Absorption, scattering and shadow by a noncommutative black hole with global monopole

In this paper, we investigate the process of massless scalar wave scattering due to a noncommutative black hole with a global monopole through the partial wave method. We computed the cross section of differential scattering and absorption at the low frequency limit. We also calculated, at the high frequency limit, the absorption and the shadow radius by the null geodesic method. We showed that noncommutativity causes a reduction in the differential scattering/absorption cross section and shadow radius, while the presence of the global monopole has the effect of increasing the value of such quantities. In the limit of the null mass parameter, we verify that the cross section of differential scattering, absorption and shadow ray approach to a non-zero value proportional to a minimum mass.


I. INTRODUCTION
Hawking [1] black hole radiation emission studies revealed a physically relevant possibility for us to enter the universe of quantum gravity.A very important point raised by Hawking is related to the final stage of black hole evaporation, so that when the mass of the black hole approaches the Planck mass, a quantum theory of gravity is needed.However, stimulated by the need to study the structure of spacetime on the Planck scale, many other approaches have been proposed considering noncommutative geometry [2][3][4].As with position and moment uncertainty in conventional quantum mechanics, we can find a noncommutative relationship in Einstein theory of general relativity.Thus, position measurements can fail to commute by suggesting a noncommutative manifold [5].Studies as proposed by Nicolini [6] showed that noncommutativity can be implemented in general relativity by modifying the matter source.This can be done by modifying the mass density by replacing a Dirac delta function with a Gaussian distribution of minimum width √ θ given by ρ θ (r) = M (4πθ) −3/2 exp (−r 2 /4θ), or even considering replacing by a Lorentzian distribution [7] tanking the following form ρ θ (r) = M √ θπ −3/2 (r 2 + πθ) −2 , where θ is the noncommutative parameter and M is the total mass that due to the uncertainty of noncommutativity is diffused throughout the linear region of size √ θ.Besides the noncommutativity, the study of topological defects has been used in different areas of physics as a way to improve understanding the behavior of the early Universe.
It is well known in the literature that global monopoles are types of topological defects that are formed by a spontaneous symmetry breaking of global symmetry and that a large part of its energy is concentrated into a point more specifically in the center [8] and, a charge of this nature is characterized by a spontaneous symmetry breaking of original global O(3) symmetry to U (1) [9].In recent years many authors have devoted themselves to studying black holes with global monopoles [10][11][12][13][14], initiated by Barriola and Vilenkin [15] which was the first study considering a metric of black hole with global static monopoles.In [16], the differential scattering cross section and the absorption of a black hole with global monopole in f (R) gravity have been computed by the partial wave method.
In previous studies [73], we investigated the effect of noncommutativity in black hole by considering a Lorentzian smeared mass distribution on the scattering problem, where we find a differential scattering cross section for small angles given approximately by dσ/dΩ ≈ 16G 2 M 2 /ϑ 2 [1 + (4/M ) θ/π].In the present work, we will see that the presence of noncommutativity and the global monopole changes the behavior of the differential scattering cross section in a nontrivial way.In addition, we also saw in [73] that for the low frequency limit the absorption cross section is always proportional to the area of the event horizon of the noncommutative Schwarzschild black hole σ ≈ 16G 2 M 2 /ϑ 2 [1 + (2/M ) θ/π] 2 .We will see that the absorption cross section increases when we add the influence of the global monopoles on the geometry.At the small mass limit M → 0, we have a result that differs from studies done for other geometries [74][75][76], and at this limit the absorption cross section has a direct relationship with the mass M .Therefore, starting from a Lorentzian distribution, analytical results are more easily examined than using a Gaussian distribution where the analysis is done numerically.In [77,78], employing the Lorentzian distribution, the authors have found logarithmic corrections for entropy as well as the condition for the black hole remnant.In [79], we have investigated the quasinormal modes and the shadow radius of a noncommutative Schwarzschild black hole.We show that, in the zero-mass limit, the shadow ray does not vanish, being proportional to a minimum mass for the finite noncommutative parameter and the black hole becomes a black hole remnant.Therefore, in the present work, we aim to explore the effect of quantum gravity corrections that contribute to the process of massless scalar wave scattering by a noncommutative black hole with a global monopole.We will also apply the null geodesic method to calculate the shadow radius in order to verify the influence of noncommutativity and the global monopole.
The paper is organized as follows.In Sec.II we consider the influence of noncommutativity and the global monopole on the Schwarzschild black hole metric to determine the differential cross section and absorption.In Sec.III we apply the null geodesic method to compute the shadow of the noncommutative black hole with a global monopole.In Sec.IV we make our final considerations.Here we adopt the natural units = c = k B = G = 1.
In order to obtain a minimum mass in the final stage of black hole evaporation, we compute the specific heat capacity using the formula Writing in terms of r s we have Note that for r s = r min = 12(1 − 8πη 2 ) θ/π, we have C = 0 and the black hole becomes a remnant.So we have r min = 2M min and from there we find the following minimum mass M min = 6(1 − 8πη 2 ) θ/π.Therefore, expressing equation (7) in terms of r s , we have Now substituting the radius r min into (10) we find the following maximum temperature of the black hole remnant In Fig. 1 we have the graph of temperature as a function of r s for Θ = 0.06 and η = 0.03, where Θ = √ θ/(M √ π).For a minimum radius r min = 0.70371 and a minimum mass M min = 0.3519, then the value for the maximum temperature is T Hmax = 0.08101.The maximum temperature value corresponds precisely to the peak of the graph.

