Surrounded Bonnor–Vaidya solution by cosmological fields
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Abstract
In the present work, we generalize our previous work (Heydarzade and Darabi in arXiv:1710.04485, 2018 on the surrounded Vaidya solution by cosmological fields to the case of Bonnor–Vaidya charged solution. In this regard, we construct a solution for the classical description of the evaporatingaccreting charged Bonnor–Vaidya black holes in the generic dynamical backgrounds. We address some interesting features of these solutions and classify them according to their behaviors under imposing the positive energy condition. Also, we analyze the timelike geodesics associated with the obtained solutions and show that some new correction terms arise in comparison to the case of standard Schwarzschild black hole. Then, we explore all these features for each of the cosmological backgrounds of dust, radiation, quintessence and cosmological constantlike fields in more detail.
1 Introduction
In 1951, Vaidya introduced a new nonstatic solution, describing a spherical symmetric object possessing an outgoing null radiation, for the Einstein field equations [1, 2]. This solution is characterized by a dynamical mass function, depending on the retarded time coordinate. Based on its dynamical nature, the Vaidya solution has been used for studying the process of spherical symmetric gravitational collapse and as a testing ground for the cosmic censorship conjecture [3, 4, 5, 6, 7], and as a dynamical generalisation of the Schwarzschild solution representing a spherically symmetric evaporating black hole, as well as studying the Hawking radiation [8, 9, 10, 11, 12, 13, 14, 15]. This solution was generalized by Bonnor and Vaidya to the charged case, well known as the Bonnor–Vaidya solution [16]. This solution and its interesting features and applications are studied in [17, 18, 19, 20] as instances. Further generalization of the original Vaidya solution were introduced in [21] by Husain for a null fluid with a particular equation of state, and in [22] by wang and Wu using the fact that any linear superposition of particular solutions is also a solution to the Einstein field equations. Using this approach, one can find other general solutions such as the Vaidya–de Sitter [23], Bonnor–Vaidya–de Sitter [18, 24, 25, 26, 27] and radiating dyon solutions [28]. The Vaidya solution and its generalizations are also studied in the context of modified theories of gravity, see for examples [4, 29, 30, 31, 32, 33].
Black holes have such an strong gravitational attraction that their nearby matter, even light, cannot escape from their gravitational field. Although, the black holes cannot be observed directly but there are some different ways to detect them in the binary systems as well as at the centers of their host galaxies. The most promising way for this detection is the accretion process. In the language of astrophysics, the accretion is defined as the inward flow of captured matter fields by a gravitating object towards its centre which leads to an increase of the mass and angular momentum of the accreting body. The observation of supermassive black holes at the center of galaxies represents that such massive black holes could have been gradually developed through the appropriate accretion processes. However, the accretion processes do not always increase the mass of the accreting bodies but they can also decrease their mass and lead them to shrink. It is shown that the accretion of phantom energy can decrease the black hole area [34, 35, 36, 37, 38, 39]. For instance, in [34], it is shown that black holes will gradually vanish as the universe approaches to a cosmological big rip state. The shrink of the black hole area during the accretion of a potentially surrounding field is an interesting phenomena in the sense that it can be considered as an alternative for the black hole evaporation through the Hawking radiation or even as an auxiliary for speeding up the evaporation process. One physical explanation for diminishing the black hole mass through the accretion process is that the accreting particles of a phantom scalar field have a total negative energy [40]. Similar particles with negative energies are created through the Hawking radiation process as well as in the process of energy extraction from a black hole by the Penrose mechanism. Thus, the accretion process into the black holes is one of the most interesting research fields in relativistic astrophysics to answer how black holes affect their cosmological surrounding fields and what are the consequences or what are the influences of these surrounding fields on the features, dynamical behaviors and abundance of black holes [41, 42, 43, 44, 45, 46, 47, 48, 49]. See also [50] for the accretion of dark energy into black holes, and [38, 39, 51, 52] for the accretion into the charged black holes.
In the present work, following the approach of [53, 54] and [55, 56], we construct a dynamical solution for the classical description of the evaporatingaccreting Bonnor–Vaidya black holes in generic dynamical backgrounds. The organization of the paper is as follows. In Sect. 2, the surrounded Bonnor–Vaidya black hole solution and some of its general features are introduced. In Sects. 2.1–2.4, the special classes of this solution named as the Bonnor–Vaidya black hole surrounded by the dust, radiation, quintessence and cosmological constant fields, as well as their properties are studied in detail. Finally, the Sect. 3 is devoted to the summary and concluding remarks.
2 Surrounded evaporatingaccreting Bonnor–Vaidya black hole solution
In this section, we generalize our previous solution [55, 56] to the surrounded charged Bonnor–Vaidya black hole solution by following the approach of [53, 54]. There are two main motivations for us for doing this generalization. The first one is that the existence of the charge can drastically change the global structure of the original spacetime [57]. For instance, we know the Reissner–Nordström black hole has a very distinct causal structure relative to the Schwarzschild case such that it predicts infinite series of parallel universes. The second reason is that a charged black hole possesses a spacetime structure almost similar to a rotating one, the Kerr black hole. Regarding that the existing spherical symmetry in the charged case makes it more easily analyzable, then understanding the structure of a charged black hole may be a suitable ground to better understanding the structure of a more realistic rotating one.

The solution by setting \(f=f(u,r)\) and \(\rho _s=\rho _s(r)\)
These considerations lead to \(M=M(u)\), \(Q=Q(u)\) and \(N_s=constant\) in the metric function f(u, r) and \(\sigma (u,r)\ne 0\) for the energy density. Then, there is no dynamics in the surrounding field and consequently the accretion of the surrounding field by the black hole cannot happen. Indeed, this case represents an evaporating charged black hole solution in a static background. The radiating charged black holes in an empty background \((\rho _s=0)\) known as the original Bonnor–Vaidya solution [16], and in (anti)de Sitter space \((\rho _s = \rho _{\Lambda }=constant)\) are special subclasses of this solution [18, 58]. Some interesting features of these black holes can be found in [15, 19, 20, 59].

The solution by setting \(f=f(r)\) and \(\rho _s=\rho _s(r)\)
These considerations lead to \(M=constant\), \(Q=constant\) and \(N_s=constant\) in the metric function f(u, r) and consequently \(\sigma (u,r)= 0\) for the radiationaccretion density. This case represents a static charged back hole in a static background and consequently, there is no radiationaccretion. The Reissner–Nordström black hole as well as its generalization to (anti)de Sitter background are special subclasses of this solution. For a general background, not just the (anti)de Sitter background, it is interesting that for a constant mass and charge black hole in a static nonempty background, using the coordinate transformationone arrives at the solution of the Reissner–Nordström black hole surrounded by a surrounding field as$$\begin{aligned} du=dt+\frac{\epsilon dr}{1\frac{2M}{r}+\frac{Q^2}{r^2}\frac{N_s}{{r}^{{3\omega _s +1}} }}, \end{aligned}$$(24)This solution is a generalization of the Kiselev solution [53] to the charged case and its interesting properties are studied in [60, 61, 62, 63]. Then, the generalized Kiselev solution is a subclass of our general dynamical solution (22) in the stationary limit.$$\begin{aligned} ds^2= & {} \left( 1\frac{2M}{r}+\frac{Q^2}{r^2}\frac{N_{s}}{{r}^{{3\omega _s +1}} } \right) dt^2\nonumber \\&+\frac{dr^2}{1\frac{2M}{r}+\frac{Q^2}{r^2}\frac{N_{s}}{{r}^{{3\omega _s +1}} }}+r^2 d\Omega ^2. \end{aligned}$$(25)

For \(\omega _s<0\), the charge contribution is dominant near the black hole. For the far distances (\(r\gg \)), the black hole charge contribution falls down faster than the black hole mass and the surrounding field contributions, respectively, i.e \(\sigma _Q<\sigma _M<\sigma _{N_s}\). Then, at large distances the surrounding field contribution is dominant.

