Surrounded Vaidya Solution by Cosmological Fields

In the present work, we study the general surrounded Vaidya solution by the various cosmological fields and its nature describing the possibility of the formation of naked singularities or black holes. Motivated by the fact that real astrophysical black holes as non-stationary and non-isolated objects are living in non-empty backgrounds, we focus on the black hole subclasses of this general solution describing a dynamical evaporating-accreting black holes in the dynamical cosmological backgrounds of dust, radiation, quintessence, cosmological constant-like and phantom fields, the so called surrounded Vaidya black hole. Then, we analyze the timelike geodesics associated with the obtained surrounded black holes and we find that some new correction terms arise relative to the case of Schwarzschild black hole. Also, we address some of the subclasses of the obtained surrounded black hole solution for both dynamical and stationary limits. Moreover, we classify the obtained solutions according to their behaviors under imposing the positive energy condition and discuss how this condition imposes some severe and important restrictions on the black hole and its background field dynamics.


Introduction
The Vaidya solution [1,2] is one of the non-static solutions of the Einstein field equations and can be regarded as a generalization of the static Schwarzschild black hole solution. This solution describes a spherical symmetric object possessing an outgoing null radiation. This null radiation may be interpreted in different contexts such as high frequency electromagnetic or gravitational waves, massless scalar particles or neutrinos. The Vaidya solution is characterized by a mass function m depending on the retarded time coordinate u, i.e m = m(u). In this regard, various shells of null matter fields can be constructed such that are bounded either by flat (m = 0) or Schwarzschild-like (m = constant = 0) vacuum regions. This property allows us to implement such solutions for studying the process of spherical symmetric gravitational collapse and as a testing ground for the cosmic censorship conjecture [3,4,5,6], as a model for describing the evaporating black hole solutions as well as studying the Hawking radiation [7,8,9,10,11,12,13], among the other applications. This solution was generalized to the charged case known as the Bonnor-Vaidya solution [14], see also its application for example in [15,16,17,18,19]. Also, a generalisation of the Vaidya solution is introduced in [5]. This generalisation is based on the fact that the total supporting energy-momentum tensor of spacetime, constructed from type I and type II energy-momentum tensors [20], is linear in term of the mass function. Consequently, any linear superposition of particular solutions to the Einstein field equations will also a solution. Then, using this approach, we can construct more general solutions such as the Bonnor-Vaidya [14], Vaidya-de Sitter [21], radiating dyon solution [22], Bonnor-Vaidya-de Sitter [23,24,25,26,27] and the Husain solution [28].
In general, black holes possess strong gravitational pull such that their nearby matter, even light, cannot escape from their gravitational field. Because the black holes cannot be observed directly, there are some different ways to detect them in binary systems as well as at the centers of their host galaxies. The most promising way is the accretion process. In the language of astrophysics, the accretion is defined as the inward flow of matter fields surrounding a compact object, such as black holes and neutron stars, due to the gravitational attraction. Then, the process of accretion into black holes is one of the most interesting research fields in relativistic astrophysics [29,30,31,32,33]. This process may be described by a perfect fluid coupled to general relativity representing a plasma which obeys the equations of ideal or resistive magnetohydrodynamics or a fluid coupled to radiation. Such accretion processes along with their detailed physical descriptions, can be found in [34] and references therein, see also [35,36,37,38,39,40,41,42]. On the other hand, there are also other kind of accretion processes related to the black holes surrounded by exotic matter fields as potential models of dark energy, whose existence and features are motivated by the problems in the standard model of cosmology. A number of theoretical and observational studies confirmed that our universe in its early stages experienced an inflation process while it is undergoing an accelerated expansion in the late time. In order to explain these events, an energy component, known as the dark energy, is required to be introduced to the framework of general theory of relativity. The cosmological constant is a leading candidate for dark energy while there are other proposals including the dynamical scalar fields such as quintessence and phantom fields. Then, by the presence of such fields around the black holes, one