We studied in detail the propagation of a massive tachyonic scalar field in the background of a five-dimensional (5D) 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 5D black string.

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 (D>4) 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 S2×S1 and spacetime is asymptotically MD-1×S1, four-dimensional (4D) 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 4D BSs. A rotating version of the charged 4D BSs [22, 23] exhibits features similar to the Kerr-Newman BH in spherical topology. The problem of analyzing GFs of scalar fields from charged 4D 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 5D 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 5D 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 5D BS spacetime belongs to the massive tachyonic scalar fields. Therefore, the current study focuses on the wave dynamics of tachyonic particles moving in dilatonic 5D BS spacetime. However, using tachyonic modes in 5D 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 4D physical trajectory is the projection of higher-dimensional worldlines. Effective 4D 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 5D 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 5D BS.

Our work is organized as follows. Following this introduction, a brief overview of the geometry of the dilatonic 5D BS is provided in Section 2. Section 3 describes the KGE of the tachyonic fields in the dilatonic 5D 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 5D 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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M22"><mml:mn>5</mml:mn><mml:mi>D</mml:mi></mml:math></inline-formula> BS in EYMBID Theory

D=d+1-dimensional action in the EYMBID theory is given by [26] (1)IEYMBID=-116πGD∫Mddx-gR-4▽ψ2D-2+4χ2e-bψ1-1+Fe2b2χ2,

where ψ is the dilaton field, χ denotes the Born–Infeld parameter [34], and b=-4/d-2α with the dilaton parameter α=1/d-1. G(D) represents the D-dimensional Newtonian constant and its relation to its 4D form G(4) is given by(2)GD=G4LD-4,where L is the upper limit of the compact coordinate ∫0Ldz=L. Furthermore, R stands for the Ricci scalar and F=Fλρ(a¯)F(a¯)λρ where the 2-form Yang–Mills field is given by(3)Fa=dAa¯+12σCb¯c¯a¯Ab¯∧Ac¯,with C(b¯)(c¯)(a¯) and σ being structure and coupling constants, respectively. The Yang–Mills potential A(a) is defined by following the Wu-Yang ansatz [35](4)Aa¯=Qr2xidxj-xjdxi,(5)r2=∑i=1d-1xi2,2≤j+1≤i≤d-1,1≤a¯≤d-1d-22,where Q is the Yang–Mills charge. The solution for the dilaton is as follows:(6)ψ=-d-22αlnrα2+1. On the other hand, the line-element of the dilatonic 5D BS is given by [26](7)ds02=-frβdt~2+βdr2rfr+rdz~2+βdθ2+sin2θdϕ2, where f(r)=r-r+ and β=4Q2/3. r+ represents the outer event horizon having the following (d+1)-dimensional form:(8)32Ld-4Q2dd-1d-2/2=r+dd-2+2/d.

Because, in our case, D=5(i.e.,d=4), the horizon becomes(9)r+=4β2/5=4.488Q4/5.After rescaling the metric (7)(10)ds2=ds02β=-frβ2dt~2+dr2rfr+rβdz~2+dθ2+sin2θdϕ2,and in sequel assigning t~ and z~ coordinates to the new coordinates(11)t~→βt,z~→βz,we get the metric that will be used in our computations:(12)ds2=-frdt2+dr2rfr+βrdz2+dθ2+sin2θdϕ2.

It is worth noting that the surface gravity [36] of the dilatonic 5D BS can be evaluated by(13)κ2=-12▿ρΥσ▿ρΥσr=r+,in which Υμ represents the timelike Killing vector:(14)Υμ=1,0,0,0,0.Then, (13) results in(15)κ=rf′2r=r+=r+2,where the prime denotes the derivative with respect to r. Furthermore, the associated Hawking temperature is expressed by(16)TH=κ2π=r+4π.

It is important to remark that the Hawking temperature of the dilatonic BS given in (40) of [26] is incorrect. The authors of [26] computed the Hawking temperature of the dilatonic 5D BS considering the metric to be symmetric, which is not the case since gtt≠1/grr. Meanwhile, it is obvious that dilatonic 5D BS (12) has a nonasymptotically flat structure. Therefore, it possesses a quasilocal mass [37–39], which can be computed as follows:(17)MQL=16βrh3/2L=43β8/5L≅2.113Q16/5L.

