Spherically symmetric solutions in a FRW background
aa r X i v : . [ g r- q c ] M a y Spherically symmetric solutions in a FRW background
H. Moradpour ∗ and N. Riazi † Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran, Physics Department, Shahid Beheshti University, Evin, Tehran 19839, Iran.
We impose perfect fluid concept along with slow expansion approximation to derive new solutionswhich, considering non-static spherically symmetric metrics, can be treated as Black Holes. We willrefer to these solutions as Quasi Black Holes. Mathematical and physical features such as Killingvectors, singularities, and mass have been studied. Their horizons and thermodynamic propertieshave also been investigated. In addition, relationship with other related works (including mcVittie’s)are described.
I. INTRODUCTION
The Universe expansion can be modeled by the so called FRW metric ds = − dt + a ( t ) [ dr (1 − kr ) + r dθ + r sin ( θ ) dφ ] , (1)where k = 0 , +1 , − k = 0) universe [1]. a ( t ) is the scale factor and for a background which is filled by aperfect fluid with equation of state p = ωρ , there are three classes of expanding solutions. These three solutions are a ( t ) = a t ω +1) (2)for ω = 0 when − < ω and , a ( t ) = a e Ht (3)for ω = − ω < −
1) is a ( t ) = a ( t − t ) ω +1) , (4)where t is the big rip singularity time and will be available, if the universe is in the phantom regime.In Eq. (3), H ( ≡ ˙ a ( t ) a ( t ) ) is the Hubble parameter and the current estimates are H = 73 +4 − kms − M pc − [1].Note that, at the end of the Phantom regime, everything will decompose into its fundamental constituents [2]. Inaddition, this spacetime can be classified as a subgroup of the Godel-type spacetime with σ = m = 0 and k ′ = 1 [3].A signal which was emitted at the time t by a co-moving source and absorbed by a co-moving observer at a latertime t is affected by a redshift ( z ) as 1 + z = a ( t ) a ( t ) . (5)The apparent horizon as a marginally trapped surface, is defined as [4] g µν ∂ µ ξ∂ ν ξ = 0 , (6)which for the physical radius of ξ = a ( t ) r , the solution will be: ξ = 1 q H + ka ( t ) . (7) ∗ [email protected] † n [email protected] The surface gravity of the apparent horizon can be evaluated by: κ = 12 √− h ∂ a ( √− hh ab ∂ b ξ ) . (8)Where the two dimensional induced metric is h ab = diag ( − , a ( t )(1 − kr ) ). It was shown that the first law of thermo-dynamics is satisfied on the apparent horizon [5–8]. The special case of ω = − ds = − (1 − H r ) dt + dr (1 − H r ) + r d Ω . (9)This metric belongs to a more general class of spherically symmetric, static metrics. For these class of sphericallysymmetric static metrics, the line element can be written in the form of: ds = − f ( r ) dt + dr f ( r ) + r d Ω , (10)where the general form of f ( r ) is: f ( r ) = 1 − mr + Q r − H r . (11)In the above expression, m and Q represent mass and charge, respectively. For this metric, one can evaluate redshift:1 + z = ( 1 − mr + Q r − H r − mr + Q r − H r ) . (12)Where, r and r are radial coordinates at the emission and the absorption points. For the horizons, the radius andthe surface gravity can be found using equations g tt = f ( r ) = 0 −→ r h (13) κ = f ′ ( r )2 | r h , where ( ′ ) denotes derivative with respect to the coordinate r [9]. From the thermodynamic laws of Black Holes (BHs)we know T = κ π , (14)which T is the temperature on the horizon [9]. Validity of the first law of the thermodynamics on the static horizonsfor the static spherically symmetric spacetime has been shown [10, 11].The BHs with the FRW dynamic background has motivated many investigations. The first approach, which isnamed Swiss Cheese, includes efforts in order to find the effects of the expansion of the Universe on the gravitationalfield of the stars [12], introduced originally by Einstein and Straus (1945) [13]. In these models, authors tried tojoin the Schwarzschild metric to the FRW metric by