Singularity Formation in Radiating Star with Dark Energy Background

Singularity formation in radiating star with dark energy background Grishma Verma,∗ Rajesh Kumar,† Keshav Ram Mishra‡ Department of Mathematics and Statistics Deen Dayal Upadhyaya Gorakhpur University,Gorakhpur,INDIA. Abstract In this paper, we considered the gravitational collapse of a spherically symmetric radiating star consisting of perfect fluid (baryonic) in the background of dark energy (DE) with equation of state ? = ld (l being non-zero constant parameter describing DE candidates). The effect of DE on the singularity formation has been discussed first separately (only DE present) and then combination of both baryonic and DE interaction. We have also showed that DE component ? play important role in the formation of Black-Hole(BH). In some cases the collapse of radiating star leads to black hole formation and in other cases it forms Naked-Singularity(or, eternally collapse). The present work is in itself remarkable to describe the effect of dark energy on singularity formation in radiating star. Key Words:- Dark Energy, Black Hole, Naked-Singularity, Radiating Star, Grav- itational collapse. 1 Introduction arXiv:2106.14220v1 [gr-qc] 27 Jun 2021 A massive star undergoes a continual gravitational collapse when the pressures inside the collapsing star become insufficient to balance the pull of gravity and the physics of it being crucial and interesting amongst astrophysicists. Using general relativistic techniques, it has been shown that the final fate of such a catastrophic collapse into singularity which may be either a black hole or a naked singularity, ∗[email protected] †[email protected] ‡[email protected] 1 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].

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