Black string corrections in variable tension braneworld scenarios Roldão da Rocha* Centro de Matemática, Computação e Cognição, Universidade Federal do ABC (UFABC) 09210-170, Santo André, SP, Brazil J.M. Hoff da Silva† Departamento de Fı́sica e Quı́mica, Universidade Estadual Paulista, Av. Dr. Ariberto Pereira da Cunha, 333, Guaratinguetá, SP, Brazil (Received 4 November 2011; published 24 February 2012) Braneworld models with variable tension are investigated, and the corrections on the black string horizon along the extra dimension are provided. Such corrections are encrypted in additional terms involving the covariant derivatives of the variable tension on the brane, providing profound consequences concerning the black string horizon variation along the extra dimension, near the brane. The black string horizon behavior is shown to be drastically modified by the terms corrected by the brane variable tension. In particular, a model motivated by the phenomenological interesting case regarding Eötvös branes is investigated. It forthwith provides further physical features regarding variable tension braneworld scenarios, heretofore concealed in all previous analysis in the literature. All precedent analysis considered uniquely the expansion of the metric up to the second order along the extra dimension, which is able to evince solely the brane variable tension absolute value. Notwithstanding, the expansion terms aftermath, further accomplished in this paper from the third order on, elicits the successive covariant derivatives of the brane variable tension, and their respective coupling with the extrinsic curvature, the Weyl tensor, and the Riemann and Ricci tensors, as well as the scalar curvature. Such additional terms are shown to provide sudden modifications in the black string horizon in a variable tension braneworld scenario. DOI: 10.1103/PhysRevD.85.046009 PACS numbers: 11.25.�w, 04.50.�h, 04.50.Gh I. INTRODUCTION Braneworld models perform a distinct branch of high- energy physics. Several of the main ideas concerning such models were inspired in formal advances in string theory [1]. It is possible to delineate the precursory works dealing with the possibility of a braneworld universe [2], and nowadays there is a great variety of alternative models dealing with several aspects regarding braneworlds. Motivated by a comprehensive program regarding gravity and black strings on braneworld scenarios, and subsequent generalizations probing extra-dimensional features [3–10], some gravitational aspects concerning black strings, aris- ing from braneworld models with variable brane tension, are introduced and widely investigated. In order to analyze a conceivable gravitational effect, arising genuinely from the brane variable tension, we analyze the corrections on the black string time-dependent horizon, in such a sce- nario. This formalism provides manageable models and their possible ramifications into some aspects of gravity in this context, cognizable corrections, and physical effects as well. Besides the black string behavior being sharply modified by the variable tension brane localized at y ¼ 0 (the extra dimension is denoted by y), unexpected addi- tional terms in a Taylor expansion of the metric along the extra dimension are elicited. This Taylor expansion regards the black string horizon behavior along the extra dimen- sion. Taking into account a variable tension brane, this expansion involves the covariant derivatives of the brane tension—besides its Lie derivative along the extra dimen- sion likewise. Such expansion, for instance, was used in [11] to investigate the gravitational collapse of compact objects in braneworld scenarios. Braneworld models, wherein a brane is endowed with variable tension � ¼ �ðx�Þ, are a prominent approach as an attempt to ascertain new signatures coming from high- energy physics [12]. In fact, due to the drastic modification of the temperature of the Universe along its cosmological evolution, a variable tension braneworld scenario is indeed demanded. The full covariant variable tension brane dy- namics was established in Ref. [13], and further explored in [14].Moreover, thevariable tensionwas implemented in the braneworld model consisting of two branes [15,16], also in the context of scalar