Singularity Formation in Radiating Star with Dark Energy Background

Home , Anzhong Wang

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( l being ‡ ,Grav- r, IA. depending on the initial conditions of gravitational collapse [1, 2, 3]. Some works also revealed the eternally collapse of star and preventing theformation anykinds of singularities [4, 5, 6, 7, 8], while dust collapse is well studied by Oppenheimer and H.Snyder[9]. Thestudiesconcentratedtothesingularityformation in gravitational collapse with the frame work of GTR and other gravity theories are the interesting problems for theoretical astrophysicist [10, 11, 12, 13, 14, 15, 16]. There are some foundtheevidenceofanothertypeofmatter, knownasdarkenergy and dark matter, play important role in the formation of large-scale structures in the universe, such as stars, galaxies, etc. Astronomical observation indicated that current universe 2 1 consist of approximately 3 dark energy(DE) and 3 dark matter(DM). The nature of dark energy as well as dark matter is unknown, and many radically different models have been proposed, such as, cosmological constant, quintessence [17, 18], DGP branes [19, 20], Gauss-Bonnet [21] and dark energy in brane worlds[22,23,24,25]. Such kinds of exotic fluids do not appear to interact with the usual standard model particles, and therefore, it is also very difficult to detect a them by the modern detectors [26],[27]. For the stars are made of baryonic matter, besides gravitational interaction, they can also interact with each other by means of strong, weak, and electromagnetic forces. The situation becomes more interesting if we give our attention to larger scales, i.e., the galactic scales and the cosmological scale, where, in fact, the dark matter/dark energy could dominate [28],[29] and references there in. Dark matter/Dark energy is supposed to interact with ordinary matter only through gravity. Therefore, only by its gravitational effects, one can gather information, such as its positions, mass, density profiles, etc. A natural question is how dark energy affects the process of the gravitational collapse of a star. It is known that dark energy exerts a repulsive force on its surrounding, and this repulsive force may prevent the star from collapse. Indeed, there are speculations that a massive star doesn’t simply collapse to form a black hole, instead, to the formation of stars that contain dark energy. Some recent works have been considered the spherically symmetric star consist of non-baryonic (DE and DM) matter and discussed its nature of singularity formation[30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. In a recent work, Mazur and Mottola [30] have suggested a solution with a final configuration without neither singularitiesnor horizons, which they called ”gravastar” (gravitational vacuum star). The dark energy, by virtue of its repulsive gravitational nature, is interesting to study gravitational collapse and formation of Black-Hole. As all massive stars do not form black holes (may be neutron stars or white dwarfs), so it is generally speculated [31],[32] that the dark energy may play an important role in the collapsing stars. Gravitational collapse and formation of black holes in the presence of dark energy have been considered in several works [33, 34, 35, 38, 39]. Cai and Wang [39] investigated the black hole formation from collapsing dust in the background of dark energy, and showed that the dark energy itself never collapses to form

2 black holes, but when both the dark energy and the dust are present, Black-Holes ,can be formed due to the condensation of the dust. It has been shown that the mass of a Black-Hole decreases due to phantom energy accretion and tends to zero when the Big Rip approaches [40],[41]. Babichev [40],[41] considered the accretion of a relativistic perfect fluid into blackholes and showed that, if the universe is dominated by DE, Black-Hole will decrease their mass due to phantom energy acceleration and tend to zero at time of big rip. Since recent observation indicated that two-third part of the total energy of the universe is to be attributed to the matter of exotic nature, usually called dark energy. Thus, with the notion of such exotic content of universe, it is necessary to introduce this new kinds of matter distribution. Motivated from above works, we considered the radiating star in the background of dark energy components. The paper is organised in the following way- in section-2 we considered a class of conformally flat spherically symmetric metric describing the interior of radiating star. In section 3 we considered the exterior space time (Vaidya metric) and established the Darmois- Israel junction conditions. In section 4, we studied the Dynamics of radiating star in the background of dark energy. Here we discussed two cases- first when only DE present and secondly a combination of both baryonic and DE and has been investigated their nature of singularities formation. Discussion and conclusion are discussed in last section.

