Jounal of Physics: Confeence Seies PAPER OPEN ACCESS Conditions fo the naked singulaity fomation in genealized Vaidya spacetime To cite this aticle: V D Vetogadov 2016 J. Phys.: Conf. Se. 769 012013 View the aticle online fo updates and enhancements. Related content - Naked singulaity explosion in highedimensional dust collapse Masahio Shimano and Umpei Miyamoto - Effects of Gauss--Bonnet tem on the final fate of gavitational collapse Hideki Maeda - High-speed collapse of a hollow sphee of type I matte Zahid Ahmad, Tomohio Haada, Ken-ichi Nakao et al. This content was downloaded fom IP addess 37.44.200.252 on 16/02/2018 at 05:36
Jounal of Physics: Confeence Seies 769 2016 012013 Conditions fo the naked singulaity fomation in genealized Vaidya spacetime V D Vetogadov Depatment of Theoetical Physics, Hezen Univesity, St. Petesbug 191186, Russia E-mail: vitalii.vetogadov@yandex.u Abstact. The gavitational collapse of genealised Vaidya spacetime is consideed. It is known that the endstate of gavitational collapse, as to whethe a black hole o a naked singulaity is fomed, depends on the mass function Mv,. Hee we give conditions fo the mass function which coesponds to the equation of the state P = αρ whee α 0, 1 3] and accoding to these conditions we obtain eithe a black hole o a naked singulaity at the endstate of gavitational collapse. Also we give conditions fo the mass function when the singulaity is gavitationally stong. Also we povide the metic which is the analogue of Vaidya metic in case of otation. 1. Intoduction In ecent aticles of A.A. Gib, Yu.V. Pavlov and V.D. Vetogadov [1, 2] popeties of geodesics fo paticles with negative enegy o Penose geodesics in Ke metic have been studied. It has been shown that all such geodesics appea in the egosphee fom a egion inside the gavitational adius. Howeve Ke black holes ae etenal ones. In eal case one needs to conside the question about the gavitational collapse to lean the natue of Penose geodesics. One of the oppotunities fo existence of such geodesics is the naked singulaity fomation duing the gavitational collapse. In the beginning we decided to conside the simplest case of gavitational collapse - the case of spheically symmetic collapse, in paticula - the case of gavitational collapse of genealized Vaidya spacetime which will be teated in this aticle. It is woth mentioning that Papapetou [3] showed that Vaidya metic also known as the adiating Schwazschild spacetime beaks the cosmic censoship pinciple. So Vaidya metic is the ealiest example of cosmic censoship violation. Joshi[4] showed that esult of gavitational collapse, as to whethe a black hole o a naked singulaity is fomed, depends on the initial data. In the esent pape [5] it has been shown that the endstate of gavitational collapse of genealized Vaidya spacetime depends on the mass function Mv,. Also in Ref. [6] the attempt has been made to show that the esult of gavitational collapse of Vaidya-DeSitte spacetime is the naked singulaity. Howeve in this papes the question about the appaent hoizon fomation has not been consideed. But in the case of Vaidya-DeSitte spacetime the time of the appaent hoizon fomation is less than the time of the singulaity fomation, so the esult of such collapse is the black hole. We conside the gavitational collapse of thin adiating shells. The fist shell collapses down at the cental singulaity at = v = 0 whee M0, 0 = 0. Duing the collapse of othe shells the Content fom this wok may be used unde the tems of the Ceative Commons Attibution 3.0 licence. Any futhe distibution of this wok must maintain attibution to the authos and the title of the wok, jounal citation and DOI. Published unde licence by Ltd 1
Jounal of Physics: Confeence Seies 769 2016 012013 mass function is gowing and when the last shell collapses down then the mass function becomes well-known Schwazschild mass. Also we ae only inteested in shell-focusing singulaities. Shellcossing singulaities ae gavitationally weak and we ae not inteested in them. If thee is a family of non-spacelike futue-diected geodesics which oiginate at the cental singulaity in past and the time of singulaity fomation is less than the time of the appaent hoizon fomation then the esult of such gavitational collapse is the naked singulaity. If thee is no such a family of geodesics o the time of the appaent hoizon fomation is less than the time of singulaity fomation then the esult is the black hole. It is woth mentioning that hee we conside only locally naked singulaities. It means that the appaent hoizon is fomed ealie than the geodesic cosses the last collapsing thin shell. If it is not so then the singulaity is global