating additionally the electromagnetic sector.In particular, we extend the total action to S = 12 κ Z { [ f ( T ) + T − ⋆ } + Z L F , (3.9)– 4 –here L F = − F ∧ ⋆ F (3.10)is the Maxwell Lagrangian, while F = dA , with A ≡ A µ dx µ , is the electromagnetic poten-tial 1-form. The action variation leads to the following field equations: δ L = δe a ∧ (cid:26)(cid:18) dfdT (cid:19) n ρ h d ⋆ de a + i a ( de b ∧ ⋆de b ) − i a ( de b ) ∧ ⋆de b i + ρ n − e a ∧ d ⋆ ( de b ∧ e b ) + 2 de a ∧ ⋆ ( de b ∧ e b ) + i a h de c ∧ e c ∧ ⋆ ( de b ∧ e b ) i − i a ( de b ) ∧ e b ∧ ⋆ ( de c ∧ e c ) o + ρ n − e b ∧ d ⋆ ( e a ∧ de b ) + 2 de b ∧ ⋆ ( e a ∧ de b )+ i a h e c ∧ de b ∧ ⋆ ( de c ∧ e b ) i − i a ( de b ) ∧ e c ∧ ⋆ ( de c ∧ e b ) oo +2 d fdT dT h ρ ⋆ de a + ρ e a ∧ ⋆ ( de b ∧ e b ) + ρ e b ∧ ⋆ ( de b ∧ e a ) i + (cid:20) f ( T ) − T dfdT (cid:21) ∧ ⋆e a − ⋆ e a −
12 [ F ∧ i a ( ⋆F ) − i a ( F ) ∧ ⋆F ] (cid:27) + δA ( d ⋆ F ) = 0 , (3.11)where i a is the interior product. Although one could investigate solution subclasses withgeneral coupling parameters ρ i , in the following, for the sake of simplicity, we restrict tothe standard case ρ = 0, ρ = − / ρ = 1 of (3.6). Let us now investigate the charged solutions of the theory. In order to extract the staticsolutions we consider the metric form ds = F ( r ) dt − G ( r ) dr − r i = D − X i =1 dx i , (4.1)which arises from the vierbein diagonal ansatz e = F ( r ) dt , e = 1 G ( r ) dr , e = rdx , e = rdx , . . . . (4.2)Let us make an important comment here concerning the vierbein choice that correspondsto the metric (4.1). In the case of linear-in- T gravity the above simple, diagonal relationbetween the metric (4.1) and the vierbeins (4.2) is always allowed. On the contrary, inthe extension of f ( T ) gravity, in general, one could have a more complicated relationconnecting the vierbein with the metric, with the vierbein being non-diagonal even for adiagonal metric [65–68]. However, in the cosmological investigations of f ( T ) gravity [3–51], as well as in its black-hole solutions [69–73], the authors still use the diagonal relationbetween the vierbeins and the metric, as a first approach to reveal the features of thetheory. Thus, in the present investigation we also impose the diagonal relation between– 5 –he vierbeins and the metric, as a first approach on the subject and in order to reveal themain features of the solution structure. However, we are aware that a detailed study of thegeneral vierbein choice (and its relation to extra degrees of freedom) is a necessary step fora deeper understanding of f ( T )-gravity foundations.Concerning the electric sector of the electromagnetic 2-form we assume F = dA = E r ( r ) e ∧ e + i = D − X i =1 E i ( r ) e ∧ e i +1 , (4.3)where E r is the radial electric field, neglecting for the moment the magnetic part. Thus,inserting the above ansatzes in the field equations (3.11), we finally obtain (cid:18) dfdT (cid:19) T − (cid:20) f ( T ) − T dfdT (cid:21) + 2Λ + 12 E r − i = D − X i =1 E i = 0 , (4.4) (cid:18) dfdT (cid:19) " − G ( r ) G ′ ( r ) r + F ′ ( r ) G ( r ) rF ( r ) − d fdT T ′ ( r ) G ( r ) r − i = D − X i =1 E i = 0 , (4.5) (cid:18) dfdT (cid:19) " − F ′′ ( r ) G ( r ) F ( r ) − F ′ ( r ) G ′ ( r ) G ( r ) F ( r ) + F ′ ( r ) G ( r ) rF ( r ) − (cid:18) dfdT (cid:19) " ( D − G ′ ( r ) G ( r ) r − ( D − G ( r ) r − d fdT T ′ ( r ) " F ′ ( r ) G ( r ) F ( r ) + ( D − G ( r ) r + 12 E r − E = 0 , (4.6) E r E j = 0 j = 1 , . . . , D − , (4.7) E i E j = 0 i, j = 1 , . . . , D − i = j ) , (4.8)where T ( r ) = 2 ( D − F ′ ( r ) G ( r ) rF ( r ) + ( D −
