Gravastar formation: What can be the evidence of a ?

Ken-ichi Nakao, Chul- Yoo, and Tomohiro Harada

Citation Physical Review D, 99(4); 044027 Issue Date 2019-02-15 Type Journal Article Textversion Publisher ©2019 American Physical Society. The following article appeared in Physical Review D and Right may be found at https://doi.org/10.1103/PhysRevD.99.044027 URI http://dlisv03.media.osaka-cu.ac.jp/il/meta_pub/G0000438repository_24700029-99-4-044027 DOI 10.1103/PhysRevD.99.044027

SURE: Osaka City University Repository http://dlisv03.media.osaka-cu.ac.jp/il/meta_pub/G0000438repository

NAKAO, K.-I., YOO, C.-M., & HARADA, T. (2019). formation: What can be the evidence of a black hole? Physical Review D. 99. 044027. DOI: 10.1103/PhysRevD.99.044027. PHYSICAL REVIEW D 99, 044027 (2019)

Editors' Suggestion

Gravastar formation: What can be the evidence of a black hole?

Ken-ichi Nakao,1 Chul-Moon Yoo,2 and Tomohiro Harada3 1Department of Mathematics and Physics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan 2Gravity and Particle Cosmology Group, Division of Particle and Astrophysical Science, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan 3Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan

(Received 2 November 2018; published 15 February 2019)

Any observer outside black holes cannot detect any physical signal produced by the black holes themselves, since, by definition, the black holes are not located in the causal past of the outside observer. In fact, what we regard as black hole candidates in our view are not black holes but will be gravitationally contracting objects. As well known, a black hole will form by a gravitationally collapsing object in the infinite future in the views of distant observers like us. At the very late stage of the , the gravitationally contracting object behaves as a blackbody due to its . Due to this behavior, the physical signals produced around it (e.g., the quasinormal ringings and the shadow image) will be very similar to those caused in the eternal black hole . However, those physical signals do not necessarily imply the formation of a black hole in the future, since we cannot rule out the possibility that the formation of the black hole is prevented by some unexpected event in the future yet unobserved. As such an example, we propose a scenario in which the final state of the gravitationally contracting spherical thin shell is a gravastar that has been proposed as a final configuration alternative to a black hole by Mazur and Mottola. This scenario implies that time necessary to observe the moment of the gravastar formation can be much longer than the lifetime of the present civilization, although such a scenario seems to be possible only if the dominant condition is largely violated.

DOI: 10.1103/PhysRevD.99.044027

I. INTRODUCTION outside J−ðλÞ cannot causally affect the observer λ. As can be seen in Fig. 1, the black hole is outside the causal past of The black hole is defined as a complement of the causal the observer λ, and hence any signal detected by the observer past of the future null infinity (see, e.g., [1,2])or,in λ (e.g., the black hole shadow, the quasinormal ringing of physical terminology, a domain that is outside the view gravitational radiation, the relativistic jet produced through of any observer located outside it. As well known, not the Blandford-Znajek effect [5]) cannot be caused by the only but also many modified theories of back hole itself, although they strongly suggest the formation gravity predict the formation of black holes through the of the black hole as can be seen in Fig. 1.Aswellknown, gravitational collapse of massive objects in our Universe. the black hole will form after infinite time has elapsed in Many black hole candidates have been found through the view of the distant observers. electromagnetic (see, for example, [3]) and gravitational The black hole is often explained as an invisible radiations [4]. , but rigorously speaking, this explan- Hereafter, any discussion in this paper will be basically ation is inappropriate. We call an object invisible if it is in based on general relativity. Figure 1 is the conformal diagram that describes the formation of a black hole our view but does not emit anything detectable to our eyes through the gravitational collapse of a spherical massive or detectors. However, the black hole is located outside the object; the region shaded by gray is the black hole, the view of the outside observer; this is the reason why the region shaded by green is the collapsing massive object, the outside observer cannot see it. In the view of the outside dark red curve λ is the of a typical observer with observer, there is a gravitationally contracting object whose finite lifetime outside the black hole, and the region shaded surface is asymptotically approaching the corresponding by blue is the causal past of the observer often denoted by . J−ðλÞ. The causal past of the observer is defined as a set of Although the black hole is a promising final configu- all events which can be connected to λ by causal curves, ration of a gravitationally collapsing object in the frame- i.e., timelike or null curves; we believe that our situation in work of general relativity, various alternatives have been our Universe is similar to the observer λ. Thus any event proposed (see, for example, [6] and references therein).

