Abstract
We studied in detail the propagation of a massive tachyonic scalar field in the background of a five-dimensional () 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 field in the geometry under consideration. The behaviors of the obtained results are graphically represented for different 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 black string.
1. Introduction
A wealth of information about quantum gravity can be obtained by studying the unique and fascinating objects known as black holes (BHs). In BH physics, greybody factors (GFs) modify black-body radiation, or predicted Hawking radiation [1, 2], within the limits of geometrical optics [3]. In other words, GFs modify the Hawking radiation spectrum observed at spatial infinity (SI), so that the radiation is not purely Planckian [4].
GF, absorption cross section (ACS), and decay rate (DR) are quantities dependent upon both the frequency of radiation and the geometry of spacetime. Currently, although there are many studies of GF, ACS, and DR (see, for example, [5–10] and the references therein), the number of analytical studies of GFs that consider modified black-body radiation of higher-dimensional () spacetimes, like the BHs in string theory and black strings [11–13], is rather limited (see, for instance, [6, 7, 14–18]). This paucity of studies has arisen from the mathematical difficulty of obtaining an analytical solution to the wave equation of the stringent geometry being considered; in fact, analytical GF computations apply to spacetimes in which the metric components are independent of time. It is also worth noting that although BSs are defined as a higher-dimensional generalization of a BH, in which the event horizon is topologically equivalent to and spacetime is asymptotically , four-dimensional () BSs are also derived. Lemos and Santos [19–21] showed that cylindrically symmetric static solutions, with a negative cosmological constant, of the Einstein–Maxwell equations admit charged BSs. A rotating version of the charged BSs [22, 23] exhibits features similar to the Kerr-Newman BH in spherical topology. The problem of analyzing GFs of scalar fields from charged BSs has recently been discussed by Ahmed and Saifullah [24]. An interesting point about GFs has been reported by [25]: BH energy loss during Hawking radiation depends, crucially, on the GF and the particles’ degrees of freedom.
As mentioned above, further study of the GFs of BSs is required. To fill this literature gap, in the current study, we considered dilatonic BS [26], which is a solution to the Einstein–Yang–Mills–Born–Infeld–dilaton (EYMBID) theory. We analytically studied its GF, ACS, and DR for massive scalar fields; however, we considered tachyonic scalar particles instead of ordinary ones. The main reason for this consideration is that using ordinary mass in the Klein–Gordon equation (KGE) of the dilatonic BS (as will be explained in detail later) leads to the diverging of GFs. Roughly speaking, this is due to the flux of the propagating waves of the ordinary massive scalar fields. Namely, once the scalar fields to be considered belong to the massive ordinary particles, the incoming SI flux becomes zero. The latter remark implies that detectable radiation emitted from a dilatonic BS spacetime belongs to the massive tachyonic scalar fields. Therefore, the current study focuses on the wave dynamics of tachyonic particles moving in dilatonic BS spacetime. However, using tachyonic modes in geometry should not be seen as nonphysical; instead they should be considered as the imaginary mass fields rather than faster-than-light particles [27]. First, Feinberg [28, 29] proposed that tachyonic particles could be quanta of a quantum field with imaginary mass. It was soon realized that excitations of such imaginary mass fields do not in fact propagate faster than light [30].
Following the idea of Kaluza–Klein [31], any physical trajectory is the projection of higher-dimensional worldlines. Effective worldlines associated with massive particles are causality constrained to be timelike. However, the corresponding higher-dimensional worldlines need not be exclusively timelike, which gives rise to a topological classification of physical objects. In particular, elementary particles in a geometry should be viewed as tachyonic modes. The existence of tachyons in higher dimensions has been thoroughly studied by Davidson and Owen [32]. Furthermore, the reader may refer to [33] to understand tachyon condensation in the evaporation process of a BS. To find the analytical GF, ACS, and DR, we have shown how to obtain the complete analytical solution to the massive KGE in the geometry of a dilatonic BS.
Our work is organized as follows. Following this introduction, a brief overview of the geometry of the dilatonic BS is provided in Section 2. Section 3 describes the KGE of the tachyonic fields in the dilatonic BS geometry; we present the exact solution of the radial equation in terms of hypergeometric functions. In Section 4, we compute the GF and consequently the ACS and DR of the dilatonic BS, respectively. We then graphically exhibit the results of the ACS and DR. Section 5 concludes with the final remarks drawn from our study.
