Absorption Cross-Section and Decay Rate of Dilatonic Black Strings

Huriye Gursel¨ ∗ and Izzet˙ Sakallı† Abstract 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.

PACS numbers: 04.20.Jb, 04.62.+v, 04.70.Dy

Keywords: Black String, Tachyon, Greybody factor, Cross-section, Decay-rate, Dilaton, Klein-Gordon

Equation, Hypergeometric Functions. arXiv:1806.03446v2 [hep-th] 25 Oct 2018

∗ [email protected] † [email protected]; [email protected]

1 CONTENTS

I. Introduction2

II. Dilatonic 5D BS in EYMBID Theory4

III. Wave Equation of a Massive Scalar Tachyonic Field in Dilatonic 5D BS7

IV. Radiation of Dilatonic 5D BS 11 A. The Flux Computation 11 B. ACS of Dilatonic 5D BS 11 C. DR of Dilatonic 5D BS 13

V. Conclusion 14

Acknowledgment 14

References 15

I. 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 pure 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

2 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 cosmo- logical 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 litera- ture gap, in the current study, we considered dilatonic 5D BS [26], which is a solution to 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 to 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 classifica- tion of physical objects. In particular, elementary particles in a 5D geometry should be

3 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 Sec. 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 Sec. 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.

II. DILATONIC 5D BS IN EYMBID THEORY

D (= d + 1)-dimensional action in the EYMBID theory is given by [26]

Z " 2 1 d p 4 (5ψ) IEYMBID = − d x −g R − + 16πG(D) M D − 2

 s  2b # − Fe 4χ2e bψ 1 − 1 +  , (1) 2χ2

4 where ψ is the dilaton ﬁeld, χ denotes the Born-Infeld parameter [34], and b = − d−2 α with the dilaton parameter α = √ 1 . G represents the D-dimensional Newtonian d−1 (D) constant and its relation to its 4D form G(4) is given by

D−4 G(D) = G(4)L , (2) R L = R where L is the upper limit of the compact coordinate 0 dz L . Furthermore, (a) (a)λρ stands for the Ricci scalar and F = Fλρ F where the 2-form Yang-Mills ﬁeld is given by 1 (a) F(a) = dA(a) + C A(b) ∧ A(c) , (3) 2σ (b)(c) (a) with C and σ being structure and coupling constants, respectively. The Yang-Mills (b)(c) potential A(a)is deﬁned by following the Wu-Yang ansatz [35]

4 Q A(a) = x dx − x dx , (4) r2 i j j i

d−1 2 2 r = ∑ xi , 2 ≤ j + 1 ≤ i ≤ d − 1, 1 ≤ a ≤ (d − 1)(d − 2)/2, (5) i=1 where Q is the Yang-Mills charge. The solution for the dilaton is as follows

(d − 2) α ln r ψ = − . (6) 2 α2 + 1 On the other hand, the line-element of the dilatonic 5D BS is given by [26]

f (r) βdr2 ds2 = − det2 + + rdz2 + β dθ2 + sin2 θdφ2 , (7) 0 β r f (r) e

4Q2 where f (r) = r − r+ and β = 3 . r+ represents the outer event horizon having the following (d + 1)−dimensional form:

d−2 32 Q2d 2 d(d−2)+2 = r d . (8) Ld−4 d − 1 + Because, in our case, D = 5 (i.e., d = 4), the horizon becomes

2 4 r+ = 4β 5 = 4.488Q 5 . (9)

After rescaling the metric (7)

ds2 f (r) dr2 r ds2 = 0 = − det2 + + dz2 + dθ2 + sin2 θdφ2, (10) β β2 r f (r) β e

and in sequel assigning et and ez coordinates to the new coordinates:

et → βt, ez → βz, (11)

we get the metric that will be used in our computations:

dr2 ds2 = − f (r)dt2 + + βrdz2 + dθ2 + sin2 θdφ2. (12) r f (r) It is worth noting that the surface gravity [38] of the dilatonic 5D BS can be evaluated by

5 2 1 ρ σ κ = − O Υ OρΥσ , (13) 2 r=r+ in which Υµ represents the timelike Killing vector:

Υµ = [1, 0, 0, 0, 0]. (14)

Then, Eq.(13) results in

√ 0 √ r f r+ κ = = , (15) 2 r=r+ 2 where the prime denotes the derivative with respect to r. Furthermore, the associated Hawking temperature is expressed by