tensor bulk gravity [17]. Furthermore, the cosmological evolution of theUniversewas investigated in a particular model in which the brane tension has an exponential dependence with the scale factor [18]. In this paper, we are concerned with analyzing the information about the black string behavior, evinced by a time variable tension on the brane, delving into the Taylor expansion outside a black hole metric along the extra dimension, where the corrections in the area of the five-dimensional black string horizon are elicited. It is shown how the variable tension and its covariant derivatives determine the variation in the area of the black string *roldao.rocha@ufabc.edu.br †hoff@feg.unesp.br;hoff@ift.unesp.br PHYSICAL REVIEW D 85, 046009 (2012) 1550-7998=2012=85(4)=046009(9) 046009-1 � 2012 American Physical Society http://dx.doi.org/10.1103/PhysRevD.85.046009 horizon along the extra dimension. It induces interesting physical effects, for instance, the preclusion of the horizon decreasing, when the brane tension varies in a model physi- cally motivated by Eötvös law. Our program throughout this paper explicitly consists of the following: the next section provides the black string behavior along the extra dimension, studying the variation in the black string horizon due to terms including the covariant derivatives of the variable brane tension. In par- ticular, it is considered the Eötvös phenomenological brane case, and the Lie derivatives of � along the extra dimension are identically zero and therefore concealed in the subse- quent analysis. Some results [19] are extended in order to encompass this case. The fine character of the expansion along the extra dimension is crucial to analyze variable tension models. The Taylor expansion along the extra dimension, concerning variable tension braneworld mod- els, provides terms of superior order and brings some more precise information about the behavior of the black string than in models with brane constant tension. Furthermore, when variable tension braneworld models are taken into account, the extra terms—from the third order on—probe physical features concerning the variable tension. For instance, such terms induce the horizon to decrease in a different rate along the extra dimension, when we analyze the physical concrete example of Eötvös branes. In Sec. III, we delve into the corrections for the Schwarzschild case analysis, and to this point the framework is completely general. In Sec. IV, the specific and physically motivated example, regarding Eötvös branes endowed with a de Sitter-like scale factor variable tension, is investigated. II. BLACK STRING BEHAVIOR ALONG THE EXTRA DIMENSION In this section, the comprehensive formalism in [19] is briefly introduced and reviewed. Hereon, f��g, � ¼ 0; 1; 2; 3 ½f�Ag; A ¼ 0; 1; 2; 3; 4� denotes a basis for the cotangent space T� xM at a point x in a 3-braneM embedded in a bulk. If a local coordinate chart is chosen, it is possible to represent �A ¼ dxA. Take now n ¼ nAeA, a timelike vector orthogonal to T� xM, and let y be the associated Gaussian coordinate, indicating how an observer upheavals out the brane into the bulk. In particular, nAdx A ¼ dy in the hyper-surface defined by y ¼ 0. Avector field v ¼ xAeA in the bulk is split into components in the brane and orthogo- nal to the brane, respectively, as v ¼ x�e� þ ye4 ¼ ðx�; yÞ. The bulk is endowed with a metric g � ABdx AdxB ¼ g��ðx�; yÞdx�dx� þ dy2. The brane metric components g�� and the bulk metric are related by g � ��¼g��þn�n�, since according to the notation in [19], as g44 ¼ 1 and gi4 ¼ 0, the bulk indices A, B effectively run from 0 to 3. It is well know that the four-dimensional gravitational constant is an effective coupling constant inherited from the fundamental coupling constant, and the four-dimensional cosmological constant is nonzero when the balance between the bulk cosmological constant and the brane tension—provided by the Randall-Sundrum bra- neworld model [20]—is broken [19]: �2 4 ¼ 1 6 ��4 5; �4 ¼ �2 5 2 � �5 þ 1 6 �2 5� 2 � ; (1) where �4 is the effective brane cosmological constant, �5 [�4] denotes the five-dimensional [four-dimensional] gravitational coupling, and � is the brane tension. The extrinsic curvature K�� ¼ 1 2Lng�� (hereon Ln denotes the Lie