A. Differential Scattering Cross Section and Absorption
Now, we start our analysis of the massless scalar field equation to examine the scattered wave in the background (1), given by Next, we use the following separation of variables in the Klein-Gordon equation ( 12) being Y lm (ϑ, φ) the spherical harmonics and ω the frequency.Hence, we find the radial equation for R ωl (r) as follows with the effective potential.
Next, we introduce a new radial function, χ(r) = F 1/2 (r)R(r), such that being V (r) the potential, given by At this point, by performing a power series expansion in 1/r in the above equation, we find The effective potential is now given by Here, due to the modification of the term 1/r 2 in the effective potential [16,73], we define Note that the suitable asymptotic behavior for the effective potential is satisfied, i.e., U ef f (r) → 0 when r → ∞.
Next, we can find the phase shift analytically, at the low frequency limit, by means of the following Ansatz [16,73] Hence, the phase shift δ l for l → 0 becomes Furthermore, with determined δ l we can calculate the differential scattering cross section using the formula [16,80,81] where P l (cos ϑ) are the Legendre polynomials, being and Therefore, from expression (24) for l = 0, we have and from relation (25), we have as a result: and because δ l≥1 = 0. Then the equation for the differential scattering cross section ( 23) can be now written as follows and that at the small angle limit equation ( 29) takes the form Moreover, at the low frequency limit, by using the result of ( 22), we find the following result for the differential scattering cross section For η = 0 and θ = 0 we obtain the result for the Schwarzschild black hole case.Now, an interesting result for the differential scattering cross section emerges when we take M → 0 and thus find a non-zero result given by where M min = 6(1 − 8πη 2 ) θ/π is the minimal mass [78].Note that the differential scattering cross section is increased by the monopole effect.At this point, we will compute the absorption cross section for a noncommutative black hole with a global monopole, at the low frequency limit, which can be obtained as follows: Now, by taking the limit ω → 0 with δ l given by ( 22) the absorption reads We can recover the result for the absorption of the Schwarzschild black hole in the absence of noncommutativity and global monopole by making η = θ = 0. Furthermore, an interesting result for the absorption arises when the mass parameter goes to zero and in this case we obtain a non-zero result given by In Fig. 2 we show the behavior of the absorbing cross section for l = 0 when varying M from 1 to the minimum mass value.Note that by varying the η parameter, the absorption has its value increased.Here, it is worth mentioning that the results obtained in (33) and (37) are not present in the usual case of the Schwarzschild black hole when the mass parameter approaches zero.Besides, for absorption we can express the result in terms of the area of the event horizon of the noncommutative black hole with global monopole, so we have We can also express Eq. ( 37) in terms of a minimum area as follows where A min = 4πr 2 min is a minimal area.Therefore, we verified that at the limit of M going to zero, the black hole becomes a remnant of black hole and for η = 0 the absorption is equal to a minimal area due to the effect of noncommutativity.Now, we will calculate the absorption at the high frequency limit.For this we rewrite the potential (19) as follows where In this case, to determine the phase shift we apply the following Ansatz For l → ∞ the phase shift is given by Therefore, at the limit of l → ∞ , the absorption is given by such that we obtain the following result for absorption at the high frequency limit Here we have successfully obtained absorption at the high frequency limit by the partial wave method.Note that for η = θ = 0, we recover the absorption result for the Schwarzschild black hole at the high frequency limit obtained usually by the null geodesic method, σ hf abs = 27πM 2 .For η = 0 and θ = 0, we find showing that the absorption is reduced when we vary θ.
For η = 0 and θ = 0, we obtain In this case the absorption is increased when we vary η, which coincides with the results obtained by the null geodesic method.