Thus, the first law of thermodynamics is satisfied:(18)dMQL=THdSBH,

where SBH denotes the Bekenstein-Hawking entropy [36], which takes the following form for the dilatonic 5D BS:(19)SBH=AH4=14βrh∫0πsinθdθ∫02πdϕ∫0Ldz=πβLrh.

3. Wave Equation of a Massive Scalar Tachyonic Field in Dilatonic <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M75"><mml:mn>5</mml:mn><mml:mi>D</mml:mi></mml:math></inline-formula> BS

As the scalar waves being studied belong to the massive scalar tachyons, the corresponding KGE is given by (20)□-iμ2Ψt,r=□+μ2Ψt,r=0.We chose the ansatz as follows:(21)Ψt,r=RrYlmθ,ϕeikze-iωt,where Ylm(θ,ϕ) is the usual spherical harmonics and k is a constant. After making straightforward calculations, we obtained the radial equation as follows:(22)rfR¨R+R˙Rf+rf˙+ω2f-k2βr+μ2-λ=0,where λ=l(l+1) and a dot mark denotes a derivative with respect to r. Multiplying each term by rβf(r)R(r) and using the ansatz y=r+-r/r+, which in turn implies r=r+-yr+, one gets(23)y1-yR′′+1-2yR′+ω2yr++k2β1-yr+-μ2+λR=0,

where prime denotes derivative with respect to y. Setting(24)ω2yr++k2β1-yr+-μ2+λ=A2y-B21-y+C,

one can obtain(25)A=-ω2κ,B=ik2κβ,C=λ-μ2.

Equation (23) can be solved by comparing it with the standard hypergeometric differential equation [40] which admits the following solution:(26)R=ξ1-yiA1-y-BFa,b;c;y+ξ2-y-iA1-y-BFα,ς;η;y,where(27)a=121+1+4C+iA-B,(28)b=121-1+4C+iA-B,(29)c=1+2iA. And(30)α=a-c+1,(31)ς=b-c+1,(32)η=2-c. According to our calculations, to have a nonzero SI incoming flux (see (53)) or nondivergent GF (see (55)), 1+4C must be imaginary. To this end, we must impose the following condition:(33)4μ2>4λ+1,

such that(34)1+4C=iτ,where(35)τ=4μ2-4λ-1,τ∈R.To obtain a physically acceptable solution, we must terminate the outgoing solution at the horizon, which can be simply done by imposing ξ1=0. Thus, the physical radial solution reduces to(36)R=ξ2-y-iA1-y-BFα,ς;γ;y.It 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 y→0, one can state(37)RNH=ξ2-y-iA,

which implies that the purely ingoing plane wave reads(38)ψNH=ξ2e-iωr^∗+teikz,where(39)r∗=∫drrf→r^∗=limr→r+r∗≃ln-yr+,r^∗≃12κln-y⇒y=-e2κr^∗.

On the other hand, for y→∞, the inverse transformation of the hypergeometric function is given by [41](40)Fα,ς;η;y=-y-αΓηΓς-αΓςΓη-α×Fα,α+1-η;α+1-ς;1y+-y-ςΓηΓα-ςΓαΓη-ς×Fς,ς+1-η;ς+1-α;1y,which yields the following asymptotic solution:(41)RSI≃ξ2-y-iA-B-αΓηΓς-αΓςΓη-α+ξ2-y-iA-B-ςΓηΓα-ςΓαΓη-ς.To express (41) in a more compact form, let us perform the following simplifications. Considering(42)-iA-B-α=-121+iτ,together with(43)-iA-B-ς=-121-iτ,