may have interest to explore some interesting facts such as whether black holes have hair or scalar wigs [43], how black holes affect these cosmological surrounding fields and what are the consequences or what are the influences of these surrounding fields on the features, behaviors and abundance of black holes. In this regard, one may find the reference [44] as a good review including various scenarios of accretion process into black holes, see also [38,45] for charged black hole accretion. Among the all of the accretion processes, the most interesting one are related to those that the accretion of the surrounding fields enforcing a black hole to shrink. These surrounding field include the scalar fields or fluid violating the weak energy condition, i.e ρ > 0 & ρ + p > 0 [44]. Specific scenarios involving the accretion of phantom energy have shown that the black-hole area decreases with the accretion [46,47,48,49]. For example, in [46], it is shown that black holes will gradually vanish as the universe approaches a cosmological big rip state. The big rip scenario for a cosmos occurs when its filling dark energy is the phantom energy with p < −ρ. In this scenario, the cosmological phantom field disrupts finally all bounded objects of the universe up to sub-nuclear scales. For the test-field approximation, one may find the accretion process of a scalar field violating the energy conditions leading the decrease in the black holes area in [48,50]. Moreover, the shrink of the black hole area through the accretion of a phantom scalar field has been confirmed in full nonlinear general relativity [51,52]. In this regard, the shrink of the black-hole area by the accretion of a potentially surrounding field is an interesting phenomena in the sense that it can be an alternative process for black-hole evaporation through the Hawking radiation or even be an auxiliary for speeding up it. One physical explanation for a black hole mass diminishing may be is that accreting particles of a phantom scalar field have a total negative energy [53]. Similar particles possessing negative energies are created through the Hawking radiation process and also in the energy extraction process from a black hole by the Penrose mechanism. The effect of phantom-like dark energy onto a charged Reissner-Nordström black hole is studied in [54] and it is found that accretion is possible only through the outer horizon. On the other hand, for scalar fields regarding the energy conditions, there is a possibility indicating that the accretion of a scalar field can be partial such that the amount of accreted scalar field depends on features of the incident wave packet, i.e. the wave number and the width of the packet. This has been studied both in the test-field approximation [55] and in full general relativity [51,52]. In this line, some studies in the test-field limit indicate that a scalar field can also be sustained by a black hole without being accreted [56].
In the present work, following the approach of [57,58] introduced for the static black holes, we wish to find a dynamical solution for the classical description of the evaporating-accreting black holes in generic dynamical backgrounds. The organization of the paper is as follows. In section 2, we introduce the general surrounded Vaidya black hole solution. Then, in subsections 2.1 to 2.5, we give the special classes of this solution as the surrounded Vaidya black hole by the dust, radiation, quintessence, cosmological constant and phantom fields, respectively. In section 3, we give our conclusion with introducing two of the our underwork researches in this line.

The General Surrounded Evaporating-Accreting Vaidya Black Hole Solution
In this section, we are looking for the general surrounded Vaidya black hole solutions by the approach of [57,58]. Then, we consider the general spherical symmetric spacetime metric in the form of where dΩ 2 = dθ 2 + sin 2 θdφ 2 is the two dimensional unit sphere and f (u, r) is a generic metric function depending on both of the advanced/retarded time coordinate u and the radial coordinate r. The cases, ǫ = −1 and ǫ = +1 represent the outgoing and ingoing flows corresponding to the effectively evaporating and accreting Vaidya black hole solutions, respectively. Using the metric (1), we obtain nonvanishing components of the Einstein tensor as where dot and prime signs represent the derivatives with respect to the time coordinate u and the radial coordinate r, respectively. Then, the total energy-momentum supporting this spacetime should have the following non-diagonal form where also must obey the symmetries in Einstein tensor G µ ν . With respect to the field equations in (2), the equalities G 0 0 = G 1 1 and G 2 2 = G 3 3 require T 0 0 = T 1 1 and T 2 2 = T 3 3 , respectively. Then, for the nature of the black hole evaporation-accretion in the presence of a dynamical background, one can consider a total energy-momentum tensor supporting the Einstein field equations in the following form where