satisfying the junction conditions on the boundary, which isan expanding timelike hypersurface. The inner spacetime is described by the Schwarzschild metric, while the FRWmetric explains the outer spacetime. These models don’t contain dynamical BHs, Because the inner spacetime is inthe Schwarzschild coordinate, hence, is static [14]. In addition, the Swiss Cheese models can be classified as a subclassof inhomogeneous Lemabitre-Tolman-Bondi models [15, 16].Looking for dynamical BHs, some authors used the conformal transformation of the Schwarzschild BH, wherethe conformal factor is the scale factor of the famous FRW model. Originally, Thakurta (1981) have used thistechnique and obtained a dynamical version of the Schwarzschild BH [17]. Since the Thakurta spacetime is a conformaltransformation of the Schwarzschild metric, it is now accepted that its redshift radii points to the co-moving radii ofthe event horizon of BH [16, 18, 19]. By considering asymptotic behavior of the gravitational lagrangian (Ricci scalar),one can classify the Thakurta BH and its extension to the charged BH into the same class of solutions [18, 19]. TheThakurta spacetime sustains an inward flow, which leads to an increase in the mass of BH [18–20]. This ingoing flowcomes from the back-reaction effect and can be neglected in a low density background [20]. In fact, for the low densitybackground, the mass will be decreased in the Phantom regime [21]. Also, the radius of event horizon increases withthe scale factor when its temperature decreases by the inverse of scale factor [18, 19].Using the Eddington-Finkelstein form of the Schwarzschild metric and the conformal transformation, Sultana andDyer (2005) have constructed their metric and studied its properties [22]. In addition, unlike the Thakurta spacetime,the curvature scalars do not diverge at the redshift singularity radii (event horizon) of the Sultana and Dyer spacetimes.Since the Sultana and Dyer spacetimes is conformal transformation of the Schwarzschild metric, it is now acceptedthat the Sultana and Dyer spacetimes include dynamic BHs [16]. Various examples can be found in [16, 23–25].Among these conformal BHs, only the solutions by M c Clure et al. and Thakurta can satisfy the energy conditions[16, 19]. Static charged BHs which are confined into the FRW spacetime and the dynamic, charged BHs were studiedin [26–33]. The Brane solutions can be found in [34–36].In another approach, mcVittie found new solutions including contracting BHs in the coordinates co-moving withthe universe’s expansion [37]. Its generalization to the arbitrary dimensions and to the charged BHs can be found in[38, 39]. In these solutions, it is easy to check that the curvature scalars diverge at the redshift singularities. In thisapproach, authors have used the isotropic form of the FRW metric along as the perfect fluid concept and could findtheir solutions which can contain BHs [40]. The mass and the charge of their BHs seem to be decreased with the scalefactor. Also, it seems that the redshift singularities does not point to a dynamic event horizon [41–44]. Unlike theSwiss Cheese models, the energy conditions are violated by these solutions [16]. These solutions can be considered asModels for cosmological inhomogeneities [15].This paper is organized as follows: in the next section, we consider the conformal transformation of a non-staticspherically symmetric metric, where conformal factor has only time dependency. In addition, we derive the generalpossible form of metric by using perfect fluid concept. In section 3, slow time varying approximation is used in order tofind the physical meaning of the parameters of metric. In continue, the mcVittie like solution and its thermodynamicproperties are addressed. In section 4, we generalized our debates to the charged spacetime, when the effects of thedark energy are considerable. In section 5, we summarize and conclude the results.