2) ( D − G ( r ) r . (4.9)The remaining field equations are equivalent to equation (4.6), that is the a = j equationis similar to (4.6), but with − E replaced by − E j − .A first observation is that from (4.4) we deduce that T has, in general, an r -dependence,which disappears for a zero electric charge. Such a behavior reveals the new featuresthat are brought in by the richer structure of the addition of the electromagnetic sector.Moreover, form (4.7) and (4.8), we deduce that we cannot have simultaneously two non-zeroelectric field components. This result is similar to the known no-go theorem of 3D GR-likegravity [74, 75], which states that configurations with two non-vanishing components ofthe Maxwell field are dynamically not allowed. However, it is not valid anymore if we addthe magnetic sector, as we will see in subsection 4.3 (it holds only for D=3). Therefore,in the following we investigate the cases of radial electric field, of non-radial electric field,and of magnetic and radial electric field, separately.– 6 – .1 Radial electric field We first consider the case where there exists only radial electric field. Thus, the Maxwellequations give E r = Qr D − , (4.10)where Q is an integration constant which, as usual, coincides with the electric charge ofthe black hole. Now, integrating equation (4.5) we find the very simple and helpful result F ( r ) = G ( r ) (cid:18) dfdT (cid:19) . (4.11)Using equations (4.11) and (4.9) we obtain dG ( r ) dr + (cid:20) ddr ln (cid:18) dfdT (cid:19) + ( D − r (cid:21) G ( r ) − rT ( r )( D −
2) = 0 , (4.12)whose solution is G ( r ) = 1 (cid:16) dfdT (cid:17) r D − " D − Z (cid:18) dfdT (cid:19) r D − T ( r ) dr + Const , (4.13)and using equation (4.11) we get F ( r ) = 1 r D − " D − Z (cid:18) dfdT (cid:19) r D − T ( r ) dr + Const , (4.14)where Const is an integration constant related to the mass of the spherical object.In order to proceed, and similar to [73], we will consider Ultraviolet (UV) correctionsof f ( T ) gravity. In particular, we examine the modifications on the solutions caused by UVmodifications of D-dimensional gravity and we consider a representative ansatz of the form f ( T ) = αT . This is the first order correction in every realistic f ( T ) gravity, in which weexpect f ( T ) ≪ T [14, 40], since T (like R ) is small in κ -units. Thus, for α = 0, equation(4.4) leads to T ( r ) = − ± p − α Λ − αQ r − D α , (4.15)with the upper and lower signs corresponding to the positive and negative branch solutionsrespectively (note that if α = 0 then (4.4) becomes linear having only one solution, which isgiven by the α → T ( r ) = − Q r − D / − dfdT = 23 ± p − α Λ − αQ r − D , (4.16)– 7 –nd therefore performing the integration that appears in (4.13) and (4.14), we obtain Z (cid:18) dfdT (cid:19) r D − T ( r ) dr = 154 α (cid:20) − αQ r − D − D − (1 + 72 α Λ) r D − D − (cid:21) ± p r D (1 − α Λ − αQ r − D )54 α (cid:20) αQ r − D D − − ( − α Λ) r − − D D − (cid:21) ∓ ( D − ( − α Λ) Q r D q αQ r − D − α Λ 2 F (cid:16) D − D − , , D − D − ; αQ r − D − α Λ (cid:17) D −
3) (2 D −
5) ( D − p r D (1 − α Λ − αQ r − D ) , (4.17)where F ( a, b, c ; x ) is the hypergeometric function. We mention that the last argument ofthis function, namely (cid:0) αQ r − D (cid:1) / (1 − α Λ), must be negative, while from (4.15) it isrequired that 1 − α Λ − αQ r − D must be positive, therefore we deduce that α shouldbe negative.In summary, inserting the integral (4.17), along with (4.16), in (4.13) and (4.14), wefind that the black-hole solution is: G ( r ) = 1 (cid:16) ± p − α Λ − αQ r − D (cid:17) r D − n D − n α h − αQ r − D − D − (1 + 72 α Λ) r D − D − i ± p r D (1 − α Λ − αQ r − D )54 α h αQ r − D D − − ( − α Λ) r − − D D − i ∓ ( D − ( − α Λ) Q r D q αQ r − D − α Λ 2 F (cid:16) D − D − , , D − D − ; αQ r − D − α Λ (cid:17) D −
3) (2 D −
5) ( D − p r D (1 − α Λ − αQ r − D ) o + Const o (4.18)and F ( r ) = 1 r D − n D − n α h − αQ r − D − D − (1 + 72 α Λ) r D − D − i ± p r D (1 − α Λ − αQ r − D )54 α h αQ r − D D − − ( − α Λ) r − − D D − i ∓ ( D − ( − α Λ) Q r D q αQ r − D − α Λ 2 F (cid:16) D − D − , , D − D − ; αQ r − D − α Λ (cid:17) D −