τ µ ν is the energy-momentum tensor associated to the Vaidya null radiation-accretion as such that σ = σ(u, r) is the measure of the energy flux or the energy density of the outgoing radiationingoing accretion flow [59] and k µ = δ 0 µ is a null vector field while T µ ν is the energy-momentum tensor of the surrounding perfect fluid defined as in [57] T 0 0 = −ρ s (u, r), where subscript "s" stands for the surrounding field which can be a dust, radiation, quintessence, cosmological constant, phantom field or even any complex field constructed by the combination of these fields. This form of energy-momentum for the surrounding fluid is implying that the spatial profile of the black hole surrounding energy-momentum tensor is proportional to the time component, describing the dynamical energy density ρ s (u, r), with the arbitrary constant parameters α and β depending the internal structure of the surrounding fields. The isotropic averaging over the angles results in [57] < since we considered < r i r j >= 1 3 δ i j r n r n . Then, we have the barotropic equation of state for the surrounding field as p s (u, r) = ω s ρ s (u, r), ω s = 1 3 α, where p s (u, r) and ω s are the dynamical pressure and the constant equation of state parameter of the surrounding field, respectively. Thus, regarding the Einstein tensor components in (2) and the total energymomentum tensor given by the equations (3)-(6), we have T 0 0 = T 1 1 and T 2 2 = T 3 3 . These exactly provide the so called principle of additivity and linearity considered in [57] in order to determine the free parameter β of the energy momentum-tensor T µ ν of the surrounding field as Then, by substituting α and β parameters in (8) and (9) into (6), the non-vanishing components of the surrounding energy-momentum tensor T µ ν will be , Now, by having the Einstein tensor components and the corresponding total energy-momentum tensor T µ ν , one can obtain the associated differential equations. Then, the G 0 0 = T 0 0 and G 1 1 = T 1 1 components of the Einstein field equations give the following differential equation Similarly, the G 1 0 = T 1 0 component leads to and G 2 2 = T 2 2 and G 3 3 = T 3 3 components read as Thus, we see that there are three unknown dynamical functions f (u, r), σ(u, r) and ρ s (u, r) which can be determined analytically by the above three differential equations. Simultaneous solving the differential equations (11) and (13), one obtains the following solution for the metric function with the energy density ρ s (u, r) of the surrounding field in the form of where M (u) and N s (u) are integration coefficients representing the black hole dynamical mass and the surrounding dynamical field structure parameter, respectively. On the other hand, respecting to the weak energy condition imposing the positivity of any kind of energy density of the surrounding field, i.e ρ s ≥ 0, demands This implies that for the surrounding field with a positive equation of state parameter ω s , it is needed to have N s (u) ≤ 0 and conversely for a negative ω s , it is required to have N s (u) ≥ 0. This condition fixes the nature of the term associated to surrounding field in the metric function f (u, r).
Regarding the metric function (14), the spacetime metric (1) reads as representing an effectively evaporating-accreting black hole in a dynamical background. One may realize the following distinct subclasses of this solution as • The solution by setting f = f (u, r) and ρ s = ρ s (r) in the field equations (11) to (13).
These considerations lead to M = M (u) and N s = constant in the metric function f (u, r) and σ = 0 for the black hole's radiation density. In this case, there is no dynamics in the surrounding field and consequently there is no accretion to the black hole. Indeed, this case represents an evaporating black hole solution with ǫ = −1 in a static background. The evaporating black hole in an empty background, i.e ρ s = 0 [1], and (anti)-de Sitter space, i.e ρ s = ρ Λ = constant, are special subclasses of this solution [21,60,61]. Some interesting physical features of these solutions can be found in the references [7,8,19,62,63,64,65,66,67,68,69,70,71,72].
• The solution by setting f = f (r) and ρ s = ρ s (r) in the field equations (11) to (13).
These considerations lead to M = constant, N s = constant in the metric function and σ = 0 for the radiation-accretion density. This case represents a non-dynamical back hole in a static background and consequently, there are no accretion and evaporation. The Schwarzschild black hole as well as its generalization to (anti)-de Sitter background are subclasses of this solution. For a general background, not just the (anti)-de Sitter background, it is interesting that using the coordinate transformation one can obtain the general static solution of the Schwarzschild black hole surrounded by a surrounding field as which was found by Kiselev [57]. Then, the Kiselev solution is a subclass of our general dynamical solution (17) in the stationary limit.