For \(0<\omega _s<1/3\), the charge contribution is dominant near the black hole. For the far distances (\(r\gg \)), the charge contributions falls down faster than the surrounding field and mass contributions, respectively, i.e \(\sigma _Q<\sigma _{N_s}<\sigma _M\). Then, at large distances the black hole mass contribution is dominant.

For \(\omega _s>1/3\), the surrounding field contribution is dominant near the black hole. For the far distances (\(r\gg \)), the the surrounding field contributions falls down faster than the charge and mass contributions, respectively, i.e \(\sigma _{N_s}<\sigma _Q<\sigma _M\). Then, at large distances the black hole mass contribution is dominant again.

The terms in the first line are exactly the same as that of the standard Schwarzschild black hole solution except the time dependance in the mass of the black hole. Here, the terms represent the Newtonian gravitational force, the repulsive centrifugal force and the relativistic correction of general relativity (which accounts for the perihelion advance of planets), respectively.
 The terms in the second line are new correction terms, in comparison to the standard Schwarzschild case, due to the charge of the central object. Here, the first term represents the Coulomb force while the second one represents a relativisticlike correction of GR through the coupling between the charge Q(u) and L angular momentum. These new correction terms may be small in general in comparison to their Schwarzschild counterparts. However, one can show that there are possibilities that these terms can be comparable or equal to them. Then, for finding the situations where these forces are comparable to the Newtonian gravitational force and the GR correction term in (36), we define the distances \(D_{q_{1}}\) and \(D_{q_{2}}\) corresponding to \(\big \frac{a_{q_1}}{a_{N}}\big \simeq 1\) and \(\big \frac{a_{q_2}}{a_{L}}\big \simeq 1\), respectively, in which \(a_N\), \(a_L\) are the Newtonian and the relativistic correction accelerations, respectively, and \(a_{q_1}\) and \(a_{q_2}\) are defined asAccordingly, we obtain the distances \(D_{q_{1}}\) and \(D_{q_{2}}\) corresponding to \(\big \frac{a_{q_1}}{a_{N}}\big \simeq 1\) and \(\big \frac{a_{q_2}}{a_{L}}\big \simeq 1\), respectively, as$$\begin{aligned} a_{q_1}=\frac{Q^2(u)}{r^3},\quad a_{q_2}=\frac{2Q^2(u)L^{2}}{r^5}. \end{aligned}$$(37)$$\begin{aligned} D_{q_{1}}=\frac{Q^2(u)}{M(u)},\quad D_{q_{2}}=\sqrt{\frac{2Q^2(u)}{3M(u)}}. \end{aligned}$$(38)
 In the third line, we have two new correction terms due to the presence of the surrounding field. Here, the first term is similar to that of Newtonian gravitational term and the second term is similar to the relativistic correction of GR through the coupling between the background filed parameter \(N_s(u)\) and angular momentum L. Then, we see that for the more realistic nonempty backgrounds, the geodesic equation of any object depends strictly not only on the mass of the central object of the system and the conserved angular momentum of the orbiting body, but also on the background field nature. Similar to the previous case, one can show that there are possibilities that the background correction terms can be comparable to their Schwarzschild counterparts. Thus, for this case, we define the distances \(D_{s_{1}}\) and \(D_{s_{2}}\) which correspond to \(\big \frac{a_{s_1}}{a_{N}}\big \simeq 1\) and \(\big \frac{a_{s_2}}{a_{L}}\big \simeq 1\), respectively, in which \(a_{s_1}\) and \(a_{s_2}\) areThen, we obtain the distances \(D_{s_{1}}\) and \(D_{s_{2}}\) as$$\begin{aligned} a_{s_1}=\frac{(3\omega _s+1)N(u)}{2r^{3\omega _s+2}},\quad a_{s_2}=\frac{3(\omega _s+1)N(u)L^{2}}{2r^{3\omega _s+4}}. \end{aligned}$$(39)$$\begin{aligned} D_{s_{1}}\simeq & {} \left( \frac{(3\omega _s +1)N_s(u)}{2M(u)}\right) ^{\frac{1}{3\omega _s}},\nonumber \\ D_{s_{2}}\simeq & {} \left( \frac{(\omega _s +1)N_s(u)}{2M(u)}\right) ^{\frac{1}{3\omega _s}}. \end{aligned}$$(40)
 The term in the fourth line is also a new nonNewtonian correction resulting from the dynamics of black hole and its surrounding field. It is associated with the radiationaccretion power of the black hole and its surrounding field.^{1} Calling this acceleration as the induced acceleration \(a_{i}\) by the dynamics, where the subscript i stands for “induced”, we havein which, following Lindquist, Schwartz and Misner [64], we define the generalized “total apparent flux” as \(\mathcal {A}_{F}=\epsilon \left( {\dot{M}}(u) \frac{Q(u){\dot{Q}}(u)}{r}+\frac{{\dot{N}}(u)}{2r^{3\omega _s}}\right) {\overset{*}{u}}^{2}={\mathfrak {L}}\frac{\mathcal {Q}}{r}+\frac{\mathfrak {N}}{2r^{3\omega _{s}}}\) where \(\mathfrak {L}, \mathcal {Q}\) and \(\mathfrak {N}\) are the apparent fluxes associated to the black hole mass, charge and its surrounding field, respectively. Using these definitions, we can rewrite (41) as$$\begin{aligned} a_{i}=\frac{1}{2}\epsilon {\dot{f}}\overset{*}{u}^{2} = \ \epsilon \left( \frac{{\dot{M}}(u)}{r}\frac{Q(u){\dot{Q}}(u)}{r^{2}} +\frac{{\dot{N}}(u)}{2r^{3\omega _s+1}}\right) \overset{*}{u}^{2}, \end{aligned}$$(41)This new correction term may also be small in general in comparison to the Newtonian term [64]. However, one can show that there are also possibilities that these two terms can be comparable. Then, we define the distance \(D_i\) which satisfies \(a_i\simeq a_N\), and it will be given by the solutions of the following equation for different values of M, \(\omega _s\) and apparent fluxes \(\mathfrak {L}, \mathcal {Q}\) and \(\mathfrak {N}\) as$$\begin{aligned} a_{i}=\frac{\mathfrak {L}}{r}+\frac{\mathcal {Q}}{r^{2}}\frac{\mathfrak {N}}{2r^{3\omega _s+1}}. \end{aligned}$$(42)It is hard to find the general solutions to this equation in terms of its generic parameters \(\mathfrak {L}, \mathcal {Q}, \mathfrak {N}, M\) and \(\omega _s\). However, we will show that there are possible solutions for the various backgrounds of dust, radiation, quintessence and cosmological constantlike fields for some particular ranges of the parameters.$$\begin{aligned} \mathfrak {L}D_i^{3\omega _s}\mathcal {Q}D_i^{3\omega 1} +\frac{1}{2}\mathfrak {N}\simeq MD_i^{3\omega _s1}. \end{aligned}$$(43)
In the following subsections, we consider the cosmological surrounding fields of dust, radiation, quintessence and cosmological constantlike fields as the special classes of the obtained general solution (22), and we will investigate some of their interesting features in more detail.
2.1 Evaporatingaccreting Bonnor–Vaidya black hole surrounded by the dust field