derivative, which in Gaussian normal coordinates reads Ln ¼ @=@y). The junction condition determines the extrinsic curvature on the brane, as K�� ¼ �1 2� 2 5½T�� þ 1 3ð�� TÞg���; (2) where T ¼ T� �. In what follows, we denoteK ¼ K� � and K2 ¼ K��K ��. Given the five-dimensional Weyl tensor C��� ¼R��� � 2 3ðg � ½��R�� þg � ½� gR���Þ� 1 6Rðg�½�g� �Þ; (3) where R��� denotes the components of the bulk Riemann tensor (R�� and R obviously are the associated Ricci tensor and the scalar curvature), the symmetric and trace-free components, respectively, denoted by E�� and B��� are known as the electric and magnetic Weyl tensor components—given by E�� ¼ C��� n �n and B��� ¼ g �g��C ���n �. The Weyl tensor represents the part of the curvature that is not determined locally by matter, and the field equations for the Weyl tensor are simulated by Bianchi identities, determining the part of the spacetime curvature that depends on the matter distribution at other points [21]. The effective field equations are complemented by a set of equations that are forthwith obtained from the five- dimensional Einstein and Bianchi equations [22]. Those equations are also obtained in [19,23], as well as in a brane with variable tension context [13]. All approaches are shown to be completely similar and hereon the following effective field equations are considered: LnK�� ¼ K��K � � � E�� � 1 6�5g��; (4) LnE��¼r�B�ð��Þþ 1 6�5ðK���g��KÞ þK��R����þ3K� ð�E�Þ��KE�� þðK��K���K��K��ÞK��; (5) LnB���¼�2r½�E���þK� �B����2B��½�K�� �; (6) LnR����¼�2R�� ½�K�� �r�B���þr�B���: (7) ROLDÃO DA ROCHA AND J.M. HOFF DA SILVA PHYSICAL REVIEW D 85, 046009 (2012) 046009-2 These equations are to be solved subject to the boundary condition B��� ¼ 2r½�K��� at the brane [19]. Those expressions are subsequently used in order to explicitly calculate the terms of the Taylor expansion of the metric along the extra dimension. Such expansion provides the black string profile and some physical con- sequences. The equations above were used to develop a covariant analysis of the weak field [22], and are used to develop a Taylor expansion of the metric along the extra dimension, which predicts the black string behavior. We intend here to perform the calculations beyond second- order term in the expansion along the extra dimension. Solely in this way, it is possible to realize how the variable tension generates additional terms concerning the cova- riant derivatives of the brane variable tension. The standard Taylor expansion along the extra dimen- sion y is given by the expression g��ðx; yÞ ¼ g��ðx; 0Þ þ ðLng��ðx; yÞÞjy¼0jyj þ ðLnðLng��ðx; yÞÞÞjy¼0 jyj2 2! þ ðLnðLnðLng��ðx; yÞÞÞÞjy¼0 jyj3 3! þ � � � þ ðLk nðg��ðx; yÞÞjy¼0 jyjk k! þ � � � : (8) The term in the first-order jyj above is immediately calcu- lated by the definition of the extrinsic curvature K�� ¼ 1 2Lng�� and by the junction condition (2). The term in y2 is proportional to LnK��, which is given by Eq. (4), and in order to calculate the term K��K � � in Eq. (4), the junction condition (2) is used forthwith. Furthermore, the coeffi- cient term of jyj3 in Eq. (8) can be expressed as 2LnðLnK��Þjyj 3 3! ¼ � LnðK��K � �Þ�LnE����5 6 Lng�� �jyj3 3 ; (9) where Eq. (4) was used. As the Lie derivatives terms in the right-hand side of this last expression are, respectively, given by Eq. (4) [where the Leibniz rule LnðK��K � �Þ ¼ LnðK��ÞK� � þ K��LnðK� �Þ is employed], by Eq. (5), and by the definition of the extrinsic curvature, one arrives further at the expression for jyj3 in Eq. (8). Finally, the term in y4 is obtained when the Lie derivative of the right-hand side of Eq. (9) is taken into account, as well as Eqs. (4)–(7) again. Explicitly, up to fourth order in the extra dimension, the Taylor expansion is given by [hereon we denote g��ðx; 0Þ ¼ g��]: g��ðx; yÞ ¼ g��ðx; 0Þ � �2 5 � T�� þ 1 3 ð�� TÞg�� � jyj þ � �E�� þ 1 4 �4 5 � T��T � � þ 2 3 ð�� TÞT�� � þ 1 6 � 1 6 �4 5ð�� TÞ2 ��5Þg�� � y2 þ ½2K��K � �K � � � ðE��K � � þ K��E� �Þ � 1 3 �5K�� �r�B�ð��Þ þ 1 6 �5ðK�� � g��KÞ þ K��R���� þ 3K� ð�E�Þ� � KE�� þ ðK��K�� � K��K��ÞK�� ��5 3 K�� � jyj3 3! þ � �5 6 � R��5 3 þ K2 � g�� þ � K2 3 ��5 � K��K � � þ ðR��5 þ 2K2ÞE�� þ ðK� �K �� þ E�� þ KK��ÞR���� � 1 6 �5R�� þ 2K��K � �K � �K � � þ K� K � KK�� þ E�� � K��K �� � 3K� �K � � þ 1 2 KK� � � þ � 7 2 KK� � � 