• The solution for ǫ = +1 with changing the background field parameters as ω s → 1 3 (2k − 1) and N s (u) → − 2g(u) 2k−1 . By this considerations, we recover the Husain solution describing a null fluid collapse [28] as with the energy density This solution is widely studied in the literature, see for instances [73,74,75]. Substituting the metric function (14) in the equation (12) gives the radiation-accretion density of the effectively evaporating-accreting black hole as where the first and second terms in RHS are the radiation-accretion density corresponding to the mass change of the black hole and the dynamics of the surrounding field, respectively. This shows that for construction of a realistic effectively evaporating-accreting black hole model, one needs to implement such a solution including a dynamical black hole in a dynamical background described by the energy-momentum (10). Considering (22), and turning off the background dynamics byṄ s (u) = 0, we recover the energy flux associated to the mass change of the central black hole corresponding to the original Vaidya solution [1], see also [59] for more discussion. Regarding (22), it is seen that ifṀ (u) andṄ s (u) have a same order of magnitude, then for the case of ω s > 0, the surrounding background contribution to the total density σ(u, r) is dominant near the black hole while at far distances from the black hole it decreases faster than the contribution of the black hole mass changing term. In contrast, for ω s < 0, the surrounding background contribution is dominant at large distances while the black hole contribution is dominant near the black hole itself. Considering the positive energy density condition (by the weak energy condition) on the total radiation-accretion density σ(u, r) in (22) requires This inequality confines the dynamical behaviours of the black hole and its background field at any time and distance (u, r). In the case of a static background, as in the Vaidya's original solution ( [1]), it is required that ǫ andṀ (u) have the same signs to have positive energy density. This shows that for a radiating black hole withṀ (u) < 0 we have ǫ = −1 which represents the outgoing null flow, while for an accreting black hole it is required to have ǫ = +1, representing the ingoing null flow. In the presence of the background dynamics, it is not mandatory that ǫ andṀ (u) take the same signs and the satisfaction of the positive energy density condition can be achieved even by their opposite signs depending on the background field parametersṄ s (u) and ω s . Based on the relation (23), the dynamical behaviour of the background field is governed by Then, at any distance r from the black hole, the background field must obey the above conditions. Interestingly, for the special case ofṄ s (u) = −2 r 3ωsṀ (u), there is no pure radiation-accretion density, i.e σ(u, r) = 0. This case corresponds to two possible physical situations. The first one is related to the situation where for any particular distance r 0 , the backgroundṄ (u) and black holeṀ (u) behave such that their contributions cancel out each others leading to σ(u, r 0 ) = 0. The second situation is related to the case that for the given dynamical behaviors of the black hole and its background, one can always find the particular 3ωs possessing zero energy density σ(u, r * (u)). For the case of constant rates ofṄ s (u) andṀ (u), the distance r * is fixed to a particular value. To have a particular distance at which the density σ(u, r * ) is zero, the positivity of r * also requires thatṀ (u) andṄ s (u) have opposite signs. For the cases in which r * is not positive, the lack of a positive value radial coordinate is interpreted as follows: the radiation-accretion density σ(u, r) never and nowhere vanishes.
In the case of positive radial coordinate r * > 0, for the given radiation-accretion behaviors of the black hole and its surrounding field, i.eṀ (u) andṄ s (u), it is possible to find a distance at which we have no any radiation-accretion 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. Beyond or within this particular distance, the various general situations can be realized in the Tables 1 and 2 for the black hole (BH) and its surrounding field (SF).
Accretion/Decay of SF by Evaporating/Vanishing BH Table 1: General BH and SF parameters for ǫ = −1.
Then, regarding these tables and Eq.(23), we observe the following points.
• The cases possessing negative values of r * (the cases I, IV, V and VIII) mean that the radiationaccretion density does not vanish somewhere and forever. Among these cases, the ones which have positive σ(u, r) are only physical, i.e the cases IV and VIII for ǫ = −1, and I and V for ǫ = +1.
• The remaining positive values of r * , corresponding to a zero radiation-accretion density, are physically viable and their corresponding physical processes are listed in the last column. These properties are determined according to the behaviours of the parameters ǫ, ω s , and quantitiesṀ (u),Ṅ s (u), and σ(u, r). Those values of r * corresponding to the negative energy density σ(u, r) represent no physical situation about the evaporation-absorption or accretion. The real features of those regions are hidden by the weak energy condition.
• For ω s > − 2 3 , the particular distance r * , where σ(u, r) vanishes, corresponds to two possible cases as r * (u) = −Ṅ s(u) 2Ṁ(u) 1 3ωs and r * = ∞. In the first case, for − 2 3 < ω s < 0 with |Ṅ s (u)| ≪ |Ṁ (u)| and for ω s ≥ 0 with |Ṁ (u)| ≪ |Ṅ s (u)|, we have r * → ∞. This means that the first situation indicates that black hole evolves very faster than its background while the second indicates that black hole evolves very slow relative to its background. By satisfaction of these dynamical conditions to hold r * → ∞, the positive energy density is respected everywhere in the spacetime. In other cases, the positive energy density will be respected in some regions while violated beyond those regions.