II. METRIC, GENERAL PROPERTIES AND BASIC ASSUMPTIONS
Let us begin with this metric: ds = a ( τ ) [ − f ( τ, r ) dτ + dr (1 − kr ) f ( τ, r ) + r dθ + r sin ( θ ) dφ ] . (15)Where a ( τ ) is the arbitrary function of time coordinate τ . This metric has three Killing vectors ∂ φ , sin φ ∂ θ + cot θ cos φ ∂ φ and cos φ ∂ θ − cot θ sin φ ∂ φ . (16)Now, if we define new time coordinate as τ → t = Z a ( τ ) dτ, (17)we will get ds = − f ( t, r ) dt + a ( t ) [ dr (1 − kr ) f ( t, r ) + r dθ + r sin ( θ ) dφ ] , (18)which possesses symmetries like as Eq. (16). From now, it is assumed that a ( t ) is the cosmic scale factor similar tothe FRW’s. For f ( t, r ) = 1, Eq. (18) is reduced to the FRW metric (1). Also, conformal BHs can be achieved bychoosing f ( t, r ) = f ( r ) where, the general form of f ( r ) is [18]: f ( r ) = 1 − mr + Q r − Λ r . (19)Therefore, conformal BHs can be classified as a special subclass of metric (18). n α = δ rα is normal to the hypersurface r = const and yields n α n α = g rr = (1 − kr ) f ( t, r ) a ( t ) , (20)which is timelike when (1 − kr ) f ( t, r ) <
0, null for (1 − kr ) f ( t, r ) = 0 and spacelike if we have (1 − kr ) f ( t, r ) > t and r , when it is absorbed at coordinates t and r simple calculations leadto 1 + z = λλ = a ( t ) a ( t ) ( f ( t, r ) f ( t , r ) ) , (21)as induced redshift due to the universe expansion and factor f ( t, r ). Redshift will diverge when f ( t , r ) goes to zeroor 1 + z −→ ∞ . This divergence as the signal of singularity is independent of the curvature scalar ( k ), unlike theMcvittie’s solution and its various generalizations [38, 39], which shows that our solutions are compatible with theFRW background. As a desired expectation, it is obvious that the FRW result is covered when f ( t , r ) = f ( t, r ) = 1.The only non-diagonal term of the Einstein tensor is G tr = − − kr f ( t, r ) a ( t ) r ( a ( t ) ˙ f ( t, r ) − f ′ ( t, r ) ˙ a ( t ) r ) , (22)which (˙) and ( ′ ) are derivatives with respect to time and radius, respectively. Using ∂f∂t = ˙ a ∂f∂a , one gets G tr = − (1 − kr ) ˙ a ( t ) f ( a ( t ) , r ) a ( t ) r ( a ( t ) ˜ f ( a ( t ) , r ) − f ′ ( a ( t ) , r ) r ) , (23)where ˜ f ( a ( t ) , r ) = ∂f∂a . In order to get perfect fluid solutions, we impose condition G tr = 0 and reach to f ( t, r ) = f ( a ( t ) r ) = X n b n ( a ( t ) r ) n . (24)Although Eq. (24) includes numerous terms, but the slow expansion approximation helps us to attribute physicalmeaning to the certain coefficients b n . Since G tr = 0, we should stress that here that there is no redial flow andthus, the backreaction effect is zero [19, 20], which means that there is no energy accretion in these solutions [45].Finally and briefly, we see that the perfect fluid concept is in line with the no energy accretion condition. The onlyanswer which is independent of the rate of expansion can be obtained by condition b n = δ n which is yielding theFRW solution. III. MCVITTIE LIKE SOLUTION IN THE FRW BACKGROUND
The mcVittie’s solution in the flat FRW background can be written as [16] ds = − ( 1 − M a ( t )˜ r M a ( t )˜ r ) dt + a ( t ) (1 + M a ( t )˜ r ) [ d ˜ r + ˜ r d Ω ] . (25)This metric possess symmetries same as metric (18). ˜ r is isotropic radius defined by: r = ˜ r (1 + M r ) . (26)There is a redshift singularity at radii ˜ r h = M a ( t ) which yields the radius r h = M a ( t ) (1 + a ( t )) [51]. In addition, ˜ r h isa spacelike hypersurface, and can not point to an event horizon [45].Consider f ( a ( t ) r ) = 1 − b − a ( t ) r . This assumption satisfies condition (24) and leads to ds = − (1 − b − a ( t ) r ) dt + a ( t ) [ dr (1 − kr )(1 − b − a ( t ) r ) + r d Ω ] . (27)For b − = 0, this metric will converge to the FRW metric when r −→ ∞ . The Schwarzschild metric is obtainable byputting a ( t ) = 1, b − = M and k = 0. Metric suffers from three singularities at a ( t ) = 0 (big bang), r = 0 and f ( a ( t ) r ) = 0 ⇒ a ( t ) r h = 2 b − . (28)Third singularity exists if b − >
