Quasi-Normal Modes of Massless Scalar Field around the 5D Ricci-flat Black String
As one candidate of the higher dimensional black holes, the 5D Ricci-flat black string is considered in this paper. By means of a non-trivial potential , the quasi-normal modes of a massless scalar field around this black string space is studied. By using the classical third order WKB approximation, we analyse carefully the evolution of frequencies in two aspects, one is the induced cosmological constant and the other is the quantum number . The massless scalar field decays more slowly because of the existences of the fifth dimension and the induced cosmological constant. If extra dimension has in fact existed near black hole, those quasi-normal frequencies may have some indication on it.
Through an additional field or perturbing the metric itself, the black hole suffers a damping oscillation phase. Usually, people call it Quasi-normal models (QNMs) or quasi-normal ringing. So, as the emission of gravitational wave (GW), a normal model oscillation is replaced by the complex frequencies where the real part represents the actual frequency and the imaginary part represents the damping of the oscillation. Those frequencies are directly connected to the black hole’s mass, charge, momentum and so on. In the view of QNMs’ evolution, there are three stages: the first one is the initial outburst from the source of perturbation, the second one is the damping (quasi-normal) oscillation and the last one is the asymptotic tails at very late time. The evolution significantly depends on the asymptotic behavior of the space. In perturbation theory, the linear perturbation of static black holes was first studied by Regge and Wheeler in 1957 Regge (). Soon after that, Vishveshwara Vishveshwara () presented the QNMs by calculating the scattering of gravitational waves (GW) around a Schwarzschild black hole. Then Press Press () gave the original term (QN) . This perturbations have been studied extensively in many literatures QNMs (). For the in-depth reviews, one can refer to Nollert () Kokkotas (). People believe that the QN frequencies could be detected by GW observatories (LIGO, VIRGO, TAMT, GEO600, and so on) in the future.
On the other hand, one kind of higher dimensional theory named induced matter theory is shown by Wesson and co-workers Wesson () Overduin () in the 90’s of the last century. They showed a non-compact fifth dimension and pointed out the 4D source is induced from an empty 5D manifold. That is, 5D manifold is Ricci-flat while 4D hypersurface is curved by the 4D matters. So this theory is also called Space-Time-Matter (STM) theory. Meanwhile, in the STM framework, there are many extensive literatures discussing Quantum Dirac Equation Macias (), Perihelion Problem Lim (), Kaluza-Klein Solitons Billyard () Liusoliton (), Black Hole Liu1 () Liu222 () Mashhoon () Liu00 (), Solar System Tests Liu333 () and so on.
In the last decade, other robust extra dimensional models have appeared in gravitational-field theory such as ADD model ADD () and Randall-Sundrum I model Randall2 ()/ II model Randall1 () with the additional spacelike dimensions. In those models, our world is a 3-brane which is embedded in the higher dimensional space (bulk). To avoid interactions beyond any acceptable phenomenological limits, standard model (SM) particles (such as fermions, gauge bosons, Higgs) are confined on a (3 + 1) dimensional hypersurface (3-brane) without accessing the transverse dimensions, except for the gravitons and scalar particles without charges under SM gauge group. In this paper, we assume the scalar field can freely propagate in the bulk.
If matter trapped on the brane undergoes gravitational collapse, a black hole will form naturally and its horizon extends into the extra dimension transverse to brane. Such higher dimensional object looked like a black hole on the brane is actually a black string in the higher dimensional brane world blackstring (). One natural candidate is a Schwarzschild-de Sitter (SdS) black hole embedded into the 5D Ricci-flat space Liu1 () Mashhoon () Liu00 () Molin1 (). It should be noticed that the STM theory is equivalent to the brane world Ponce () Seahra () Liu_plb ().
Meanwhile, one of exciting predictions in large extra dimensional model ADD () is that the CERN Large Hadron Collider (LHC) will produce black holes by the colliding of highly energetic particles when the scale of quantum gravity is near TeV LHC (). Naturally, the detectable QNMs are studied widely in higher dimensional background higherQNMs (). In this paper, we calculate the QN frequency of massless scalar field around a 5D Ricci-flat black string space.
This paper is organized as follows: In Section II, the 5D Ricci-flat black string metric and the time-dependent radial equation about are represented. In section III, by a tortoise coordinate transformation, the propagating master equation of scalar field is obtained. In section VI, by using the third order WKB method, the QN frequencies is obtained in Table I, II and III. Section V is a conclusion. We adopt the signature and put , ,and equal to unity. Lowercase Greek indices will be taken to run over 0, 1, 2, 3 as usual, while capital indices A, B, C, run over all five coordinates (0, 1, 2, 3, 4).
Ii Klein-Gordon Equation in the 5D Ricci-flat Black String Space
A class of 5D black holes solutions have been presented by Mashhoon, Wesson and Liu Liu1 () Mashhoon () Wesson () under STM scenario. Briefly, the static, three-dimensional spherically symmetric line element takes the form
where is the open non-compact extra dimension coordinate. The part of this metric inside the square bracket is exactly the same line-element as the 4D Schwarzschild-de Sitter solution, which is bounded by two horizons — an inner horizon (black hole horizon) and an outer horizon (one may call this cosmological horizon).
