1 Introduction

Emission of Massive Scalar Fields by a Higher-Dimensional Rotating Black-Hole

Emission of Massive Scalar Fields

[2mm] by a Higher-Dimensional Rotating Black-Hole



P. Kanti and N. Pappas


Division of Theoretical Physics, Department of Physics,

University of Ioannina, Ioannina GR-45110, Greece


Abstract

We perform a comprehensive study of the emission of massive scalar fields by a higher-dimensional, simply rotating black hole both in the bulk and on the brane. We derive approximate, analytic results as well as exact numerical ones for the absorption probability, and demonstrate that the two sets agree very well in the low and intermediate-energy regime for scalar fields with mass 1 TeV in the bulk and 0.5 TeV on the brane. The numerical values of the absorption probability are then used to derive the Hawking radiation power emission spectra in terms of the number of extra dimensions, angular-momentum of the black hole and mass of the emitted field. We compute the total emissivities in the bulk and on the brane, and demonstrate that, although the brane channel remains the dominant one, the bulk-over-brane energy ratio is considerably increased (up to 33%) when the mass of the emitted field is taken into account.

1 Introduction

The postulation of the existence of additional spacelike dimensions in nature, that can be as large as a few micrometers [1] or even infinite in size [2], has led to the idea of a higher-dimensional gravitational theory with a fundamental energy scale much smaller than the traditional Planck scale . If this can be realised with close to the TeV scale, present of future experiments may accelerate particles at energies beyond this new gravity scale. This will unavoidably lead to the occurence of strong gravity effects in particle collisions and the production of heavy final states, including miniature black holes [3].

The lifetime of these black holes is expected to be very short as they instantaneously decay via the emission of Hawking radiation [4] (for detailed reviews of their properties, see [5, 6]). Since these black holes will be created and decay in front of our detectors, it is anticipated that the emission of Hawking radiation will be the main obervable signature of their creation and, at the same time, a manifestation of the existence of additional spacelike dimensions in nature. As a result, the study of the emission of radiation by higher-dimensional black holes has been the subject of an intensive research activity over the last years. This includes the emission from both spherically-symmetric [7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and rotating [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] black holes in the form of zero and non-zero spin fields.

In order to simplify the analysis, the emitted fields are assumed to be minimally-coupled to gravity but otherwise free as well as massless. Nevertheless, in the context of the four-dimensional analysis [33] it was found that for certain particles and mass of the black hole, the particle mass can significantly (up to 50%) suppress the emission rate. Recently, a set of works [34] has addressed the question of the role of the mass of the emitted field (as well as that of the charge) for emission on the brane by a higher-dimensional black hole. Here, we extend this analysis by considering the case of a higher-dimensional black hole with a non-vanishing angular momentum emitting massive scalar fields. We perform a comprehensive study of the absorption probability and energy emission rate for a range of values of the mass of the emitted field, number of extra dimensions, and angular momentum of the black hole. By integrating over the entire frequency range, we compute the total emissivities and obtain the suppression factors in each case. We also consider the cases of both bulk and brane emission, and pose the additional question of whether the presence of the mass of the emitted field can affect the bulk-over-brane energy ratio and threaten the dominance of the brane channel.

The outline of this paper is as follows: In section 2, we study the emission of massive scalar fields by a higher-dimensional, simply rotating black hole in the bulk; we compute the value of the absorption probability both analytically and numerically and compare the two sets of results; finally, we derive the exact energy emission spectra and discuss their behaviour. In section 3, we turn to the brane and perform the same tasks. The total emissivities for bulk and brane emission are derived in section 4, and the bulk-over-brane ratio is computed for a large number of values of the parameters of the theory. We close with our conclusions in section 5.