0. In this manner, Eq. (21) will diverge at r = r h . In addition and in contrast to theGao’s solutions, the radii of the redshift singularity ( r h ) in our solutions is independent of the background curvature( k ), while for the flat case our radius is compatible with the previous works [16, 37, 39]. Also, metric changes itssign at r = r h just the same as the schwarzschild spacetime. In addition, curvature scalars diverge at this radius aswell as the mcVittie spacetime. Accordingly, this singularity point to a naked singularity which can be consideredas alternatives for BHs [46, 47]. In continue, we will point to the some physical and mathematical properties of thissingularity which has the same behaviors as event horizon if one considers slow expansion approximation. The surfacearea integration at this radius leads to A = Z √ σdθdφ = 4 πr h a ( t ) = 16 π ( b − ) . (29)The main questions that arise here are: what is the nature of b − ? and can we better clarify the meaning of r h ? Forthese purposes, we consider the slow expansion approximation ( a ( t ) ≈ c ), define new coordinate η = cr and get ds ≈ − (1 − b − η ) dt + dη (1 − k ′ η )(1 − b − η ) + η dθ + η sin ( θ ) dφ , (30)where k ′ = kc . In these new coordinates, ( t, η, θ, φ ), and from Eq. (20) it is apparent that for b − >
0, hypersurfacewith equation η = η h = 2 b − is a null hypersurface. When our approximation is broken, then η h may not be actuallya null hypersurface, despite its resemblance to that. We call this null hypersurface a quasi event horizon which issignalling us an object like a BH and we refer to that as a quasi BH. From now, we assume b − >
0, the reason of thisoption will be more clear later, when we debate mass. Therefore by the slow expansion approximation, r h (= b − c )plays the role of the co-moving radius of event horizon and it is decreased with time. In order to find an answer tothe first question about the physical meaning of b − , we use Komar mass: M = 14 π Z S n α σ β ▽ α ξ βt dA, (31)where ξ βt is the timelike Killing vector of spacetime. Since the Komar mass is only definable for the stationary andasymptotically flat spacetimes [48], one should consider the flat case ( k = 0) and then by bearing the spirit of thestationary limit in mind (the slow expansion approximation) tries to evaluate Eq. (31).Consider n α = q − b − a ( t ) r δ tα and σ β = a ( t ) r − b − a ( t ) r δ rβ as the unit timelike and unit spacelike four-vectors, respectively.Now using Eq. (31) and bearing the spirit of the slow expansion approximation in mind, one gets M = 14 π Z S n α σ β Γ βαt dA = b − , (32)which is compatible with the no energy accretion condition ( G tr = 0). In addition, we will find the same result asEq. (32), if we considered the flat case ( k = 0) of metric (30) and use n α = q − b − η δ tα and σ β = q − b − η δ ηβ . Sincethe integrand is independent of the scale factor ( a ( t )), the slow expansion approximation does not change the result ofintegral. But, the accessibility of the slow expansion approximation is necessary if one wants to evaluate the Komarmass for dynamical spacetimes [48]. Indeed, this situation is the same as what we have in the quasi-equilibriumthermodynamical systems, where the accessibility of the quasi-equilibrium condition lets us use the equilibrium for-mulation for the vast thermodynamical systems [49]. It is obvious that for avoiding negative mass, we should have b − >
0. Relation to the Komar mass of the mcVittie’s solution can be written as [16, 39] M mcV ittie = Ma ( t ) . (33)In addition, some studies show that the Komar mass is just a metric parameter in the mcVittie spacetime [41, 42, 45].Indeed, Hawking-Hayward quasi-local mass satisfies ˙ M = 0, which is compatible with G rt = 0 and indicates thatthere is no redial flow and thus the backreaction effect, in the mcVittie’s solution [19–21, 45]. In order to clarifythe mass notion in the mcVittie spacetime, we consider the slow expansion approximation of the mcVittie spacetimewhich yields ds ≈ − ( 1 − M η M η ) dt + (1 + M η ) [ dη + η d Ω ] . (34)This metric is signalling us that the M may play the role of the mass in the mcVittie spacetime. In addition, bydefining new radii R as R ( t, r ) = a ( t )˜ r (1 + M r ) , (35)one can rewrite the mcVittie spacetime in the form of ds = − (1 − MR − H R ) dt − HR q − MR dtdR + dR − MR + R d Ω , (36)where H = ˙ aa [50]. This form of the mcVittie spacetime indicates these facts that the Komar mass is a metricparameter and M is the physical mass in this spacetime [50]. Finally, we see that the results of the slow expansionapproximation (Eq. (34)) and Eq. (36) are in line with the result of the study of the Hawking-Hayward quasi-localmass in the mcVittie spacetime [41, 42, 45, 50]. For the flat case ( k = 0) of our spacetime (Eq. (27)), by consideringEq. (33) and following the slow expansion approximation, we reach at ds ≈ − (1 − Mη ) dt + dη (1 − Mη ) + η dθ + η sin ( θ ) dφ . (37)Also, if we define new radius R as r = Ra (1 + M R ) , (38)we obtain ds = − ( (1 − M R ) (1 + M R ) − R H (1 + M R ) (1 − M R ) ) dt − RH (1 + M R ) (1 − M R ) dtdR (39)+ (1 + M R ) [ dR + R d Ω ] . Both of the equations (37) and (39) as well as the no energy accretion condition suggest that, unlike the mcVittie’sspacetime, the Komar mass may play the role of the mass in our solution. From Eq. (39) it is apparent that R = M points to the spacelike hypersurface where, in the metric (36), R = 2 M points to the null hypersurface. In thenext subsection and when we debate thermodynamics, we will derive the same result for the mass notion in ourspacetime.Only in the a ( t ) = 1 limit (the Schwarzschild limit), Eqs. (39) and (25) will be compatible which showsthat our spacetime is different with the mcVittie’s. Let us note that the obtained metric (Eq. (39)) is consistent withEq. (36), provided we take M = 0 (the FRW limit). Horizons, energy and thermodynamics
There is an apparent horizon in accordance with the FRW background which can be evaluated from Eq. (6):(1 − kr ap )(1 − Ma ( t ) r ap ) − r ap ˙ a ( t ) = 0 . (40)This equation covers the FRW results in the limit of M −→ a ( t ) = 0, which supports our previous definition for b − . Calculations for the flat case yield foursolutions. The only solution which is in full agreement with the limiting situation of the FRW metric (in the limit ofzero M ) is r ap = 1 + √ − HM a . (41)Therefore, the physical radius of apparent horizon ( ξ ap = a ( t ) r ap ) is ξ ap = 1 + √ − HM H , (42)which is similar to the conformal BHs [19]. It is obvious that in the limit of M −→
0, the radius for the apparenthorizon of the flat FRW is recovered. For the surface gravity of apparent horizon, one can use Eq. (8) and gets: κ = κ F RW (1 − Ma ( t ) r ap ) + Ma ( t ) [ 1 r ap + 1(1 − Ma ( t ) r ap ) (¨ a ( t ) + 2 ˙ a ( t ) a ( t ) )] , (43)where h ab = diag ( − − Ma ( t ) r , − Ma ( t ) r a ( t ) ), r ap is the apparent horizon co-moving radius (41) and κ F RW is the surfacegravity of the flat FRW manifold κ F RW = − ˙ a ( t ) + a ( t )¨ a ( t )2 a ( t ) ˙ a ( t ) . (44)The schwarzschild limit ( κ = M ) is obtainable by inserting a ( t ) = 1 in Eq. (43). In the limiting case M −→ ξ for this spherically symmetric spacetime is defined as [52]: M MS = ξ − h ab ∂ a ξ∂ b ξ ) . (45)Because this definition does not yield true results in some theories such as the Brans-Dicke and scalar-tensor gravities,we are pointing to the Gong-Wang definition of mass [53]: M GW = ξ h ab ∂ a ξ∂ b ξ ) . (46)It is apparent that, for the apparent horizon, Eqs. (45) and (46) yield the same result as M GW = M MS = ξ ap . Inthe limit of M −→
0, the FRW’s results are recovered and we reach to M GW = M MS = ρV as a desired result [10].Using Eqs. (45) and (46) and taking the slow expansion approximation into account, we reach to M GW = M MS ≃ M as the mass of quasi BH. Also, this result supports our previous guess about the Komar mass as the physical massin our solution, and is in line with the result of Eqs. (37) and (39). For the Mcvittie metric, Eqs. (45) and (46) yield M GW = M MS ≃ M as the confined mass to radius ξ h = a ( t )˜ r h = M . Also, Eqs. (32), (45) and (46) leads to the sameresult in the Schwarzschild’s limit ( M = M GW = M MS = M ). For the flat background, using metric (30), Eq. (13)and inserting results into Eq. (14), one gets T ≃ πM , (47)for the temperature on the surface of quasi horizon. The same calculations yield similar results, as the temperatureon the horizon of the Mcvittie’s solution. For the conformal Schwarzschild BH, the same analysis leads to T ≃ πa ( t ) M , (48)which shows that the a ( t ) M plays the role of mass, and is compatible with the energy accretion in the conformal BHs[19–21, 51]. Again, we see that the temperature analysis can support our expectation from M as the physical massin our solutions. For the area of quasi horizon, we have A = Z √ σdθdφ = 4 πa ( t ) r h = 16 πM . (49)In the mcVittie spacetime, this integral leads to A = 16 πM . In order to vindicate our approximation, we consider S = A for the entropy of quasi BH. In continue and from Eq. (47), we get T dS ≃ dM = dE. (50)Whereas, we reach to T dS ≃ dM = dE for the mcVittie spacetime. In the coordinates ( t, η, θ, φ ), we should remindthat, unlike the mcVittie spacetime, E = M GW = M MS ≃ M is valid for quasi BH and the work term can beneglected as the result of slow expansion approximation ( dW ∼
