Research Article Absorption Cross Section and Decay Rate of Dilatonic Black Strings

Hindawi Advances in High Energy Physics Volume 2018, Article ID 8504894, 7 pages https://doi.org/10.1155/2018/8504894 Research Article Absorption Cross Section and Decay Rate of Dilatonic Black Strings Huriye Gürsel and Ezzet SakallJ Physics Department, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North Cyprus, Mersin 10, Turkey Correspondence should be addressed to ˙Izzet Sakallı; [email protected] Received 25 October 2018; Accepted 4 December 2018; Published 19 December 2018 Academic Editor: Piero Nicolini Copyright © 2018 Huriye Gursel¨ and ˙Izzet Sakallı. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3. We studied in detail the propagation of a massive tachyonic scalar feld in the background of a fve-dimensional (5�) Einstein–Yang–Mills–Born–Infeld–dilaton black string: the massive Klein–Gordon equation was solved, exactly. Next we obtained complete analytical expressions for the greybody factor, absorption cross section, and decay rate for the tachyonic scalar feld in the geometry under consideration. Te behaviors of the obtained results are graphically represented for diferent values of the theory’s free parameters. We also discuss why tachyons should be used instead of ordinary particles for the analytical derivation of the greybody factor of the dilatonic 5� black string. 1. Introduction of time. It is also worth noting that although BSs are defned as a higher-dimensional generalization of a BH, in which 2 1 A wealth of information about quantum gravity can be the event horizon is topologically equivalent to � ×� and �−1 1 obtained by studying the unique and fascinating objects spacetime is asymptotically � ×�,four-dimensional known as black holes (BHs). In BH physics, greybody factors (4�) BSs are also derived. Lemos and Santos [19–21] showed (GFs) modify black-body radiation, or predicted Hawking that cylindrically symmetric static solutions, with a negative radiation [1, 2], within the limits of geometrical optics [3]. cosmological constant, of the Einstein–Maxwell equations In other words, GFs modify the Hawking radiation spectrum admit charged 4� BSs. A rotating version of the charged 4� observed at spatial infnity (SI), so that the radiation is not BSs [22, 23] exhibits features similar to the Kerr-Newman BH purely Planckian [4]. insphericaltopology.TeproblemofanalyzingGFsofscalar GF, absorption cross section (ACS), and decay rate felds from charged 4� BSs has recently been discussed by (DR) are quantities dependent upon both the frequency of Ahmed and Saifullah [24]. An interesting point about GFs radiation and the geometry of spacetime. Currently, although has been reported by [25]: BH energy loss during Hawking there are many studies of GF, ACS, and DR (see, for example, radiation depends, crucially, on the GF and the particles’ [5–10] and the references therein), the number of analytical degrees of freedom. studies of GFs that consider modifed black-body radiation As mentioned above, further study of the GFs of BSs of higher-dimensional (�>4) spacetimes, like the BHs in is required. To fll this literature gap, in the current study, string theory and black strings [11–13], is rather limited (see, we considered dilatonic 5� BS [26], which is a solution forinstance,[6,7,14–18]).Tispaucityofstudieshasarisen to the Einstein–Yang–Mills–Born–Infeld–dilaton (EYMBID) from the mathematical difculty of obtaining an analytical theory. We analytically studied its GF, ACS, and DR for solution to the wave equation of the stringent geometry massive scalar felds; however, we considered tachyonic being considered; in fact, analytical GF computations apply to scalar particles instead of ordinary ones. Te main reason spacetimes in which the metric components are independent for this consideration is that using ordinary mass in the 2 Advances in High Energy Physics Klein–Gordon equation (KGE) of the dilatonic 5� BS (as will Newtonian constant and its relation to its 4� form (�(4)) is be explained in detail later) leads to the diverging of GFs. given by Roughly speaking, this is due to the fux of the propagating � =� ��−4, waves of the ordinary massive scalar felds. Namely, once (�) (4) (2) the scalar felds to be considered belong to the massive � � (∫ �� = ordinary particles, the incoming SI fux becomes zero. Te where istheupperlimitofthecompactcoordinate 0 latter remark implies that detectable radiation emitted from a �).Furthermore,R stands for the Ricci scalar and �= �(�)�(�)�� dilatonic 5� BS spacetime belongs to the massive tachyonic �� where the 2-form Yang–Mills feld is given by scalar felds. Terefore, the current study focuses