√ κ r+ T = = . (16) H 2π 4π

It is important to remark that the Hawking temperature of the dilatonic BS given in Eq. (40) of Ref. [26] is incorrect. The authors of Ref. [26] computed the Hawking temperature of the dilatonic 5D BS considering the metric to be symmetric, which is not the case since g 6= 1 . Meanwhile, it is obvious that dilatonic 5D BS (12) has a non-asymptotically ﬂat tt grr structure. Therefore, it possesses a quasilocal mass [39–41], which can be computed as follows

1 3 4 8 16 M = βr 2 L = β 5 L =∼ 2.113Q 5 L. (17) QL 6 h 3

Thus, the ﬁrst law of thermodynamics is satisﬁed:

dMQL = THdSBH, (18)

where SBH denotes the Bekenstein-Hawking entropy [38], which takes the following form for the dilatonic 5D BS

Z π Z 2π Z L AH 1 SBH = = βrh sin θdθ dφ dz = πβLrh. (19) 4 4 0 0 0

6 III. WAVE EQUATION OF A MASSIVE SCALAR TACHYONIC FIELD IN DILATONIC 5D BS

As the scalar waves being studied belong to the massive scalar tachyons, the corresponding KGE is given by

h 2i h 2i − (iµ) Ψ(t, r) = + µ Ψ(t, r) = 0. (20)

We chose ansatz as follows:

m ikz −iωt Ψ(t, r) = R(r)Yl (θ, φ)e e , (21)

m where Yl (θ, φ) is the usual spherical harmonics and k is a constant. After making straightforward calculations, we obtained the radial equation as follows:

·· · R R · ω2 k2 r f + f + r f + − + µ2 − λ = 0, (22) R R f βr where λ = l(l + 1) and a dot mark denotes a derivative with respect to r. Multiplying r f (r)R(r) y = r+−r r = r − each term by β and using the ansatz r+ , which in turn implies + yr+, one gets

ω2 k2 y(1 − y)R00 + (1 − 2y)R0 + + − µ2 + λ R = 0, (23) yr+ β(1 − y)r+ where prime denotes derivative with respect to y. Setting

ω2 k2 A2 B2 + − µ2 + λ = − + C, (24) yr+ β(1 − y)r+ y 1 − y one can obtain

ω A = − , 2κ ik B = p , (25) 2κ β C = λ − µ2.

Equation (23) can be solved by comparing it with the standard hypergeometric differ- ential equation [42] which admits the following solution:

7 iA −B R = ξ1 (−y) (1 − y) F (a, b; c; y) + −iA −B ξ2 (−y) (1 − y) F (α, ς; η; y) , (26) where

1 √ a = 1 + 1 + 4C + iA − B, (27) 2

1 √ b = 1 − 1 + 4C + iA − B, (28) 2

c = 1 + 2iA. (29)

and

α = a − c + 1, (30)

ς = b − c + 1, (31)

η = 2 − c. (32)

According to our calculations, to have a non-zero SI incoming ﬂux [see Eq. (53)] or non- √ divergent GF [see Eq.(55)] , 1 + 4C must be imaginary. To this end, we must impose the following condition:

4µ2 > 4λ + 1, (33)

such that

√ 1 + 4C = iτ, (34)

where

q τ = 4µ2 − 4λ − 1, τ ∈ R. (35)

8 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

−iA −B R = ξ2 (−y) (1 − y) F (α, ς; γ; y) . (36)

It should be noted that checking the forms of Eq. (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:

−iA RNH = ξ2 (−y) , (37)

which implies that the purely ingoing plane wave reads

−iω(r∗+t) ikz ψNH = ξ2e b e , (38) where

Z ∗ dr ∗ ln (−y) r = √ → br∗ = lim r ' √ , r f r→r+ r+

1 2κr∗ r∗ ' ln (−y) =⇒ y = −e b . (39) b 2κ

On the other hand, for y → ∞, the inverse transformation of the hypergeometric func- tion is given by [43]

− Γ(η)Γ(ς − α) F (α, ς; η; y) = (−y) α × Γ(ς)Γ(η − α) F (α, α + 1 − η; α + 1 − ς; 1/y) + Γ(η)Γ(α − ς) (−y)−ς × Γ(α)Γ(η − ς) F (ς, ς + 1 − η; ς + 1 − α; 1/y) , (40)

which yields the following asymptotic solution

9 Γ(η)Γ(ς − α) R ξ (−y)−iA−B−α + SI w 2 Γ(ς)Γ(η − α) − − − Γ(η)Γ(α − ς) ξ (−y) iA B ς . (41) 2 Γ(α)Γ(η − ς)