naked one. But hee we won t conside global naked singulaities. In this pape we give conditions fo the mass function and coesponding to these conditions we obtain eithe a naked singulaity o a black hole as a esult of gavitational collapse. We conside the matte which satisfies the equation of the state P = αρ whee α 0, 1 3 ]. Also we give conditions fo mass function and coesponding to these conditions we obtain the gavitationally stong singulaity. If we follow Tiple definition which was given in the pape[7]: a singulaity is temed to be gavitationally stong o simply stong if it destoys by stetching o cushing any object which falls into it. If it does not destoy any object this way then the singulaity is temed to be gavitationally weak. In Sec. 2 we give basic infomation about genealized Vaidya spacetime. In Sec. 3 we pesent conditions fo the mass function when we obtain eithe a naked singulaity o a black hole. In Sec. 4 we give conditions fo the mass function when the singulaity is gavitationally stong. In sec.4 we povide the metic which is the analogue of Vaidya metic in the case of otation. The system units G = c = 1 will be used in this pape. Dash and dot denote patial deivatives d d, d dv espectively. Values g αβ, Γ α βγ, R αβ ae metic, Chistoffel and Ricci tensos components espectively. Geek lettes ae equal to 0, 1, 2, 3. 2. Genealized Vaidya Spacetime Genealized Vaidya spacetime coesponds to the combination of two matte fields I and II types and in geneal case is given by:[8] ds 2 = e 2ψv, 1 2Mv, dv 2 + 2εe ψv, dvd + 2 dω 2, dω 2 = dθ 2 + sin 2 θdϕ 2, 1 hee Mv, - the mass function depending on coodinates and v which coesponds to advanced/etaded time, ε = ±1 - ingoing/outgoing adiating thin shells espectively. So we ae inteested in gavitational collapse then we put ε = +1. Also with suitable choice of coodinates we can put ψv, = 0. So 1 now has the fom: ds 2 = 1 2Mv, dv 2 + 2εdvd + 2 dω 2. 2 Now let us wite down non vanishing covaiant and contavaiant metic components: g 00 = 1 g 01 = 1, g 11 = 2Mv,, g 01 = 1, g 22 = 2, g 33 = 2 sin 2 θ. 3 1 2Mv,, g 22 = 1 2, g33 = 1 2 sin 2 θ. 4 2
Jounal of Physics: Confeence Seies 769 2016 012013 Now we can wite down non-vanishing Chistoffel components: Γ 1 00 = Γ 0 00 = M M 2, Γ 0 22 =, Γ 0 33 = sin 2 θ, 1 2M M M + Ṁ 2, Γ 1 10 = γ00 0, Γ 1 22 = 2M, Γ 1 33 = sin 2 θ 2M, Γ 2 12 = γ13 3 = 1, Γ 2 33 = sinθ cosθ, Γ 3 23 = ctgθ. 5 Now let us wite down non-vanishing Ricci components: R 01 = M, R 00 = 2M M + 2Ṁ 2, Non-vanishing components of Einstein tenso ae: R 22 = 2M, R 33 = sin 2 θ2m. 6 G 0 0 = G 1 1 = 2M 2, G 1 0 = 2Ṁ 2, G 2 2 = G 3 3 = M. 7 We can wite down the enegy momentum tenso in the following fom[5, 8]: T µν = T n µν + T m µν, 8 whee the fist tem coesponds to the matte field I type and the othe one coesponds to the matte field II type[11]. Now let us wite down the expession of the enegy momentum tenso: T n µν = µl µ L ν, T m µν = ρ + P L µ N ν + L ν N µ + P g µν, µ = 2Ṁ 2, ρ = 2M 2, 3
Jounal of Physics: Confeence Seies 769 2016 012013 N µ = 1 2 P = M, L µ = δµ 0, 1 2M δµ 0 εδµ 1, L µ L µ = N µ N µ = 0, L µ N µ = 1. 9 Hee P - pessue, ρ - density and L, N - two null vectos. This model must be physically easonable so the enegy momentum tenso should satisfies weak, stong and dominant enegy conditions. It means that ρ must be positive and fo any non-spacelike vecto v α : and the vecto T αβ v α must be timelike. Stong and weak enegy conditions demand: T αβ v α v β > 0, 10 µ 0, ρ 0, P 0. 11 The dominant enegy condition imposes following conditions on the enegy momentum tenso: µ 0, ρ P 0. 12 Also let us intoduce following equations and notation. The equation of adial null geodesic has fom: dv d = 2 2M. 13 The existence of a family of non-spacelike futue-diected geodesics which oiginate at the cental singulaity in past is defined by sign dv d. If the sign is + then such family exists if the sign is - then such family does not exist. Also let us intoduce the following notation X 0 as: x 0 = v v 0, 0. 14 3. The Endstate of Gavitational Collapse In the beginning let us conside the simplest case when the equation of the state is: Then if we use 9 then we obtain the mass function: Pessue and density ae given by: P = ρ 3. 15 3M v, + 2M v, = 0, Mv, = Cv + Dv 1 3. 16 ρ = 2 Dv, 3 8 3 P = 2 Dv. 17 9 8 3 4