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nothing is emitted from the . Almost the same quasinormal mode spectrum of gravitational waves as that of an eternal black hole spacetime will be generated around the contracting object in the very late stage and detected by distant observers. However, it should be noted that we cannot conclude from these observables that the black hole must form, since there is always the possibility that the formation of the black hole is prevented by some unexpected events and the contracting object settles into some alternative to the black hole in the future. In this paper, we revisit a very simple model which describes the gravitational collapse of an infinitesimally thin spherical shell and offer a scenario of the gravitational collapse accompanied by the formation of not a black hole but a gravastar that has been proposed as a final configu- ration of a gravitationally collapsing object alternative to a black hole by Mazur and Mottola [7]. Our model shows that it is observationally very important when the gravastar formation begins. If the gravastar formation occurs in the FIG. 1. The conformal diagram of the black hole formation very late stage of the gravitational collapse, the observers through the gravitational collapse of a spherical object is depicted. like as λ will get shadow images and a quasinormal mode spectrum of gravitational waves which are almost the same

null to the center as those of the maximally extended Schwarzschild space- of shadow image time with the boundary condition under which nothing null to the rim 1 of shadow image emerges from the white hole. In this scenario, the unex- p pected event to prevent the black hole formation is the black hole gravastar formation. This paper is organized as follows. In Sec. II, we briefly review the basic equations to treat an infinitesimally thin spherical massive shell. In Sec. III, based on the analyses of null rays in the spacetime with a spherical massive shell in white hole Appendix A, we discuss why a massive object without the event horizon is regarded as a black hole candidate in the very late stage of its gravitational collapse in the view of null to the off-center points distant observers. Then, we give a model which represents a of shadow image decay of the dust shell into two concentric timelike shells in FIG. 2. Typical null geodesics emanated from the event p in Sec. IV. In Sec. V, we show a scenario in which a gravastar past direction toward the shadow image observed at p are forms in the very late stage of the gravitational collapse of depicted in the conformal diagram of the maximally extended the dust shell; the gravastar formation is triggered by the Schwarzschild black spacetime. From this figure, we see that the decay of the dust shell. Section VI is devoted to summary shadow image is not produced by the absorption of photons into and discussion. In Appendix B, we show that the Bianchi the black hole but is an image of the white hole with no radiation. identity leads to the conservation of the four-momentum at If the white hole emits photons, the shadow image can be colored. the decay event. In this paper, we adopt the abstract index notation, the We usually think that if a black hole candidate is not a sign conventions of the metric and Riemann tensors in black hole, it should be a static or stationary compact Ref. [2], and basically the geometrized unit in which object and believe that we will find differences from the Newton’s gravitational constant and the speed of black hole in observational data [6]. As mentioned in the are one. If convenient, we adopt natural units with notice. above, any observer sees not black holes but gravitation- ally contracting objects and regards them as black holes. 1 In the very late stage of the gravitational collapse of In the case of the maximally extended Schwarzschild space- time, the so-called black hole shadow is the image of the white the massive object, distant observers can take a photo of hole in the sense that if the white hole emits photons whose color the same shadow image as that in the eternal black hole is blue to distant observers, the shadow images are blue (see spacetime with the boundary condition under which Fig. 2).