3) (2 D −
5) ( D − p r D (1 − α Λ − αQ r − D ) o + Const o . (4.19)The special point in the parameter space Λ = 1 / (24 α ) needs to be analyzed separately,– 8 –ince in this point we obtain the solution Z (cid:18) dfdT (cid:19) r D − T ( r ) dr = − r − − D α " r D D − − αQ r D D − ± √ (cid:0) − αQ r D (cid:1) / D − , (4.20)and thus G ( r ) = 1 (cid:16) ± p − α Λ − αQ r − D (cid:17) r D − n D − n − r − − D α h r D D − − αQ r D D − ± √ (cid:0) − αQ r D (cid:1) / D − io + Const o , (4.21)and F ( r ) = 1 r D − n D − n − r − − D α h r D D − − αQ r D D − ± √ (cid:0) − αQ r D (cid:1) / D − io + Const o . (4.22)Finally, the case D = 3 has to be analyzed separately. Taking properly the limit D = 3 of the above expressions we obtain the solutions extracted in [73] for 3D Maxwell- f ( T ) gravity. Lastly, one can straightforwardly check that in the limit α → Let us for the moment assume that we have zero radial field. In this case equation (4.8)implies that we can have at most one non-zero component of the electric field along thenon-radial (transversal) directions. However, as we mentioned below equation (4.9), for
D > − E replacedby − E j − , therefore subtracting these equations we acquire the conditions E i = E j , with i and j running from 1 to D −
2. These conditions, along with equation (4.8), yield E i = 0( i = 1 , ..., D −
2) for
D >
3, that is the electric field is completely zero. The only caseswhere zero radial electric field does not lead to a disappearance of the total electric fieldis for D = 3 (where a non-zero azimuthal electric field is possible) which was analyzedin detail in [73], or if we consider simultaneously non-zero non-radial electric field withmagnetic field, case which lies beyond the scope of the present investigation. For completeness we also examine the case where magnetic field is present. While in D = 3we deduce that electric field must be absent [73], for D > F = E r ( r ) e ∧ e + B ( r ) e ∧ e , (4.23)– 9 –hat is we consider a radial electric field E r and a magnetic field B both depending on theradial coordinate r only. From the Maxwell equations in four dimensions for the electricfield we immediately obtain E r ( r ) = Qr , (4.24)while incorporating the equations of motion analogous to (4.4)-(4.8) we can see that asolution is obtained by B ( r ) = Pr , (4.25)leading to the metric coefficients (4.19) and (4.20) with Q + P in place of Q (and for D = 4). Let us now investigate the singularities and the horizons of the above solutions. Thefirst step is to find at which r do the functions G ( r ) and F ( r ) become zero or infinity.However, since these singularities may correspond to coordinate singularities, the usualprocedure is to investigate various invariants, since if these invariants diverge at one pointthey will do that independently of the specific coordinate basis, and thus the correspondingpoint is a physical singularity (note that the opposite is not true, that is the finiteness of aninvariant is not a proof that there is not a physical singularity there). In standard black-holeliterature of curvature-formulated gravity (either General Relativity or its modifications),one usually studies the Ricci scalar, the Kretschmann scalar, or other invariants constructedby the Riemann tensor and its contractions.In teleparallel description of gravity, one has, in principle, two approaches of findinginvariants. The first is to use the solution for the vierbein and the Weitzenb¨ock connectionin order to calculate torsion invariants such as the torsion scalar T . The second is to use thesolution for the corresponding metric in order to construct the Levi-Civita connection, andthen use it to calculate