2 Emission of Massive Scalars in the Bulk

In this work, we will consider the case of a higher-dimensional, neutral, simply rotating black hole whose gravitational background is described by the following form of the Myers-Perry solution [35]

(1)

where

(2)

and is the line-element on a unit -sphere. The above line-element is expected to describe black holes created by an on-brane collision of particles that acquire only one non-zero angular momentum component, parallel to our brane. The black hole’s mass and angular momentum are then related to the parameters and , respectively, as follows

(3)

with being the -dimensional Newton’s constant, and the area of a -dimensional unit sphere given by . The black hole’s horizon radius follows from the equation , and may be written as , where .

A massive scalar field, with mass , propagating in the gravitational background (1) will satisfy the equation of motion

(4)

where the higher-dimensional metric tensor and its determinant satisfying the relation

(5)

Even in the presence of the mass term, the above equation can be separated [17, 36] by assuming the factorised ansatz

(6)

where are the hyperspherical harmonics on the -sphere that satisfy the equation [37, 38]

(7)

The functions and in turn satisfy the following decoupled radial and angular equation

(8)
(9)

respectively. In the above,

(10)

For the above decoupling to take place, the angular function needs to satisfy a modified higher-dimensional spheroidal harmonics equation: compared to the massless case [39], it has the energy replaced by the momentum . Then, the massive angular eigenvalue is related to the massless one by merely a shift of its argument: . Here, we will employ the power-series expression of the angular eigenvalue [39] in terms of the parameter which, under the aforementioned shift and up to 5th order, takes the form

(11)

The analytic form of the angular eigenvalue was studied in detail in the context of previous works focusing on the emission of massless scalars [29, 30] and gravitons [31] in the bulk. It was found that its value, when terms up to 5th order or higher are kept, is remarkably close to the exact numerical value and that considerable deviations appear only for a very large angular momentum of the black hole or energy of the emitted particle, that lie beyond the range of values considered in this work. For this reason, the analytic form (11) of the angular eigenvalue will be employed in the derivation of the absorption probability in both an analytic and numerical method. We should still demand of course the convergence of the power series by imposing restrictions on the allowed values of the integer parameters that specify the emission mode : , that denotes the angular momentum of the mode along our brane, may take any integer value while and – the angular momentum number in the -sphere and total angular momentum number, respectively – may take any positive or zero integer value provided [39] that and .

2.1 The Absorption Probability in the Bulk

For the derivation of the absorption probability we need the solution for the radial function . We will first solve Eq. (8) analytically by using an approximate method, and we will derive an analytic expression for the absorption probability which in principle is valid in the low-energy and low-angular-momentum limit. We will then solve the same equation numerically to derive the exact value of , that will subsequently be used to derive the Hawking radiation spectrum. The two sets of results will be compared, and the validity of the approximate method will be studied in terms of the value of the angular-momentum parameter , number of extra dimensions and mass of the emitted particle .

The approximate analytic method amounts to solving the radial equation in the two asymptotic regimes, those of the black-hole horizon and far away from it, and then matching them in an intermediate regime. Apart from the appearance of the mass parameter , the analysis is very similar to the one for the emission of massless scalar fields in the bulk which has already appeared in the literature [29]. Therefore, here we briefly present the analysis and results giving emphasis to the differences arising due to the presence of the mass term.

In terms of the new radial variable [27, 28], the radial equation (8) near the horizon () takes the form

(12)

where , , while and are defined as

(13)

respectively. By employing the transformation the above equation takes the form of a hypergeometric differential equation [40] as long as

(14)

The radial function must satisfy the boundary condition that no outgoing modes exist near the black-hole horizon which then reduces the general solution of the hypergeometric equation to the physically acceptable one

(15)

with , , and an integration constant. Indeed, we may easily check that in the limit (or equivalently ), and by making the choice , we obtain

(16)

that has a form of an incoming plane-wave, as expected, in terms of a tortoise-like coordinate defined by . In the above, is given by

(17)

where is the rotation velocity of the black hole.

