1. Introduction
Many efforts from the astrophysical community have been made to explain the gamma-ray bursts (GRBs) phenomenon with enough of an energy source. GRBs are abrupt increases in gamma-ray intensity coming from an identifiable point in the sky. After the Big Bang, a GRB is the most powerful celestial explosion. Investigating the GRBs' energy source has been a focus of the astrophysics community. One suitable theory is that the heated accretion disk can annihilate neutrinos and antineutrinos into electrons and positrons, $\nu +\bar{\nu }\to {e}^{-}+{e}^{+}$. This mechanism has the potential to produce the energy of GRBs [1–14]. A newborn stellar-mass black hole that is accreting matter at hyperaccretion rates powers this system. Originating from the standpoint of the annihilation process, high-energy gamma-rays are released when electron–positron couples close to the neutron star's surface collapse, resulting in a supernova explosion. There is an obvious explosion like this one. The efficiency of the annihilation of neutrino–antineutrino couples into electron–positron pairs is more than four times that of the Newtonian case in the background structure of a type II supernova, as demonstrated by [15]. This rate of annihilation increases by 30 times on the surface of a collapsing neutron star. A variety of black holes that have been expanded upon by general relativity have been examined recently. For a Kerr black hole with a thin accretion disk, [16, 17] examined the relativistic impact of the annihilation process $\nu +\bar{\nu }\to {e}^{-}+{e}^{+}$ on the energy deposition rate close to the rotation axis; [18, 19] addressed the implications of a strong off-axis. Additional generalizations of general relativity, including higher derivative gravity, Einstein dilation Gauss–Bonnet, charged Galileon, Brans–Dicke, Born–Infeld generalization of the Reissner–Nordstrom solution and Eddington-inspired Born–Infeld, all dramatically increase the energy deposition rate in neutron star and supernova envelopes [20, 21]. When the quintessence field was taken into account, the authors in [22] found that quintessence strengthened the energy of GRBs, which could perhaps explain how GRBs originate. The annihilation process of neutrinos around a massive source with a f(R) global magnetic monopole is investigated in this study, and it is found that this sort of black hole can be an even superior supplier of GRBs than the earlier kind of explanation [23].
Black holes have attracted a lot of attention from the physical community for several decades. As solutions to Einstein's field equation in general relativity, black holes are also the end results of some stars during the process of gravitational collapse [24–27]. In the case of black holes, the classical description of spacetime can not be used to define the surroundings of singularities inside the event horizon [24–27]. In order to overcome the plague from the singularity in the background, it is necessary to generalize general relativity up to the high-energy scale [28–37]. The generalization can lead to quantum gravity which could resolve the unphysical black hole singularities [28–37]. Several proposals such as loop quantum gravity [28–32], Horava–Lifshitz gravity [33–37], etc have been put forward. Here, we are going to focus on the powerful issue called the asymptotic safety (AS) scenario which is utilized to get rid of the singularities of black holes [37]. The technique of the functional renormalization group (FRG) was used to limit the results of the theory at trans-Planckian energies, making the physical quantities avoid divergences at all scales [37]. The quantum effects are always shown as corrections to the classical metrics [38–46]. A series of corrected metrics include Schwarzschild [38], Kerr [39], Schwarzschild-AdS [40], Kerr-AdS [41] and Reissner–Nordstrom-AdS [41–46] without the classical singularity. More attention has been given to RGI-Schwarzschild black holes. The evaporation of the RGI-Schwarzschild black hole was investigated [47, 48], as was mini-black hole production in colliders [49]. The radial accretion of matter onto a RGI-Schwarzschild black hole was also explored [50–52]. The shadows of black holes have been studied in the AS scenario for quantum gravity [53–57]. Recently, the quantum gravity effects on the radiation properties of a thin accretion disk surrounding an RGI-Schwarzschild black hole within the frame of the IR limit of the AS theory were discussed [58].
The metric of a Schwarzschild black hole modified by quantum effects needs to be studied in different directions. The motion of a test particle around an RGI-Schwarzschild black hole was described [59]. Electromagnetic ray reflection spectroscopy for the corrected black hole was also considered [27, 60–63]. The quantum gravity effects on radiation characteristics of thin accretion disks around an RGI-Schwarzschild black hole were shown [58]. Besides the black hole features mentioned above, it is important to probe the annihilation energy deposition rate in the background of this kind of black hole to show the implications of asymptotic safety introducing quantum corrections to the spacetime structures of the gravitational sources. However, to the best of our knowledge, little attention has been paid to this topic. In section 2 , we compute the orbits of the surrounding photons and the geodesics of the RGI-Schwarzschild black hole. Here, we will explore the shadow and the energy deposition rate by the neutrino pair annihilation process around a Schwarzschild black hole in asymptotic safety. In section 3 , our analysis and plotting of the black hole accretion disc's temperature is performed. In section 4 , the ratio of the Newtonian total energy deposition to the total energy deposition adjusted by the spacetime geometric quantum effect parameter ξ is calculated, after the integral form of the neutrino pair annihilation efficiency is obtained. In the asymptotically safe scenario, we will address the effect of RGI-Schwarzschild black holes in the IR limit on the likelihood of GRBs produced by the annihilation process, supported by numerical estimates. In section 5 , the conclusions and perspective are presented.