To express Eq.(41) in a more compact form, let us perform the following simpliﬁcations. Considering

1 − iA − B − α = − (1 + iτ) , (42) 2 together with

1 − iA − B − ς = − (1 − iτ) , (43) 2 and letting x = −y, the radial equation for r → ∞ takes the form

1 − iτ Γ(η)Γ(ς − α) iτ Γ(η)Γ(α − ς) R = √ ξ x 2 + ξ x 2 . (44) SI x 2 Γ(ς)Γ(η − α) 2 Γ(α)Γ(η − ς) One can express x in terms of the tortoise coordinate at SI as

Z ∗ dr ∗ 2 r = √ → r∗ = lim r ' −√ , (45) r f b r→∞ r such that

−2br∗ x = r − r+ x|r→∞ ' r = 4e , (46)

∗ where br∗ = ln r . Therefore, we have

1 h i R = √ Λ eibr∗τ + Λ e−ibr∗τ , (47) SI r 1 2 where

Γ(η)Γ(ς − α) Λ = 2−iτξ , (48) 1 2 Γ(ς)Γ(η − α)

Γ(η)Γ(α − ς) Λ = 2−iτξ . (49) 2 2 Γ(α)Γ(η − ς) Thus, the asymptotic wave solution becomes

10 eikz h i ψ = √ Λ ei(br∗τ−ωt) + Λ e−i(br∗τ+ωt) . (50) SI r 1 2

IV. RADIATION OF DILATONIC 5D BS

A. 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 [36, 37]

ABH zNH = ψ ∂r∗ψNH − ψNH∂r∗ψ , (51) 2i NH NH

which, after a few manipulations, can be written as

2 zNH = −4πβ |ξ2| r+. (52)

The incoming ﬂux at SI is computed via

ABH zSI = ψ ∂r∗ψSI − ψSI∂r∗ψ . (53) 2i SI SI Having performed the steps to evaluate the derivatives with respect to the tortoise coordinate, the incoming ﬂux at SI takes the form

2 zSI = −4πβ |Λ2| τ. (54)

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.

B. ACS of Dilatonic 5D BS

The GF of the dilatonic 5D BS is obtained by the following expression [6,8]

11 −4πβ |ξ |2 r l,k = zNH = 2 + γ 2 , (55) zSI −4πβ |Λ2| τ which is nothing but

|ξ |2 r l,k = 2 + γ 2 . (56) |Λ2| τ After a few manipulations, with the following (see [42])

π |Γ(iy)|2 = , (57) y sinh (πy)

πy |Γ(1 + iy)|2 = , (58) sinh (πy)

2 1 π Γ( + iy) = , (59) 2 cosh (πy) Eq. (56) can be presented as

l k κr+ 2πω γ , = e κ − 1 Ξ, (60) ω where

e2πτ − 1 Ξ = " #" #. (61) π τ+ ω − √k π τ+ ω + √k e κ κ β + 1 e κ κ β + 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:

4π (l + 1)2 σl,k = γl,k, (62) ω3 which, in our case, becomes

2 l k 4π (l + 1) κr+ 2πω σ , = e κ − 1 Ξ. (63) ω4 Furthermore, one can also get the total ACS as follows [45]

∞ Total l,k σabs = ∑ σ . (64) l=0

12 In Fig. 1., the relationship between absorption ACS and frequency is examined; the ﬁgure is drawn based on Eq. (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 analyzes, which means that superradiance does not occur [46], as expected (as the dilatonic 5D BS (12) does not rotate).

FIG. 1: Plots of the ACS (σl,k) versus frequency ω. The plots are governed by Eq. (63). The conﬁguration of the

dilatonic 5D BS is as follows µ = 3 and Q = 0.2.

C. DR of Dilatonic 5D BS

The ﬁnal step follows from the ACS evaluation. The DR of the dilatonic 5D BS can be computed via [8]

σl,k 4π (l + 1)2 κr l,k = = + ΓDR 2πω 4 Ξ. (65) e κ − 1 ω

Fig. 2. shows how the DR behaves with respect to the frequency. By taking Eq. (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.

13 l,k FIG. 2: Plots of the DR (ΓDR) versus frequency ω. The plots are governed by Eq. (65). The conﬁguration of the dilatonic 5D BS is as follows µ = 3 and Q = 0.2.

V. 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 .

ACKNOWLEDGMENT

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

14 Mediterranean University through the project: BAPC-04-18-01.

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