3K� �K � � � E�� � 13 2 K��E� �K � � þ ð3K� �K � � � K��K ��ÞE�� � K��K��E�� � 4K��R�� �K � � 7 6 K��K� �R����� y 4 4! þ � � � : (10) Such expansion was analyzed in [19] only up to the second order. This procedure does not suffice to evince the addi- tional terms, arising from the variable tension in the brane. For an alternative method that does not take into account the Z2 symmetry, and some subsequent applications, see [24]. There are still some additional terms that were con- cealed in the expression above, that are going to be em- phatically focused on the next sections, concerning the derivatives of the variable tension. The additional terms coming from the variable brane tension are shown to be essential for the subsequent analysis on the brane tension influence on the black string behavior along the extra dimension. A. Additional terms elicited from the variable tension Although the brane tension is variable, up to terms in y2 at the expansion (10) there are no additional terms, which appear only from the order jyj3 on, regarding (10). The BLACK STRING CORRECTIONS IN VARIABLE TENSION . . . PHYSICAL REVIEW D 85, 046009 (2012) 046009-3 unique term to contribute to the derivatives of the variable tension � comes from the coefficientLnE�� of jyj3 in (10), given by r�B�ð��Þ, wherein one can substitute the bound- ary condition B��� ¼ 2r½�K��� [19,22]. Apart from the term related to the energy-momentum tensor in K�� ¼ � 1 2� 2 5½T�� þ 1 3 ð�� TÞg���, since we are concerned only about the extra terms coming from the variable tension, such extra terms read r�B�ð��Þ ¼ �2 3� 2 5ððr�r��Þg�� � ðrð�r�Þ�ÞÞ: (11) Terms in order y4 can be obtained immediately, besides the additional terms coming from the variable tension brane, since we are interested on the further effects of the variable brane tension on the black string character. Such extra terms in the order y4 of expansion (10) are given by �1 3� 2 5½hðh�Þg�� �rð�r�Þðh�Þ� þ � 1 3� 2 5 þ 2K � ½ðh�ÞEð��Þ � r�ððrð��ÞE�Þ�Þ� þ 6½ðh�ÞKð��E� �Þ � r�ððrð��ÞE�Þ�Þ� þ 2 � K þ 7 3� 2 5 � ½ðh�ÞKK�� �r�ððrð��ÞKK��ÞÞ� þ 1 3� 2 5½ðh�ÞR�� �r�ððrð��ÞR��ÞÞ� � 2K��½ðh�ÞRð���Þ� �r�ððrð��ÞR���Þ�Þ� þ � 2K� K � � 1 3�5 � ½ðh�Þg�� �rð�r�Þ�� þ 1 3� 2 5½ðh�ÞðKð��K�Þ�K�� � ðK� K � ÞKð��ÞÞ � r�ððrð��ÞðK��K�Þ � � KK��ÞÞÞ� (12) and are obtained when one substitutes the Lie derivative of each term of such equation in the expression for LnðLnE��Þ, taking into account once more Eqs. (4)–(7). We shall consider in this paper the brane tension as a time-dependent function � ¼ �ðtÞ. B. Black string corrections for the vacuum case Now, we focus on the situation where Eqs. (11) and (12) provide the further extra terms in the expansion given by Eq. (10) in the vacuum on the brane. In this case, T�� ¼ 0, and consequently Eq. (2), is led to K�� ¼ �1 6� 2 5�g��; (13) which provides the expansion at Eq. (10) to be written straightforwardly as g��ðx; yÞ ¼ g�� � 1 3 �2 5�g��jyj þ � �E�� þ � 1 36 �4 5� 2 � 1 6 �5 � g�� � y2 þ �� � 193 216 �3�6 5 � 5 18 �5� 2 5� � g�� þ 1 6 �2 5E�� þ 1 3 �2 5ðE�� þ R��Þ � jyj3 3! þ � 1 6 �5 �� R� 1 3 �5 � 1 18 �2�4 5 � þ 7 324 �4�8 5 � g�� þ � R��5 þ 19 36 �2�4 5 � E�� þ � 37 216 �2�4 5 � 1 6 �5 � R�� þ E��R����� y 4 4! þ � � � : (14) As the coefficients above concern uniquely quantities in the brane, we evince the property that when there is vacuum on the brane the brane field equations R�� ¼ �E��; R � � ¼ 0 ¼ E� �; r�E�� ¼ 0 (15) hold. It induces the last term of jyj3 in Eq. (14) equals zero. Hence, Eq. (14) is written as g��ðx; yÞ ¼ g�� � 1 3 �2 5�g��jyj þ � �E�� þ � 1 36 �4 5� 2 � 1 6 �5 � g�� � y2 � �� 193 216 �3�6 5 þ 5 18 �5� 2 5� � g�� þ 1 6 �2 5R�� � jyj3 3! þ � 1 6 �5 �� R� 1 3 �5 � 1 18 �2�4 5 � þ 7 324 �4�8 5 � g�� þ � Rþ 5 6 �5 � 77 216 �2�4 5 � R�� � R��R���� � y4 4! þ � � � : (16) Furthermore, in the vacuum Eq. (12)—corresponding to the additional terms arising from the derivatives of variable brane tension �—is led to ROLDÃO DA ROCHA AND J.M. HOFF DA SILVA PHYSICAL REVIEW D 85, 046009 (2012) 046009-4 � 1 3� 2 5ðhðh�Þg�� �rð�r�Þh�Þ þ � �1 3�5 þ 8 9� 2�4 5 � ððh�Þg�� �rð�r�Þ�Þ þ 1 9� 2�4 5ð�g�� þrð�r�Þ�Þ � 4 3� 2 5½ðh�ÞEð��Þ � r�ððrð��ÞE�Þ�Þ� þ 1 3� 2 5ðh�Þ � 5 216� 3�6 5g�� � þ 1 3�� 2 5½ðh�ÞRð��Þ � r�ððrð��ÞR��ÞÞ� þ 6½ðh�ÞKð��E� �Þ � r�ðrð��ÞE�Þ��; where on the brane R�� ¼ �E�� holds