The radial-dependent metric function takes the form
where is the induced cosmological constant and is the central mass. The metric (1) satisfies the 5D vacuum equation . Therefore, there is no cosmological constant when viewed from 5D. But when viewed from 4D, there is an effective cosmological constant . So one can treat this as a parameter which comes from the fifth dimension. This solution has been studied in many works Mashhoon11 () Wesson_1 () Liu_2 () Mashhoon_1 () focusing mainly on the induced constant , the extra force and so on.
We redefine the fifth dimension . With this redefinition, the metric (1) can be rewritten as
Using the line element (1), the metric function (2) and above new extra dimension, a Randall-Sundrum (RS) type brane model is built up. Now, let us show the configuration in detail. There are two branes in this model: one brane is at and the other brane is at . So the fifth dimension becomes finite. It could be very large as RS II model Randall2 () or very small as RS I model Randall1 (). The 4D line-element represents exactly the Schwarzschild-de Sitter black hole on a hypersurface ( or = ). However, viewing from the 5D space, the horizon does not form a 4D sphere — it looks like a black string lying along the extra dimension. So, we call the solution (1) a 5D Ricci-flat black string solution.
The metric function (2) can be expressed by the horizons as follows
The null hypersurface of this black string space is determined by its singularity . Obviously, the solutions to this equation are inner horizon , outer horizon and a negative solution . The last one has no physical significance. Here we only consider the real solutions. and are given as
where with . The real physical solutions are accepted only if satisfy Liu1 ().
The massless scalar field in the 5D black string space, satisfies the Klein-Gordon equation , where is the 5D d’Alembertian operator. We assume that the separable solutions are of the form
where is the radial time-dependent function, is the usual spherical harmonic function. The dependent equation about in QNMs aspect is,
where is the eigenvalue of function . The fifth dimensional equation about is
which is discussed carefully in Liu00 (). In the Randall-Sundrum double branes system, the modes along the extra dimension are quantized by means of stable standing waves, and then the eigenvalue is naturally discretized. The discrete spectra of is
and the quantum parameter is
where and is the thickness of the bulk.
Iii The Master Equation for Propagation of Scalar Field in The Bulk
It is known that radial direction determines the evolution of black hole radiation. The time variable of Eq. (7) can be removed by the Fourier component via
whose potential function is given by
Now we introduce the tortoise coordinate
The tortoise coordinate can be expressed with the surface gravity as follows
So under the tortoise coordinate transformation (14), the radial perturbation equation is obtained as
It is evident that Eq. (17) is exactly the same as Ragge-Wheeler equation in QNMs. The incoming or outgoing particle flowing between inner horizon and outer horizon is reflected and transmitted by the potential . Substituting the quantum parameters (10) into Eq. (13), the quantum potentials are obtained as follows
which are illustrated in Fig. 1 and Fig. 2. The 5D potential contains the quantum number which is higher and thicker than the 4D’s when = 0. Also, the height and the thickness of the former increase with bigger . Meanwhile, for increasing cosmological constant , the potential also becomes higher and thicker, and the interval between and is larger, too.
Iv The QN frequency of Massless Scalar Field with the third order WKB Method
Numerical WKB approximation is an effective method to obtain the complex QN frequencies by using the well-known Bohr-Sommerfeld rule. It was originally shown by Schutz et al Schutz () and was later developed to the third order by Iyer et al Iyer1 () Iyer2 () and to the sixth order by Konoplya Konoplya (). Then after that this method is extensively used in various spaces WKB (). The third order WKB formula for QN frequencies has the form Iyer1 () Iyer2 (),
where and symbol is the various overtones. The primes and superscript denote differentiation with respect to the tortoise coordinate . The subscript on a variable denotes the value at , which is the position of maximum , namely, . Substituting potential (18) into and , we can obtain the vital QN frequencies for the massless scalar field in the 5D black string space. Meanwhile, it is known that the WKB approximation is accurate for the low-lying QNM modes, but it fails to calculate the higher-order modes. Therefore, the condition is employed and the QN frequencies of fundamental key cases: (), () and () are listed in the Table I, II and III, respectively. Meanwhile, potential (13) illustrate clearly that when the Regge-Wheeler equation (17) is naturally reduced to 4D SdS case. So those tables also include a comparison with the results of 4D case. Here we should notice that the denotation does not indicates . To avoid the confusion about parameters and , we provide some explanation in the conclusion part. Here, we adopt and and analyse the QN frequencies from two aspects: cosmological constant and quantum number .
Firstly, for a given cosmological constant , it is shown that the real parts of QNMs () increase with bigger quantum number . But the absolute value of the imaginary parts () decrease for bigger . In general, the actual frequencies in 5D are larger than 4D’s, and the scalar field in 5D decays more slowly than the one in 4D. With increasing , the QN frequencies become larger and the scalar field decays more slowly.