2. The effective potential and photon orbits of the RGI-Schwarzschild black hole
We adopt the Schwarzschild metric modified by the quantum gravity in the infrared limit as follows [64–68]:4 ). When the extreme points are substituted into the second-order derivative,7 ) allow us to set M = 1 for the later simulation.
$\begin{eqnarray}{\rm{d}}{s}^{2}=f(r){\rm{d}}{t}^{2}-\displaystyle \frac{{\rm{d}}{r}^{2}}{f(r)}-{r}^{2}\left({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\varphi }^{2}\right),\end{eqnarray}$
where $\begin{eqnarray}f(r)=1-\displaystyle \frac{2{MG}(r)}{r}\end{eqnarray}$
with mass M. For the quantum gravity-corrected Schwarzschild metric in the infrared limit, the running coupling G(r) takes the form [65]: $\begin{eqnarray}G(r)={G}_{0}\left(1-\displaystyle \frac{\xi }{{r}^{2}}\right),\end{eqnarray}$
where ξ is a parameter with dimensions of length squared associated to the scale identification between the momentum scale and the radial coordinate. The general function of radial coordinate G(r) as the generalization of the gravitational constant comes as a result of quantum effects. It should be pointed out that the running coupling G(r) contains the parameter ξ standing for the renormalization group improvement and the parameter will change the spacetime structure [38, 57]. There are some upper bounds on ξ derived from a series of gravitational and nongravitational measurements. We can choose c = G0 = 1 in natural units. The radius of the event horizon is a root of f(r) = 0. Naturally, we observe that the Schwarzschild radius for ξ = 0 is rAS = rS = 2M. Determining the limit of the dimensionless parameter for the quantum gravity is crucial to preserve the two-horizon metric [58]. Since the first-order derivative is equal to zero, $\begin{eqnarray}\displaystyle \frac{{\rm{d}}f(r)}{{\rm{d}}r}=\displaystyle \frac{2M\left({r}^{2}-3\xi \right)}{{r}^{4}}=0.\end{eqnarray}$
The extrme point is ${r}_{0}=\sqrt{3\xi }$ which is also the root of equation ( $\begin{eqnarray}{\left.\displaystyle \frac{{{\rm{d}}}^{2}f(r)}{{\rm{d}}{r}^{2}}\right|}_{r=\sqrt{3\xi }}=\displaystyle \frac{4M}{3\sqrt{3{\xi }^{3}}}\gt 0.\end{eqnarray}$
This is a minimum point, as can be observed. Given the two-horizons requirement, there ought to be two roots for f(r) = 0, $\begin{eqnarray}f\left({r}_{0}\right)=1-\displaystyle \frac{4M}{3\sqrt{3\xi }}\leqslant 0.\end{eqnarray}$
The mass M and the dimensionless parameter ξ should be constrained in accordance with the criteria rAS ≥ 0 and computation, as stated in [58]: $\begin{eqnarray}\left\{\begin{array}{l}0\leqslant \xi \leqslant \displaystyle \frac{16}{27}{M}^{2}\\ M\gt \sqrt{\displaystyle \frac{27}{16}\xi }.\end{array}\right.\end{eqnarray}$
The findings of [58] are in line with it. The boundary conditions (It is necessary to describe the accretion disk of Schwarzschild black holes in asymptotic safety because only sufficiently hot accretion disks can emit the neutrinos [69–72]. We plan to discuss the geodesics for a particle motion in the background specified by metric (1 ). We limit the particle motion to the equatorial plane with θ = π/2 to write the Lagrangian as the description of the planar motion given by [16, 73],
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal L } & = & \displaystyle \frac{1}{2}{g}_{\mu \nu }\displaystyle \frac{{\rm{d}}{x}^{\mu }}{{\rm{d}}\tau }\displaystyle \frac{{\rm{d}}{x}^{\nu }}{{\rm{d}}\tau }\\ & = & \displaystyle \frac{1}{2}\left[f(r){\left(\displaystyle \frac{{\rm{d}}t}{{\rm{d}}\tau }\right)}^{2}-\displaystyle \frac{1}{f(r)}{\left(\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\tau }\right)}^{2}-{r}^{2}{\left(\displaystyle \frac{{\rm{d}}\varphi }{{\rm{d}}\tau }\right)}^{2}\right],\end{array}\end{eqnarray}$
then the components of the particle's momentum are $\begin{eqnarray}E=f(r)\displaystyle \frac{{\rm{d}}t}{{\rm{d}}\tau },\end{eqnarray}$
$\begin{eqnarray}L={r}^{2}\displaystyle \frac{{\rm{d}}\varphi }{{\rm{d}}\tau }.\end{eqnarray}$
The geodesics introduce E and L are constants and the momenta satisfy PμPμ = 0 for photons [15]. The equation for radial motion is $\begin{eqnarray}{\left(\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\tau }\right)}^{2}+{V}_{\mathrm{eff}}(r)=\displaystyle \frac{1}{{b}^{2}},\end{eqnarray}$
where the effective potential is $\begin{eqnarray}{V}_{\mathrm{eff}}(r)=\displaystyle \frac{1}{{r}^{2}}\left[1-\displaystyle \frac{2M}{r}\left(1-\displaystyle \frac{\xi }{{r}^{2}}\right)\right].\end{eqnarray}$