• For ω s ≤ − 2 3 , the particular distance r * is given as r * (u) = −Ṅ s(u) 2Ṁ (u) 1 3ωs . Then, for a rapidly evolving black hole relative to its background, i.e |Ṅ s (u)| ≪ |Ṁ (u)|, we have r * → ∞. This case implies an evolving black hole in an almost static background in which the positive energy condition is respected everywhere in this spacetime.
In the following subsections, the metric of the Vaidya radiating-accreting black hole surrounded by the dust, radiation, quintessence, cosmological constant and phantom fields, as the special classes of the general solution (17), as well as their interesting features are studied in more detail.

Radiating-Accreting Vaidya Black Hole Surrounded by the Dust Field
For the dust surrounding field, we set ω d = 0 [57,77]. Then, the metric (17) takes the following form It is seen that the effectively radiating-accreting black hole in the dust background appears as an effectively radiating-accreting black hole with an effective mass M ef f (u) = 2M (u) + N d (u). In this case, the presence of new mass term changes the thermodynamics, causal structure and Penrose diagrams just up to a re-scaling in the original Vaidya solution.
The radiation-accretion density in the dust background is given by For the Vaidya's original solution in an empty background, i.e N d (u) = 0, or even in a static background, i.ė N d (u) = 0, the positive energy density condition, i.e σ(u, r) ≥ 0, requires that ǫ andṀ (u) always have the same signs. This means that for ǫ = +1, M (u) is a monotone increasing mass function while for the case of ǫ = −1, M (u) is a monotone decreasing mass function. In our general solution for the Vaidya black hole in the dust background, the condition σ(u, r) ≥ 0 imposed on (26) is satisfied for more general situations indicated in the Table 3.
Absorbtion of BH's radiation by SF +1 + + No Condition Accretion of BH and SF Table 3: BH and its surrounding dust field parameters for ǫ = ±1. For these cases, the positive energy condition is satisfied everywhere in spacetime.
Interestingly, for the special case ofṄ d (u) = −2Ṁ(u), there is no pure radiation-accretion density, i.e σ(u, r) = 0, and the energy-momentum tensor (4) will be diagonalized. This means that the black hole and its surrounding background completely cancel out the effects of each others. ForṄ d (u) = −2Ṁ(u), regarding (26), we find that for r * → ∞, the radiation-accretion density vanishes, i.e σ(u, r) → 0. This means that for the effective emission case, the out going radiation can penetrate through the dust background so far from the black hole and for the effective accretion case by the black hole, the black hole affect its so far surrounding objects. Regrading the conditions in the Table 3 for ǫ = −1 and ǫ = +1, the behaviour of radiation-accretion density σ in (26) is plotted for some typical values ofṀ andṄ d in the Figures 1 and  2, respectively. Using these plots, one can compare the radiation-accretion density values for the various situations.

Evaporating-Accreting Vaidya Black Hole Surrounded by the Radiation Field
For the radiation surrounding field, we set ω r = 1 3 [57,77]. Then, the metric (17) takes the following form Regarding the positive energy condition on the surrounding radiation field, represented by the relation (16), it is required that N r (u) 0. Then, by defining the positive parameter N r (u) = −N r (u), we have This metric looks like a radiating charged Vaidya black hole with the dynamical charge Q(u) = N r (u). This result can be interpreted as the positive contribution of the characteristic feature of the surrounding radiation field to the effective charge term of the Vaidya black hole with the 1 r 2 gravitational contribution.   In this case, the total radiation-accretion density is given by Then, we see that there is no positive r * (u) forṀ (u) andṄ r (u) having opposite signs, and consequently σ(u, r) never vanishes except at infinity. But as r * → ∞, the radiation-accretion density again vanishes, i.e σ(u, r * ) → 0. This means that for the emission case, the out going radiation can penetrate through the radiation background so far from the black hole and for the accretion case by the black hole, the black hole affects its so far surrounding radiation filed. The positivity condition of σ(u, r) is satisfied everywhere for the situations present in the Table 4.
ǫṀṄ r Physical Process -1 -+ Absorbtion of BH's radiation by SF +1 + -Accretion of SF by BH Table 4: BH and its surrounding radiation field parameters for ǫ = ±1. For these cases, the positive energy condition is satisfied everywhere in spacetime. For any other behaviour of theṀ (u) andṄ r (u) parameters, the positive energy condition will be violated.
Regrading the Table 4, the behaviour of radiation-accretion density σ in (29) is plotted for some typical values ofṀ andṄ r in Figures 3 and 4. Using these plots, one can compare the radiation-accretion densities for the various situations. Figure 4: The radiation-accretion density σ versus the distance r for some typical constantṀ andṄ r values for ǫ = +1 in the radiation background. Here, σ(r) is a decreasing function but is positive, and consequently the positive energy condition is satisfied everywhere in spacetime.