0) [51]. Finally and unlike the mcVittie’s horizon,we see that
T dS ≃ dE is valid on the quasi event horizon. This result points us to this fact that the first law ofthe BH thermodynamics on quasi event horizon will be satisfied if we use either the Gong-Wang or the Misner-Sharpdefinitions for the energy of quasi BH. T dS ≃ dE is valid for the conformal Schwarzschild BH, too [51]. For the flatbackground, we see that the surface area at redshift singularity in our spacetime is equal to the mcVittie metric whichis equal to the Schwarzschild metric. In continue and by bearing the slow expansion approximation in mind, we sawthat the temperature on quasi horizon is like the Schwarzschild spacetime [19]. In addition, we saw that the qualityof the validity of the first law of the BH thermodynamics on quasi event horizon is like the conformal SchwarzschildBH’s and differs from the mcVittie’s solution.In another approach and for the mcVittie spacetime, if we use the Hawking-Hayward definition of mass as the totalconfined energy to the hypersurface ˜ r = M a ( t ) , we reach to T dS ≃ dM = dE, (51)where we have considered the slow expansion approximation. In addition, Eq. (51) will be not valid, if one usesthe Komar mass (33). Finally, we saw that the first law of thermodynamics will be approximately valid in themcVittie’s solution, if one uses the Hawking-Hayward definition of energy. Also, none of the Komar, Misner-Sharpand Gong-Wang masses can not satisfy the first law of thermodynamics on the mcVittie’s horizon. IV. OTHER POSSIBILITIES
According to what we have said, it is obvious that there are two other meaningful sentences in expansion (24). Thefirst term is due to n = − n = 2 and it is related tothe cosmological constant. Therefore, the more general form of f ( t, r ) can be written as: f ( t, r ) = 1 − Ma ( t ) r + Q ( a ( t ) r ) −
13 Λ( a ( t ) r ) , (52)where we have considered the slow expansion approximation and used these definitions b − ≡ Q and b ≡ − Λ.Imaginary charge ( b − <
0) and the anti De-Sitter (Λ <
0) solutions are allowed by this scheme, but these possibilitiesare removed by the other parts of physics. Consider Eq. (52) when Λ = 0, there are two horizons located at r + = M + √ M − Q a ( t ) and r − = M − √ M − Q a ( t ) . These radiuses are same as the Gao’s flat case [39]. In the low expansionregime ( a ( t ) ∼ c ), these radiuses point to the event and the Coushy horizons, as the Riessner-Nordstorm metric [9].Hence, we refer to them as quasi event and quasi Coushy horizons. The case with Q = 0, M = 0 and Λ > a ( t ) ≃ c ), one can rewrite this case as ds ≈ − (1 − Λ3 η ) dt + dη (1 − Λ3 η ) + η d Ω . (53)This is nothing but the De-Sitter spacetime with cosmological constant Λ, which points to the current accelerationera. Horizons and temperature
Different f ( t, r ) yield apparent horizons with different locations, and one can use Eqs. (6) and (8) in order to findthe location and the temperature of apparent horizon. For every f ( t, r ), using the slow expansion regime, we get: ds ≈ − f ( η ) dt + dη f ( η ) + η d Ω . (54)Now, the location of horizons and their surface gravity can be evaluated by using Eq. (13). Their temperature isapproximately equal to Eq. (14), or briefly: T i ≃ f ′ ( η )4 π | η hi , (55)where ( ′ ) is derivative with respect to radii η and η hi is the radii of i th horizon. V. CONCLUSIONS
We considered the conformal form of the special group of the non-static spherically symmetric metrics, where itwas assumed that the time dependence of the conformal factor is like as the FRW’s. We saw that the conformal BHscan be classified as a special subgroup of these metrics. In order to derive the new solutions of the Einstein equations,we have imposed perfect fluid concept and used slow expansion approximation which helps us to clarify the physicalmeaning of the parameters of metric. Since the Einstein tensor is diagonal, there is no energy accretion and thusthe backreaction effect is zero. This imply that the energy (mass) should be constant in our solutions. These newsolutions have similarities with earlier metrics that have