Regarding (45), for \({\dot{N}}_d(u)\ne \frac{2}{r} (Q(u){\dot{Q}}(u)r{\dot{M}}(u) )\), we find that the radiationaccretion density vanishes only for \(r_*\rightarrow \infty \). This means that for the emission case, the outgoing charged radiation can penetrate through the dust background so far from the black hole horizon and for the accretion case by the black hole, the black hole affects the so far surrounding dust field.

Regarding (47), for the case of constant rate for \({\dot{N}}_{d}(u)\), \({\dot{M}}(u)\) and \({\dot{Q}}(u)\), the distance \(r_{*}\) is fixed to a particular value. In general case where \({\dot{N}}_{d}(u)\) and \({\dot{M}}(u)\) and \({\dot{Q}}(u)\) have no constant rates, the \(r_*\) is a dynamical position with respect to the time coordinate u, i.e \(r_*=r_*(u)\).

Regarding (47), to have a particular distance at which the energy density \(\sigma (u,r_{*})\) is zero, the positivity of \(r_*(u)\) also requires that \(Q(u){\dot{Q}}(u)\) and \(2{\dot{M}}_{eff}=2{\dot{M}}(u)+{\dot{N}}_d(u)\) have the same signs. For the cases in which \(r_*(u)\) is not positive, the lack of a positive value radial coordinate is interpreted as follows: the total radiationaccretion density \(\sigma (u,r)\) never and nowhere vanishes.

Regarding (47), demanding that \({\dot{Q}}(u)\) and \({\dot{M}}(u)\) have the same signs for both of the radiation and accretion processes, the positivity condition of \(r_*(u)\) requires the condition \(2{\dot{M}}(u)\ge \,{\dot{N}}_d(u)\) when \({\dot{N}}_d (u)\) takes opposite sign.

In the case of \(r_*(u)\) being the positive radial distance, for the given radiationaccretion behaviors of the black hole and its surrounding dust field, i.e \({\dot{M}}(u)\), \({\dot{Q}}(u)\) and \({\dot{N}}_{d}(u)\), it is possible to find a distance at which we have no any radiationaccretion energy density contribution. In other words, it turns out that the rate of outgoing radiation energy density of the black hole is exactly balanced by the rate of ingoing absorption rate of surrounding field at the distance \(r_{*}\) and vice versa.

Regarding (47), for the case of \(Q(u){\dot{Q}}(u) \ll {\dot{M}}_{eff} \), we have \(r_*\rightarrow \infty \). Considering the unit charge gauge, for the extremal case \({\dot{Q}}(u)\approx {\dot{M}}(u)\), for \(r_*\rightarrow \infty \), we find that black hole evolves very slow relative to its background. Then, by satisfaction of these dynamical conditions to have \(r_*\rightarrow \infty \), the positive energy density condition is respected everywhere in the spacetime. In other cases, the positive energy density is respected in some regions while it is violated beyond those regions.

Another interesting situation happens when \({\dot{M}}_{eff}=0\), i.e \(2{\dot{M}}(u)=\ {\dot{N}}_d(u)\). In this case, regarding (45) and (47), the radiationaccretion density is only resulting from the charge contribution in the form of \(\sigma (u,r)= \ \epsilon \frac{2Q(u){\dot{Q}}(u)}{r^3}\) and consequently \(r_*\rightarrow \infty \). Also, in order to respect to the positive energy condition here, it is required that \(\epsilon \) and \(Q(u){\dot{Q}}(u)\) have opposite signs.

Regarding (45), for both of the cases of neutral black hole (\(Q(u)=0\)) and black hole with static charge (\({\dot{Q}}(u)=0\)), if \({\dot{M}}_{eff}\ne 0\), we have \(r_*\rightarrow \infty \).
General Bonnor–Vaidya BH and its dust SF parameters for \(\epsilon = \ 1\). ECM denotes external charged matter which may contribute to the accretion
\({\dot{M}}\)  \({\dot{Q}}\)  \({\dot{N}}_d\)  \(r_*\)  \(\sigma _{({r<r_*})}\)  \(\sigma _{({r=r_*})}\)  \(\sigma _{({r>r_*})}\)  Condition  Physical process  

I  +  +  +  +  +  0  −  No  Accretion of BHSF and ECM 
II  +  +  −  +  +  0  −  \(2{\dot{M}}>{\dot{N}}_d\)  Accretion of SFECM by BH 
III  +  +  −  −  +  +  +  \(2{\dot{M}}<{\dot{N}}_d\)  Accretion of SFECM by BH 
IV  +  0  +  \(\infty \)  −  −  −  No  Not physical 
V  +  0  −  \(\infty \)  −  −  −  \(2{\dot{M}}>{\dot{N}}_d\)  Not physical 
VI  +  0  −  \(\infty \)  +  +  +  \(2{\dot{M}}<{\dot{N}}_d\)  Accretion of SF by BH 
VII  −  −  −  +  −  0  +  No  Accretion/decay of SF by evaporating/vanishing BH 
VIII  −  −  +  +  −  0  +  \(2{\dot{M}}>{\dot{N}}_d\)  Absorbtion of BH’s radiation by SF 
VIIII  −  −  +  −  −  −  −  \(2{\dot{M}}<{\dot{N}}_d\)  Not physical 
X  −  0  −  \(\infty \)  +  +  +  No  Accretion/decay of SF by evaporating/vanishing BH 
XI  −  0  +  \(\infty \)  +  +  +  \(2{\dot{M}}>{\dot{N}}_d\)  Absorbtion of BH’s radiation by SF 
XII  −  0  +  \(\infty \)  −  −  −  \(2{\dot{M}}<{\dot{N}}_d\)  Not physical 
General Bonnor–Vaidya BH and its dust SF parameters for \(\epsilon =+ \ 1\)
\({\dot{M}}\)  \({\dot{Q}}\)  \({\dot{N}}_d\)  \(r_*\)  \(\sigma _{({r<r_*})}\)  \(\sigma _{({r=r_*})}\)  \(\sigma _{({r>r_*})}\)  Condition  Physical process  