as one of the field equations in (15). III. BLACK STRING SCHWARZSCHILD CORRECTIONS As the case of interest to be investigated is exactly the corrections on the black string horizon along the extra dimension, we shall focus on the term g��—corresponding to the square of the black string horizon—of the expansion at Eq. (10). Clearly, a time-dependent brane tension shall modify the black string Schwarzschild background. The complete solution is, however, hugely difficult to accom- plish. Therefore, we adopt an effective approach, studying the horizon variation with tension variation corrections only in the Taylor expansion. As shall be shown, even in this approximative case interesting results are accomplished. A static spherical metric on the brane can be expressed as g��dx �dx� ¼ �FðrÞdt2 þ ðHðrÞÞ�1dr2 þ r2d�2; (18) where d�2 denotes the line element of a two-dimensional unit sphere. The projected Weyl term on the brane is given by [19] E �� ¼ �1þH þ r 2 H � F0 F þH0 H � ¼ 0; (19) for the Schwarzschild metric FðrÞ ¼ HðrÞ ¼ ð1� 2M r Þ, where we denoteM � GM=c2. As the black string horizon variation along the extra dimension is analyzed, the term g��ðx; yÞ in (14) above is given by g��ðx; yÞ ¼ r2 � r2 3 �2 5�jyj þ � 1 36 �4 5� 2 � 1 6 �5 � r2y2 � � 193 216 �3�6 5 þ 5 18 �5� 2 5� � r2jyj3 3! � 1 18 �5 �� �5 þ 1 6 �2�4 5 � þ 7 324 �4�8 5 � � r2y4 4! þ � � � : (20) Note that g�� ¼ g��ðx; yÞ coming from Eq. (14) is a com- ponent of the Taylor expanded metric along the extra dimension, while g�� in Eq. (16) depends only on the brane variables, as usual [19]. Now, as � ¼ �ðtÞ, the terms r�r�� for the Schwarzschild metric are computed and the additional terms in jyj3 for g�� in Eq. (20) are given by � 2 3� 2 5� 00r2; (21) where �0 denotes the derivative with respect to the time t—and the additional terms in y4 for g�� are given by � 1� 2M r �� � 1 3 �2 5 � 1� 2M r � �0000r2 þ �00r2 �� 1 3 �5 � 8 9 �2�4 5 � þ 5 648 �3�8 5 �� þ 1 9 �2�4 5�r 2: (22) All the expressions obtained are so far the most general. In order to better understand the physical implications of a variable tension in the event horizon along the extra di- mension, let us particularize our analysis to a specific physical motivated case. A specific example regarding a brane variable tension In general, there are two distinct approaches to imple- ment a variable tension brane. On the one hand, one may realize the brane tension as a (fundamental) scalar field appearing in the Lagrangian. This comprehensive picture is widely assumed in the context of string theory [25] and supersymmetric branes [26]. Instead, to emulate such ap- proach the brane tension may be understood as an intrinsic property of the brane as in, e.g., [13–16,18]. We delve into this point of view in our subsequent analysis. From the braneworld picture, the functional form of the variable tension on the brane is an open issue. However, taking into account the huge variation of the Universe temperature during its cosmological evolution, it is indeed plausible to implement the brane tension as a variable function of spacetime coordinates. In particular, in such case as a function of time coordinate. Although it lacks a complete scenario still, the phenomenological interesting case regarding Eötvös standard fluid membranes [27] is useful to extract deep physical results. The phenomenological Eötvös law asserts that the fluid membrane depends on the temperature as � ¼ �ðTc � TÞ; (23) where � is a constant and Tc represents a critical tempera- ture denoting the highest temperature for which the mem- brane exists. We heuristically depict in what follows how the brane tension varies, in full compliance with Eötvös membranes models. BLACK STRING CORRECTIONS IN VARIABLE TENSION . . . PHYSICAL REVIEW D 85, 046009 (2012) 046009-5 If there are no stresses in the bulk—apart the cosmo- logical constant—there is no exchange of energy momen- tum between the bulk and the brane [19]. Herewith, it is possible to assert that there is no exchange of heat between the brane and the bulk—the regarded interaction is purely gravitational. Therefore, the well-known expression dQ ¼ dEþ pdV ¼ 0 holds for the brane, and concerning pho- tons from the cosmological microwave background the explicit formulas E ¼ E ¼ �T4V and p ¼ E=3 ¼ �T4 