Greybody factors for higher-dimensional non-commutative geometry inspired black holes

Non-commutative (NC) space-time geometry has allowed insight into the quantum nature of gravity. Within the effective theory for NC black holes, the point-like sources in the energy-momentum tensor, that are normally represented by Dirac delta functions of position, are replaced by Gaussian smeared matter distributions of width $\sqrt{2\theta}$. Effective theories for non-commutativity have enable calculations of black hole properties distinctly different from those of classical gravity. The NC black hole has a finite maximum temperature. A minimum mass and horizon radius exist at which the temperature is zero and the heat capacity vanishes which may terminate Hawking evaporation. These properties are in clear contradistinction to the classical black hole with temperature becoming infinite as it approaches zero mass and horizon radius.

Many studies have elucidated the nature of NC geometry inspired black holes but little attention has been devoted to the calculation of greybody factors and their subsequent use in studying absorption cross sections, and particle and energy spectra. Models for NC geometry inspired chargeless, non-rotating black holes were developed by Nicolini, Smailagic and Spallucci [1]. The model was extended to the case of charge in four dimensions [2], generalized to higher dimensions by Rizzo [3], and then to charge in higher dimensions [4]. A review of the developments can be found in Ref. [5]. The Hawking effect and other thermodynamic aspects of NC black holes have been studies in Ref. [6, 7, 8]. Phenomenological considerations for searches with the Large Hadron Collider (LHC) experiments appeared in Ref. [9]. Graybody factor calculations for massless scalar emission were presented in Ref. [10].

The aim of this paper is to present the emission spectra from non-rotating NC inspired black holes in higher dimensions for massless fields of spin 0, 1/2, 1, and 2. The results are compared with the higher-dimensional Schwarzschild-Tangherlini (ST) black hole. One of our goals is to separate the temperature characteristics from the transmission factors in the comparison of emission by performing calculations of the graybody factors for different black hole masses.

Throughout, we will work in units of $\hbar=c=k=1$, and use the PDG [11] definition for the higher-dimensional ADD Planck scale $M_{D}$. Note that $M_{*}^{n+2}=8\pi/(2\pi)^{n}M_{D}^{n+2}$ is often used in the literature. When $M_{*}\sim 1$ TeV is taken, the value of $M_{D}$ will be more than four times beyond the current experimental lower bounds on $M_{D}$ for $n\geq 3$, and ruled out for $n=1$ and 2 extra dimensions. To allow comparisons with the literature, we have taken the common values of $M_{D}\sim~{}\sqrt{\theta}^{-1}\sim 1$. For $n=0$, the units are $M_{D}\sim 10^{16}$ TeV and $\sqrt{\theta}\sim 10^{-35}$ m, and for $n>0$, and units can be chosen as $M_{D}\sim 1$ TeV and $\sqrt{\theta}\sim 10^{-4}$ fm.

A nice review of NC geometry inspired black holes already exists [5]. We will only present the mathematical results used here. The $g_{00}$ component of the NC inspired metric is

For the effective potentials to be discussed soon, we will also require the derivative of the metric function with respect to $r$:

The horizon radius for non-commutative inspired black holes is given by

For given $M$, $n$, $M_{D}$, and $\theta$, there can be one, two, or no horizon. For reasons related to positive temperature given later, only the outer horizon radius is relevant to the work presented here. Figure 1 shows $h(r)$ for the case of a single event horizon for $n=7$, $M=20.4M_{D}$, and $M_{D}=\sqrt{\theta}=1$, and a comparison with the ST black hole.

Because there is a minimum mass, there are masses below which the black hole will not form, and above the minimum mass the horizon radius is double valued. As the mass increases, the inner horizon radius shrinks to zero, while the outer horizon radius approaches the commutative value. The situation thus depicted in Figure 1 represents the minimum mass case in which the values of the horizon radii for NC and ST black holes are maximally different. This will be a useful condition when considering maximum differences in the transmission coefficients between NC and ST black holes.

Hawking radiation from black holes is typically studied by examining the response to perturbations. Hence, understanding modifications of Hawking radiation due to NC geometry requires one to solve the equations of motion for various spin perturbations on the NC inspired black hole metric.

The Teukolsky equation describes spin 0, 1/2, 1, and 2 field perturbations in the background metric due to the black hole [12, 13, 14]. The partial differential equations can be separated. The angular equation can be numerically solved to obtain the energy eigenvalues or separation constant. The energy eigenvalues are then used in the radial equation. The radial equation can be solve to find transmission coefficients for fields emitted from the black hole horizon. The transmission coefficients describe the probability that a particle, generated by quantum fluctuations at the horizon of a black hole, escapes to spatial infinity.