I  +  +  +  +  −  0  +  No  Accretion of BHSF and ECM 
II  +  +  −  +  −  0  +  \(2{\dot{M}}>{\dot{N}}_d\)  Accretion of SFECM by BH 
III  +  +  −  −  −  −  −  \(2{\dot{M}}<{\dot{N}}_d\)  Not physical 
IV  +  0  +  \(\infty \)  +  +  +  No  Accretion of BH and SF 
V  +  0  −  \(\infty \)  +  +  +  \(2{\dot{M}}>{\dot{N}}_d\)  Accretion of SF by BH 
VI  +  0  −  \(\infty \)  −  −  −  \(2{\dot{M}}<{\dot{N}}_d\)  Not physical 
VII  −  −  −  +  +  0  −  No  Accretion/decay of SF by evaporating/vanishing BH 
VIII  −  −  +  +  +  0  −  \(2{\dot{M}}>{\dot{N}}_d\)  Absorbtion of BH’s radiation by SF 
VIIII  −  −  +  −  +  +  +  \(2{\dot{M}}<{\dot{N}}_d\)  Absorbtion of BH’s radiation by SF 
X  −  0  −  \(\infty \)  −  −  −  No  Not physical 
XI  −  0  +  \(\infty \)  −  −  −  \(2{\dot{M}}>{\dot{N}}_d\)  Not physical 
XII  −  0  +  \(\infty \)  +  +  +  \(2{\dot{M}}<{\dot{N}}_d\)  Absorbtion of BH’s radiation by SF 
Regarding Table 1, we see that for the cases I, II, VII and VIII, there are regions in spacetime that the positive energy condition is respected, while beyond these regions it is violated. The cases IV, V, VIIII and XII are not physical in the sense that the positive energy condition is violated in the whole spacetime. The cases III, VI and XI as well as X represent the situations that the positive energy condition is respected in the whole spacetime with and without a priory condition on the black hole and its surrounding dust field dynamics, respectively.
Regarding Table 2, we see that for the cases I, II, VII and VIII, there are regions in spacetime that the positive energy condition is respected, while beyond these regions it is violated. The cases III, VI, X and XI are not physical in the sense that the positive energy condition is violated in the whole spacetime. The cases IV as well as V, VIIII and XII represent the situations that the positive energy condition is respected in the whole spacetime without and with a priory condition on black hole and its surrounding dust filed dynamics, respectively.
Finally, considering the timelike geodesic equations, for this case, we have \(D_{s_1}=D_{s_2}\) and both the situations of \(\big \frac{a_{s_1}}{a_{N}}\big \simeq 1\) and \(\big \frac{a_{s_2}}{a_{L}}\big \simeq 1\) can be met for \(M(u)=\frac{N_d(u)}{2}\) in the whole spacetime. In the Fig. 4, we have plotted the possibility of being these particular situations for some typical ranges of M(u) and \(N_d(u)\) parameters. Then, regarding this figure, we realize the possibility of the equality of the Newtonian force as well as GR correction terms to the corresponding contributions of the dust background.
2.2 Evaporatingaccreting Bonnor–Vaidya black hole surrounded by the radiation field

Regarding (52), for \(\mathcal {{\dot{N}}}_{r}(u)\ne 2 ( r{\dot{M}}(u)Q(u){\dot{Q}}(u) )\), we find that the radiationaccretion density vanishes only for \(r_*\rightarrow \infty \). This means that for the emission case, the outgoing charged radiation can penetrate through the radiation background so far from the black hole horizon and for the accretion case by the black hole, the black hole affects the so far surrounding radiation.

Regarding (54), for the case of constant rate for \(\mathcal {{\dot{N}}}_{r}(u)\), \({\dot{M}}(u)\) and \({\dot{Q}}(u),\) the distance \(r_{*}\) is fixed to a particular value. In general case where \(\mathcal {{\dot{N}}}_{r}(u)\) and \({\dot{M}}(u)\) and \({\dot{Q}}(u)\) have no constant rates, the \(r_*\) is a dynamical position with respect to the time coordinate u, i.e \(r_*=r_*(u)\).

Regarding (54), for having a particular distance at which the density \(\sigma (u,r_{*})\) is zero, the positivity of \(r_*\) also requires that \({\dot{M}}(u)\) and \(2Q_{eff}(u){\dot{Q}}_{eff}(u)=2Q{\dot{Q}} (u)+ \mathcal {{\dot{N}}}_r(u)\) have the same signs. For the cases in which \(r_*\) is not positive, the lack of a positive value radial coordinate is interpreted as follows: the radiationaccretion density \(\sigma (u,r)\) never and nowhere vanishes.

Regarding (54), demanding that \({\dot{Q}}(u)\) and \({\dot{M}}(u)\) have the same signs for both of the radiation and accretion processes, the positivity condition of \(r_*(u)\) requires the condition \(2Q(u){\dot{Q}}(u)\ge \mathcal {{\dot{N}}}_r(u)\) when \(\mathcal {{\dot{N}}}_r(u)\) takes opposite sign.

In the case of \(r_*\) being the positive radial distance, for the given radiationaccretion behaviors of the black hole and its surrounding field, i.e \({\dot{M}}(u)\), \({\dot{Q}}(u)\) and \(\mathcal {{\dot{N}}}_r(u)\), it is possible to find a distance at which we have no any radiationaccretion energy density contribution. In other words, it turns out that the rate of outgoing radiation energy density of the black hole is exactly balanced by the rate of ingoing absorption rate of surrounding field at the distance \(r_{*}\) and vice versa.

Regarding (54), for the case of \( {\dot{M}}(u) \ll Q_{eff}(u){\dot{Q}}_{eff}(u)\), we have \(r_*\rightarrow \infty \). Considering the unit charge gauge, for the extremal case \({\dot{Q}}(u)\approx {\dot{M}}(u)\), we find that black hole evolves very slow relative to its radiation background. Then, by satisfaction of these dynamical conditions to have \(r_*\rightarrow \infty \), the positive energy density is respected everywhere in the spacetime. In other cases, the positive energy density will be respected in some regions, while it is violated beyond those regions.

Another interesting situation happens for two different cases as \(Q_{eff}(u)=0\) and \({\dot{Q}}_{eff}(u)=0\) corresponding to \(Q=\mathcal {N}_{r}=0\) and \(2Q(u){\dot{Q}}(u)=\mathcal {{\dot{N}}}_r(u)\), respectively. In these cases, regarding (52) and (54), the radiationaccretion density is only resulting from the black hole mass contribution in the form of \(\sigma (u,r)=\epsilon \frac{2{\dot{M}}(u)}{r^2}\) and consequently \(\epsilon \) must have the same sign as \({\dot{M}}(u)\) to have a positive energy density. In this case, the radiationaccretion density looks like the original neutral Vaidya solution in an empty space, while the black hole and its background here is completely different, and vanishes as \(r_*\rightarrow \infty \).
 Regarding (52), for both of the cases of neutral black hole (\(Q(u)=0\)) and black hole with static charge (\({\dot{Q}}(u)=0\)), we haveThen, the positivity of \(r_*\) demands that \(\mathcal {{\dot{N}}}_{r}(u)\) and \({\dot{M}}(u)\) have same signs, and for \(2{\dot{M}}(u)\ll \mathcal {{\dot{N}}}_{r}(u)\), we have \(r_*\rightarrow \infty \).$$\begin{aligned} r_{*}(u)=\frac{\mathcal {{\dot{N}}}_{r}(u)}{2{\dot{M}}(u)}. \end{aligned}$$(55)
General Bonnor–Vaidya BH and its radiation SF parameters for \(\epsilon = \ 1\)
\({\dot{M}}\)  \({\dot{Q}}\)  \(\mathcal {{\dot{N}}}_r\)  \(r_*\)  \(\sigma _{({r<r_*})}\)  \(\sigma _{({r=r_*})}\)  \(\sigma _{({r>r_*})}\)  Condition  Physical process  