curvature invariants such are the Ricci and Kretschmann scalars(a calculation of curvature scalars using straightaway the Weitzenb¨ock connection leads tozero by construction). The comparison of both approaches is a main subject of interest ofthe present work, capable of pointing out differences between curvature and torsion gravity.In particular, we are going to investigate whether one can formulate everything interms of vierbens, Weitzenb¨ock’s connection and torsion invariants, as one can do with themetric, the Levi-Civita connection and curvature invariants. Perhaps one could say thatthe use of curvature invariants, instead of torsion ones, is better justified by the fact that ina realistic theory matter is coupled to the gravitational sector through the metric and notthrough the vierbeins (with the interesting exception of fermionic matter), and particlesfollow geodesics defined by the Levi-Civita connection. On the other hand, one could saythat the two approaches are equivalent only with a suitable, non-diagonal, relation betweenthe vierbeins and the metric. In any case, while at the classical level the above approachescould look equivalent or alternative, for the quantization procedure it would be crucialto determine whether the metric or the vierbein is the fundamental field. Therefore, thefollowing analysis can enlighten this subject.– 10 –he torsion invariant T , that is the torsion scalar, that arises from the vierbein solution(4.2) with the use of Weitzenb¨ock’s connection is (4.9), which in the examined case becomesjust (4.15). The curvature invariants that arise from the metric solution (4.1) through thecalculation of the Levi-Civita connection are R = − G ( r ) F ′′ ( r ) F ( r ) − G ( r ) G ′ ( r ) F ′ ( r ) F ( r ) − D − G ( r ) F ′ ( r ) rF ( r ) − D − G ( r ) G ′ ( r ) r − ( D −
2) ( D − G ( r ) r , (5.1) R µν R µν = G ( r ) F ( r ) r (cid:2) rF ′′ ( r ) G ( r ) + rF ′ ( r ) G ′ ( r ) + ( D − G ( r ) F ′ ( r ) (cid:3) + G ( r ) F ( r ) r (cid:2) rF ′′ ( r ) G ( r ) + rF ′ ( r ) G ′ ( r ) + ( D − G ′ ( r ) F ( r ) (cid:3) + ( D − G ( r ) F ( r ) r (cid:2) rG ( r ) F ′ ( r ) + rG ′ ( r ) F ( r ) + ( D − G ( r ) F ( r ) (cid:3) , (5.2) R µνρσ R µνρσ = 4 G ( r ) F ( r ) (cid:2) F ′′ ( r ) G ( r ) + F ′ ( r ) G ′ ( r ) (cid:3) + 4 ( D − G ( r ) F ′ ( r ) r F ( r ) +4 ( D − G ′ ( r ) G ( r ) r + 2 ( D −
2) ( D − G ( r ) r , (5.3)being respectively the Ricci scalar, the Ricci tensor square and the Kretschmann scalar.Note that, using (4.9), the Ricci scalar is given by R = − T − G ( r ) F ′′ ( r ) F ( r ) − G ( r ) G ′ ( r ) F ′ ( r ) F ( r ) − D − G ( r ) G ′ ( r ) r , which is just relation (2.5) calculated for the vierbeins (4.2).Observing the form of the torsion scalar T in (4.15) we deduce that in the chargedcase it diverges only at r = 0. This can be alternatively verified examining the form (4.9)along with the expressions (4.18),(4.19) (or (4.21),(4.22) for the special solution).Observing the forms of Ricci and Kretschmann scalars in (5.1), (5.3) we deduce thatin the charged case the possible divergence points are at r = 0, at the points where G ( r ) → ∞ , or at the roots of F ( r ). From the solutions for G ( r ) and F ( r ) of (4.18),(4.19)we straightforwardly obtain that r = 0 indeed leads to divergent Ricci and Kretschmannscalars. From the form of G ( r ) in (4.18) along with (4.16) we observe that G ( r ) → ∞ at1 + dfdT = 0, that is at r s = (cid:18) − α Λ2 αQ (cid:19) − D , (5.4)a relation which holds only for the negative branch, since for the positive branch 1 + dfdT hasno roots (additionally since α < < − / (8 α )). Indeedone can straightforwardly see that the Ricci and Kretschmann scalars do diverge at r = r s .