3 V accrue. Using these relations, it is straightforwardly verified that 1 T dT dt ¼ � 1 3V dV dT : (24) Now, relating the volume to the Friedmann-Robertson- Walker scale factor—V ¼ a3ðtÞ—it follows that T � 1 aðtÞ , and therefore one may use a physically motived input regarding the tension variable functions. A complete cosmological setup must be forthwith ob- tained from the solution of the full cosmological brane equations taking into account variable tension, which is in general obviously a hard task. Notwithstanding, we are assuming that the variability of tension may be outlined by considering the relation T � a�1ðtÞ valid in some approxi- mation [14]. This conservative approach, although not the most complete one, is certainly in full compliance with the standard cosmological model. Therefore, considering the analogy to Eötvös membranes and the expression T � a�1ðtÞ we have � ¼ �0ð1� amin=aÞ, where �0 is a constant related to the four-dimensional coupling constants [14], and amin denotes the minimum scale factor under which the brane cannot exist, since its tension would become negative. For the qualitative analysis that we de- velop, it is sufficient to consider the general shape of � given by � ¼ 1� 1 aðtÞ ; (25) with normalized tension and scale factor. In order to sup- plement our assumption in (25), we stress that this type of tension variation may be useful to reconcile supersymme- try and inflationary cosmology. Indeed, as first noticed in Ref. [14], the time-dependent brane tension gives rise to a time-dependent four-dimensional effective cosmological constant. In particular for a tension varying as Eq. (25), it is possible to show that �4D � 1� 1 aðtÞ � 1� 1 aðtÞ � : (26) Hence, the cosmological constant starts at high negative values and as the brane universe expands it converges to a positive small value. It is indeed a remarkable character- istic of this type of tension.1 To finalize, a de Sitter-like brane behavior is assumed, by setting a� e�t with positive � [29], in such a way that �ðtÞ ¼ 1� e��t: (27) It is important, in view of the assumption (27), to assert few remarks concerning its phenomenological viability. It is well known that the projected gravitational constant de- pends linearly on the brane tension [19,22]. Therefore, a time variable tension engenders a variation on the Newtonian constant G� �ðtÞ. The variation of a dimen- sional constant may always be incorporated in a suitable redefinition of length, time, and energy [30]. Nevertheless, it is possible to pick up some arbitrary value defining it as the standard one, and study its possible fractional variation. The recent astrophysical data indicate the constancy of the Newtonian constant (for an up-to-date review see [31]). In fact, the best model-independent bound on _G=G is given by lunar laser ranging measurements whose upper limit is ð4� 9Þ � 10�13 yr�1. Hence, taking into account Eq. (27), the following condition, e�t > 1þ 1013�; (28) must hold. In this qualitative analysis, we shall not fix the value of �, but as it is not desirable to have a huge departure from the standard scenario, it is expected to have ��H0, the Hubble parameter [19]. Thus, in princi- ple, 1013� is not so far from 1. An important characteristic of the constraint (28) is that there is a particular time, say, �t, given by �t ¼ lnð1þ 1013�Þ=� below which this constraint is violated. This apparent difficulty may be circumvented by the fact that it is possible to introduce free parameters on the functional form of � such that �t is small enough, making the condition t < �t to belong to an early time range, when the assumption T � 1=aðtÞ—possibly—no longer holds anymore. Besides, it is possible to interpret the violation of (28) as a change variation faster than 10�13 yr�1, which could in principle be scrutinized at intermediate redshifts [32]. Having explicitly presented the type of variable tension, let us particularize our analysis taking into account Eq. (27). For the Eötvös brane scenario, the following figures show the black string behavior along the extra dimension y in the Schwarzschild picture, as predicted by Eqs. (20)–(22). The graphics below for each figure illustrate how the black string horizon varies along the extra dimension. Equation (20) is the landmark