In the far-field regime (), the substitution brings Eq. (8) into the form of a Bessel equation [40]

(18)

in terms of , with solution

(19)

In the above, and are the Bessel functions of the first and second kind, respectively, and .

We now need to smoothly match the two asymptotic solutions (15) and (19) in an intermediate regime. The near-horizon solution (15) must first be shifted, so that its argument changes from to , and subsequently expanded in the limit. Then, it takes the polynomial form

(20)

with and defined as

(21)

The far-field solution (19) is in turn expanded to small values of leading to

(22)

The two polynomial forms match perfectly if we take the small and limit in the power coefficients of . In that case we can ignore terms of order or higher, and obtain , , and . We then demand the matching of the corresponding multiplicative coefficients, which leads to a constraint for the far-asymptotic integration constants and , namely

(23)

that guarantees the existence of a smooth, analytic solution for the radial part of the wavefunction for all , valid in the low-energy and low-rotation limit. We stress that, in order to achieve a higher level of accuracy in our analysis, no expansion is performed in the arguments of the Gamma functions. This method has been used in the literature before to derive analytic solutions for brane [27] and bulk [29] massless scalar fields. In both cases, the analytic results were shown to be in excellent agreement with the exact numerical ones in the low-energy regime and quite often at the intermediate-energy regime too.

In the presence of the mass term, though, there is one more constraint that needs to be satisfied for the perfect match to take place. In the low-energy and low-angular-momentum limit, the expression for the parameter , Eq. (14), becomes

(24)

For and , we thus need to satisfy . In order to derive some quantitative results, let us assume that TeV and TeV. If we ignore for a moment the angular momentum of the black hole and use the mass-horizon radius relation for a higher-dimensional Schwarzschild black hole, we find fm for , respectively [5]. Then, the aforementioned constraint on the mass of the bulk scalar field translates to

(25)

If we reinstate the angular momentum of the black hole, then the value of the black-hole horizon, for the same mass, becomes smaller since ; therefore, the upper bound on the mass of the scalar field increases further and becomes easier to satisfy.

In order to define the absorption probability, we finally expand the far-field solution (19) for , and obtain

(26)

which readily leads to

(27)

The above expression, in conjunction with Eq. (23), is our final analytic result for the absorption probability for massive scalar fields emitted in the bulk by a higher-dimensional, simply-rotating black hole. Summarizing all of the aforementioned assumptions, it is valid as long as the energy and mass of the emitted particle and the angular-momentum of the black hole stay below unity (in units of and , respectively). Its range of validity will be shortly investigated in terms of the values of the above parameters, as well as that of the number of extra dimensions .

Equation (27) is also useful for studying analytically various aspects of the absorption probability such as its behaviour in the superradiant regime and the asymptotic limit . If we expand Eq. (27) in the low-energy limit, a more convenient form may be derived for both purposes – a similar analysis was presented in all detail in [27] where the emission of massless scalar fields on the brane by the same type of black hole was studied. From Eq. (23) we see that, in that limit, we obtain , and therefore

(28)

where

(29)
(30)

and

(31)

The term above stands for the complex conjugate of the corresponding expression. As the energy of the emitted mode decreases, moving towards the asymptotic limit , for modes with , we meet the value . From Eqs. (14) and (17), it is clear that for that value , in which case Eq. (28) gets simplified to

(32)

In the above expression, all terms are positive definite, including the one, apart from whose sign, as expected, defines the sign of the absorption probability: for , takes a negative value signalling the occurence of superradiance.