Evaporating-Accreting Vaidya Black Hole Surrounded by the Quintessence Field
In the cosmological context, the quintessence filed is known as the simplest scalar field dark energy model without having theoretical problems such as Laplacian instabilities or ghosts. The energy density and the pressure profile of the quintessence filed are generally considered to vary with time and depend on the scalar field and the potential, which are given by ρ = 1 2φ 2 + V (φ) and p = 1 2φ 2 − V (φ), respectively. Then, the associated equation of state parameter for quintessence field lies in the range −1 < ω q < − 1 3 . The static Schwarzschild black hole solution surrounded by a quintessence field was found by Kiselev [57]. This solution was generlaiazed to the charged case and studied in [88,89,90].
For the quintessence surrounding field, we set ω q = − 2 3 [57,77]. Then, the metric (17) takes the following form This result shows a non-trivial contribution of the characteristic feature of the surrounding quintessence field to the metric of the Vaidya black hole. The presence of the background quintessence filed changes the causal structure and Penrose diagrams of this black hole solution in comparison to the black hole in an empty background. A rather similar effect happens when one immerses an static Schwarzschild in a (anti)-de Sitter background with the difference that here the spacetime tends asymptotically to quintessence rather than (anti)-de Sitter asymptotics. Regarding the positive energy condition for the quintessence background, represented by the relation (16), it is required to have N q (u) 0. The radiation density is given by Then, the dynamical behaviour of the background quintessence field is governed by Consequently, at any distance r from the black hole, the surrounding quintessence field must obey the above conditions. Interestingly, for the special case ofṄ q (u) = − 2Ṁ(u) r 2 , there is no pure radiation-accretion density, i.e σ(u, r) = 0. This case corresponds to two possible physical situations. The first one is related to the situation where observer can be located at any distance r such that the quintessence background's and black hole's contributions cancel out each others leading to σ(u, r) = 0 for a moment or even a period of time. Then, it is required that for an evaporating black hole, we have an equal absorbing quintessence background or for an accreting black hole we have an equal accreted quintessence background. The second situation is related to the case that for the given dynamical behaviors of the black hole and its quintessence background, one can find the particular distance r * = − 2Ṁ(u) Nq(u) possessing zero energy density. For |Ṅ q (u)| ≪ |Ṁ (u)|, we have r * → ∞. This indicates that for an evolving black hole in an almost static quintessence background, the positive energy condition is satisfied everywhere. Also, the positivity of r * also requires thatṀ (u) anḋ N q (u) have opposite signs. Then, if one realize the black and its surrounding quintessence filed behaviors, i.ė M (u) andṄ q (u) values, he can find a distance at which we have no any radiation-accretion energy density contribution. Based on these possibilities, the various situations in the Table 5 can be realized.  Table 5: BH and its surrounding quintessence field parameters for ǫ = ±1. For the quintessence background, the positive energy condition may be completely or partially respected regarding to the above situations.
Then, regarding this table, the positive values of r * are physically viable and their corresponding physical processes are listed in the last column. These properties are determined according to the behaviours of the parameters ǫ, ω q , and quantitiesṀ (u),Ṅ q (u), and σ(u, r). Those values of r * corresponding to the negative energy density σ(u, r) represent no physical situation about the evaporation-absorption or accretion. The real features of those regions are hidden by the weak energy condition. In the reference [76], the accretion into a static Kiselev black hole with a static exterior spacetime surrounded by a quintessence field without the back-reaction effect is studied. The obtained results in [76] are implying that the accretion rate and the critical points depend on the background quintessence parameter N q . Then, these features deserve to be incorporated in astrophysical studies of the accretion processes. Regrading the Table 5, the behaviour of radiation-accretion density σ in (31) is plotted for some typical values ofṀ andṄ q in Figures 5 and 6. Using these plots, one can compare the radiation-accretion densities for the various situations.