Secondly, for a given we can also read that and decrease with larger . It means that with the increasing cosmological constant the actual frequency becomes smaller and the scalar field decays more slowly. These results are in agreement with the results of 4D SdS case with the sixth order WKB method Zhidenko (). As mentioned above, the two circumstances for given and can be manifested in Fig. 3 and Fig. 4 which are obtained by Table I.
In order to study the relationship between the actual frequency and the damping of the oscillation, we also plot the versus graph in Fig. 5. Obviously, the absolute value of the imaginary parts increase entirely with the larger real parts. However, when the quantum number becomes larger, there is a break point in the curve. In other words, does not monotonously increase with for bigger , especially in the case of .
To explain the reason, we should refer to this black string’s reflection (or transmission) Molin1 (). In the square barrier model Molin1 (), the reflection or transmission coefficients have the analytic forms. As one knows that the reflection should be stronger with higher barriers. The reflection coefficients of would be larger than the cases of or in the usual viewpoint. On the contrary, the values of reflection coefficients of are smaller than the cases of or . If the resonant effect of quantum mechanics exists in the barriers, the peculiar behavior is easily to be gotten. Mathematically, an oscillatory cosine function is contained in the expression of reflection coefficients or transmission coefficients Molin1 (). (For detailed discussion of those behavior, see Ref. Molin1 ().) The scatting potential of also have the peculiar QN frequencies by the same resonant effect. From Fig. 1, we can read that with larger the peak of potential slips the cosmological horizon and the potential becomes higher and wider, especially in the case of . Hence with the enhancing of resonant effect the QN frequencies decay more slowly, even though the QN frequencies become larger. So the anomalous values of are the result of resonant effect of quantum mechanics in the barriers.
In this paper, we have used the third-order WKB approximation to calculate the quasi-normal frequencies of massless scalar field outside a 5D black string. We summarize what has been achieved.
1. In this 5D Ricci-flat black string space, the QNMs is studied by fixing either the cosmological constant or the quantum number . From the result we find that the scalar field decays more slowly with the increasing or . For a given cosmological constant , the 5D actual frequency becomes bigger with increasing . While for a given , the frequency becomes smaller with bigger cosmological constant . In other words, the 5D QN frequencies are larger than 4D’s ().
2. As one candidate of the higher dimensional black hole, the 5D Ricci-flat black string implies us something interesting. The quantum number depicts a new wave solution of Schrdinger wavelike equation. The spectrum of original potential is discrete for the existence of quantum number . The non-trivial radiation can reveal much of valuable information that characterizes the higher dimensional background, such as the dimensionality of space, the topological structure and so on. Here, the QN frequencies of 5D Ricci-flat black string are determined by the black hole mass , the effective cosmological constant , the quantum number and the thickness of the bulk . The information about extra dimension is encoded in those QN frequencies such as the magnitude of extra dimension and the thickness of the bulk. It is known that the best method to probe black hole is the detectable QN spectrum. If extra dimension does exist and is visible near black hole, maybe those QN frequency can prove its existence.
3. To ensure the validity of results, tables and figures, we use Mathematica software to design a program and calculate carefully those QN frequencies. The induced four dimensional results () are exactly identical with 4D SdS black hole’s Zhidenko (). Otherwise, just as 4D case Zhidenko (), the QN frequencies decrease with increasing cosmological constant . Of course, this program is tested in some other black holes such as 4D Schwarzschild black hole Iyer2 (), and the same results are got which is not presented in this paper. This method gets desired effects and hence believable. .
4. The reason why we discuss, not the gravitational perturbation but a test scalar field, is that this paper is a continuation of previous work Liu00 () to a certain extent. As we known from the spirit of string theory, the standard model fields are confined on 3-brane except for gravitons and scalar particles. The original goal to introduce the scalar field is to examine the effect of an extra dimension on black hole radiation. Surprisingly, the radial component of 5D Klein-Gordon equation can be rewritten exactly as the Regge-Wheeler form. Considering the QNMs boundary condition, we have studied the spectrum of QN frequencies by usual third order WKB method. After all, the QNMs of higher dimensional black hole is very attractive and interesting. Certainly, the basic gravitation field is easier to be detected by gravitational wave than other fields. Anyway, it is interesting to study this case and further work is needed.
5. It should be noticed that the parameter is not the same as . In fact, parameter is introduced to separate the variables , and in this paper. But parameter is a particular eigenvalue which is deduced from the original under the standing wave condition Liu00 (),
In other words, is a free parameter, while is constrained by , and . Meanwhile, the quantum number is a positive integer i.e. according to the condition (24) Liu00 (). Furthermore, cosmological constant is nonzero in the de Sitter universe. So considering the conditions and , we can get the eigenvalue . By the way, when the steady stand wave (9) can not be formed and this case must be abandoned. On the other hand, the potential function (13) indicates that the case (not n = 0) corresponds to usual 4D SdS. Hence, the notations in figures and tables just show the one of 4D SdS. Of course, some other notation could be used in principle.
Acknowledgements.Project supported by the National Basic Research Program of China (2003CB716300), National Natural Science Foundation of China (10573003) and National Natural Science Foundation of China (10573004). We are grateful to Feng Luo for useful help.
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