2 Dynamical equations for collapsing radiating star

Consider the spherically symmetric gravitational collapsing of radiating star with baryonic matter(perfect fluid) in the background of dark energy. The system here is divided into three different regions (surface of star), −(interior region ) and + (exterior region) of star. The interior of star is described by spherically symmetric line element considered here is a classÍ of conformallyÍ flat space-time metric Í

2 1 2 2 2 2 2 2 2 3B − = [−3C + 3A + A 3\ + A sin \3q ] (1) '(C,A)2 The space time (1)Í is also conformal to the FLRW metric. The conformally flat spacetime, in context of gravitational collapse of radiating star were first studied by Som and Santos[42]. The interior of the star is considered here perfect fluid undergoing dissipation in the form of heat flux with the dark-energy components 8 represented by the energy momentum tensor )9 , 8 = 8 8 )9 )9 (") + )9 () (2) here the coordinate G8 = (C,A,\,q) are taken to be co-moving and 8 = 8 8 8 8 )9 (") (? + d)E E 9 + ?X 9 + @ E 9 + E @ 9 (3)

3 8 = 8 8 )9 () (? + d )E E 9 + ? X 9 (4) respectively are energy momentum tensor for baryonic(dissipative perfect fluid) and dark energy fluids. The quantities d, ?, @8 are respectively the energy density, pressure, heat flux of radiation, and d, ? are the energy density and pressure of dark energy components, while E8 is the co-moving velocity of the fluid. For the line element (1) the four vectors E8 and @8 are satisfy,

8 8 E E8 = −1, E = (', 0, 0, 0) 8 8 (5) @ = (0,@, 0, 0), E @8 = 0 where @ is a function of (C,A) represents heat flux of radiation. The kinematical quantities- the four acceleration 08 and expansion scalar Θ are given by, = 9 Θ= 8 08 E8; 9 E , E;8 (6) For the space-time metric (1), we have '′ 0 = − X1 (8 = 0, 1, 2, 3) (7) 8 ' 8 and the expansion scalar, Θ= −3'¤ (8) Since in this work we are mainly concerned with the phenomena of gravitational collapse, then Θ < 0 ⇐⇒ '¤ > 0 (9) 8 = 8 8c Now, the Einstein’s field equation 9 :)9 (where k= 24 ) for metric (1) with matter field (2)-(4) yield the following equations 2' : (d + d ) = 3 '¤2 − '′2 + (2'′ + A'′′) (10) A :@ = −2'¤′' (11) ' : (? + ? ) = 2A'¥ − 4'′ − 3 '¤2 − '′2 (12) A 2' : (? + ? ) = A'¥ − '′ − A'′′ − 3 '¤2 − '′2 (13) A here dot (.) and dash (’) represent the derivative with respe ct to C and A respectively. The mass function m(t,r) of a collapsing star for any instant (C,A) is given by ([43, 44]), 1 A3 < (C,A) = [2''′′ + ('¤2 − '′2)] (14) 2 '3

4 3 Exterior space time metric and junction condition

The matter ﬁelds are co-moving inside the spherical surface − with space time described by the metric (1), the surface of the collapsing star is chosen by time like 3-space A = A (2>=BC0=C) and the intrinsic metric is, Í Í Í A2 3B2 = W 3b83b 9 = −3g2 + 3\2 + sin2 \3q (15) 8 9 2 '0Í(g) Í 8 where b =(g,\,q) is the intrinsic coordinate and '0 (g) is intrinsic geometrical radius of star. The region of star + is considered as Vaidya metric which describe the exterior space-time of a radiating star [45, 46, 47]. Í