been presented by others [37–39]. A related metric which issimilar to the special class of our solutions was introduced by mcVittie [37, 39]. These similarities are explicit in theflat case (temperature and entropy at the redshift singularity), but the differences will be more clear in the non-flatcase ( k = 0), and we pointed to the one of them, when we debate the redshift. In addition and in the flat case, wetried to clear the some of differences between our solution and the mcVittie’s. We did it by pointing to the behavior ofthe redshift singularity in the various coordinates, the mass notion, and thermodynamics. Meanwhile, when our slowexpansion approximation is broken then there is no horizon for our solutions. Indeed, these objects can be classifiedas naked singularities which can be considered as alternatives for BHs [46, 47].For the our solutions and similar with earlier works [37–39], the co-moving radiuses of the redshift singularitiesare decreased by the expansion of universe. Also, unlike the previous works [37–39], the redshift singularities in oursolutions are independent of the background curvature. By considering the slow expansion approximation, we wereable to find out BH’s like behavior of these singularities. We pointed to these objects and their surfaces as quasi BHsand the quasi horizons, respectively. In continue, we introduced the apparent horizon for our spacetime which shouldbe evaluated by considering the FRW background.In order to compare the mcVittie’s solution with our mcVittie’s like solution, we have used the three existingdefinitions of mass including the Komar mass, the Misner-Sharp mass ( M MS ) and the Gong-Wang mass ( M GW ). Wesaw that the notion of the Komar mass of quasi BH differs from the mcVittie’s solution. Also, in our spacetime, weshowed that the M MS and M GW masses yield the same result on the apparent horizon and cover the FRW’s resultin the limiting situations. In addition, using the slow expansion approximation, we evaluated M MS and M GW on thequasi event horizon of our mcVittie’s like solution, which leads to the same result as the Komar mass. In addition,we should express that, the same as the mcVittie spacetime, the energy conditions are not satisfied near the quasihorizon.In addition, we have proved that, unlike the mcVittie’s solution, the first law of thermodynamics may be satisfiedon the quasi event horizon of our mcVittie’s like solution, if we use the Komar mass or either M MS or M GW as theconfined mass and consider the slow expansion approximation. This result is consistent with previous studies aboutthe conformal BHs [51], which shows that the thermodynamics of our solutions is similar to the conformal BHs’. Inorder to clarify the mass notion, we think that the full analysis of the Hawking-Hayward mass for our solution isneeded, which is out of the scope of this letter and should be considered as another work, but our resolution makesthis feeling that the predictions by either the slow expansion approximation or using the suitable coordinates fordescribing the metric for mass, may be in line with the Hawking-Hayward definition of energy, and have reasonableaccordance with the Komar, M MS and M GW masses of our solutions. Indeed, this final remark can be supportedby the thermodynamics considerations and the no energy accretion condition ( G tr = 0). Moreover, we think that, indynamic spacetimes, the thermodynamic considerations along as the slow time varying approximation can help us toget the reasonable assumptions for energy and thus mass. Finally, we saw that the first law of thermodynamics willbe approximately valid in the mcVittie’s and our solutions if we use the Hawking-Hayward definition of the mass inthe mcVittie spacetime and the Komar mass as the physical mass in our solution, respectively. In continue, the moregeneral solutions such as the charged quasi BHs and the some of their properties have been addressed.Results obtained in this paper may help achieving a better understanding of black holes in a dynamical background.From a phenomenological point of view, this issue is important since after all, any local astrophysical object lives inan expanding cosmological background. Finally, we tried to explore the concepts of mass, entropy and temperaturein a dynamic spacetime. VI. ACKNOWLEDGMENTS
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