Evaporating-Accreting Vaidya Black Hole Surrounded by the Cosmological Constant
For the cosmological constant surrounding field, we set ω c = −1 [57,77]. Then, the metric (17) takes the following form This result indicates the non-trivial contribution of the characteristic feature of the surrounding cosmological constant to the metric of the Vaidya black hole. The presence of the background cosmological constant changes the causal structure and Penrose diagrams of this black hole solution in comparison to the black hole in an empty background. This is similar to the case of the static Schwarzschild black hole in a static de Sitter background such that the Penrose diagram changes from Schwarzschild to Schwarzschild-(anti) de Sitter. Then, in our case, the Penrose diagram changes from Vaidya to Vaidya-de Sitter case with dynamical cosmological causal boundaries.  Regarding the positive energy condition for this case, represented by the relation (16), it is required to have N c (u) 0. In this case, N c (u) plays the role of a positive dynamical cosmological constant. Then, this case may describes the dynamical black holes in more general cosmological scenarios considering a time varying cosmological term, which have been recently proposed in the literature. The main purpose of these cosmological models is to provide an explanation for the recent accelerating phase of the universe [78]- [87]. These models are well known as the Λ(t), where t is the cosmic time. For the case of N c = constant = Λ, we recover the solution of the Vaidya black hole embedded in a de Sitter space obtained in [21]. The evolutionary behaviour of such an evaporating black hole including the structures, locations and dynamics of the apparent and event horizons are studied in [60].
In this case, the radiation-accretion density is given by Then, the dynamical behaviour of the background cosmological constant field is governed by Consequently, at any distance r from the black hole, the surrounding cosmological field must obey the above conditions. Similar to the previous solution, for the special case ofṄ c (u) = − 2Ṁ(u) r 3 , there is no pure radiation-accretion density, i.e σ(u, r) = 0. This case corresponds to two possible physical situations. The first one is related to the situation where observer can be located at any distance r such that the cosmological background's and black hole's contributions cancel out each others leading to σ(u, r) = 0 for a moment or even a period of time. Then, it is required that for a radiating black hole, we have an equal absorbing cosmological background or for an accreting black hole we have an equal accreted cosmological background. The second situation is related to the case that for the given dynamical behaviors of the black hole and its cosmological background, one can find the particular distance r * = − 2Ṁ(u) Nc(u) 1 3 possessing zero energy density. For |Ṅ c (u)| ≪ |Ṁ (u)|, we have r * → ∞. This indicates that for an evolving black hole in an almost static cosmological background, the positive energy condition is respected everywhere. Here also, the positivity of r * also guarantees thatṀ (u) andṄ c (u) have opposite signs. Then, if one realize the black and its surrounding cosmological filed behaviors, i.eṀ (u) andṄ c (u) values, he can find a distance at which we have no any radiation-accretion energy density contribution. Based on these possibilities, the various situations in the Table 6 can be realized.  Table 6: BH and its surrounding cosmological field parameters for ǫ = ±1. For the cosmological background, the positive energy condition may be completely or partially respected regarding to the above situations.
Regrading the Table 6, the behaviour of radiation-accretion density σ in (34) is plotted for some typical values ofṀ andṄ c in Figures 7 and 8. Using these plots, one can compare the radiation-accretion densities for the various situations.

Evaporating-Accreting Vaidya Black Hole Surrounded by the Phantom Field
For the phantom surrounding field, we set ω p = − 4 3 [77]. Then, the metric (17) takes the following form Similarly, this result is interpreted as the non-trivial contribution of the characteristic feature of the surrounding phantom field to the metric of the Vaidya black hole. The presence of the background phantom filed changes the causal structure and Penrose diagrams of this black hole solution in comparison to the Vaidya black hole in an empty background.
Regarding the weak energy condition for this case, represented by the relation (16), it is required to have N p (u) 0. In this case, the radiation-accretion density is given by Then, the dynamical behaviour of the background field is governed by Consequently, at any distance r from the black hole, the surrounding phantom field must obey the above conditions. Similar to the previous solutions, for the special case ofṄ p (u) = − 2Ṁ (u) r 4 , there is no pure radiation-accretion density, i.e σ(u, r) = 0. This case corresponds to two possible physical situations. The first one is related to the situation where observer can be located at any distance r such that the phantom background's and black hole's contributions cancel out each others leading to σ(u, r) = 0 for a moment or even a period of time. Then, it is required that for a radiating black hole, we have an equal absorbing phantom background or for an accreting black hole we have an equal accreted phantom background. The second situation is related to the case that for the given dynamical behaviors of the black hole and its phantom background, one can find the particular distance r * = − 2Ṁ(u) Np(u) Similarly, for |Ṅ p (u)| ≪ |Ṁ (u)|, we have r * → ∞. This indicates that for an evolving black hole in an almost static phantom background, the positive energy condition is satisfied everywhere. Also, the positivity of r * also requires thatṀ (u) andṄ p (u) must have opposite signs. Then, if one realize the black and its surrounding phantom filed behaviors, i.eṀ (u) andṄ p (u) values, he can find a distance at which we have no any radiation-accretion energy density contribution. Based on these possibilities, the various situations in the Table 7 can be realized.  Table 7: BH and its surrounding phantom field parameters for ǫ = ±1. For the phantom background, the positive energy condition may be completely or partially respected regarding to the above situations. Specific scenarios involving the accretion of phantom energy and resulting in the area decrease of black hole [46,47,48,49] are related to the first case in the above table. For example, in [46], it is shown that the black holes will gradually vanish as the universe approaches a cosmological big rip state with a phantom field. Regrading the Table 7, the behavior of radiation-accretion density σ in (37) is plotted for some typical values ofṀ andṄ p in Figures 9 and 10. Using these plots, one can compare the radiation-accretion densities for the various situations.