– 11 –inally, concerning the roots of F ( r ), due to (4.11), namely F ( r ) = G ( r ) (cid:16) dfdT (cid:17) , theyare just the roots of G ( r ), since F ( r ) remains finite and non-zero at 1 + dfdT = 0 since atthis point G ( r ) → ∞ . Taking the corresponding limits and using (4.18),(4.19), we cansee that the roots of G ( r ) leads always to finite Ricci and Kretschmann scalars. All theabove hold also for the special solution (4.21),(4.22). In summary, in the charged case theRicci and Kretschmann scalars diverge at r = 0 and at r = r s given by (5.4). Lastly, wemention that in the uncharged case the Ricci scalar vanishes, however the Kretschmannscalar behaves as in the charged case.From the above analysis we are led to the very interesting result that in some casesthe singularities obtained by the torsion scalar analysis are less than those obtained bythe curvature scalar analysis. In particular, this happens for the negative branch of thesolutions, for Q = 0 and for Λ < − / (8 α ), in which case the curvature invariants possess anadditional physical singularity at r = r s given by (5.4). We stress that when f ( T ) = 0, thatis in the case of usual teleparallel gravity, the negative branch disappears as we mentionedabove, thus the singularity analyses of the two approaches coincide. Additionally, in theuncharged case, that is when Q = 0, the extra singularity at r s disappears too, and thesingularity analyses of the two approaches coincide too. In conclusion, we deduce thatthe above difference in the physical singularities of the torsion and curvature analysis, isa result of both the non-linear f ( T ) structure and of the non-zero electric charge, whichreveals the novel features that are brought in in the theory in this case.Let us discuss on the horizons of the above solutions. Although we showed that theroots of G ( r ) at r > G ( r ), say at r = r H ,are just coordinate singularities, we consider the Painlev´e-Gullstrand coordinates [76–79]through the transformation dt = dτ + g ( r ) dr , with g ( r ) a function of the radial coordinate.Therefore, the metric (4.1) becomes ds = F ( r ) dτ + 2 g ( r ) F ( r ) drdτ − (cid:20) G ( r ) − F ( r ) g ( r ) (cid:21) dr − r i = D − X i =1 dx i . (5.5)Choosing g ( r ) = F ( r ) h G ( r ) − i and defining h ( r ) = F ( r ) /G ( r ) = h p − α Λ − αQ r − D i / ds = F ( r ) dτ + 2 h ( r ) q − G ( r ) drdτ − dr − r i = D − X i =1 dx i , (5.7)which is regular at r = r H . Therefore, r = r H , if they exist, are just coordinate singularities,that is horizons.From the above analysis it is implied that the black-hole solutions of the charged f ( T )gravity may possess a horizon at r H that shields the physical singularities. However, firstly– 12 –t is not guaranteed that r H exists, since there could be parameter choices for which G ( r )has no roots, that is the physical singularity at r = 0 becomes naked. Secondly, evenif r H exists it is not guaranteed that it will shield the second physical singularity of thecharged negative branch at r = r s given by (5.4), since this will depend on the specificparameter choice. In particular, we can see that if F ( r s ) < r H exists and shieldsthe singularity at r s , that is r H > r s , otherwise r s is a naked singularity. This is not thecase for f ( T ) → Q →
0, in which, as we mentioned, r s disappears. Therefore, weconclude that the cosmic censorship theorem, namely that there are always horizons thatshield the physical singularities, does not always hold for f ( T ) gravity, a result that wasalready found in the 3D case too [73].Before proceeding to the numerical elaboration of the obtained solutions, we makethe following comment. In curvature gravity there can be cases where the Ricci scalar isfinite at one point although there is a physical singularity there, which is revealed throughthe use of the Kretschmann scalar, and that is why people usually examine