for all graphics hereon, which we imposed r ¼ 1 ¼ �5 ¼ �5, in order to make our analysis straightforward. It is clear that such rescaling does not affect the shape of the graphics below. In Figs. 1–3 the black string horizon behavior is obvi- ously different for differing values of �t. In Fig. 1, for the sake of completeness we depict the black string horizon behavior along the extra dimension, but now considering only terms up to the order y2 in Eq. (10), commonly 1For another model presenting such a behavior of the effective cosmological constant see [28]. ROLDÃO DA ROCHA AND J.M. HOFF DA SILVA PHYSICAL REVIEW D 85, 046009 (2012) 046009-6 approached in the literature, for instance, in [19]. Figures 2 and 3 show the variable tension brane correction including all terms up to order y4, respectively, without and with the terms elicited in Eqs. (21) and (22), regarding the deriva- tives of the brane variable tension. Those graphics show the paramount importance of considering more terms in the metric expansion given by Eq. (8), as accomplished heretofore. It also robustly illustrates that those terms drastically modify the black string horizon along the extra dimension, in the model here presented. For the three-dimensional graphics, Fig. 4 does not take into account the time derivative terms, while Fig. 5 does. We shall draw some remarks in detail on the general behavior encoded in the figures in the next section. 0.0 0.2 0.4 0.6 0.8 1.0 y 0.2 0.4 0.6 0.8 1.0 1.2 1.4 g FIG. 3. Graphic of the brane effect-corrected black string horizon g�� for the Schwarzschild metric with variable tension, along the extra dimension y and also as function of the time t. The brane tension is given by �ðtÞ ¼ 1� expð��tÞ. This graphic does take into account the extra terms given by Eqs. (21) and (22). For the dashed light-gray line: �t ¼ 0:25; for the dashed thick dark-gray line: �t ¼ 0:5; for the gray thick line: �t ¼ 0:75; for the dash-dotted line: �t ¼ 1; for the black line: �t ¼ 1:5; for the dash-dotted light-gray line: �t ¼ 2:5. 0.0 0.2 0.4 0.6 0.8 1.0 y 0.2 0.4 0.6 0.8 1.0 1.2 1.4 g FIG. 2. Graphic of the brane effect-corrected black string horizon g�� for the Schwarzschild metric with variable tension, along the extra dimension y and also as function of the time t. The brane tension is given by �ðtÞ ¼ 1� expð��tÞ. This graphic does not take into account the extra terms given by Eqs. (21) and (22). For the dashed light-gray line: �t ¼ 0:25; for the dashed thick dark-gray line: �t ¼ 0:5; for the gray thick line: �t ¼ 0:75; for the dash-dotted line: �t ¼ 1; for the black line: �t ¼ 1:5; for the dash-dotted light-gray line: �t ¼ 2:5. 0.2 0.4 0.6 0.8 1.0 y 0.2 0.4 0.6 0.8 1.0 1.2 1.4 g FIG. 1. Graphic of the brane effect-corrected black string horizon g�� for the Schwarzschild metric with variable tension, along the extra dimension y and also as function of the time t. The brane tension is given by �ðtÞ ¼ 1� expð��tÞ. We in- cluded, for comparison criteria, merely terms up to the order y2 in the expansion (20). For the long-dashed gray line (lower curve): �t ¼ 0:25; for the dashed thick dark-gray line: �t ¼ 0:5; for the thick gray line: �t ¼ 0:75; for the dash-dotted line: �t ¼ 1; for the black line: �t ¼ 1:5; for the dash-dotted light-gray line: �t ¼ 2:5. FIG. 4 (color online). Graphic of the brane effect-corrected black string horizon g�� for the Schwarzschild metric with variable tension, along the extra dimension y and also as function of the time t. The brane tension is given by �ðtÞ ¼ 1� expð��tÞ. This graphic does not take into account the extra terms given by Eq. (12) and (21), that for the case considered are encrypted in Eq. (22). BLACK STRING CORRECTIONS IN VARIABLE TENSION . . . PHYSICAL REVIEW D 85, 046009 (2012) 046009-7 IV. CONCLUDING REMARKS In this paper, branes whose variable tension is only dependent on the time are focused. The Taylor expansion of the metric along the extra dimension for this case is provided. Albeit terms up to the second order already evince the variable tension character, when terms from the third order on are investigated, it is thoroughly possible to unveil the real character of the variable tension concerning the chang- ing of the event horizon. It accrues additional and unex- pected properties, since