on the wave (�) (�) 1 (�) (�) (�) dynamics of tachyonic particles moving in dilatonic 5� BS � =�� + � (� ∧� ) , (3) 2� (�)(�) spacetime. However, using tachyonic modes in 5� geometry (�) should not be seen as nonphysical; instead they should be with � and � being structure and coupling constants, considered as the imaginary mass felds rather than faster- (�)(�) �(�) than-light particles [27]. First, Feinberg [28, 29] proposed respectively. Te Yang–Mills potential is defned by that tachyonic particles could be quanta of a quantum feld followingtheWu-Yangansatz[35] with imaginary mass. It was soon realized that excitations of � �(�) = (� �� −�� ), such imaginary mass felds do not in fact propagate faster than �2 � � � � (4) light [30]. Following the idea of Kaluza–Klein [31], any 4� physical �−1 �2 = ∑�2, trajectory is the projection of higher-dimensional worldlines. � �=1 Efective 4� worldlines associated with massive particles are (5) causality constrained to be timelike. However, the corre- (�−1)(�−2) 2≤�+1≤�≤�−1,1≤�≤ , sponding higher-dimensional worldlines need not be exclu- 2 sively timelike, which gives rise to a topological classifcation � of physical objects. In particular, elementary particles in a 5� where is the Yang–Mills charge. Te solution for the dilaton geometry should be viewed as tachyonic modes. Te exis- is as follows: tence of tachyons in higher dimensions has been thoroughly (�−2) � ln � �=− . (6) studied by Davidson and Owen [32]. Furthermore, the reader 2 �2 +1 may refer to [33] to understand tachyon condensation in the On the other hand, the line-element of the dilatonic 5� BS is evaporation process of a BS. To fnd the analytical GF, ACS, given by [26] andDR,wehaveshownhowtoobtainthecompleteanalytical solution to the massive KGE in the geometry of a dilatonic 5� � (�) ��2 ��2 =− �̃�2 + +��̃2 BS. 0 � �� (�) Our work is organized as follows. Following this intro- (7) 2 2 2 duction, a brief overview of the geometry of the dilatonic +�(�� + sin �� ), 5� BS is provided in Section 2. Section 3 describes the KGE 5� 2 of the tachyonic felds in the dilatonic BS geometry; we where �(�) = � − �+ and �=4�/3. �+ represents the outer present the exact solution of the radial equation in terms event horizon having the following (�+1)−dimensional form: of hypergeometric functions. In Section 4, we compute the 2 (�−2)/2 GF and consequently the ACS and DR of the dilatonic 5� 32 � � (�(�−2)+2)/� ( ) =� . (8) BS, respectively. We then graphically exhibit the results of ��−4 �−1 + the ACS and DR. Section 5 concludes with the fnal remarks �=5( ,� = 4) drawn from our study. Because, in our case, i.e. , the horizon becomes 2/5 4/5 2. Dilatonic 5� BS in EYMBID Theory �+ =4� = 4.488� . (9) �(=�+1)-dimensional action in the EYMBID theory is Afer rescaling the metric (7) given by [26] ��2 ��2 = 0 1 � � =− ∫ �� −� �� √ (10) 16��(�) M 2 � (�) 2 �� 2 2 2 2 =− �̃� + + ��̃ +�� + sin �� , 2 (1) �2 �� (�) � 4(��) ��2� ⋅ [R − +4�2�−�� (1 − √1+ )] , �−2 2�2 and in sequel assigning ̃� and �̃ coordinates to the new [ ] coordinates where � is the dilaton feld, � denotes the Born–Infeld ̃��→��, parameter [34], and � = −(4/(� − 2))� with the dilaton (11) √ ��→��,̃ parameter �=1/ �−1. �(�) represents the �-dimensional Advances in High Energy Physics 3 � we get the metric that will be used in our computations: where �� (�, �) is the usual spherical harmonics and � is a constant. Afer making straightforward calculations, we ��2 ��2 =−�(�) ��2 + + ��2 +��2 + 2 ��2. obtained the radial equation as follows: �� (�) sin (12) ̈ ̇ 2 2 � � ̇ � � 2 �� + (�+��) + − +� −�=0, (22) It is worth noting that the surface gravity [36] of the � � � �� dilatonic 5� BS can be evaluated by where �=�(�+1)and a dot mark denotes a derivative with 1 � �2 =− ��Υ�� Υ � , respect to �. Multiplying each term by ��(�)�(�) and using � ��=� (13) 2 + the ansatz �=(�+ − �)/�+, which in turn implies �=�+ −��+, � one gets in which Υ represents the timelike Killing vector: �� (1−�)� +(1−2�)� Υ = [1, 0, 0, 0, 0] . (14) �2 �2 (23) Ten, (13) results in +[ + −�2 +�]�=0, ��+ �(1−�)�+ � � √�� √�+ �= � = , � 2 � 2 (15) where prime denotes derivative with respect to .Setting ��=�+ �2 �2 �2 �2 � [ + −�2 +�]= − +�, where the prime denotes the derivative with respect to .Fur- �� (1−�)� � 1−� (24) thermore, the associated Hawking temperature is expressed + + by one can obtain � � √�+ � = = . (16) �=− , � 2� 4� 2� �� ItisimportanttoremarkthattheHawkingtemperatureof �= , (25) the dilatonic BS given in (40) of [26] is incorrect. Te authors 2�√� of [26] computed the Hawking temperature of the dilatonic 2 5� BS considering the metric to be symmetric, which is not �=�−�. the case since �� =1/�̸

Load more