both scalarssimultaneously. Thus, one could ask whether one should use additional torsion scalars too,defined as various contractions of the torsion tensor. In particular, according to (2.3), thetorsion scalar T contains three separate scalars, corresponding to different contractions ofthe torsion tensor, namely I = T µνρ T µνρ , I = T ρµν T νµρ , I = T ρρµ T νµν , and thus onecould additionally examine their behavior in order to reveal the singularities. However,it is well known that these separate combinations are not invariant under local Lorentztransformations, and that was the reason that the teleparallel Lagrangian (torsion scalar) T was defined as their specific combination which becomes Lorentz invariant [2]. Therefore,one cannot use other torsion scalars apart from T in order to investigate the singularities(the explicit calculation of I i ’s for the obtained vierbein solutions shows that they acquiredifferent values in different coordinates, and thus they are not invariants, however theirspecific combination in T does acquire the same value independently of the coordinatebasis and thus it is a well-defined invariant).In order to provide a more transparent picture of the above singularity and horizon be-havior, we proceed to the numerical elaboration of specific examples. Since the asymptoticbehavior of G ( r ) is given by G ( r ) = − Λ eff r + . . . , (5.8)where . . . correspond to sub-leading terms andΛ eff = 954 α ( D − D − ± √ − α Λ) h α Λ ∓ (1 − α Λ) / i , (5.9)we can distinguish three subclasses, namely the asymptotically AdS one (Λ eff < eff >
0) and the limiting Λ eff = 0 one. Note that the second andthird solution subclasses can never be obtained by the negative solution branch. Withoutloss of generality we consider D = 4, α = − Const = −
1, while we suitably choose Λin order to lie in the above three subclasses, which we investigate separately.– 13 –
Case Λ eff < − /
25, which satisfies the condition Λ eff <
0. In Fig. 1 from upto down we depict G ( r ) , the Ricci scalar, the Kretschmann scalar and the torsionscalar T ( r ) as a function of r , for the positive and negative branches of black-holesolutions. The left graphs correspond to charged solutions ( Q = 1), while the rightgraphs correspond to uncharged solutions ( Q = 0).The positive branch in the charged case exhibits only one physical singularity at r = 0, in which the torsion scalar and the Ricci and Kretschmann scalars diverge,however it is shielded by two horizons at r H = r − and r H = r + , since in this case G ( r ) has two roots. In order to examine whether r + is a Killing horizon we see thatthe timelike Killing vector of the metric is ǫ µ ∂ µ = ∂ t , with norm ǫ µ ǫ µ = g tt = F ( r ) which vanishes at r = r + . Inside the horizon the Killing vector field is spacelike,while outside it is timelike, and thus it corresponds to a null hypersurface. Finally,for the uncharged solutions we can see that the Ricci scalar and the torsion scalar areconstants, while the Kretschmann scalar diverge at r = 0 (similarly to usual GeneralRelativity).For the negative branch in the charged case, the torsion scalar possesses only onedivergence, namely at r = 0, however the Ricci and Kretschmann scalars possesstwo divergence points, namely at r = 0 and r = r s as described above. However,in this specific numerical example, both these physical singularities are shielded bythe horizon at r H > r s , in which all invariants remain regular. In particular, r H isa Killing horizon, corresponding to an event horizon since the Killing vector field istimelike outside the horizon and spacelike inside. Finally, for the uncharged solutionsthe Ricci and torsion scalars are constants, but the Kretschmann scalar diverges at r = 0. However, this physical singularity is shielded by the horizon at r = r H inwhich G ( r ) becomes zero.In summary, we indeed verify that in the charged case and for the negative branchthe curvature invariants contain an extra divergence at r = r s , that does not appearin the torsion invariant, revealing the novel features of charged f ( T ) gravity. • Case Λ eff > /