only in this case the covariant derivatives of the variable tension appear, in the metric expansion along the extra dimension. Furthermore, it is prominently necessary to consider such terms, since in some regions of spacetime described by a brane the value of the tension can be very small, and at the same time the brane tension can bluntly vary along the spacetime coor- dinates. Additional terms in the expansion analyzed mod- ify the rate of the horizon variation along the extra dimension, when the brane tension varies in a braneworld model based upon the Eötvös law. This unexpected feature on this effective model is ex- plicitly shown in Figs. 1–5. In the first figure, the brane effect-corrected black string horizon g�� for the Schwarzschild metric with variable tension is depicted along the extra dimension y. In Fig. 1, the black string horizon behavior along the extra dimension is illustrated, regarding only terms up to the order y2, commonly approached in the literature, for instance, in [19]. Figure 2 shows that the black string behavior is obviously different for different fixed moments without taking into account terms of time derivative of the tension, while Fig. 3 does take into account such terms. By comparing Figs. 4 and 5 , it is possible to see that in Fig. 5, for all t 0, the black string presents no singular- ities. It is precluded by the extra terms in the Taylor expansion of the metric in the brane-corrected black string horizon in our model. Those terms do not appear in models with constant brane tension, since they encode the cova- riant derivatives of the brane variable tension �. The general result encoded in the figures is exhaustive: whenever the time derivative terms are taken into account the variation of the horizon along the extra dimension is such that the horizon does not tend to zero. This effect is, presumable, naively interpreted within this (eminently) classical framework. Still, one could speculate that it would be interpreted in terms of fluctuations around the brane. In fact, combing the fact that a completely rigid object cannot exist in the general relativity framework with the presence of a scalar field representing the brane position into the bulk, one arrives at the possibility of a spontaneous symmetry breaking of the bulk diffeomor- phism. In this way, a perturbative spectrum of scalar par- ticles, the so-called branons, may appear if the tension scale is much smaller than the higher-dimensional mass scale [33]. Now, the brane tension being a variable quantity, a nontrivial contribution to the branons production is ex- pected. Besides, the tension derivatives (computing the rate of tension variation) may also have an important role in the branons production. Following this reasoning, one could guess that such fluctuations could (in principle) supply the black hole horizon along the extra dimension horizon, which makes its approach to the singularity diffi- cult. Obviously, this interpretation must be enforced by a critical analysis of the precise influence of a variable tension on the branons production, as well as the branons influence on the black string behavior. These issues to- gether with possible quantum effects [34] are currently under investigation. ACKNOWLEDGMENTS R. da Rocha is grateful to Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico (CNPq) Grants No. 476580/2010-2 and No. 304862/2009-6 for financial support. FIG. 5 (color online). Graphic of the brane effect-corrected black string horizon g�� for the Schwarzschild metric with variable tension, along the extra dimension y and also as function of the time t. The brane tension is given by �ðtÞ ¼ 1� expð��tÞ. This graphic does take into account the extra terms given by Eq. (12) and (21), that for the case considered are encrypted in Eq. (22). The black string normalized horizon varies from R=ð2GMÞ to R ¼ 3=ð2GMÞ approximately. ROLDÃO DA ROCHA AND J.M. HOFF DA SILVA PHYSICAL REVIEW D 85, 046009 (2012) 046009-8 [1] P. Horava and E. Witten, Nucl. Phys. B460, 506 (1996); P. Horava and E. Witten, Nucl. Phys. B475, 94 (1996). [2] V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. 125B, 136 (1983); V.A. Rubakov and M. E. Shaposhnikov, Phys. Lett. 125B, 139 (1983); K. 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