Jounal of Physics: Confeence Seies 769 2016 012013 Stong, weak and dominant enegy conditions demand: Also the condition M0, 0 = 0 demands: Dv 0, Ċv + Ḋv 1 3 0, The equation of the appaent hoizon is given by: Cv 0. 18 C0 = 0. 19 = 2Cv + 2Dv 1 3. 20 Now we can see that when v = 0 is the time of the singulaity fomation then the equation of the appaent hoizon is given by: = 2D0 1 3. 21 If D0 > 0 then > 0 and we have that the time of the appaent hoizon fomation is less than the time of singulaity fomation and in this case we have a black hole as a esult of gavitational collapse. Hence naked singulaity is fomed only then when D0 = 0. Now let us conside the question about existence of a family of non-spacelike futue-diected geodesics which oiginate at cental singulaity = v = 0 in the past. Let us substitute the mass function 16 into 13: dv d = 2 2Cv + Dv 1 3. 22 Now if we conside the it at 0, v 0 in 22 we obtain the following conditions fo mass function and accoding to these conditions we have naked singulaity as a esult of gavitational collapse: v 0, 0 v 0, 0 D0 = 0, Cv = a 0, Dv = b 0, 2 3 2a + 2b < 1, 23 whee a, b - abitay constants. So when conditions 23 ae satisfied then we have naked singulaity as a esult of collapse. Now let us conside moe geneal case when the equation of the state is given by: P = αρ, α 0, 1. 24 3 Hee we use 9 and obtain the mass function: Mv, = Cv + Dv 1 2α. 25 5
Jounal of Physics: Confeence Seies 769 2016 012013 Now if we use 25 we obtain expession fo the pessue and the density: P = 2α 1 2α Dv 2α+2, µ = ρ = 2 1 2α Dv 2α+2, 2Ċv + Ḋv1 2α 2. 26 We can easily see that in geneal case stong, weak and dominant enegy conditions impose the same equiements as in case when the equation of state is P = 1 3ρ. Also it is not difficult to show that conditions fo the mass function ae the same lake in pevious case. So we have given conditions fo the mass function when the equation of the state is P = αρ, α 0, 1 3 ] and accoding to these conditions we obtain the naked singulaity as a esult of gavitational collapse. Now we give conditions fo mass function when the singulaity is gavitationally stong. 4. The Stength of the Singulaity In this section we use the definition which was given in the pape[9]. gavitationally stong if: The singulaity is τ 2 ψ > 0, τ 0 ψ = R αβ K α K β, K α = dxα dτ, 27 hee τ - affine paamete, K α is tangent vecto to geodesic at the singulaity. In the pape[5] it has been shown that in ou case the equation 27 can be witten in the following fom: τ 2 ψ = 1 τ 0 4 X2 0 χ 4αµ2 2α 1 X2α 1 0, 28 whee χ is an abitay constant and Expession fo X 0 was given in 14. Now let us substitute 25 into 28 we obtain: χ 4α 2Cv+Dv 1 2α 2 2 1 4 X2 0 2α 1 X 2α 1 0. 29 Now it is not difficult to obtain conditions fo the mass function 25 when the singulaity is stong: χ 0, v 0 Ḋv = +, v 0 Ċv = +, Ċv + Ḋv1 2α v 0, 0 2 = +. 30 Now let us give an explicit example when the singulaity is stong. When eithe Cv o Dv in the fom λ 2 v γ, whee λ is eal constant, γ 0, 1. 6
Jounal of Physics: Confeence Seies 769 2016 012013 5. The Analogue of Vaidya Metic In this section we povide the metic which is the analogue of Vaidya metic in the case of otation. Fo this pupose we conside Ke metic which is given by: ds 2 = 1 2M ρ 2 dt 2 4am sin2 θ ρ 2 dtdϕ+ ρ2 δ d2 +ρ 2 dθ 2 + 2 +a 2 +2a 2 M sin 2 θ sin 2 θdϕ 2, 31 hee M-the mass of the black hole, a-its angula momentum, ρ 2 δ = 2 2M + a 2. Now let us go to a new coodinate v by using following fomula: Then we obtain the metic which is given by: = 2 + a 2 cos 2 θ and v = t + 3 d. 32 2Mδ ds 2 = 1 2Mv, ρ 2 dv 2 2 1 4aMv, sin 2 θ ρ 2 dϕ 2Mv, 3 ρ 2 2Mv, δ dvd dv + 3 2M, vδ d + a 2 cos 2 θ d 2 + ρ 2 dθ 2 + 2 + a 2 + 2a 2 Mv, sin 2 θ sin 2 θdϕ 2. 33 δ Now it is not had to see that when a 0 then the metic becomes Vaidya metic: ds 2 = 1 2Mv, dv 2 2dvd + 2 dω 2. 34 We can use this metic to conside the gavitational collapse in the case of otation but we obtain vey difficult equations. 6. Conclusion In this pape we have consideed the endstate of gavitational collapse of genealised Vaidya spacetime in tems of eithe a black hole o a naked singulaity. Also we have given conditions fo the mass function when the naked singulaity is fomed and also conditions when the singulaity is stong. The naked singulaity fomation at the end state of gavitational collapse of spheically symmetic object is one of the oppotunities fo explanation of Penose geodesics natue. If the naked singulaity was fomed in case of gavitational collapse with otation then we would can explain emegence in the egosphee fom a egion inside gavitational adius not only Penose geodesics but so-called white hole geodesics which was classified in the pape[10]. Also we povided the new metic which is the analogue of Vaidya metic in the case of otation. Acknowledgments The autho says thanks to pofesso D.Ph.-M.Sc. A.A Gib fo scientific discussion and this wok was suppoted by RFBR gant 15-02-06818-a and Dynasty Foundation. 7
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