The Teukolsky radial equation can be numerically solved directly, but the convergence of the solution at the integration boundaries is not clear. The difficulty with the radial equation in the context of a scattering problem is that the first order radial derivatives create complex $is\omega$ terms which have a $1/r$ behaviour at infinity. This can eventually lead to problems in the numerical computations.

Another approach uses a Chandrasekhar transformation [15] to cast the radial equation into an effective Schrödinger equation with a short-range barrier potential different for each spin field, allowing a more careful numerical treatment. The equation takes the form

The purpose of this transformation is that the potentials $V_{s}(r)$ are now short-ranged. They vanish faster than $1/r$ which is advantageous for numerical computations. It should be noted that these potentials contain a dependence on $\omega$ through the connection coefficient to the angular equation. Working with real-valued potentials has benefits.

Arbey et al. [16] have derived general potentials $V_{s}$ for spherical symmetric metrics for different spins $s$ of massless fields. The effective potentials seen by $s=0,1/2,1$, and 2 massless fields can be written as

For spin 1/2, one must take the positive sign in Eq. (8b) to get a positive potential for all $r$. For the spin-2 case, for high $n$ and $\ell=2$, it is possible to get a negative potential. We will discuss this further below when describing absorption cross section results. These potentials differ from those in Ref. [16] in that they are missing a $1/\sqrt{2}$ factor in the second term of Eq. (5.1d), and Eq. (5.9c) should have a first constant of 4 rather than 2 in the second term and a constant of 3 rather than 1 in the last term.

Figure 2 shows two representative cases for the NC black hole potentials with $M_{D}=\sqrt{\theta}=1$ for spin 0, 1/2, 1, and 2; the case of $l=s$ is shown.

The Schrödinger Eq. (7) is in the tortoise coordinate $r_{*}$ while the potential is in $r$; or one can view $r=r(r_{*})$. The relationship between the coordinates is defined by

One thus needs to solve differential Eq. (8i) for $r_{*}$ and also invert it to obtain $r$. While the result is well known for the four-dimensional Schwarzschild metric it is less apparent for others. Analytic expressions exist for $r_{*}$ for the higher dimensional ST case but we are unaware of such analytic expressions for the NC case. We have thus numerically integrated Eq. (8i). For the initial condition, we take $r=r_{*}$ for a very large value (approximating $+\infty$), and integrate backwards. The procedure has been validated in the ST case by comparing the numerical integrations with the analytic results. The analytic formulae are taken from Ref. [17], to which we have added the $n=7$ case:

We integrate the tortoise equation using a variable step size over the range $r/r_{\mathrm{h}}=[\epsilon,350]$, where $\epsilon\approx 10^{-16}$. To obtain the inverse relation for $r$ in terms of $r_{*}$, we have integrated the tortoise equation and numerically inverted it using linear interpolation.

Gray and Visser [18] showed that the Bogoliubov coefficients relating incoming and outgoing waves on a potential barrier can be directly obtained from the following path-ordered exponential

The product integral can be approximated numerically by

The transmission probabilities $\Gamma(\omega)$ are related to the Bogoliubov coefficients by

In studying Hawking emission from NC geometry inspired black holes, we consider the following. The absorption cross section is an observable acting as an effective area representing the likelihood of a particle to be scattered by the black hole:

One of the most interesting aspects of NC inspired black holes is their temperature properties. The temperature in terms of the horizon radius is given by

Of interest to us will be the NC black hole maximum temperature $T_{\mathrm{max}}$ and mass $M_{\mathrm{e}M}$ at which the maximum temperature occurs. In addition, we will use values of the ST black hole mass $M_{\mathrm{eT}}$ at which the temperature is the same as the NC black hole maximum temperature. These values are shown in Table 1.

We now introduce the main physical variables that can be formulated using the transmission coefficients. The number of particles emitted per unit time and per unit frequency is

The black body radiation leaving the horizon sees an effective potential barrier due to the geometry of the space-time surrounding the black hole. The potential barrier attenuates the radiation such that an observer at spatial infinity away from the black hole will measure a different emission spectrum than the one at the horizon by a factor $\Gamma_{s,l}(\omega)$ called the greybody factor. Thus the grey body factor $\Gamma_{s,l}(\omega)$ represents the fraction of the black body emission which penetrates through the potential barrier and escapes to spatial infinity. The expressions for absorption cross section Eq. (8p) and emission rates Eq. (8r) and Eq. (8s) are only applicable to radiation on the brane.