Here, the impact factor is defined as b = L/E. In view of the conditions of the photon sphere like dr/dτ = 0 and d2r/dτ2 = 0, equation (12 ) leads to1 ), the outer horizon r+, radius of photon sphere rp and impact factor bp relate to the dimensionless parameter ξ associated with the quantum gravity. The relations are listed in the table 1 and the values of ξ are restricted according to equation (7 ). We plot the effective potential as function of radial coordinate in figure 1. The figure shows that the quantum gravity effect amends the features of the compact source, although the shapes of the curves corresponding to the different values of parameter ξ are similar. By combining equation (10 ), equation (11 ) and equation (12 ), we have the trajectory equation as [15], 17 ) into
$\begin{eqnarray}{V}_{\mathrm{eff}}(r)=\displaystyle \frac{1}{{b}^{2}},\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{V}_{\mathrm{eff}}}{{\rm{d}}r}=0,\end{eqnarray}$
or equivalently, $\begin{eqnarray}{r}_{p}^{2}={b}_{p}^{2}f({r}_{p}),\end{eqnarray}$
$\begin{eqnarray}{\left.{r}_{p}^{3}\displaystyle \frac{{\rm{d}}f(r)}{{\rm{d}}r}\right|}_{r={r}_{p}}=2{b}_{p}^{2}{f}^{2}\left({r}_{p}\right),\end{eqnarray}$
where rp is the radius of photon sphere and bp is the relevant impact factor. According to the discussions on metric ( $\begin{eqnarray}\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\varphi }=\pm {r}^{2}\sqrt{\displaystyle \frac{1}{{b}^{2}}-\displaystyle \frac{1}{{r}^{2}}\left[1-\displaystyle \frac{2M}{r}\left(1-\displaystyle \frac{\xi }{{r}^{2}}\right)\right]}.\end{eqnarray}$
By setting u ≡ 1/r, we can transform ( $\begin{eqnarray}\displaystyle \frac{{\rm{d}}u}{{\rm{d}}\varphi }=\sqrt{\displaystyle \frac{1}{{b}^{2}}-{u}^{2}\left[1-2{Mu}\left(1-\xi {u}^{2}\right)\right]}\equiv G(u).\end{eqnarray}$
Figure 1. The effective potential profiles with M = 1 for ξ = 1/3 (left panel) and ξ = 5/9 (right panel). The photon sphere rp has radii indicated by the dashed lines. |
Table 1. The event horion r+, the photon sphere rp and the impact parameter bp of the photon sphere with different ξ. |
| ξ = 0 | ξ = 1/9 | ξ = 2/9 | ξ = 1/3 | ξ = 4/9 | ξ = 5/9 | |
|---|---|---|---|---|---|---|
| r+ | 2.00000 | 1.94102 | 1.87336 | 1.79252 | 1.68806 | 1.51749 |
| rp | 3.00000 | 2.93553 | 2.86460 | 2.78514 | 2.69375 | 2.58397 |
| bp | 5.19615 | 5.12975 | 5.05818 | 4.98013 | 4.89356 | 4.79502 |
For the sake of distinguishing among the light trajectories, it is useful to establish orbital fractions n = φ/2π [15, 73]. The direct emission of these rays corresponds to n < 3/4, the lensing ring to 3/4 < n < 5/4 and the photon ring to n > 5/4. These rays intersect the equatorial plane once, twice and more than twice. The range of the incident parameter b of the photon ring, lens ring and direct image for the situations with ξ = 1/3 and ξ = 5/9 in the range as equation (7 ) is given below,
$\begin{eqnarray}\xi =\displaystyle \frac{1}{3}\Rightarrow \left\{\begin{array}{cc}\mathrm{Direct}\,\mathrm{emission:}\,n\lt \displaystyle \frac{3}{4},\quad & b\lt 4.74314\quad \mathrm{and}\quad b\gt 6.04424\\ \mathrm{Lensing}\,\mathrm{ring:}\,\displaystyle \frac{3}{4}\lt n\lt \displaystyle \frac{5}{4},\quad & 4.74314\lt b\lt 4.96569\quad \mathrm{and}\quad 5.02321\lt b\lt 6.04424\\ \mathrm{Photon}\,\mathrm{ring:}\,n\gt \displaystyle \frac{5}{4},\quad & 4.96569\lt b\lt 5.02321,\end{array}\right.\end{eqnarray}$
$\begin{eqnarray}\xi =\displaystyle \frac{5}{9}\Rightarrow \left\{\begin{array}{cc}\mathrm{Direct}\,\mathrm{emission:}\,n\lt \displaystyle \frac{3}{4},\quad & b\lt 4.43019\quad \mathrm{and}\quad b\gt 5.95531\\ \mathrm{Lensing}\,\mathrm{ring:}\,\displaystyle \frac{3}{4}\lt n\lt \displaystyle \frac{5}{4},\quad & 4.43019\lt b\lt 4.76382\quad \mathrm{and}\quad 4.85521\lt b\lt 5.95531\\ \mathrm{Photon}\,\mathrm{ring:}\,n\gt \displaystyle \frac{5}{4},\quad & 4.76382\lt b\lt 4.85521.\end{array}\right.\end{eqnarray}$
Figure 2 describes a series of photon trajectories with coordinates (r, φ) on the equatorial plane. In figure 2, the solid disks stand for black holes and the photon orbit is represented by the lines in color. When b < bp, the electromagnetic rays directly enter the black hole. The photons orbit the gravitational source with the constraint b = bp. The potential shown in figure 1 causes the light to deflect, and photons pass through and escape from the black hole if the impact parameter is larger, such as b > bp. Some photons escape the compact body and the remaining ones are either absorbed by the body or go around the source if the light is directed towards the black hole.