I  +  +  +  +  +  0  −  No  Accretion of BHSF and ECM 
II  +  +  −  +  +  0  −  \(2Q(u){\dot{Q}}(u)\ge \mathcal {{\dot{N}}}_r(u)\)  Accretion of SFECM by BH 
III  +  +  −  −  −  −  −  \(2Q(u){\dot{Q}}(u)\le \mathcal {{\dot{N}}}_r(u)\)  Not physical 
IV  +  0  +  +  +  0  −  No  Accretion of BH and SF 
V  +  0  −  −  −  −  −  No  Not physical 
VI  −  −  −  +  −  0  +  No  Accretion/decay of SF by evaporating/vanishing BH 
VII  −  −  +  +  −  0  +  \(2Q(u){\dot{Q}}(u)\ge \mathcal {{\dot{N}}}_r(u)\)  Absorbtion of BH’s radiation by SF 
VIII  −  −  +  −  +  +  +  \(2Q(u){\dot{Q}}(u)\le \mathcal {{\dot{N}}}_r(u)\)  Absorbtion of BH’s radiation by SF 
VIIII  −  0  −  +  −  0  +  No  Accretion/decay of SF by evaporating/vanishing BH 
X  −  0  +  −  +  +  +  No  Absorbtion of BH’s radiation by SF 
General Bonnor–Vaidya BH and its radiation SF parameters for \(\epsilon =+\ 1\)
\({\dot{M}}\)  \({\dot{Q}}\)  \(\mathcal {{\dot{N}}}_r\)  \(r_*\)  \(\sigma _{({r<r_*})}\)  \(\sigma _{({r=r_*})}\)  \(\sigma _{({r>r_*})}\)  Condition  Physical process  

I  +  +  +  +  −  0  +  No  Accretion of BHSF and ECM 
II  +  +  −  +  −  0  +  \(2Q(u){\dot{Q}}(u)\ge \mathcal {{\dot{N}}}_r(u)\)  Accretion of SFECM by BH 
III  +  +  −  −  +  +  +  \(2Q(u){\dot{Q}}(u)\le \mathcal {{\dot{N}}}_r(u)\)  Accretion of SFECM by BH 
IV  +  0  +  \(+\)  −  0  +  No  Accretion of BH and SF 
V  +  0  −  −  +  +  +  No  Accretion of SF by BH 
VI  −  −  −  +  +  0  −  No  Accretion/decay of SF by evaporating/vanishing BH 
VII  −  −  +  +  +  0  −  \(2Q(u){\dot{Q}}(u)\ge \mathcal {{\dot{N}}}_r(u)\)  Absorbtion of BH’s radiation by SF 
VIII  −  −  +  −  −  −  −  \(2Q(u){\dot{Q}}(u)\le \mathcal {{\dot{N}}}_r(u)\)  Not physical 
VIIII  −  0  −  \(+\)  +  0  −  No  Accretion/decay of SF by evaporating/vanishing BH 
X  −  0  +  −  −  −  −  No  Not physical 
Regarding Table 1, we see that for the cases I, II, IV, VI, VII and VIIII, there are regions in spacetime where the positive energy condition is respected, while beyond these regions it is violated. The cases IV, V, VIIII and XII are not physical in the sense that the positive energy condition is violated in the whole spacetime. The cases VIII and X represent the situations where the positive energy condition is respected in the whole spacetime with and without a priory condition on the black hole and its surrounding radiation field dynamics, respectively.
Regarding Table 2, we see that for the cases I, II, IV, VI, VII and VIIII, there are regions in spacetime where the positive energy condition is respected, while beyond these regions it is violated. The cases VIII, X are not physical in the sense that the positive energy condition is violated in the whole spacetime. The cases IIII as well as V represent the situations where the positive energy condition is respected in the whole spacetime with and without a priory condition on the black hole and its surrounding radiation filed dynamics, respectively.
Regrading the conditions in the Tables 3 and 4 for \(\epsilon =\ 1 \) and \(\epsilon =+\ 1\), the behaviour of radiationaccretion density \(\sigma (u,r)\) in (52) is plotted for some typical values of \({\dot{M}}(u)\), \({\dot{Q}}(u)\) and \( {\dot{N}}_{d}(u)\) in the Figs. 6 and 7, respectively. Using these plots, one can compare the radiationaccretion density values for the various situations.
2.3 Evaporatingaccreting Bonnor–Vaidya black hole surrounded by the quintessence field
In the context of cosmology, the quintessence filed is known as the simplest scalar field dark energy model free of the theoretical problems such as Laplacian instabilities or ghosts. The energy density and the pressure profile of the quintessence field are generally supposed as time varying quantities and depend on the scalar field and its associated potential given by \(\rho = \frac{1}{2}{{\dot{\phi }}}^2 + V(\phi )\) and \(p = \frac{1}{2}{\dot{\phi }}^2 V(\phi )\), respectively. Thus, the corresponding quintessence equation of state parameter lies in the range \(\ 1<\omega _q<\ \frac{1}{3}\). The static Schwarzschild black hole solution surrounded by a quintessence field was first introduced by Kiselev [53]. Then, this solution was generalized to the Reissner–Nordström case and investigated in [60, 61, 62].
General Bonnor–Vaidya BH and its quintessence SF parameters for \(\epsilon = \ 1\)
\({\dot{M}}\)  \({\dot{Q}}\)  \({\dot{N}}_q\)  \(r_*\)  \(\sigma _{({r<r_*})}\)  \(\sigma _{({r=r_*})}\)  \(\sigma _{({r>r_*})}\)  Condition  Physical process  

I  +  +  −  +  +  0  +  No  Accretion of SFECM by BH 
II  +  0  +  Imaginary  −  −  −  No  Not physical 
III  +  0  −  +  −  0  +  No  Accretion of SF by BH 
IV  −  −  +  +  −  0  −  No  Not physical 
V  −  0  +  +  +  0  −  No  Absorbtion of BH’s radiation by SF 
VI  −  0  −  Imaginary  +  +  +  No  Accretion/decay of SF by evaporating/vanishing BH 
General Bonnor–Vaidya BH and its quintessence SF parameters for \(\epsilon =+ \ 1\)
\({\dot{M}}\)  \({\dot{Q}}\)  \({\dot{N}}_q\)  \(r_*\)  \(\sigma _{({r<r_*})}\)  \(\sigma _{({r=r_*})}\)  \(\sigma _{({r>r_*})}\)  Condition  Physical process  

I  +  +  −  +  −  0  −  No  Not physical 
II  +  0  +  Imaginary  +  +  +  No  Accretion of BH and SF 
III  +  0  −  +  +  0  −  No  Accretion of SF by BH 
IV  −  −  +  +  +  0  +  No  Absorbtion of BH’s radiation by SF 
V  −  0  +  +  −  0  +  No  Absorbtion of BH’s radiation by SF 
VI  −  0  −  Imaginary  −  −  −  No  Not physical 

Regarding (60), for \({\dot{N}}_{q}(u)\ne \frac{2}{{r}^{{3}}} (Q(u){\dot{Q}}(u) r{\dot{M}}(u) )\), we find that in contrast to the cases of the Bonnor–Vaidya black hole surrounded by the dust and radiation fields, here the radiationaccretion density does not vanish at \(r_*\rightarrow \infty \). This is due to the fact that the spacetime here has the quintessence asymptotic rather than an empty Minkowski.