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II. EQUATION OF MOTION OF A shell is a spacetime singularity, we can derive its equation SPHERICAL SHELL of motion from the Einstein equations through Israel’s formalism [9], since the singularity is so weak that its In this section, we give basic equations to study the intrinsic metric on the world hypersurface of the shell exists motion of a spherically symmetric massive shell which is and the extrinsic curvature defined on each side of the infinitesimally thin and generates a timelike hypersurface world hypersurface is finite. Hence, hereafter, we do not through its motion. We will refer to this hypersurface as the regard the shell as a spacetime singularity. world hypersurface of the shell. The world hypersurface We cover the neighborhood of the world hypersurface of of the shell divides the spacetime into two domains. These the shell by the Gaussian normal coordinate λ, where ∂=∂λ domains are denoted by Dþ and D−. The situation is is a unit vector normal to the shell and directs from D− to understood by Fig. 3. Dþ. Then, the sufficient condition to apply Israel’s for- D The geometry of the domains are assumed to be malism is that the stress-energy tensor is written in the form described by the Schwarzschild–de Sitter spacetime; the infinitesimal world interval is given by ab ¼ abδðλ − λ Þ ð Þ T S s ; 3 1 2 ¼ − ð Þ 2 þ 2 þ 2ð θ2 þ 2θ ϕ2Þ where the shell is located at λ ¼ λ , δðxÞ is Dirac’s delta ds F r dt ð Þ dr r d sin d ; s F r function, and Sab is the surface stress-energy tensor on ð1Þ the shell. We impose that the metric tensor gab is continuous even a with at the shell. Hereafter, n denotes the unit normal vector to the shell, instead of ∂=∂λ. The intrinsic metric of the world hypersurface of the shell is given by 2M Λ 2 FðrÞ¼1 − − r ; ð2Þ r 3 hab ¼ gab − nanb; ð4Þ where M and Λ are the parameter, the charge parameter, and the cosmological constant in the domains and the extrinsic curvature is defined as D, respectively. We should note that the time coordinate is ¼ − c d ∇ðÞ ð Þ not continuous at the shell and hence it is denoted by tþ in Kab h ah b c nd; 5 the domain Dþ and by t− in the domain D−, whereas r, θ, ðÞ and ϕ are everywhere continuous. where ∇c is the covariant derivative with respect to the Since the finite energy and the finite momentum con- metric in the domain D. This extrinsic curvature describes centrate on the infinitesimally thin region, the stress-energy how the world hypersurface of the shell is embedded into tensor diverges on the shell. This fact implies that the shell the domain D. In accordance with Israel’s formalism, the is categorized into the so-called scalar polynomial singu- Einstein equations lead to larity [8] through the Einstein equations. Even though the 1 Kþ − K− ¼ 8π S − h trS ; ð6Þ temporal ab ab ab 2 ab

u a where trS is the trace of Sab. Equation (6) gives us the condition of the metric junction. n a By the spherical symmetry of the system, the surface stress-energy tensor of the shell should be the following D− D+ form;

World hypersurface S ¼ σu u þ Pðh þ u u Þ; ð7Þ of the shell ab a b ab a b radial where, if we assume that the shell is composed of the perfect fluid type , σ, P, and ua are the energy per unit area, the tangential , and the four-velocity, respectively. Due to the spherical symmetry of the system, the motion of the shell is described in the form of t ¼ FIG. 3. A schematic diagram of the situation considered in TðτÞ and r ¼ RðτÞ, where τ is the of the shell. Sec. II. The vertical direction is timelike, whereas the horizontal The four-velocity is given by direction is spacelike. The world hypersurface of the shell is a a thick curve. The four-velocity u is the timelike unit tangent to μ ¼ð_ _ 0 0Þ ð Þ and na is the unit normal to the world hypersurface of the shell. u T; R; ; ; 8

044027-3 NAKAO, YOO, and HARADA PHYS. REV. D 99, 044027 (2019) where a dot represents a derivative with respect to τ. Then, dm dR2 þ 4πP ¼ 0; ð20Þ nμ is given by dτ dτ _ _ nμ ¼ð−R; T; 0; 0Þ: ð9Þ where m is the proper mass of the shell defined as

2 Together with uμ and nμ, the following unit vectors form an m ≔ 4πσR : ð21Þ orthonormal frame, By dividing both sides of Eq. (20) by dR=dτ, we have 1 θˆμ ¼ 0 0 0 ð Þ ; ; ; ; 10 dm r þ 8πPR ¼ 0: ð22Þ dR 1 ϕˆ μ ¼ 0; 0; 0; : ð11Þ r sin θ By giving the to determine P, Eq. (22) determines the dependence of m on R. Hereafter, we σ The extrinsic curvature is obtained as assume is positive so that the energy density is positive in the rest frame of the shell, and hence m is also positive. 1 F0 In general, the energy cannot be uniquely defined a b ¼ ̈þ ð Þ Kabu u _ R ; 12 within the framework of general relativity. However, in FT 2 the case of the spherically symmetric spacetime, quasi-