Figure 2. The way photons behave in the RGI-Schwarzschild spacetime depends on the impact parameter b. |
By comparing the aforementioned equations with those of the RGI-Schwarzschild black hole, it is clear that the parameter ξ decreases the regions of the direct emission, the lensing ring and the photon ring. The black hole's accretion disk is about to form and develop and neutrinos are released from the hotter disk.
3. The temperature of the relativistic thin accretion disk
High-energy astronomical events like GRBs are thought to have their source of energy in this neutrino-cooling-dominated accretion flow (NDAF) hypothesis [11, 74–81]. Comprehensive numerical computations and examinations of the physical properties of NDAFs, encompassing annihilation luminosity, peak energy and detection rate, have been conducted and published in [82–84]. Relativistic jets are created when the heated material inside the accretion disk releases energy via the neutrino–antineutrino annihilation process. Thus, the medium-energy deposition rate and, consequently, the neutrino flux are directly influenced by the temperature of the accretion disk. Neutrinos interact with surrounding matter after exiting the accretion disk, eventually annihilating to generate gamma-rays. These gamma-rays can then be further accelerated and recombined by internal excitations or other mechanisms to form the high-energy radiation jets that are detected. The Novikov–Thorne model [60, 85], a relativistic extension of the Shakura–Sunyaev model [86], describes geometrically thin and optically thick accretion disks. The transmission of energy and angular momentum outward via radiation and viscous magnetic or turbulent strains makes accumulation feasible. According to the model, a quasi-steady disk is situated in the equatorial plane of an axisymmetric background spacetime geometry and the material within the disk orbits in a manner that approximates a geodesic circle. The physical quantities are integrated vertically since the disk is thin, with a maximum half-thickness H of H/R ≪ 1 with a characteristic radius of R. The quantities describing the thermal properties of the disk are the averages of the azimuthal angle φ = 2π, the altitude H and the time scale Δt needed for the gas to flow inwards for 2H. The heat produced by stress and dynamic friction is efficiently emitted from the surface of the disk, primarily in the form of radiation. Furthermore, the large-scale magnetic fields and the flow of energy and angular momentum between distant regions of the disk are disregarded. Under these assumptions, the conservation rules of rest mass, energy and angular momentum may be used to determine the disk's time-averaged radial structure. The integral of the mass conservation equation can be used to determine the mass accretion rate's constancy,
$\begin{eqnarray}\dot{M}=-2\pi r{\rm{\Sigma }}(r){u}^{r}={\rm{constant}},\end{eqnarray}$
where Σ(r) is the surface density and ur is the radial velocity. The energy flow that the disk's surface emits on average throughout time is [60] $\begin{eqnarray}{ \mathcal F }(r)=-\displaystyle \frac{\dot{M}}{4\pi \sqrt{-g}}\displaystyle \frac{1}{{\left(E-{\rm{\Omega }}L\right)}^{2}}\displaystyle \frac{{\rm{d}}{\rm{\Omega }}}{{\rm{d}}r}{\int }_{{r}_{\mathrm{ISCO}}}^{r}{\left(E-{\rm{\Omega }}L\right)}^{2}\displaystyle \frac{{\rm{d}}L}{{\rm{d}}r}{\rm{d}}r.\end{eqnarray}$
To be more precise, E, L and Ω stand for the conserved specific energy, the angular velocity of the equatorial circular geodesic and the axial component of the conserved specific angular momentum, respectively. With the radial component of the geodesic equation satisfied and dr/dτ = θ/dτ = d2r/dτ2 = 0, the following angular velocity relation may be obtained [87], $\begin{eqnarray}{\rm{\Omega }}=\sqrt{\displaystyle \frac{1}{2r}\displaystyle \frac{{\rm{d}}f(r)}{{\rm{d}}r}}=\sqrt{\displaystyle \frac{M\left({r}^{2}-3\xi \right)}{{r}^{5}}}.\end{eqnarray}$
$\sqrt{-g}=r$ is the determinant of the near-equatorial plane metric. The radius of the accretion disk's innermost stable circular orbit (ISCO) is known [58].Considering that thermodynamic equilibrium is assumed for the disk, the radiation released may be classified as blackbody radiation, with a temperature of
$\begin{eqnarray}T(r)={\left[\displaystyle \frac{{ \mathcal F }(r)}{\sigma }\right]}^{\tfrac{1}{4}},\end{eqnarray}$
where σ is the Stefan–Boltzmann constant.Using equation (27 ) from [58], one may approximate the increase in the accretion disk's differential luminosity peak, which is about 10.38% of its highest value in the situation, ξ = 5/9. The temperature of the accretion disk is displayed in figure 3 for various values of ξ. It is evident that when ξ grows, the radius of the temperature peak tends to decrease, indicating a higher internal temperature within the disk. The peak likewise increases as ξ increases; at ξ = 5/9, it is raised by about 8.76% over the classical case. Neutrino radiation can cool the accretion disk when its temperature reaches a certain level.