2 2" (a) 2 2 2 2 2 3B + = − 1 − 3a − 23a3Ω + Ω 3\ + sin \3q (16) Ω Í 8 in the coordinate G+ = (a, Ω,\,q), M(a) represent the mass of the star inside the surface i.e. Newtonian mass of gravitating star as measured by distant observer and a is the retarded time. The metric (16) is the unique solution of the spherically symmetricÍ ﬁeld for the radiating star and the Einstein’s tensors for that are 2 3" = − XC XC , 8,9 = 0, 1, 2, 3 (17) 8 9 Ω2 3E 8 9 The energy momentum tensor for radiation,

+ = )8 9 nF8F 9 (18) where n is energy density of radiation and F8 = (1, 0, 0, 0) the four velocity vector. Thus from (17) and (18),

2 3" 3a 2 :n = − (19) Ω2 3E 3C Í which is energy density of radiation measured by an observer on . Since the star 3" is radiating (n > 0) to the exterior space time, therefore it must have 3a ≤ 0 that is, mass of the star diminished during radiation [3], [45, 46, 47]. Í For junction condition, we employed the Israel and Darmois condition [48, 49], here we merely stated the results of the matching of line element (1) and (16) because these condition have been extensively studied by many authors (see [3] ,[45], etc). The junction condition require that the metrics (1) and (16) match smoothly across the boundary which imply-

Í 5 • The continuity of the ﬁrst fundamental form over

2 2 2 (3B−) = 3B = (3B+) Í (20)

Í Í Í ± • The continuity of the second fundamental form (extrinsic curvature 8 9 ) over

[ − ] = [ + ] (21) Í 8 9 8 9 where Í Í m2 j: m j< m j= ± = −[± ± − [±Γ: ± ± 8 9 : mb8mb 9 : <= mb8 mb 9 ± 8 and [: represent the extrinsic curvature and normal vector to the surface in j± coordinate respectively. Using the junction condition (20) with the metric (1), (15) and (16), weÍ obtain 3C = '(A ,C) (22) 3g Í Í

' A ,C = '0 (g) (23) Í A Ω (Ía) = (24) Í '0Í(g) 3a 2 3Ω 2" (a) −1 = 2 + 1 − (25) 3g 3a Ω Í The extrinsic curvature for the line element (1) and exterior metric (16) have following the non-vanishing components

− ′ gg = ' (A ,C) (26) Í A A2 Í − = − '′ ) = − 2>B422\ (27) \\ ' '2 qq Í and 32a 3a −1 " 3a + = − (28) gg 3g2 3g Ω2 3g Í 3Ω 3a 2" + = Ω + Ω 1 − = + 2>B422\ (29) \\ 3g 3g Ω qq Í Depending on the choice of space time outside the radiating star which is non- ± ± empty, the induced metric W8 9 and extrinsic curvature 8 9 are continuous across . However, such dependence may not be always consistent with (20)-(21). If the exterior of the collapsing star is empty, then there is no way to do the matching Í 6 of equations (26) and (28) because DE component ? ≠ 0. Since the present studies concerned with the radiating star, then the matching of conditions (26) and (28) is stable here. Using equations (27) and (29), the condition (21) gives A A'′ 2" 3a 3Ω 1 − = 1 − Ω + Ω ' ' Ω 3g 3g Í with the help of equations (22)-(25) in above equation, we get,

1 A3 " (a) = '¤2 − '′2 + 2''′′ = <(C,A ) (30) 2 '3 Í Using equations (26), (28) the conditions (21) gives Í

32a 3a −1 3a " '′ = − (31) 3C2 3g 3g Ω2 Í with the help of equation (22), (23), (24) and (25) gives

'¤ '′ −1 ''¤ ′ '¤′ '¤2 '¥ '′2 2'′ '′ = 1 − A − A A' − + − 3 + + − ' ' '2 ' '2 ' 2'2 A' " # Using ﬁeld equations (11) - (12) in above, one obtained Í