For modes with , there is no superradiance effect, and we may thus approach the asymptotic limit . From the coefficient in the expression of it is clear that, in the massive case, too, it is the lowest partial modes that dominate the value of the absorption probability in the low-energy regime. We will therefore focus our attention on the dominant mode , and derive the behaviour of the absorption probability in the above asymptotic limit. Although for massive modes with the parameter never becomes exactly zero, it acquires its smallest possible value as . Equation (32) therefore remains approximately valid, and, for and , it is simplified further to give

(33)

We may also compute the absorption cross-section for the dominant massive scalar bulk mode in the asymptotic low-energy regime by using the formula [41, 29]

(34)

that relates the absorption cross-section with the absorption probability for a scalar mode propagating in the background of a higher-dimensional, simply rotating black hole. In the above

(35)

are the multiplicity of the -th partial wave in the expansion of the wave function over the hyperspherical harmonics on the -sphere [29], and the horizon area of the -dimensional rotating black hole, respectively. Substituting for the absorption coefficient, we obtain

(36)

For and , the above reduces to the horizon area of a higher-dimensional, spherically-symmetric black hole, as was found in [8]. For and , it was shown in [29, 23] that the value of remains very close to the area of the corresponding rotating black hole as long as is not large. For , we observe significant deviations from this behaviour as the value of the absorption cross-section for the lowest partial mode is not only energy-dependent but deviates as – this is in accordance with previous results derived in the cases of a massive field propagating in the background of a 4-dimensional Kerr [33] or of a -dimensional, spherically-symmetric black hole [23]. This behaviour is observed only in the case of the lowest mode; higher modes have a leading factor in their absorption probability, and a dependence for their absorption cross-section – for , this leads to a vanishing value in the asymptotic limit .

Figure 1: Absorption probabilities for the bulk scalar mode , for (a) , (from left to right) and various , and (b) , (from left to right) and various .

For the derivation of the value of the absorption probability, that would be valid for arbitrary values of the energy of the emitted particle and angular momentum of the black hole, we need to solve Eq. (8) numerically. To this end, a MATHEMATICA code was constructed that numerically solved for the value of the radial function from the horizon outwards. The boundary conditions for the second order differential equation was the value of and its first derivative at the horizon. The asymptotic solution (16) was used for that purpose, with the boundary conditions at having the form

(37)

The first condition was imposed to ensure that since no outgoing mode is allowed to exist at the horizon. The second follows readily from the asymptotic solution (16) and the use of the first condition. The integration proceeds until we reach radial infinity (in practice, this happens for ) where, according to Eq. (26), the radial function is a superposition of incoming and outgoing modes. The corresponding amplitudes are then isolated and the value of the absorption coefficient follows by use of the definition (27).

As a consistency check, we have succesfully reproduced the numerical results presented in [30] for the case of massless scalar fields emitted in the bulk by a simply rotating black hole - the case with is also included in our plots for the easy comparison with the massless case. In Fig. 1 we plot the absorption probability for the dominant mode as a function of the three parameters, , and , respectively. Figure 1(a) was drawn for fixed angular-momentum parameter (), and depicts the dependence of on the value of mass of the field and number of extra dimensions: we observe that as increases the value of the absorption probability decreases as expected, since a larger amount of energy is necessary for the emission of an increasingly more massive field. This pattern holds independently of the value of , nevertheless, the suppression with becomes less important as the number of extra dimensions gets larger. Figure 1(a) reveals also that the suppression of the absorption probability with the number of extra dimensions, found previously for massless scalar fields in the bulk [29, 30], holds also for massive fields. In Fig. 1(b), we keep fixed the number of extra dimensions () and vary and : again the suppression with the mass of the field is evident - contrary to what happens with , the suppression is more prominent as increases, particularly in the low- and intermediate-energy regimes. The enhancement of the absorption probability as itself increases, found again previously in [29, 30], persists also in the massive case.

Figure 2: Comparison of the analytical (solid lines) and exact numerical (data points) results for the absorption probability for the bulk scalar mode , for (a) , and , (b) , and , and (c) , and .