Figure 3. The curves of temperature T(r) as a function of the ratio R for quantum gravity factors ξ = 0, 1/9, 1/3 and 5/9. |
4. The energy deposition rate for the neutrino annihilation process
We start to consider the energy deposition in the spacetime governed by equations (2 ) and (3 ). The ratio of neutrino annihilation is roughly 1%. The energy deposition per unit time and per unit volume for the neutrino pair annihilation process is given by [15], 9 ) can be chosen as [15], 8 ), the expression of deposition energy per unit time and unit volume is given by [15],
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}E({\boldsymbol{r}})}{{\rm{d}}t{\rm{d}}V}=2{{KG}}_{F}^{2}F(r)\iint n({\varepsilon }_{\nu })n({\varepsilon }_{\bar{\nu }})({\varepsilon }_{\nu }+{\varepsilon }_{\bar{\nu }}){\varepsilon }_{\nu }^{3}{\varepsilon }_{\bar{\nu }}^{3}{\rm{d}}{\varepsilon }_{\nu }{\rm{d}}{\varepsilon }_{\bar{\nu }},\end{eqnarray}$
where $\begin{eqnarray}K=\displaystyle \frac{1}{6\pi }\left(1\pm 4{\sin }^{2}{\theta }_{W}+8{\sin }^{4}{\theta }_{W}\right)\end{eqnarray}$
with the Weinberg angle ${\sin }^{2}{\theta }_{W}=0.23$. For various neutrino pairs, the forms of equation ( $\begin{eqnarray}K({\nu }_{\mu },{\bar{\nu }}_{\mu })=K({\nu }_{\tau },{\bar{\nu }}_{\tau })=\displaystyle \frac{1}{6\pi }\left(1-4{\sin }^{2}{\theta }_{W}+8{\sin }^{4}{\theta }_{W}\right)\end{eqnarray}$
and $\begin{eqnarray}K({\nu }_{e},{\bar{\nu }}_{e})=\displaystyle \frac{1}{6\pi }\left(1+4{\sin }^{2}{\theta }_{W}+8{\sin }^{4}{\theta }_{W}\right),\end{eqnarray}$
respectively. Here, the Fermi constant is GF = 5.29 ×10−44 cm2MeV−2. The angular integration factor is represented by [15], $\begin{eqnarray}\begin{array}{rcl}F(r) & = & \iint {\left(1-{{\boldsymbol{\Omega }}}_{{\boldsymbol{\nu }}}\cdot {{\boldsymbol{\Omega }}}_{\bar{{\boldsymbol{\nu }}}}\right)}^{2}{\rm{d}}{{\rm{\Omega }}}_{\nu }{\rm{d}}{{\rm{\Omega }}}_{\bar{\nu }}\\ & = & \displaystyle \frac{2{\pi }^{2}}{3}{\left(1-x\right)}^{4}\left({x}^{2}+4x+5\right),\end{array}\end{eqnarray}$
where $\begin{eqnarray}x=\sin {\theta }_{r}.\end{eqnarray}$
The angle θr is between the particle trajectory and the tangent vector to a circle orbit at radius r. For one kind of neutrino and antineutrino, ${{\rm{\Omega }}}_{\nu }({{\rm{\Omega }}}_{\bar{\nu }})$ is the unit direction vector and ${\rm{d}}{{\rm{\Omega }}}_{\nu }({\rm{d}}{{\rm{\Omega }}}_{\bar{\nu }})$ is a solid angle. At temperature T, n(ϵν) and $n({\varepsilon }_{\bar{\nu }})$ are number densities for neutrino and antineutrino, respectively, in the phase space and satisfy the Fermi–Dirac distribution [15], $\begin{eqnarray}n({\varepsilon }_{\nu })=\displaystyle \frac{2}{{h}^{3}}\displaystyle \frac{1}{\exp \left(\tfrac{{\varepsilon }_{\nu }}{{kT}}\right)+1},\end{eqnarray}$
where h is the Planck constant and k is the Boltzmann constant. With integrating equation ( $\begin{eqnarray}\displaystyle \frac{{\rm{d}}E}{{\rm{d}}t{\rm{d}}V}=\displaystyle \frac{21\zeta (5){\pi }^{4}}{{h}^{6}}{{KG}}_{F}^{2}F(r){\left({kT}\right)}^{9}.\end{eqnarray}$
It is significant to derive the expression of dE/dtdV which is used to further the research on the converted energy rate on the different compact bodies. The expression has something to do with the position, as has the temperature T = T(r), which is called local temperature [15].The local temperature measured by a local observer is defined as $T({\boldsymbol{r}})\sqrt{f({\boldsymbol{r}})}={\rm{constant}}$. The neutrino temperature at the neutrinosphere can be described as33 ), equation (34 ) and equation (35 ) and substitute them into equation (32 ) to obtain36 ). We can calculate the radiation energy power in the background of the gravitational source by means of the deposition energy density over time from equation (36 ). In order to compute the angular integration F(r) from equation (29 ), we should further discuss the variable x in equation (30 ). We follow the procedure of [15] and solve the null geodesic in the spacetime of a spherically symmetric gravitational object to show [20]2 ). It is useful to relate the variable $x=\sin {\theta }_{r}{| }_{{\theta }_{R}}=0$ to the environment structure around the gravitational object [20]. According to equation (29 ), the angular integration factor becomes the function of the metric. We can proceed the integration of rate per unit time and unit volume from equation (36 ) on the spherically symmetric volume around the gravitational source [22], 38 ) with the Newtonian quantities as [15, 20, 22],1 ). According to [15], we can present ${\rm{d}}\dot{Q}/{\rm{d}}r$ as a function of radial coordinate r