? + ? = @ (32) Í The equations (30) and (32) are required junction conditions. The important result is (32) relating the radiation heat ﬂux to the dark energy component(d). Itisto be noted that for non-dissipative system(@ = 0), the pressure is balanced by dark energy component (? ) on surface . The condition (32) then shows the eﬀect of dark energy on the radiation of the star. As anticipated here the analysis being persued here included as initial epoch,Í this conclusion is important as a class and have been discussed extensively and exclusively in next section. In view of equ.(19), the luminosity observed on the surface can be read as,

1 3< 3a 2 Í ! = :Ω2n = − (33) 2 3a 3g Í Í The total luminosity for an observer at inﬁnity is

−1 1 2 3< 3< 3C 3a !∞ = lim :Ω n = − = − (34) Ω→∞ 2 3a 3C 3g 3g 3Ω " # 3g →0 Í Í 7 The boundary redshift / is

Í 3a 1 + / = (35) 3g Í with the help of (22)-(25) and (30), this becomesÍ

A' − A2 '′ − '¤ 1 + / = (36) ' (A − 2<') " # Í using equations (22), (35)-(36) into equations (34), oneÍ obtain

−<' ¤ 2 (A − 2<') !∞ = (37) A' − A2 '′ − '¤ Í 4 Dynamics of radiating star in the background of dark energy

From equation (12)-(13), one obtained

'′ = A'′′ (38) which gives by integrating

' (C,A) = _ (C) A2 + ` (C) (39) where _ and ` are orbitrary function of C. To determine orbitrary constant _ and `, we apply the boundary conditions (23) and (30). Then we have,

1 ' 2"' '∗2 _(C) = 0 1 ± 1 − 0 + A2 0 (40) 2 A2 v A '4  t 0   Í   Í  Í    '∗2  1  2"'0 2 0  `(C) = '0 1 ∓ 1 − + A (41) 2 v A '4  t 0   Í   Í  Thus equation (39) produced      2 2 '∗2 1 A A 2"'0 2 0 ' (C,A) = '0 1 + + 1 − 1 − + A (42) 2 A2 A2 v A '4  ! ! t 0   Í   Í   Í Í   8    for collapsing conﬁguration A

Í 2 2 '∗2 1 A A 2" (a)'0 2 0 ' (C,A) = '0 1 + + − 1 1 − + A (43) 2 A2 A2 A '4  ! ! tv 0   Í   Í Í Í  for expanding conﬁguration A >A . Here '0 is the intrinsic geometrical radius   ∗ = 3'0 of star and '0 3g . Since the presentÍ study concerns with the phenomena of gravitational collapse then further analysis done with equation (42). Using equ. (42) into (11, we obtain

2A '∗ 2"' '∗2 @ = − [ 0 (1 − 1 − 0 + A2 0 ) 2 ' A 4 :A 0 tv '0 Í Í ∗ 2 ∗3 ∗ ∗∗ Í ' 2' "¤ 2"' 4A '0 ' ' − 0 (− 0 − 0 − + 2A2 0 0 )] 6 5 '∗2 A '0A Í' ' 2 1 − 2"'0 + A2 0 0 0 A '4 Í r 0 Í Í Í Í 1 A2 A2 2"' '∗2 [ ' {(1 + ) + (1 − ) 1 − 0 + A2 0 }] (44) 2 0 2 2 A 4 A A tv '0 Í In view of equations (10) and (38),Í the equationÍ (14) canÍ also be written as 1 :A3 < (C,A) = (d + d ) (45) 6 '3 Also, taking time derivative of equation (14) and using field equation (11)-(12), we obtain 1 :A2 A'′ <¤ (C,A) = ?A'¤ + ? A'¤ − @' 1 − (46) 2 '4 ' The equation (46) represent the variation of total energy of star inside collapsing surface. Here we consider the collapsing star consists of perfect fluid(boryonic) and dark energy as the fluid distribution. As mentioned before, the existence of dark energy fluids comes from the observation of accelerating universe and such fluids are characterized by violation of certain energy condition- weak, dominant and strong. The denomination of dark energy is applied to fluid which violate only the strong energy condition and phantom associated to fluid violating both strong and weak energy condition[50, 51, 52]. To study the effect of such kinds of exotic fluids on the gravitational collapse of radiating star and singularity formation, we consider the equation of state,

? = ld (47)

9 1 where l is non-zero constant parameter characterized as l < − 3 (dark energy) and l < −1(phantom).