25, which satisfies the condition Λ eff > eff > G ( r ) , the Ricci scalar, the Kretschmann scalarand the torsion scalar T ( r ) as a function of r , for the positive branch of black-holesolutions. The left graphs correspond to charged solutions ( Q = 1), while the rightgraphs correspond to uncharged solutions ( Q = 0).The charged case possesses a physical singularity at r = 0, where the torsion scalarand the Ricci and Kretschmann scalars diverge, however it is shielded a horizon Note that none of the metric coefficients depends on time and thus the manifold has a timelike Killingvector ∂ t , and similarly since none of the metric coefficients depends on x i there exist ( D −
2) spacelikeKilling vector fields ∂ x i . – 14 – - - - r G H r L Negative BranchPositive Branch 0 1 2 3 4 5 6 - - - r G H r L Negative BranchPositive Branch0 1 2 3 4 5 6 - - r R H r L Negative BranchPositive Branch 0 1 2 3 4 5 6 - - r R H r L Negative BranchPositive Branch0 1 2 3 4 5 6020406080100 r R Μ Ν Ρ Σ R Μ Ν Ρ Σ H r L Negative BranchPositive Branch 0 1 2 3 4 5 6020406080100 r R Μ Ν Ρ Σ R Μ Ν Ρ Σ H r L Negative BranchPositive Branch0 1 2 3 4 5 6 - - - r T H r L Negative BranchPositive Branch 0 1 2 3 4 5 6 - - - r T H r L Negative BranchPositive Branch
Figure 1 . The solutions for G ( r ) of (4.13), for the Ricci scalar R ( r ) of (5.1), for the Kretschmannscalar R µνρσ R µνρσ ( r ) of (5.3) and for the torsion scalar T ( r ) of (4.9), as a function of r , for thepositive (thick solid curve) and negative (thin dashed curve) branch of the AdS solution subclass,for D = 4 , α = − , Const = − and Λ = − / . Left graphs correspond to charged solutions with Q = 1 , while right graphs correspond to uncharged solutions with Q = 0 . The thin -line is depictedfor convenience. – 15 – - - - r G H r L - - - r G H r L - - r R H r L - - r R H r L r R Μ Ν Ρ Σ R Μ Ν Ρ Σ H r L r R Μ Ν Ρ Σ R Μ Ν Ρ Σ H r L - - - r T H r L - - - r T H r L Figure 2 . The solutions for G ( r ) of (4.13), for the Ricci scalar R ( r ) of (5.1), for the Kretschmannscalar R µνρσ R µνρσ ( r ) of (5.3) and for the torsion scalar T ( r ) of (4.9), as a function of r , for thepositive (thick solid curve) branch of the dS solution subclass, for D = 4 , α = − , Const = − and Λ = 1 / . Left graphs correspond to charged solutions with Q = 1 , while right graphs correspond touncharged solutions with Q = 0 . The thin -line is depicted for convenience. – 16 –t r = r H , where G ( r ) becomes zero. In order to examine whether r H is a Killinghorizon, and similarly to the previous case, we observe that the timelike Killing vectorof the metric is ǫ µ ∂ µ = ∂ t , with norm ǫ µ ǫ µ = g tt = F ( r ) which vanishes at r = r H .Since outside the horizon the Killing vector field is timelike, and inside it is spacelike,it is implied that it corresponds to a null hypersurface, that is a cosmological Killinghorizon.In the uncharged case we observe that the Ricci scalar vanishes while the torsionscalar is constant, however the Kretschmann scalar diverges at r = 0. However,note that in this case G ( r ) has no roots, that is there is not a horizon to shield thephysical singularity at r = 0, which is therefore a naked one. • Case Λ eff = 0We consider Λ = 0, which satisfies the condition Λ eff = 0 for the positive branch(as we mentioned below (5.9) the negative branch cannot lead to Λ eff = 0). InFig. 