We present calculations of transmission coefficients, absorption cross sections, and spectra for spin 0, 1/2, 1, and 2 fields. We have validated the method by comparing with well known results for four dimensional Schwarzschild and Kerr black holes, as well as the ST black holes [19, 20] that we use for comparison with the NC results.

The model of NC geometry inspired black holes in higher dimensions has three unknown parameters $n$, $M_{D}$, and $\sqrt{\theta}$. Typically we present results for each extra dimension $n$. Usually it is necessary to fix the other two parameters. One possibility for fixing the parameters is to be guided by experimental constraints. Updating the approach taken in Ref. [9] (see the Appendix) restricts the values of $\sqrt{\theta}M_{D}$ that can be probed from 0.25 to 0.98, different for each number of extra dimensions. The allowed range in $\sqrt{\theta}M_{D}$ for any given number of extra dimensions is severely restricted.

Calculations using values of $\sqrt{\theta}M_{D}<1$ begin to probe the details of the matter smearing distribution and become model dependent. However, the primary goal in this paper is to study the differences in Hawking emission from NC and ST black holes so we choose the usual condition $M_{D}=\sqrt{\theta}=1$. This implies that our phenomenological predictions will not have particular consequence for the physics at the LHC.

Transmission coefficients for spin 0, 1/2, 1, and 2 fields from NC geometry inspired black holes of extra dimension from 0 to 7 have been calculated. The NC black hole transmission coefficients are similar to the ST black hole transmission coefficients when their horizon radius are similar. However, there are major differences when the black hole masses are similar but the horizon radius are significantly different. The major difference in transmission coefficients is that the NC black hole transmission coefficients turn-on at slightly higher frequency. The differences between NC black hole and ST black hole transmission coefficients are about the same for all spins.

The absorption cross section of different spin fields from NC geometry inspired black holes of different number extra dimensions have been calculated. Significant differences in NC black hole and ST black hole absorption cross sections occur at low frequencies while the cross sections at high frequencies approach the geometrical optics limits. For masses near the minimum NC black hole mass, the differences are more apparent, particularly for higher dimensions.

The particle and energy spectra on the brane of different spin fields from NC geometry inspired black holes of different number of extra dimensions have been calculated. For equal masses, the NC black hole spectra are significantly lower than for ST black holes, mainly due to the lower temperature. However, at equal temperature the NC black hole spectra are still significantly lower than for ST black hole spectra.

When integrating the particle spectra over frequency the particle flux from NC black holes is significantly less than that from ST black holes. For spin-0 fields, the reduction can be from 0.70-0.80 depending on if the black holes have equal mass (lower number) or equal temperature (higher number). The dependence on the number of dimensions is small. For spin-1/2 fields, the ratio is from 0.65-0.81, also not too dependent on number of extra dimensions. For spin-1 fields, the ratio is 0.54-0.78. But for spin-2 fields, the reduction is most significant from 0.40-0.72, increasing with increasing number of dimensions. The general trends in the power are similar to the trends in the particle flux.

Considering the NC geometry inspired black hole emission on its own, it is common to compare the fluxes to the spin-0 case or the $n=0$ case. Increases in particle flux and power relative to the $n=0$ case are observed for increasing number of dimensions. The increase is most prominent for spin 2 and smallest for spin 0. The particle flux and power of spin-1/2 fields relative to spin-0 fields is less, with $n=0$ being the lowest and $n=3$ the highest for the particle flux and $n=2$ for power. The particle flux of spin-1 fields relative to spin-0 fields is less and decreases significantly with decreasing number of dimensions. The power of spin-1 fields relative to spin-0 fields is less or greater, depending on the number of dimensions; being significantly less for $n=0$ and slightly greater for $n=7$. The spin-2 fields particle fluxes and power are always significantly less than for spin-0 fields, being 1% for $n=0$.

We have presented greybody factors, absorption cross sections, and particle and energy spectra for all spin fields from higher-dimensional non-commutative geometry inspired black holes for the first time. The calculations are numerical and thus valid over the entire frequency range. The emission of higher spin fields, particularly graviton emission, could be useful for relating possible future observations of high-temperature black hole radiation to theory. The reduction in emission due to the greybody factors, not temperature, that we observe are hopefully model independent. This work represents another step towards possibly elucidating some aspects of quantum gravity.

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Nous remercions le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG) de son soutien.