4.1 Eﬀect of Dark energy in singularity formulation In this section, we consider the radiating star contains only dark energy (? = 0 = d,@ ≠ 0,? ≠ 0,d ≠ 0) obeying equation of state (47). From equations (10), (12) and (38), we have 1 :d = '¤2 − '′2 + 2''′′ (48) 3

′′ 2 ′2 :? = 2' '¥ − 2' − 3 '¤ − ' (49) Case(i): l = −1 1 The dark energy is described by equation of state (47) and a value l < − 3 is required for the cosmic-acceleration [33, 34, 39]. The simplest explanation of dark energy is cosmological constant for which l = −1. It follows from equations (48) and (49) that

'¥ + '′′ = 0 (50)

Now, using equation (50) into (14) one obtain 2<' 2A2 '¤2 A2 1 − = 1 + '¥ − A2 + '′2 (51) A ' '2 '2 Making use of equation (42) it gives 2<' '∗∗ '∗2 1 − = [1 + 2A2 0 − 3A2 + (1 − U)2] (52) A 3 4 Í '0 '0 Í Í ∗2 where U= 1 − 2"'0 + A2 '0 . A '4 r 0 Now with (52) equationsÍ (36)Í and (37) yield

'∗∗ ∗2 ¤ 2 0 2 ' 2 −"'0 1 + 2A 3 − 3A 4 + (1 − U) '0 '0 ! = (53) ∞ Í '∗ Í A 2 + U '0 Í and ∗ A ' + U '2 1 + / = 0 (54) ∗∗ Í ' ∗2 1 + 2A2 0 − 3A2 ' + (1 − U)2 Í '3 '4 0 0 Í Í 10 One can observe from equations (53)-(54) that the collapse of radiating star will leads to Naked-Singularity (or, eternally collapse) when the quantity in square bracket is non zero. Otherwise the collapse end as Black-Hole !∞ → 0, / →∞ and the mass of Black-Hole Í A '∗∗ '∗2 1 " = A2 0 − 0 + '2 ± A2 3'∗2 − 2' '∗∗ − '4 (55) 3 2 0 0 0 0 0 ' '0 ' 2 Í0 " 0 ! r # Í Í In view of equations (42) and (50), we have from (48) that

1 A2 1 A2 '∗2 :? = '2(1−U)2 − [A2 (1− )(2"'4 −A3 0 )+2A A2'3(1−U)] 3 4 0 5 2 0 2 0 A '0A A '0 Í Í Í ∗ 2 2 2 ∗ ∗∗ ∗2 1 ' Í A A Í 1 Í A ' ' ' − [ 0 (1 + ) + U(1 − ) + {A2 (1 − )(2A3 0 ( 0 − 0 ) 2 ' 2 2 3 3 2 ' ' 2 0 A A 4UA '0 A 0 0 '0 Í Í ∗3 Í Í Í Í ' − 2"'¤ 5 − 2"'3'∗ − 3A3 0 )}]2 (56) 0 0 0 3 '0 Í Now, from equ. (46) we also have 1 :A2 A'′ <¤ = ? A'¤ − @' 1 − (57) 2 '4 ' Since '¤ > 0 , @ > 0 and ? < 0 (DE component). Initially when collapse starts, ? contributing in heat ﬂux (32)and in equ. (57) both terms in R.H.S. are negative showing the total energy of radiating star being decreasing and DE- component (? ) increases the rate of diminishing energy. = 2 Case(ii): l − 3 2 Soma et al.[34] obtained the singularity formation for the range −1 − 3 leads to Black-Hole formation where as for l < − 3 , the = 2 collapse end as Naked-Singularity. However for l − 3 , the end state of collapse may be either Black-Hole or Naked-Singularity depending on initial density.