3 from up to down we show G ( r ) , the Ricci scalar, the Kretschmann scalarand the torsion scalar T ( r ) as a function of r , for the positive branch, with leftgraphs corresponding to charged solutions ( Q = 1) and right graphs correspondingto uncharged solutions ( Q = 0).The charged solutions possess a physical singularity at r = 0, where all invariantsdiverge, however it is shielded a horizon at r = r H , where G ( r ) becomes zero.Examining the Killing vector, and similarly to the previous cases, we deduce that r H is a cosmological Killing horizon. In the uncharged case we observe that both theRicci and torsion scalars vanish, however the Kretschmann scalar diverges at r = 0.Notice that in this case G ( r ) has no roots, that is there is not a horizon to shieldthe physical singularity at r = 0, which is therefore a naked one. In this work we considered D-dimensional f ( T ) gravity including the Maxwell field. Weextracted exact charged black-hole solutions depending on the functional form of f ( T ), onthe electric charge and on the number the dimensionality D. Finally, we investigated thesingularities and the horizons of the obtained solutions, following two different approaches.Firstly, by studying the torsion invariants constructed using the Weitzenb¨ock’s connectionand the vierbein solutions, and secondly by studying the curvature invariants constructedusing the Levi-Civita connection and the metric solutions.The main result is that in Maxwell- f ( T ) gravity the curvature invariants possess morephysical singularities than the torsion ones, in some particular solution subclasses. Thisdifference disappears in the uncharged case, or in the case where f ( T ) gravity becomes theusual linear-in- T teleparallel gravity, thus it reveals the novel behavior that is introducedby the combined complication of the non-trivial f ( T ) structure with the electromagneticsector. It seems that curvature and torsion invariants behave very differently dependingon the presence of the Maxwell field. More generally, extending gravity in terms of f ( T ) or– 17 – - - - r G H r L - - - r G H r L - - r R H r L - - r R H r L r R Μ Ν Ρ Σ R Μ Ν Ρ Σ H r L r R Μ Ν Ρ Σ R Μ Ν Ρ Σ H r L - - - r T H r L - - - r T H r L Figure 3 . The solutions for G ( r ) of (4.13), for the Ricci scalar R ( r ) of (5.1), for the Kretschmannscalar R µνρσ R µνρσ ( r ) of (5.3) and for the torsion scalar T ( r ) of (4.9), as a function of r , for thepositive (thick solid curve) branch, for D = 4 , α = − , Const = − and Λ = 0 . Left graphscorrespond to charged solutions with Q = 1 , while right graphs correspond to uncharged solutionswith Q = 0 . The thin -line is depicted for convenience. – 18 – ( R ) formulations could give very different results as soon as matter fields are taken intoaccount.Finally, we have to note that, in the scenario we have considered, the physical singular-ities are not always shielded by horizons. Thus, the cosmic censorship does not always holdfor D-dimensional Maxwell- f ( T ) gravity. From a cosmological point of view such a featurecould be extremely relevant in order to investigate the early phases of cosmic evolution.On the other hand, considering astrophysical structures in strong field regimes, derivedfrom torsion or curvature representation of gravity, could give rise to very deep differencesin dynamics [22, 80]. Acknowledgments
S.C. is supported by INFN (Sez. di Napoli). The research of E.N.S. is implemented withinthe framework of the Action Supporting Postdoctoral Researchers of the Operational Pro-gram “Education and Lifelong Learning” (Actions Beneficiary: General Secretariat forResearch and Technology), and is co-financed by the European Social Fund (ESF) and theGreek State. Y. V. is supported by FONDECYT grant 11121148, and by Direcci´on deInvestigaci´on y Desarrollo, Universidad de La Frontera, DIUFRO DI11-0071.
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