$\begin{eqnarray}T(r)\sqrt{f(r)}=T(R)\sqrt{f(R)},\end{eqnarray}$
where R is the radius of a gravitational source. It is elegant to replace the local temperature T(r) in the future calculation according to the identity. The luminosity relating to the redshift can be selected as $\begin{eqnarray}{L}_{\infty }=f({R}_{0})L({R}_{0}),\end{eqnarray}$
where the luminosity for a single neutrino species at the neutrinosphere is $\begin{eqnarray}L(R)=4\pi {R}_{0}^{2}\displaystyle \frac{7}{4}\displaystyle \frac{{ac}}{4}{T}^{4}(R),\end{eqnarray}$
where a is the radiation constant and c is the speed of light in vacuum. In order to replace the temperature associated with the observer's position, we combine equation ( $\begin{eqnarray}\displaystyle \frac{{\rm{d}}E({\boldsymbol{r}})}{{\rm{d}}t{\rm{d}}V}=\displaystyle \frac{21\zeta (5){\pi }^{4}}{{h}^{6}}{{KG}}_{F}^{2}{k}^{9}{\left(\displaystyle \frac{7}{4}\pi {ac}\right)}^{-\tfrac{9}{4}}{L}_{\infty }^{\tfrac{9}{4}}F(r)\displaystyle \frac{{\left[f(R)\right]}^{\tfrac{9}{4}}}{{\left[f(r)\right]}^{\tfrac{9}{4}}}{R}_{0}^{-\tfrac{9}{2}}.\end{eqnarray}$
In addition to the radial coordinate, the metric components for the massive source surface also appear in the expression of deposition energy per unit time and unit volume as in equation ( $\begin{eqnarray}\begin{array}{rcll}{x}^{2} & = & {\sin }^{2}{\theta }_{r}{| }_{{\theta }_{R}=0}= & 1-\displaystyle \frac{{R}^{2}}{{r}^{2}}\displaystyle \frac{f(r)}{f(R)}.\end{array}\end{eqnarray}$
Here, f(r) = f(r), a component of metric ( $\begin{eqnarray}\begin{array}{rcl}\dot{Q} & = & \displaystyle \frac{{\rm{d}}E}{\sqrt{f}{\rm{d}}t}\\ & = & \displaystyle \frac{84\zeta (5){\pi }^{5}}{{h}^{6}}{{KG}}_{F}^{2}{k}^{9}{\left(\displaystyle \frac{7}{4}\pi {ac}\right)}^{-\tfrac{9}{4}}{L}_{\infty }^{\tfrac{9}{4}}{\left[f(R)\right]}^{\tfrac{9}{4}}{R}^{-\tfrac{9}{2}}\\ & & \times {\displaystyle \int }_{{R}_{0}}^{\infty }\displaystyle \frac{{r}^{2}\sqrt{-f{\left(r\right)}^{-1}}F(r)}{f(r)}{\rm{d}}r.\end{array}\end{eqnarray}$
The metrics of the curved spacetime have a considerable influence on the deposition energy rate. Here, $\dot{Q}$ can reflect the total amount of energy converted from neutrinos to electron–positron pairs per unit time at any radius. The conversion may become an explosion with extremely large values of $\dot{Q}$. It is significant to compare the energy deposition rate ( $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\dot{Q}}{{\dot{Q}}_{{\rm{Newt}}}} & = & 3{\left[f(R)\right]}^{\tfrac{9}{4}}{\displaystyle \int }_{1}^{\infty }{\left(x-1\right)}^{4}\left({x}^{2}+4x+5\right)\\ & & \times \displaystyle \frac{{y}^{2}\sqrt{-f{\left({Ry}\right)}^{-1}}}{f{\left({Ry}\right)}^{\tfrac{9}{2}}}{\rm{d}}y,\end{array}\end{eqnarray}$
with dimensionless variable y = r/R and f(r), $f{\left(r\right)}^{-1}$ components of metric ( $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}\dot{Q}}{{\rm{d}}r} & = & 4\pi \left(\displaystyle \frac{{\rm{d}}{E}}{{\rm{d}}{t}{\rm{d}}{V}}\right)\sqrt{-f{\left(r\right)}^{-1}}{r}^{2}\\ & = & \displaystyle \frac{168\zeta (5){\pi }^{7}}{3{h}^{6}}{{KG}}_{F}^{2}{k}^{9}{\left(\displaystyle \frac{7}{4}\pi {ac}\right)}^{-\tfrac{9}{4}}{L}_{\infty }^{\tfrac{9}{4}}\\ & & \times {\left(x-1\right)}^{4}\left({x}^{2}+4x+5\right)\\ & & \times {\left[\displaystyle \frac{f(R)}{f(r)}\right]}^{\tfrac{9}{4}}{R}^{-\tfrac{5}{2}}\sqrt{-f{\left(r\right)}^{-1}}{\left(\displaystyle \frac{r}{R}\right)}^{2}.\end{array}\end{eqnarray}$
Here, the derivative is a function of the radial coordinate with the origin at the centre of the gravitational source while involving the metric functions. It is necessary to consider how the structure of the compact body in asymptotic safety affects the neutrino annihilation. The derivative function could tell us which kind of stars attracting the annihilation could become source of GRBs.It is important to discuss the ratio (39 ) in the quantum gravity-corrected Schwarzschild spacetime in the IR limit. We can relate that f(r) = f(r) and $f{\left(r\right)}^{-1}=-1/f(r)$ for metric (1 ). The ratio (39 ) can be reformed as [15, 64–68],2 ) and the running coupling (3 ), we rewrite the variable (37 ) as