Regarding (62), for the case of constant rates for \({\dot{N}}_{q}(u)\), \({\dot{M}}(u)\) and \({\dot{Q}}(u)\), the distance \(r_{*}\) is fixed to a particular value. In general case which \({\dot{N}}_{q}(u)\) and \({\dot{M}}(u)\) and \({\dot{Q}}(u)\) have no constant rates, the \(r_*\) has a dynamical position with respect to the time coordinate u, i.e \(r_*=r_*(u)\).

Regarding (62), to have a particular distance at which the density \(\sigma (u,r_{*})\) is zero, the positivity of \(r_*\) also requires that \({\dot{N}}_q (u)\) takes an opposite sign of \({\dot{M}}(u)\) (and \({\dot{Q}}(u)\)), see (78). This is in agreement with our primary consideration for the signs of dynamical parameters (\({\dot{N}}_q (u)\), \({\dot{M}}(u)\) and \({\dot{Q}}(u)\)) for the radiation and accretion processes in the previous sections. For the cases in which \(r_*\) is not positive, the lack of a positive value radial coordinate is interpreted as follows: the radiationaccretion density \(\sigma (u,r)\) never and nowhere vanishes.

In the case of \(r_*\) being the positive radial distance, for the given radiationaccretion behaviors of the black hole and its surrounding field, i.e \({\dot{M}}(u)\), \({\dot{Q}}(u)\) and \({\dot{N}}_q(u)\), it is possible to find a distance at which we have no any radiationaccretion energy density contribution. In other words, it turns out that the rate of outgoing radiation energy density of the black hole is exactly balanced by the rate of ingoing absorption rate of surrounding field at the distance \(r_{*}\) and vice versa.

Regarding (62) and (78), for both of the black holes with \({\dot{M}}(u)\ll Q(u){\dot{Q}}(u)\) and \({\dot{M}}(u)\rightarrow 0\), we have \(r_*\rightarrow \infty \) and \({\dot{N}}_q(u)\rightarrow 0\). This means that for a black which is almost active only due to its dynamical charge, one can find that (i) there is a nonzero radiation density even at far distance from the black hole and (ii) positive energy condition is respected everywhere.
 Regarding (60), for both of the cases of neutral black hole (\(Q(u)=0\)) and black hole with static charge (\({\dot{Q}}(u)=0\)), we haveThen, for \( {\dot{N}}_{q}(u) \ll 2{\dot{M}}(u)\), we have \(r_*\rightarrow \infty \). This means that for an almost static background (the background with negligible dynamics relative to the black hole mass), the zero of the radiationaccretion density lies at infinity and the positive energy density is respected everywhere in the spacetime. In other cases, one can find a finite value for \(r_*\) representing the zero radiationaccretion density in which the positive energy density will be respected in some regions while it is violated beyond those regions, see our previous work on the neutral black hole case for more details [55, 56].$$\begin{aligned} r_{*}(u)=\left( \frac{2{\dot{M}}(u)}{{\dot{N}}_{q}(u)}\right) ^{\frac{1}{2}}. \end{aligned}$$(64)

Regarding both the solutions (62) and (64), the signs of \({\dot{M}}(u)\) and \({\dot{N}}_q (u)\) should be opposite to have a zero radiationaccretion density for both of the radiationaccretion processes.
Regarding Table 5, we see that for the cases III and V, there are regions is spacetime that the positive energy condition is respected, while beyond these regions it is violated. The case II is not physical in the sense that the positive energy condition is violated in the whole spacetime. The case IV is also not physical in the sense that the positive energy condition is violated in whole spacetime except at the zero density point. The cases I and VI represent the situations that the positive energy condition is respected in the whole spacetime without a priory condition on the black hole and its surrounding quintessence filed dynamics.
2.4 Evaporatingaccreting Bonnor–Vaidya black hole surrounded by the cosmological field
The positive energy density condition on the surrounding cosmological field, represented by the relation (21), requires \(N_c(u)\geqslant 0\). Then, in this case, \(N_{c}(u)\) plays the role of a positive dynamical cosmological field. This case may represents the dynamical black holes in more general cosmological models proposing a time varying cosmological term. The main purpose of these cosmological scenarios is to provide an explanation for the recent observed accelerating expansion of the universe, see [66, 67, 68, 69, 70, 71, 72] as some instances. For the case of \(N_c=constant=\Lambda >0\), we recover the Bonnor–Vaidya black hole embedded in a de Sitter space obtained by Patino and Rago [18]. The solution in [18] was generalized to the case of the rotating radiating charged black hole in a static de Sitter space in [25]. In [73], the causal structure of the solution obtained in [25] is studied.

Regarding (68), for \({\dot{N}}_{c}(u)\ne \frac{2}{{r}^{{4}}} (Q(u){\dot{Q}}(u) r{\dot{M}}(u) )\), we find that in contrast to the cases of Bonnor–Vaidya black hole surrounded by dust and radiation fields, the radiationaccretion density does not vanish for \(r_*\rightarrow \infty \). This is due to the fact that here the spacetime has the de Sitter asymptotic rather than an empty Minkowski.

Regarding (70), for the case of constant rates for \({\dot{N}}_{c}(u)\), \({\dot{M}}(u)\) and \({\dot{Q}}(u)\), the distance \(r_{*}\) is fixed to a particular value. In general case which \({\dot{N}}_{c}(u)\) and \({\dot{M}}(u)\) and \({\dot{Q}}(u)\) have no constant rates, the \(r_*\) has a dynamical position with respect to the time coordinate u, i.e \(r_*=r_*(u)\).

Regarding (70) and (84), to have a particular distance at which the density \(\sigma (u,r_{*})\) is zero, the positivity of \(r_*\) also requires that \({\dot{N}}_c(u)\) takes an opposite sign of \(Q{\dot{Q}} (u)\) (and \({\dot{M}}(u)\)). This is in agreement with our primary consideration for the signs of dynamical parameters (\({\dot{N}}_c (u)\), \({\dot{M}}(u)\) and \({\dot{Q}}(u)\)) for the radiation and accretion processes in the previous sections. Similarly, for the cases in which \(r_*\) is not positive, the lack of a positive value radial coordinate is interpreted as follows: the radiationaccretion density \(\sigma (u,r)\) never and nowhere vanishes.