In this case, one can obtain from equations (48)-(49) that '¤2 − '′2 = 2''¥ (58) Taking use of equations (14) and (58) into equations (36)-(37), we obtain the red-shift and luminosity, '∗ U + A 0 '2 1 + / = 0 (59) ÍA2 '∗ ∗ 2U − 1 − 2 0 Í 2 '0 'Í0 11 A2 ∗ ∗ '0 −"'¤ 0 2U − 1 − 2 '2 '0 = Í0 !∞ ∗ (60) '0 U + A 2 '0 The equations (59)-(60) show that the collapsingÍ star will end as Black-Hole if

2 A '∗ 1 0 = U − (61) '2 ' 2 Í0 0 and its mass 3 2 3 A 1 A '∗2 '∗ ∗ A '∗ ∗ " = + 0 − 0 1 + 0 (62) 3 2 2 8 '0 2 ' ' '0 ' '0 Í Í0 " 0 ( Í0 )# otherwise, the collapse leads to Naked-Singularity(or, eternally collapse). Case(iii): %ℎ0=C><5;D83 The matter distribution with l < −1 known as "phantom energy" has recieved = 3 increased attention amongst researchers recently. For example l − 2 , the phantom energy density increases with time which also violates the dominant energy condition [53, 54]. = 3 Considering l − 2 , then it follows from equations (48)-(49) that

3 '¤2 − '′2 = −2' 2'¥ + 5'′′ (63) In view of equations (14) and (63), we obtained from equation s (36) and (37) that

'∗ 3 U + A 0 '2 1 + I = 0 (64) A2 '∗Í ∗ 7 + 4 0 − 3U Í 2 '0 'Í0 and A2 ∗ ∗ 1 '0 − "'¤ 0 7 + 4 − 3U 3 '2 '0 = Í0 !∞ ∗ (65) '0 U + A 2 '0 Thus, it can seen from (64)-(65) that the collapseÍ will lead to the formation of A2 '∗ ∗ Black-Hole if 7 + 4 0 − 3U = 0, otherwise it will form Naked-Singularity 2 '0 'Í0 (or, eternally collapse). The mass of formed Black-Hole is 3 1 '∗2 20 A 4 A '∗ ∗ '∗ ∗ " = A3 0 − − 0 7'2 + 2A2 0 (66) 5 5 0 2 ' 9 '0 9 ' '0 '0 0 Í Í0 Í Í 12 4.2 Combined effect of perfect fluid and dark energy in singularity formation In this case the collapsing star is in combination of dissipative perfect fluid (@ ≠ 0,? ≠ 0,d ≠ 0) obeying equation of state ? = nd (n being constant parameter) and dark energy components (? ≠ 0,d ≠ 0). Consider l = −1, then from (10) and (12)

: (? + d) = 2' '¥ + '′′ (67) where ? + d = 0. The initial density d0 = d C,A , luminosity and red-shift for diﬀerent values of n are given in Table-1. Í In dust ﬂuid (n = 0) case, the luminosity decreases with time and DE- component ? play major role in the loss of luminosity during collapse ( 2 ¤ 1 A since? " > 0, 3 d0 ' − 1 > 0). The collapse will end as Black-Hole when Í0 2 3'0 : (d0 − ? ) = (68) A2

It is also to be noted that junction condition (32)Í violated in this case because of ? < 0, @ ≥ 0. = 1 = For n 3 and n 1, it also observed from table-1 that ? increase the rate of luminosity decay which indicated that it contributes in the formation of black hole(!∞ → 0, IΣ →∞).