B. Numerical analyses
In order to investigate the behavior of the radial equation ( 14) in the asymptotic limits we define the tortoise coordinate as follows: Thus, the radial equation ( 14) can be written as Note that, when applying the limits r → r η and r → ∞, the effective potential defined in (15) tends to zero.Consequently, we can examine the scattering problem by exploring the radial solution at these asymptotic limits.In this way, we have the following boundary conditions: Next, we determine the absorption by applying equation (34) with the phase shift given by Here we present the numerical results that were obtained by numerically solving the radial equation (14).For this purpose we have adopted the numerical procedure performed in [69].
In the table I we show the comparison between the analytical and numerical results for ω → 0 and l = 0.In Fig. 3, we plot the partial absorption cross section for the l = 0 mode for different values of Θ = √ θ/(M √ π) and η.We can see by comparing the curves (Fig. 3(a), Fig. 3(b), Fig. 3(c) and Fig. 3(d)) for different values of Θ = √ θ/(M √ π) and η that the partial absorption is increased in relation to the Schwarzschild black hole when we vary η with fixed Θ. Now, in Fig. 3(e) and Fig. 3(f) when vary Θ with fixed η the partial absorption is decreased due to the noncommutative effect.For Θ = 0 and η = 0 the graph shows the result of the partial absorption for the Schwarzschild black hole.
In Fig. 4 we plot the partial absorption for l = 0 mode by setting the values of Θ = 0.06 and η = 0.03.We can observe that when we reduce the mass value, the absorption amplitude is not null as we can see from equation (37).The partial absorption cross section graphs for l = 0, 1, 2, 3, 4 modes are shown in Fig. 5(a).
In Fig. 6 we present the graphs of the total absorption cross section.The results for fixed Θ and varying η values are shown in figures 6(a) and 6(c).Graphs for fixed η and varying Θ values are shown in figures 6(b) and 6(d).The lines in Fig. 6 represent the values for high frequency absorption, by applying the null geodesic method presented in the next section.
Finally in Fig. 7 we plot the result of the differential scattering cross section for M ω = 1.0 and in Fig. 8 for M ω = 2.0.The effect of noncommutativity and the global monopole on the differential scattering cross section is more significant at large angles.

III. NULL GEODESICS AND SHADOW
In this section, we will apply the null geodesic method to compute the absorption at the high frequency limit and we will also determine the shadow radius of the noncommutative black hole with global monopole.Therefore, at this limit, we check the validity of our numerical results.