2. Dilatonic BS in EYMBID Theory
-dimensional action in the EYMBID theory is given by [26]
where is the dilaton field, denotes the Born–Infeld parameter [34], and with the dilaton parameter . represents the -dimensional Newtonian constant and its relation to its form is given bywhere is the upper limit of the compact coordinate . Furthermore, stands for the Ricci scalar and where the 2-form Yang–Mills field is given bywith and being structure and coupling constants, respectively. The Yang–Mills potential is defined by following the Wu-Yang ansatz [35]where is the Yang–Mills charge. The solution for the dilaton is as follows: On the other hand, the line-element of the dilatonic BS is given by [26] where and . represents the outer event horizon having the following dimensional form:
Because, in our case, , the horizon becomesAfter rescaling the metric (7)and in sequel assigning and coordinates to the new coordinateswe get the metric that will be used in our computations:
It is worth noting that the surface gravity [36] of the dilatonic BS can be evaluated byin which represents the timelike Killing vector:Then, (13) results inwhere the prime denotes the derivative with respect to . Furthermore, the associated Hawking temperature is expressed by
It is important to remark that the Hawking temperature of the dilatonic BS given in of [26] is incorrect. The authors of [26] computed the Hawking temperature of the dilatonic BS considering the metric to be symmetric, which is not the case since . Meanwhile, it is obvious that dilatonic BS (12) has a nonasymptotically flat structure. Therefore, it possesses a quasilocal mass [37–39], which can be computed as follows:
Thus, the first law of thermodynamics is satisfied:
where denotes the Bekenstein-Hawking entropy [36], which takes the following form for the dilatonic BS:
3. Wave Equation of a Massive Scalar Tachyonic Field in Dilatonic BS
As the scalar waves being studied belong to the massive scalar tachyons, the corresponding KGE is given by We chose the ansatz as follows:where is the usual spherical harmonics and is a constant. After making straightforward calculations, we obtained the radial equation as follows:where and a dot mark denotes a derivative with respect to . Multiplying each term by and using the ansatz , which in turn implies , one gets
where prime denotes derivative with respect to . Setting
one can obtain
Equation (23) can be solved by comparing it with the standard hypergeometric differential equation [40] which admits the following solution:where And According to our calculations, to have a nonzero SI incoming flux (see (53)) or nondivergent GF (see (55)), must be imaginary. To this end, we must impose the following condition:
such thatwhereTo obtain a physically acceptable solution, we must terminate the outgoing solution at the horizon, which can be simply done by imposing . Thus, the physical radial solution reduces toIt should be noted that checking the forms of (36) both at the horizon and at SI is essential. Section 3 shows that both are needed for the evaluation of the GF. For the near horizon (NH) where , one can state
which implies that the purely ingoing plane wave readswhere
On the other hand, for , the inverse transformation of the hypergeometric function is given by [41]which yields the following asymptotic solution:To express (41) in a more compact form, let us perform the following simplifications. Consideringtogether with
and letting , the radial equation for takes the formOne can express in terms of the tortoise coordinate at SI as such that where . Therefore, we havewhere Thus, the asymptotic wave solution becomes
4. Radiation of Dilatonic BS
4.1. The Flux Computation
In this section, we compute the ingoing flux at the horizon and the asymptotic flux for the SI region . The evaluation of these flux values will enable us to calculate the GF and, subsequently, the ACS and DR.
The NH-flux can be calculated via [42, 43]which, after a few manipulations, can be written asThe incoming flux at SI is computed viaHaving performed the steps to evaluate the derivatives with respect to the tortoise coordinate, the incoming flux at SI takes the form
It is important to remark that if we were dealing with the standard particles rather than tachyons, (35) would be imaginary, i.e., , and therefore SI incoming wave would lead this flux evaluation (53) to be zero. This would indicate the existence of a divergent GF.
4.2. ACS of Dilatonic BS
The GF of the dilatonic BS is obtained by the following expression [6, 8]:which is nothing butAfter a few manipulations, with (see [40])(56) can be presented aswhereTo evaluate the ACS of the dilatonic BS concerned, we follow the study of [44]. Thus, one can get the ACS expression in as follows:which, in our case, becomes
Furthermore, one can also get the total ACS as follows [45]:
In Figure 1, the relationship between absorption ACS and frequency is examined; the figure is drawn based on (63). In the high frequency regime, all ACSs tend to vanish by following the same curve. Unlike the high frequency regime, ACSs diverge in the low frequency regime as . As a final remark, negative behavior has not been observed in our graphical analyses, which means that superradiance does not occur [46], as expected (as the dilatonic BS (12) does not rotate).
4.3. DR of Dilatonic BS
The final step follows from the ACS evaluation. The DR of the dilatonic BS can be computed via [8]
Figure 2 shows how the DR behaves with respect to the frequency. By taking (65) as the reference, the plots for increasing are illustrated. In the high frequency regime, all DRs fade in the same way. In the low frequency regime, DRs tend to diverge. However, it can be observed that when has larger values, the corresponding DR diverges when is much closer to zero.
5. Conclusion
This article evaluated the GF, ACS, and DR for the dilatonic BS geometry arising from the EYMBID theory. As a result of the analytical method we followed, it was shown that the radiation of the dilatonic BS spacetime can only be caused by tachyons. The crucial point here is that if standard scalar particles had been used rather than tachyonic ones, zero incoming flux at SI would have been obtained, which would lead to the diverging of the GF. Therefore, in a way, we were forced to use tachyons to solve this problem, and this carries great importance as it implies that the fifth dimension could be directly linked to tachyons. In short, according to our analytical method, we obtained results (compatible with boundary conditions) when the radiation of the dilatonic BS was provided by the tachyons.
In future study, we want to extend our analysis to the Dirac equation for the geometry of the dilatonic BS. Hence, we are planning to undertake similar analysis for fermions and compare the results with scalar ones.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
We wish to thank Prof. Dr. Mustafa Halilsoy for drawing our attention to this problem and for his helpful comments and suggestions. This work is supported by Eastern Mediterranean University through the project BAPC-04-18-01.