13 Table 1: Initial density, Luminosity, Redshift for various values of n n kd0(Initial Density) !∞(Luminosity) 1+/ (Redshift) A2 A2 ∗ ¤ 3 1 1 ' "' : d0 −1− : ? ÍU+A 0 ∗ ∗ 2 0 3 '2 3 '2 '2 '0 '0 Í0 Í0 ! 0 0 (Dust Fluid) 2 + (1 − U) Í '0 A2 U'2+A '∗ A2 0 0 1 : (? −d )+1 3 '2 0 Í Í Í0 ∗ A2 '∗ ∗ 1 ¤ 3 0 2 ' − "' (1+U)− ' − :? 2 U+A 0 '∗ ∗ '2 2 0 '2 0 3 '2 1 3 0 0 " Í0 # 0 (radiation fluid) + (1 − U) Í 3 2 '0 A2 U'2+A '∗ A2 '∗ ∗ 0 0 0 2 (1+U)− 2 ' − 3 :? Í ' 0 Í " Í0 # A2 '∗ ∗ ∗ ¤ 3 0 ' −"' (2+U)+ :? − ' 3 U+A 0 '∗ ∗ '2 0 '2 0 '2 0 0 " Í0 # 0 1 (stiff fluid) + (1 − U) Í '0 A2 3 U'2+A '∗ A2 '∗ ∗ ( 0 0) 0 (U+2)+ 2 :? − ' Í ' 0 Í " Í0 # 5 Discussion and Concluding Remark

The evolution of structure formation in universe with the background of DE and DM have been taken considerable interest among scientific community and has been applied for understanding the astrophysical phenomena [22, 23, 24, 25]. Gen- erally, it is believed that stars and Black-Holes are the consequence of gravitational collapse of baryonic matter cloud. In recent decade, the study of gravitational collapse has been carried out in the background of non-baryonic contents (DE and DM) too [34, 35, 38, 39]. The main motivation of this work is to investi- gate the singularities formation in gravitational collapse of spherically symmetric radiating star in dark energy background. We consider a class of conformally flat spherically symmetric inhomogeneous fluids consists of dissipative perfect fluid with dark energy component and exterior region of star is described by the famous Vaidya metric. By employing the Darmois-Israel junction conditions for the smooth matching of interior( −) and exterior ( +) spacetime at the boundary , we obtained that the radiating heat flux (@) depend on the DE component Í Í ? over the hypersurface 32. The total luminosity of the radiating star as visibleÍ to a distant observer, depends on the mass of collapsing matter. Using the boundary conditions weÍ obtained the exact solution of field equations for a given mass(Newtonion) and geometric radius of collapsing star. Two cases have been considered for the discussion of dark energy effect in collapsing process and singularity formation. In first case, when the collapsing fluid in the form of DE with equation of state ? = ld we obtained - (i) for l = −1, the collapsing star end as black- hole under the condition (55) otherwise it will formed naked singularity. (ii) for

14 = 3 phantom fluid (l − 2 < −1), the collapsing system end as black-hole when A2 ∗ ∗ '0 4 2 ' − 3U + 7 ≤ 0 and star is eternally collapse or, form Naked-Singularity 'Í0 0 A2 '∗ ∗ when 4 0 − 3U + 7 > 0 (iii) for l = − 2 , the Black-Hole will form under 2 '0 3 'Í0 the condition (61). In each of three cases (i), (ii), (iii) we obtained the mass of the formed black hole in (55), (66) and (62) respectively. We have also found that dark energy affect the total energy of radiation and it increases therate of energyloss. In second case, considering the effect of combined matter distribution (perfect fluid and DE both) in the collapsing process. The perfect fluid follows linear equation of state ? = nd whereas dark energy obeying equation (47). We obtained the initial = = 1 density (d0), luminosity and red-shift for dust fluid (n 0), radiation(n 3 ) and stiff fluid (n = 1) with dark energy candidate l = −1 in table-1 . The obtained results here shows that DE affects the final out come of collapsing star and has important role in deciding the nature of singularities. Thus the presented work conclude that dark energy has an effect on the collapsing process and play the essential role in the singularity formation (Black holes or Naked-Singularities) of radiating star.

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