A. Null Geodesics
As a starting point we consider the following Lagrangian density: being "." the derivative with respect to an affine parameter.Hence, by applying the line element (1), we have At this point, we will consider the motion on the equatorial plane by taking ϑ = π/2.Hence, we can find the equations of motion from the Hamilton-Jacobi equation such that where E (energy) and L (angular momentum) are the conserved quantities.In the null geodesics analysis we have, g µν ẋµ ẋν = 0 and so we find Now, by introducing a new variable u = 1/r to write the orbital equation in terms of the radii r η and r θ , we obtain here b = L/E is the impact parameter defined as the perpendicular distance (measured at infinity) between the geodesic and a parallel line that passes through the origin.Hence, by differentiating Eq. ( 57), we find The behavior of the geodesic lines for different values of the impact parameter b can be obtained by solving the equations ( 57) and ( 58) numerically.In Fig. 9 we present the change of the geodesic lines for different values of the parameters b, Θ and η.In the graphics, we have a black disk that represents the limit of the event horizon, the internal dotted circle is the radius for the photon sphere (critical radius), and the external dashed circle is the critical impact parameter (shadow).However, by fixing η and increasing the values of the parameter Θ we see a reduction in the effect of the black hole on the light beam, while for fixed Θ and varying η we have an increase in this effect.
To determine the critical radius (r c ) and the critical impact parameter (b c ), we impose conditions du/dφ = 0 and d 2 u/dφ 2 = 0, to obtain Therefore, for high frequencies the absorption is determined by where Observe that making η = θ = 0, we have σ hf abs = 27πM 2 which is the result for the Schwarzschild black hole.For η = 0 and θ = 0, the absorption becomes Hence by varying the values of η, the absorption value at the high frequency limit has its value increased as shown in Fig. 6(a).For η = 0 and θ = 0, we obtain Here the absorption at the high frequency limit has its value reduced when we vary θ as shown in Fig. 6(b) .Now, we will calculate the size of the black hole shadow using the geodetic study and expressing it via celestial coordinates.
where (r o , θ o ) is the observer's position at infinity.In this case, for an observer in the equatorial plane, that is in θ o = π/2, we have the following relation Then For θ and η small, we get Note that the global monopole causes an increase in the shadow radius when we increase the η parameter as we can see in Fig. 10.On the other hand, noncommutativity has the effect of reducing the shadow radius.Besides, for θ = 0 in (70), we have Here the shadow radius has its value increased when we vary the η parameter.
For η = 0 in (70), we obtain Hence the shadow radius has its value reduced when we vary the Θ parameter.Next, taking the null mass limit in (69) we find that the shadow radius is non-zero, given by Therefore, we can see that by varying the η parameter with a fixed Θ the shadow radius increases as we can see in Fig. 11 and Fig. 12.

B. Classic Analysis
Rewriting the equation of the orbit (57) in terms of r, we have To calculate the total angle deflected by the beam we have to integrate the above equation.One way to integrate analytically is to take the limit M/r << 1.Before doing this expansion let us organize the equation by making

changes of variables
where we define θ = 8 M θ π and δ = 1 − 8πη 2 , so the equation ( 74) is rewritten in this new variable as Now expanding the above equation at the limit M/r << 1 and organizing the terms, we find We can identify the y in terms of M/r and a y 2 in M 2 /r 2 , so we have The total angle deflected by the beam is given by ∆φ = 2 (82) Knowing that at the limit M/r << 1 we have that y 0 → δ/b and the result becomes Finally, the deflection angle χ = ∆φ − π in terms of √ θ and η 2 is given by With the relationship of the deflection angle and the impact parameter, we can combine this result with the classical differential scattering section equation In Fig. 14 we show the behavior of the curves for the classical and numerical case.As expected the results for classical analysis fit well in the numerical plot for small angles.

IV. CONCLUSIONS
In summary, in this paper, we investigate the process of massless scalar wave scattering due to a noncommutative black hole with a global monopole through the partial wave method.We have obtained, at the low frequency limit, the phase shift analytically.The effects of the noncommutativity cause a reduction in the differential scattering/absorption cross section and shadow radius, while the presence of the global monopole has the effect of increasing the value of such quantities.In addition, we show that these quantities are non-zero when we take the limit of the zero mass parameter.Rather, they are proportional to a minimum mass.Moreover, we have also checked the analytically obtained results by numerically solving the radial equation for arbitrary frequencies.

FIG. 1 :
FIG. 1:We have the Hawking temperature for the non-commutative case with monopoles compared to the usual Schwarzschild case.

M = 1 FIG. 2 :
FIG. 2:Here we see the behavior of the absorption cross section for l = 0 when varying M from 1 to the minimum mass value, corresponding to the values of the parameters Θ and η.

FIG. 9 :
FIG. 9: Null geodesics in polar coordinates for different values of Θ and η, assuming values for the impact parameter between 4 and 7.The radius of the horizon corresponds to the radius of the disk in black, the inner black ring is the critical radius or the radius of the sphere of photos while the outer ring corresponds to the critical impact parameter.

FIG. 10 :
FIG.10: Influence of the Θ and η parameters on the shadow radius.

FIG. 14 :
FIG. 14: Classical and numerical scattering differential cross section.For the numerics we assume M ω = 1 in (a) and M ω = 2 in (b).