In the case of \(r_*\) being a real and positive radial distance, for the given radiationaccretion behaviors of the black hole and its surrounding field, i.e \({\dot{M}}(u)\), \({\dot{Q}}(u)\) and \({\dot{N}}_c(u)\), it is possible to find a distance at which we have no any radiationaccretion energy density contribution. In other words, it turns out that the rate of outgoing radiation energy density of the black hole is exactly balanced by the rate of ingoing absorption rate of surrounding cosmological field at the distance \(r_{*}\) and vice versa.

Regarding (70) and (84), for both of the black holes with \({\dot{M}}(u)\ll Q(u){\dot{Q}}(u)\) and \({\dot{M}}(u)\rightarrow 0\), we have \(r_*\rightarrow \infty \) and \({\dot{N}}_c(u)\rightarrow 0\). This means that for a black which is almost active only due to its dynamical charge, one can find (i) a nonzero radiation density even at far distance from the black hole and (ii) the respected positive energy condition at everywhere.
 Regarding (70) and (84), working in the unit charge gauge, for the extremal case \({\dot{Q}}(u)\approx {\dot{M}}(u)\), we findThen, regarding (63) and (71), comparing the quintessence and cosmological background fields for the extremal case, we have$$\begin{aligned} r_*\rightarrow \frac{4}{3},\quad {\dot{N}}_c(u)\rightarrow \frac{27}{128} {\dot{M}}(u). \end{aligned}$$(71)$$\begin{aligned} r_{*c}<r_{*q},\quad {\dot{N}}_c(u)<{\dot{N}}_q(u). \end{aligned}$$(72)
 Regarding (68), for both of the cases of neutral black hole (\(Q(u) = 0\)) and black hole with static charge (\({\dot{Q}}(u) = 0\)), we haveThen, for \( {\dot{N}}_{c}(u) \ll 2{\dot{M}}(u)\), we have \(r_*\rightarrow \infty \). This means that for an almost static background (the background with negligible dynamics relative to the black hole), the zero of the radiationaccretion density lies at infinity and the positive energy density is respected everywhere in the spacetime. In other cases, one can find a finite value for \(r_*\) representing the total zero accretion density in which the positive energy density will be respected in some regions while violated beyond those regions, see our previous work on the neutral black hole case for more details [55, 56].$$\begin{aligned} r_{*}(u)=\left( \frac{2{\dot{M}}(u)}{{\dot{N}}_{c}(u)}\right) ^{\frac{1}{3}}. \end{aligned}$$(73)

Regarding both the solutions (70) and (73), the signs of \({\dot{M}}(u)\) and \({\dot{N}}_c (u)\) should be opposite to have a zero radiationaccretion density for both of the radiationaccretion processes.

Regarding (64) and (73), in the case that the quintessence and cosmological backgrounds have a same behavior (\({\dot{N}}_{q}(u)={\dot{N}}_{c}(u)\)), for \( {\dot{N}}_{c,q}(u) < 2{\dot{M}}(u)\), we have \(r_{*c}(u)<r_{*q}(u)\) while for \({\dot{N}}_{c,q}(u) >2{\dot{M}}(u)\), we have \(r_{*c}(u)>r_{*q}(u)\).

Regarding (64) and (73) for both of the cases of neutral black hole (\(Q(u) = 0\)) and black hole with static charge (\({\dot{Q}}(u) = 0\)), when \({\dot{M}}(u)\) and \({\dot{N}}_q(u)\) as well as \({\dot{N}}_c(u)\) have the same signs, \(r_*(u)\) is imaginary and negative for the quintessence and cosmological fields, respectively. Then, for these cases, the radiationaccretion density never be zero and the positive energy condition is respected or violated in the whole spacetime.
General Bonnor–Vaidya BH and its cosmological SF parameters for \(\epsilon = \ 1\)
\({\dot{M}}\)  \({\dot{Q}}\)  \({\dot{N}}_c\)  \(r_*\)  \(\sigma _{({r<r_*})}\)  \(\sigma _{({r=r_*})}\)  \(\sigma _{({r>r_*})}\)  Condition  Physical process  

I  +  +  −  +  +  0  +  No  Accretion of SFECM by BH 
II  +  0  +  −  −  −  −  No  Not physical 
III  +  0  −  +  −  0  +  No  Accretion of SF by BH 
IV  −  −  +  +  −  0  −  No  Not physical 
V  −  0  +  +  +  0  −  No  Absorbtion of BH’s radiation by SF 
VI  −  0  −  −  +  +  +  No  Accretion/decay of SF by evaporating/vanishing BH 
General Bonnor–Vaidya BH and its cosmological SF parameters for \(\epsilon =+ \ 1\)
\({\dot{M}}\)  \({\dot{Q}}\)  \({\dot{N}}_c\)  \(r_*\)  \(\sigma _{({r<r_*})}\)  \(\sigma _{({r=r_*})}\)  \(\sigma _{({r>r_*})}\)  Condition  Physical process  

I  +  +  −  +  −  0  −  No  Not physical 
II  +  0  +  −  +  +  +  No  Accretion of BH and SF 
III  +  0  −  +  +  0  −  No  Accretion of SF by BH 
IV  −  −  +  +  +  0  +  No  Absorbtion of BH’s radiation by SF 
V  −  0  +  +  −  0  +  No  Absorbtion of BH’s radiation by SF 
VI  −  0  −  −  −  −  −  No  Not physical 
Regarding Table 7, we see that for the cases III and V, there are regions in spacetime that the positive energy condition is respected, while beyond these regions it is violated. The case II is not physical in the sense that the positive energy condition is violated in the whole spacetime. The case IV is also not physical in the sense that the positive energy condition is violated in the whole spacetime except at the zero density point. The cases I and VI represent the situations that the positive energy condition is respected in the whole spacetime without a priory condition on the black hole and its surrounding cosmological filed dynamics.
Regarding Table 8, we see that for the cases III and V, there are regions in spacetime that the positive energy condition is respected, while beyond these regions it is violated. The cases I and VI are not physical in the sense that the positive energy condition is violated in the whole spacetime. The cases II and IV represent the situations that the positive energy condition is respected in the whole spacetime without a priory condition on the black hole and its surrounding cosmological filed dynamics.
3 Summary and concluding remarks

For \(\omega _s<0\), the charge contribution is dominant near the black hole while for the far distances, we have \(\sigma _Q<\sigma _M<\sigma _{N_s}\), meaning that the surrounding field contribution is dominant at large distances.

For \(0<\omega _s<1/3\), the charge contribution is dominant near the black hole while for the far distances, we have \(\sigma _Q<\sigma _{N_s}<\sigma _M\), meaning that the black hole mass contribution is dominant at large distances.

For \(\omega _s>1/3\), the surrounding field contribution is dominant near the black hole while for the far distances, we have \(\sigma _{N_s}<\sigma _Q<\sigma _M\) meaning that the black hole mass contribution is dominant again at large distances.