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\dot{Q}}{{\dot{Q}}_{{\rm{Newt}}}} & = & 3{\left[f(R)\right]}^{\tfrac{9}{4}}{\displaystyle \int }_{1}^{\infty }{\left(x-1\right)}^{4}\\ & & \times \left({x}^{2}+4x+5\right)\displaystyle \frac{{y}^{2}}{{\left[f({Ry})\right]}^{5}}{\rm{d}}y,\end{array}\end{eqnarray}$
where $\begin{eqnarray}f(R)=1-\displaystyle \frac{2M}{R}+\displaystyle \frac{2M\xi }{{R}^{3}},\end{eqnarray}$
$\begin{eqnarray}f({Ry})=1-\displaystyle \frac{2M}{R}\displaystyle \frac{1}{y}+\displaystyle \frac{2M\xi }{{R}^{3}}\displaystyle \frac{1}{{y}^{3}}.\end{eqnarray}$
According to the metric function ( $\begin{eqnarray}{x}^{2}=1-\displaystyle \frac{1}{{y}^{2}}\displaystyle \frac{1-\tfrac{2M}{R}\tfrac{1}{y}+\tfrac{2M\xi }{{R}^{3}}\tfrac{1}{{y}^{3}}}{1-\tfrac{2M}{R}+\tfrac{2M\xi }{{R}^{3}}}.\end{eqnarray}$
In order to show the influence of quantum gravity, we should quantify the integral expression of ratio $\dot{Q}/{\dot{Q}}_{{Newt}}$ in equation (41 ) and depict the dependence on ${R}_{0}/M$ with the effect parameter ξ in figures. Although the metric is corrected by the quantum gravity, it is difficult for us to estimate the influence on the energy deposition rate according to equations (41 )–(43 ). We have to perform complicated calculations to scrutinize the impact of neutrino annihilation associated with the spacetime structure around compact bodies. It is necessary to gather data for the constraints over the values of ξ [57]. When plotting the data, we have to magnify its value a huge number of times in order to reveal the differences between the curves of the ratio $\dot{Q}/{\dot{Q}}_{{Newt}}$ for different magnitudes of factor ξ from the quantum gravity effects on the neutrino pair annihilation around the black hole in asymptotic safety gravity because of the extremely tiny values of ξ from [57]. In figure 4, the shape of ratio curves from different values of parameter ξ are similar and here we choose the values of parameter ξ within the region (7 ). The influence from this kind of source consist of two parts, mass M and gravitational coupling G(r).. As a gravitational source, the values of ratio $\dot{Q}/{\dot{Q}}_{{\rm{Newt}}}$ are much more than one, so this kind of stellar object attracting the neutrino pair annihilation process may provide a source of GRBs. For the sake of showing the asymptotic behaviour of ratio $\dot{Q}/{\dot{Q}}_{{Newt}}$ more clearly, we plot some curves to indicate that the ratio is a decreasing function of ξ from quantum corrections within the region belonging to the factor ξ due to equation (7 ) in figure 5. It should be pointed out that the quantum gravity influence limited by equation (7 ) may weaken the power converted from neutrinos to electron–positron pairs around the gravitational object even considerably, but the object can also trigger GRBs. In contrast to the case of the accretion disk around a stellar object in asymptotic safety, the quantum gravity effect can enhance the disk's thermal properties [58] while the more considerable influence will reduce the ratio $\dot{Q}/{\dot{Q}}_{{Newt}}$. Equation (38 ) in [88] may be expressed as follows:
$\begin{eqnarray}{\dot{Q}}_{51}=1.09\times {10}^{-5}{{DL}}_{51}^{\tfrac{9}{4}}{R}_{6}^{-\tfrac{3}{2}},\end{eqnarray}$
where the radius in units of 10 km is denoted by ${R}_{6}$, and the total energy deposition and luminosity are expressed as ${\dot{Q}}_{51}$ and L51, respectively, in 1051 erg/s. The neutrinos are emitted from the neutrinosphere. We assume that D=1.23, R = 20 km and ∼1053 erg/s for the neutrino luminosity at infinity. Table 2 shows that, in comparison to the Schwarzschild case, the energy deposition rate decreases when the quantum gravity parameter ξ is introduced into the background spacetime. For the investigation of neutrino luminosity, particular numerical simulations and techniques for astronomical observation are provided in [82–84].