It would be interesting to compare the exact numerical results for the value of the absorption probability with the ones following from the analytical expression (27) with given by Eq. (23). In Fig. 2 we plot both sets of results for a range of values of the parameters , and – we consider again the dominant scalar bulk mode . Figure 2(a) reveals that the agreement between numerical and analytical results holds for a wide range of values of the mass parameter below unity (in units of , as indeed expected from the discussion below Eq. (23) regarding the values of . On the other hand, in terms of the number of extra dimensions, the agreement is case-dependent: as we see from Fig. 2(b), it is remarkably good for , for it is limited in the lower part of the curves while for it stops abruptly as the analytical result suffers from the existence of poles in the arguments of the Gamma functions that force the value of to dip towards smaller values and eventually vanish. The expression for , Eq. (23), is clearly the result of an approximation method valid for small values of the angular-momentum parameter, and thus we expect the agreement between the two sets of results to become worse as the value of increases gradually; however, in Fig. 2(c), we see that the agreement is actually improving as the angular-momentum parameter increases reaching values even beyond unity, a result that holds only in the presence of the mass term of the scalar field.

Figure 3: The dependence of superradiance for bulk massive scalar modes on (a) the mass, for , and (from bottom to top), (b) on the angular momentum numbers, for , , and (from left to right), and (c) on the angular momentum parameter, for , and .

Let us finally comment on the behaviour of the superradiance [42] on the parameters of the theory. In the context of the general suppression of the value of the absorption probability as the mass of the field increases, we expect that the effect of the superradiance will also be suppressed – this is indeed depicted in Fig. 3(a) where the value of is plotted for various values of for the indicative modes and . Despite the observed dominance of the superradiance effect for the mode over the one for , this pattern does not hold indefinitely as the angular momentum numbers increase: in fact, from Fig. 3(b), where we plot the superradiant regime for the modes for , and , it is clear that the mode is the dominant superradiant one, a result that was also found in the massless case [30]. The suppression of the superradiance with the number of extra dimensions observed in [30] for massless bulk scalar modes holds also in the massive case, and thus we do not comment further. A feature that has not been noted before is the non-monotonic behaviour of both the magnitude of the superradiance effect and the extent of the superradiant regime in terms of the angular-momentum parameter : in Fig. 3(c), we see that, as increases from zero to 1.5, the superradiance effect is indeed enhanced, however, this behaviour is reversed when increases further. In addition, superradiance occurs for frequencies : the latter restriction is imposed by the vanishing of the value of the absorption probability; the former by the demand that its value is a real number, and signifies the fact that no particles of mass can be created if energy less than that is available. Interestingly enough the width of the superradiance regime, , does not monotonically grow with the increase of the angular momentum of the black hole, as one could instictively expect. Indeed, its value reaches a maximum for a particular value of the angular-momentum parameter, namely , which is in fact independent of the mass and angular momentum numbers of the mode as well as of the number of extra dimensions 1. For the case depicted in Fig. 3(c), where we have fixed the horizon value at and considered only positive values of , the superradiant regime takes its maximum value at , beyond which it starts to shrink, for both modes and .

2.2 Energy Emission Rate in the Bulk

We will next compute the rate of energy emission in the bulk in the form of massive scalar fields by using the exact numerical results for the absorption probability found in the previous section. The emission of energy per unit frequency and unit time in the bulk is given by the expression [33, 29, 30]

(38)

The multiplicity of states from the expansion of the wavefunction of the field in the -dimensional sphere is given in Eq. (35) and the parameter is defined in Eq. (17). Finally, the temperature of the higher-dimensional, simply-rotating black hole (1) is

(39)

Equation (38) is identical in form with the expression for the emission of massless scalar fields in the bulk, nevertheless, there are two major differences: the calculation of the spectrum starts from instead of zero, and the value of the absorption probability depends, apart from the spacetime parameters, on the characteristics of the emitted field including its mass.

  

Figure 4: Energy emission rates for bulk scalar fields for: (a) for , and (from top to bottom in each set of curves with fixed ), and (b) , and (from top to bottom in each set of curves with fixed ).