For the dust background, we have the Bonnor–Vaidya black hole with the effective dynamical mass \(2M_{eff}= 2M(u)+N_d(u)\). To have a particular distance \(r_*\) at which \(\sigma (u,r_{*})=0\), the condition \(2{\dot{M}}(u)\ge \,{\dot{N}}_d(u)\) is required. For the case of \(Q(u){\dot{Q}}(u) \ll {\dot{M}}_{eff} \), we have \(r_*\rightarrow \infty \). In the unit charge gauge, for the extremal case (\({\dot{Q}}(u)\approx {\dot{M}}(u)\)), for \(r_*\rightarrow \infty \), we find that black hole evolves very slow relative to its dust background. Then, by satisfaction of these dynamical conditions to have \(r_*\rightarrow \infty \), the positive energy density condition is respected everywhere in the spacetime. In other cases, the positive energy density is respected in some regions while it is violated beyond those regions. Another interesting situation happens when \({\dot{M}}_{eff}=0\). In this case, the radiationaccretion density is only resulting from the charge contribution with \(r_*\rightarrow \infty \).

For the radiation background, we have the Bonnor–Vaidya black hole with the effective dynamical charge \(Q_{eff}(u)=\sqrt{Q^{2}(u)+\mathcal {N}_r(u)}\). To have a particular distance \(r_*\) at which \(\sigma (u,r_{*})=0\), the condition \(2Q(u){\dot{Q}}(u)\ge \mathcal {{\dot{N}}}_r(u)\) is required. For the case of \( {\dot{M}}(u) \ll Q_{eff}(u){\dot{Q}}_{eff}(u)\), we have \(r_*\rightarrow \infty \). In the unit charge gauge, for the extremal case (\({\dot{Q}}(u)\approx {\dot{M}}(u)\)), we find that black hole evolves very slow relative to its radiation background. Then, by satisfaction of these dynamical conditions to have \(r_*\rightarrow \infty \), the positive energy density is respected everywhere in the spacetime. In other cases, the positive energy density will be respected in some regions, while it is violated beyond those regions. Another interesting situation happens for two different cases as \(Q_{eff}(u)=0\) and \({\dot{Q}}_{eff}(u)=0\) corresponding to \(Q=\mathcal {N}_{r}=0\) and \(2Q(u){\dot{Q}}(u)= \ \mathcal {{\dot{N}}}_r(u)\), respectively. In these cases, the radiationaccretion density is only resulting from the black hole mass contribution. For both of the cases of neutral black hole (\(Q(u)=0\)) and black hole with static charge (\({\dot{Q}}(u)=0\)), for \( 2{\dot{M}}(u) \ll \mathcal {{\dot{N}}}_{r}(u) \), we have \(r_*\rightarrow \infty \).

For the quintessence background, the positive energy density condition demands \(N_q(u)\geqslant 0\). To have a particular distance \(r_*\) at which \(\sigma (u,r_{*})=0\), the positivity of \(r_*\) requires that \({\dot{N}}_q (u)\) takes an opposite sign of \({\dot{M}}(u)\) (and \({\dot{Q}}(u)\)). For both of the black holes with \({\dot{M}}(u)\ll Q(u){\dot{Q}}(u)\) and \({\dot{M}}(u)\rightarrow 0\), we have \(r_*\rightarrow \infty \) and \({\dot{N}}_q(u)\rightarrow 0\). This means that for a black which is almost active only due to its dynamical charge, one can find that (i) there is a nonzero radiation density even at far distance from the black hole and (ii) positive energy condition is respected everywhere. Here, in the unit charge gauge, for the extremal case (\({\dot{Q}}(u)\approx {\dot{M}}(u)\)), we find \(r_*\rightarrow \frac{3}{2}\) and \({\dot{N}}_q(u)\rightarrow \frac{8}{27} {\dot{M}}(u)\). For both of the cases of neutral black hole (\(Q(u)=0\)) and black hole with static charge (\({\dot{Q}}(u)=0\)), for \( {\dot{N}}_{q}(u) \ll 2{\dot{M}}(u)\), we have \(r_*\rightarrow \infty \). This means that for an almost static background (the background with negligible dynamics relative to the black hole mass), the zero of the radiationaccretion density lies at infinity and the positive energy density is respected everywhere in the spacetime.

For the cosmological background, the positive energy density condition demands \(N_c(u)\geqslant 0\) representing (dynamical) de Sitter space. To have a particular distance \(r_*\) at which \(\sigma (u,r_{*})=0\), the positivity of \(r_*\) requires that \({\dot{N}}_c(u)\) with respect to \(Q{\dot{Q}} (u)\) (and \({\dot{M}}(u)\)) takes an opposite sign. For both of the black holes with \({\dot{M}}(u)\ll Q(u){\dot{Q}}(u)\) and \({\dot{M}}(u)\rightarrow 0\), we have \(r_*\rightarrow \infty \) and \({\dot{N}}_c(u)\rightarrow 0\). This means that for a black which is almost active only due to its dynamical charge, one can find (i) a nonzero radiation density even at far distance from the black hole and (ii) the respected positive energy condition at everywhere. In the unit charge gauge, for the extremal case (\({\dot{Q}}(u)\approx {\dot{M}}(u)\)), we find \(r_*\rightarrow \frac{4}{3}\) and \({\dot{N}}_c(u)\rightarrow \frac{27}{128} {\dot{M}}(u)\). Then, comparing the quintessence and cosmological background fields for the extremal case, we have \(r_{*c}<r_{*q}\) and \({\dot{N}}_c(u)<{\dot{N}}_q(u)\). Also, for both of the cases of neutral black hole (\(Q(u) = 0\)) and black hole with static charge (\({\dot{Q}}(u) = 0\)), for \( {\dot{N}}_{c}(u) \ll 2{\dot{M}}(u)\), we have \(r_*\rightarrow \infty \). This means that for an almost static background (the background with negligible dynamics relative to the black hole), the zero of the radiationaccretion density lies at infinity and the positive energy density is respected everywhere in the spacetime. In the case that the quintessence and cosmological backgrounds have a same behavior (\({\dot{N}}_{q}(u)={\dot{N}}_{c}(u)\)), for \( {\dot{N}}_{c,q}(u) < 2{\dot{M}}(u)\), we have \(r_{*c}(u)<r_{*q}(u)\) while for \({\dot{N}}_{c,q}(u) >2{\dot{M}}(u)\), we have \(r_{*c}(u)>r_{*q}(u)\). Finally, for both of the cases of neutral black hole (\(Q(u) = 0\)) and black hole with static charge (\({\dot{Q}}(u) = 0\)), when \({\dot{M}}(u)\) and \({\dot{N}}_q(u)\) as well as \({\dot{N}}_c(u)\) have the same signs, \(r_*(u)\) is imaginary and negative for the quintessence and cosmological fields, respectively. Then, for these cases, the radiationaccretion density never be zero and the positive energy condition is respected or violated in the whole spacetime.
In our next work, we aim to report elsewhere on the causal structures, horizon and thermodynamical properties of our obtained solutions.
Footnotes
Notes
Acknowledgements
This work has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project no. 1/523717.
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