Figure 4. The curves of the ratio $\dot{Q}/{\dot{Q}}_{{Newt}}$ as a function of the ratio R/M for quantum gravity factors ξ = 0, 1/9, 1/3 and 5/9. |
Figure 5. The curves of the ratio $\dot{Q}/{\dot{Q}}_{{Newt}}$ as a function of the quantum gravity factor ξ with R = 3M, 3.5M, 4M and 4.5M. |
Table 2. Rate of energy emission due to neutrino pair annihilation for GRBs in different spacetime. |
| R/M | $\dot{Q}(\,\mathrm{erg}/{\rm{s}})$ | |
|---|---|---|
| Newtonian | 0 | 1.5 × 1050 |
| ξ = 0 | 3 | 1.2 × 1052 |
| ξ = 0 | 4 | 3.1 × 1051 |
| ξ = 1/9 | 3 | 1.1 × 1052 |
| ξ = 1/9 | 4 | 3.0 × 1051 |
| ξ = 1/3 | 3 | 8.9 × 1051 |
| ξ = 1/3 | 4 | 2.9 × 1051 |
| ξ = 5/9 | 3 | 7.5 × 1051 |
| ξ = 5/9 | 4 | 2.8 × 1051 |
We also describe the derivative ${\rm{d}}\dot{Q}/{\rm{d}}r$ as a function of radius for several stellar masses denoted as r/R with R = 3M to show the promotion of e−e+ pair energy from the neutrino annihilation in figure 6. It is similar in that a slightly larger variable ξ from quantum gravity decreases the derivative ${\rm{d}}\dot{Q}/{\rm{d}}r$. Figure 6 also indicates that the structure with a smaller r/R leads to the greater ${\rm{d}}\dot{Q}/{\rm{d}}r$. It is interesting that the increase is much larger near the surface of the neutron star similar to the case of [15], although the compact source surrounded by the annihilation includes the quantum gravity influence.
Figure 6. The curves of the derivative ${\rm{d}}\dot{Q}/{\rm{d}}r$ as a function of the ratio r/R with R = 3M for quantum gravity factors ξ = 0, 1/9, 1/3 and 5/9. |
5. Conclusion
For the RGI-Schwarzschild black hole, we computed the geodesic equations and photon orbits. The connection between the dimensionless parameter ξ and the halo radius was also examined. The thin accretion disk was then simulated using the Novikov–Thorne model, and its variation was computed with parameter ξ. The temperature profile was then shown. We discovered that, contrary to what general relativity predicts, an increase in the quantum effect parameter ξ results in the following: (a) a shift towards smaller values of the black hole shadow's radius; (b) an increase in the disk's temperature; and (c) a shift towards smaller values of the radial coordinates for the peak of the thermal properties' radial profile.
Neutrinos surrounding Schwarzschild black holes are asymptotically secure from annihilation $\nu +\bar{\nu }\to {e}^{-}+{e}^{+}$. There are several ways to investigate how black holes are affected by quantum gravity. In the framework of the quantum gravity-corrected massive source, we developed and computed the emission energy rate $\dot{Q}$ as well as the emission energy rate ratio $\dot{Q}/{\dot{Q}}_{{\rm{Newt}}}$. In contrast to the case of an accretion disk under identical conditions, it was discovered that the quantum gravity effect reduces the energy released per unit of time $\dot{Q}$ rather than increasing the ratio $\dot{Q}/{\dot{Q}}_{{\rm{Newt}}}$. The energy deposition rate is adequate to sustain the burst, but it will decrease with increasing parameter ξ, which encodes the quantum gravity influence on the spacetime structure. GRBs could originate from the annihilation of $\nu +\bar{\nu }\to {e}^{-}+{e}^{+}$ surrounding a quantum gravity-corrected black hole, according to numerical evidence provided by the ratio $\dot{Q}/{\dot{Q}}_{{\rm{Newt}}}$. Nuclear star collapses or type II supernovae may be the source of such destruction. This annihilation mechanism is a qualified potential source of GRBs inside the ξ area.
The physics at the surface of a hot (T ∼ MeV) neutron star is significant because it involves particle interactions involving baryons, leptons, photons and neutrinos, among others, as has been reported in [15]. Neutrino–environment interactions, such neutrino–electron scattering, are also involved, as is the way in which these processes influence the final gamma-ray emission. Furthermore, taking into consideration the effects of oscillation events and neutrino mass on the results will bring about a unique set of challenges. Neutrinos can have multiple energy states due to the presence of neutrino masses, and oscillations allow them to switch between three different flavors (electron neutrino νe, muon neutrino νμ and tauon neutrino ντ). This may alter the distribution and behaviour of neutrinos on the accretion disc, resulting in deviations from the external environment's observed neutrino composition. The probability that they will interact with matter is impacted by each of these factors, which perhaps in turn influences the annihilation rate. This work focusses on the neutrino pair annihilation process, which makes up a very minor portion of the processes mentioned above. Finally, either the neutrino oscillations' transition probability or the temperature of the neutron star merger would need to be considered for a more accurate estimation of the energy deposition. Nevertheless, this is just one aspect of quantum gravity research; in the future, more realistic spacetime settings, like the RGI-Kerr black hole, can be examined from an alternative viewpoint by fusing astronomical measurements with asymptotic safety.