Conclusion
In the present work, following the approach of [57,58] for static black holes, we have constructed a solution describing the evaporating-accreting black holes in a generic dynamical background, named as the surrounded Vaidya black hole. We have shown that the original Vaidya solution can be recovered by turning off the background field, and that the Kiselev static solution can be obtained as another subclass of our general solution in the stationary limit. Also, we have demonstrated that the Husain solution can be recovered by appropriate changing of the background field paprameters here. Then, we have studied the positive energy condition on the radiation-accretion density. This condition gives some severe restrictions on the background field dynamics. Based on this condition, we have found a particular distance at which there is no effective radiation-accretion density contribution. Moreover, we have found that for the background fields having ω s > − 2 3 , the particular distance r * , where σ(u, r) vanishes, corresponds to two possible cases as r * (u) = −Ṅ s (u) 2Ṁ (u) 1 3ωs and r * = ∞. In the first case, for − 2 3 < ω s < 0 with |Ṅ s (u)| ≪ |Ṁ (u)| and for ω s ≥ 0 with |Ṁ (u)| ≪ |Ṅ s (u)|, we have r * → ∞. This means that in the first situation the black hole evolves very faster than its background while in the second situation the black hole evolves very slow relative to its background. By satisfaction of these dynamical conditions to hold at r * → ∞, the positive energy density is respected everywhere in the spacetime. The surrounded Vaidya black hole by the dust and radiation fields are two instances for this possibility. For the background fields possessing ω s ≤ − 2 3 , the particular distance r * is given by r * (u) = −Ṅ s (u) 2Ṁ (u) 1 3ωs . Then, for a rapidly evolving black hole relative to its background, i.e |Ṅ s (u)| ≪ |Ṁ (u)|, we have r * → ∞. This case implies an evolving black hole in an almost static background in which the positive energy condition is respected everywhere in spacetime. One may consider the evaporating-accreting black holes in an almost constant cosmological backgrounds (quintessence, cosmological constant or phantom fields) responsible for the accelerating expansion of the universe as the instances for this case. In the cases which these conditions are not met, the positive energy condition is Figure 10: The radiation-accretion density σ versus the distance r for some typical constantṀ andṄ p values for ǫ = +1 in the phantom background. In the upper panel, the accretion density is a decreasing function from positive to negative values. In the lower panel, the radiation density is an increasing function from negative to positive values. Then, for a dynamical phantom background, if the condition |Ṅ p (u)| ≪ |Ṁ (u)| is not met, the positive energy condition is violated in some regions of spacetime.
violated in some regions of spacetime. We also investigated the cases in which the positive energy condition is violated in some regions of spacetime for the Vaidya black hole surrounded by quintessence, cosmological constant and phantom fields in terms of the parameters ǫ = ±1, ω s ,Ṁ (u) andṄ s (u).
To better understanding the solutions, we also have given some plots for the radiation-accretion density σ versus the distance r for some typical constant values ofṀ andṄ s for ǫ = ± in the different backgrounds. Comparing the plots with common values of the parameters, we observe that the radiation-accretion density σ(r) for the radiation background is larger than the dust background at any distance r, i.e σ r (r) > σ d (r). Similarly, for the quintessence, cosmological constant and phantom backgrounds, we have σ q (r) < σ c (r) < σ p (r) for the radiation-accretion density.
We aim to report elsewhere on the causal structures and thermodynamical properties of our obtained solutions and the generalization of this work to the charged case.