and letting x=-y, the radial equation for r→∞ takes the form(44)RSI=1xξ2x-iτ/2ΓηΓς-αΓςΓη-α+ξ2xiτ/2ΓηΓα-ςΓαΓη-ς.One can express x in terms of the tortoise coordinate at SI as(45)r∗=∫drrf→r^∗=limr→∞r∗≃-2r, such that(46)x=r-r+xr→∞≃r=4e-2r^∗, where r^∗=lnr∗. Therefore, we have(47)RSI=1rΛ1eir^∗τ+Λ2e-ir^∗τ,where(48)Λ1=2-iτξ2ΓηΓς-αΓςΓη-α,(49)Λ2=2-iτξ2ΓηΓα-ςΓαΓη-ς. Thus, the asymptotic wave solution becomes(50)ψSI=eikzrΛ1eir^∗τ-ωt+Λ2e-ir^∗τ+ωt.

4. Radiation of Dilatonic <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M118"><mml:mn>5</mml:mn><mml:mi>D</mml:mi></mml:math></inline-formula> BS4.1. The Flux Computation

In this section, we compute the ingoing flux at the horizon r→r+ and the asymptotic flux for the SI region r→∞. 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](51)ϝNH=ABH2iψ¯NH∂r∗ψNH-ψNH∂r∗ψ¯NH,which, after a few manipulations, can be written as(52)ϝNH=-4πβξ22r+.The incoming flux at SI is computed via(53)ϝSI=ABH2iψ¯SI∂r∗ψSI-ψSI∂r∗ψ¯SI.Having performed the steps to evaluate the derivatives with respect to the tortoise coordinate, the incoming flux at SI takes the form(54)ϝSI=-4πβΛ22τ.

It is important to remark that if we were dealing with the standard particles rather than tachyons, τ (35) would be imaginary, i.e., τ→iτ, 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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M127"><mml:mn mathvariant="normal">5</mml:mn><mml:mi>D</mml:mi></mml:math></inline-formula> BS

The GF of the dilatonic 5D BS is obtained by the following expression [6, 8]:(55)γl,k=ϝNHϝSI=-4πβξ22r+-4πβΛ22τ,which is nothing but(56)γl,k=ξ22r+Λ22τ.After a few manipulations, with (see [40])(57)Γiy2=πysinhπy,(58)Γ1+iy2=πysinhπy,(59)Γ12+iy2=πcoshπy,(56) can be presented as(60)γl,k=κr+ωe2πω/κ-1Ξ,where(61)Ξ=e2πτ-1eπτ+ω/κ-k/κβ+1eπτ+ω/κ+k/κβ+1.To evaluate the ACS of the dilatonic 5D BS concerned, we follow the study of [44]. Thus, one can get the ACS expression in 5D as follows:(62)σl,k=4πl+12ω3γl,k,which, in our case, becomes (63)σl,k=4πl+12κr+ω4e2πω/κ-1Ξ.

Furthermore, one can also get the total ACS as follows [45]:(64)σabsTotal=∑l=0∞σl,k.

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 ω→0. As a final remark, negative σl,k behavior has not been observed in our graphical analyses, which means that superradiance does not occur [46], as expected (as the dilatonic 5D BS (12) does not rotate).

Plots of the ACS (σl,k) versus frequency ω. The plots are governed by (63). The configuration of the dilatonic 5D BS is as follows: μ=3 and Q=0.2.

4.3. DR of Dilatonic <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M147"><mml:mn mathvariant="normal">5</mml:mn><mml:mi>D</mml:mi></mml:math></inline-formula> BS

The final step follows from the ACS evaluation. The DR of the dilatonic 5D BS can be computed via [8](65)ΓDRl,k=σl,ke2πω/κ-1=4πl+12κr+ω4Ξ.

Figure 2 shows how the DR behaves with respect to the frequency. By taking (65) as the reference, the plots for increasing l 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 l has larger values, the corresponding DR diverges when ω is much closer to zero.

Plots of the DR (ΓDRl,k) versus frequency ω. The plots are governed by (65). The configuration of the dilatonic 5D BS is as follows: μ=3 and Q=0.2.

5. Conclusion

This article evaluated the GF, ACS, and DR for the dilatonic 5D 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 5D 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 5D 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 5D 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.