In order to derive the energy emission spectrum, we need to sum over a significantly large number of partial waves labeled by the set of () quantum numbers. For each value of , and , we aimed at deriving the complete spectrum, i.e. to reach values of the energy parameter where the corresponding value of the energy emission rate would be less than . At the same time, the number of partial waves summed had to be large enough so that the derivation of the energy spectrum would be as close as possible to the real one – especially for the computation of the total emissivity presented in section 4. Taking all these constraints into account, we were able to sum the contribution of all bulk scalar modes up to , that brings the total number of summed modes to . According to our estimates, the contribution of all modes higher than should be less than , for the higher values of parameters considered, namely and , an error that falls below 0.001 for the lowest values considered, i.e. and .

In Fig. 4, we depict the energy emission rate on the brane in the form of massive scalar fields in terms of the number of extra dimensions, value of the angular-momentum parameter, and mass of the emitted field itself. Thus, Fig. 4(a) shows the energy emission rate for fixed () and variable and , while Fig.4(b) plots the same quantity but for fixed () and variable . In terms of the spacetime parameters and , these plots confirm the behaviour found in the case of massless fields [29, 30]: the power spectrum is enhanced as the number of extra dimensions increases while its dependence on the angular momentum parameter is not monotonic but differs as and/or varies. More detailed features, like the oscillatory pattern of the emission curves for low values of and , that are replaced by more smoother curves as the values of these parameters increase, are also recovered.

In terms of the mass of the scalar field, we observe the expected suppression of the emission rate, for fixed and , as increases – the suppression is more prominent in the low- and intermediate-energy regimes whereas the effect of the mass becomes negligible at the high-energy regime. Compared to the case of the emission of masless scalar fields, the suppression in the low-energy regime becomes even more significant if the disappareance of the frequency range with is taken into account. The magnitude of the suppression with depends strongly on the particular value of and – the exact effect will be computed in section 4 where the total emissivities in bulk and brane will be calculated.

3 Emission of Massive Scalars on the Brane

In this section, we turn our attention to the emission of massive scalar fields by a higher-dimensional simply-rotating black hole on the brane. The analysis for the derivation of the absoprtion probability, both analytical and numerical, is quite similar to the one performed for the emission in the bulk; aspects of it have also been recently addressed in a set of publications [34] that appeared while this work was still in progress. For the sake of comparison and completeness of the analysis, we will still present in this section the most important points of our calculation on the brane and focus our discussion to aspects not covered before; these include, for example, the analytic study of the low-energy asymptotic behaviour of the absorption probability and cross-section, the role of the angular momentum of the black hole, that was ignored in [34], and the form of the energy emission spectrum, instead of the number flux that was studied in the same work.

Let us start with the form of the gravitational background that a massive scalar field sees as it propagates on the brane and its corresponding field equation. The 4-dimensional induced background will be the projection of the higher-dimensional one (1) onto the brane, and follows by fixing the values of the angular variables of the -sphere. Then, the induced-on-the-brane line-element takes the form

(40)

which is very similar to the usual 4-dimensional Kerr one but carries an explicit dependence on the number of additional spacelike dimensions . The field equation is still given by the covariant form (4) but with the higher-dimensional metric tensor replaced by the 4-dimensional one defined above. The field factorization

(41)

leads again to the decoupling of variables and to the following set of radial and angular equations

(42)
(43)

respectively. In the above, we have defined , while is again given by and by Eq. (10). The angular function satisfies again a modified spheroidal harmonics equation with . The corresponding massive eigenvalue is thus related to the massless one through the same shift, and in terms of a power series [43] is given by

(44)

The above form will be used in the computation of the absorption probability both analytically and numerically.

3.1 The Absorption Probability on the Brane

The approximation method employed in section 2 can again be used to solve the radial equation (42) analytically. The same change of variable , in the near-horizon regime (), leads to an equation of the form (12) where now