HawkingS. W.Black hole explosions?HawkingS. W.Particle creation by black holesCreekS.EfthimiouO.KantiP.TamvakisK.Greybody factors for brane scalar fields in a rotating black hole backgroundBoonsermP.Rigorous bounds on Transmission, Reflection, and Bogoliubov coefficients2009, https://arxiv.org/abs/0907.0045HoyosC.KoroteevP.On the null energy condition and causality in lifshitz holographyHarmarkT.NatarioJ.SchiappaR.Greybody Factors for d-Dimensional Black HolesKalloshR.LindeA.VercnockeB.Natural inflation in supergravity and beyondSánchez-BretañoA.CallejoM.MonteroM.Alonso-GómezÁ. L.DelgadoM. J.IsornaE.Performing a hepatic timing signal: glucocorticoids induce gper1a and gper1b expression and repress gclock1a and gbmal1a in the liver of goldfishPanotopoulosG.RinconA.Greybody factors for a minimally coupled massless scalar field in Einstein-Born-Infeld dilaton spacetimeZhangY.LiC.ChenB.Greybody factors for a spherically symmetric Einstein-Gauss-Bonnet–de Sitter black holeKunzJ.Black holes in higher dimensions (black strings and black rings)Proceedings of the MG13 Meeting on General Relativity2015SwedenStockholm University56858110.1142/9789814623995_0027GregoryR.LaflammeR.The instability of charged black strings and p-branesGregoryR.LaflammeR.Black strings and p-branes are unstableKlebanovI. R.MathurS. D.Black hole greybody factors and absorption of scalars by effective stringsCallanJ.GubserS. S.KlebanovI. R.TseytlinA. A.Absorption of fixed scalars and the D-brane approach to black holesCveticM.LarsenF.General rotating black holes in string theory: greybody factors and event horizonsKlebanovI. R.KrasnitzM.Fixed scalar greybody factors in five and four dimensionsGubserS. S.KlebanovI. R.Emission of charged particles from four- and five-dimensional black holesLemosJ. P.Two-dimensional black holes and planar general relativityLemosJ. P. S.Three dimensional black holes and cylindrical general relativitySantosN. O.Solution of the vacuum Einstein equations with nonzero cosmological constant for a stationary cylindrically symmetric spacetimeLemosJ. P. S.ZanchinV. T.Rotating charged black strings and three-dimensional black holesCaiR. G.ZhangY. Z.Black plane solutions in four-dimensional spacetimesAhmedJ.SaifullahK.Greybody factor of scalar fields from black stringsAkhmedovE. K.PulidoJ.Solar neutrino oscillations and bounds on neutrino magnetic moment and solar magnetic fieldMazharimousaviaS. H.HalilsoyM.Non-Abelian magnetic black strings versus black holesSenA.Rolling TachyonFeinbergG.Possibility of faster-than-light particlesWolfensteinL.Neutrino oscillations in matterAharonovY.KomarA.SusskindL.Superluminal Behavior, Causality, and InstabilityKleinO.Quantentheorie und fünfdimensionale relativitätstheorieDavidsonA.OwenD. A.Elementary particles as higher-dimensional tachyonsHorowitzG.Tachyon condensation and black stringsBornM.InfeldL.Foundations of the new field theoryWuT. T.YangC. N.WaldR. M.BrownJ. D.YorkJ. W.Jr.Quasilocal energy and conserved charges derived from the gravitational actionBoseS.DadhichN.Brown-York quasilocal energy, gravitational charge, and black hole horizonsChakrabortyS.DadhichN.Brown-York quasilocal energy in Lanczos-Lovelock gravity and black hole horizonAbramowitzM.StegunI. A.SlavyanovS. Y.LayW.FernandoS.Greybody factors of charged dilaton black holes in 2 + 1 dimensionsAndrewsM.LewandowskiM.TroddenM.WesleyD.Distinguishing k-defects from their canonical twinsKantiP.GrainJ.BarrauA.Bulk and brane decay of a-dimensional Schwarzschild-de SITter black hole: scalar radiationUnruhW. G.Absorption cross section of small black holesChandrasekharS.