F local proposed by many researchers agree with θˆaθˆb ¼ ϕˆ aϕˆ b ¼ − a∂ j ¼ − _ ð Þ K K n a ln r T 13 the so-called Misner-Sharp energy (see, for example, ab ab D R Ref. [10]).TheMisner-Sharpenergyjustoneachside and the other components vanish, where of the shell is given as R F ¼ FðRÞ; ð14Þ M ¼ ð1 − Þ ð Þ 2 F : 23 and a prime represents a derivative with respect to its argument, i.e., Hence, the Misner-Sharp energy included by the shell is given by

dFðRÞ F0 ¼ : ð15Þ R dR M ¼ ð − Þ ð Þ 2 F− Fþ : 24 μ By the normalization condition u uμ ¼ −1, we have From Eq. (18),wehave qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 _ ¼ _ 2 þ ð Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T R F; 16 _ 2 m _ 2 F R þ F ¼ R þ F∓; ð25Þ R where we have assumed that the shell exists outside the black hole and ua is future directed. Substituting Eq. (16) where we have used Eq. (21). By taking the square of into Eq. (13), we have Eq. (25), we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 _ 2 m ˆa ˆb _ 2 R þ F ¼ E ∓ ; ð26Þ K θ θ ¼ − R þ F: ð17Þ 2 ab R R From Eqs. (6) and (17), we have where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 R M _ 2 _ 2 ¼ ð − Þ¼ ð Þ − R þ Fþ þ R þ F− ¼ 4πσ: ð18Þ E F− Fþ 27 R R 2m m Hereafter, we assume the weak energy condition σ ≥ 0. is the specific energy of the shell. By taking the square of Then, Eq. (18) leads to Eq. (26), we obtain the energy equations for the shell as follows: F− >Fþ: ð19Þ R_ 2 þ UðRÞ¼0 ð28Þ From the u-u component of Eq. (6), we obtain the following relations: with

044027-4 GRAVASTAR FORMATION: WHAT CAN BE THE EVIDENCE … PHYS. REV. D 99, 044027 (2019) m 2 static in the causal past of the observer, the observer can see UðRÞ¼F − E ∓ : ð29Þ 2R the reflected or transmitted null rays from some sources around it without suffering strong redshift: the distant Here note that unless P ¼ 0, m and E in Eq. (29) depend observer will merely see very old distorted images of on R. sources. By contrast, it is a subtle problem whether easily Since the left-hand side of Eq. (26) is positive, the right- appreciable differences appear in the quasinormal modes of hand side should also be positive, collapsing and static shells due to the long delay of echoes if the radius of the shell is very close to the gravitational radius. m E − > 0: ð30Þ 2R IV. DECAY OF A TIMELIKE SHELL: CONSERVATION LAW By substituting Eq. (26) into Eq. (16), we have In this section, we consider the decay process of a _ 1 m T ¼ E ∓ : ð31Þ spherical massive shell into two daughter spherical shells F 2R concentric with the parent shell. In the next section, this decay process is regarded as a trigger of the gravastar Here note again that Eq. (18) is obtained under the formation. assumption that the shell is located outside the black hole. We call the parent shell shell 0 and assume that shell 0 If the shell is in the black hole, Eq. (30) is not necessarily _ initially contracts but decays just before the formation of a satisfied, and accordingly, T is not necessarily positive. black hole. One of two daughter shells called shell 1 is located outside the other one called shell 2 (see Fig. 4). III. VERY LATE STAGE OF THE Shell 0, shell 1, and shell 2 divide the spacetime into three GRAVITATIONALLY CONTRACTING domains: D0 is the domain whose boundary is composed of SHELL shell 0 and shell 2, D1 is the domain whose boundary is composed of shell 0 and shell 1, and D2 is the domain In Appendix A, by studying null rays in the spacetime whose boundary is composed of shell 1 and shell 2. with a spherical shell, we show that the contracting shell The infinitesimal world intervals of the three domains Di with the radius very close to its gravitational radius (i ¼ 0, 1, 2) are given as effectively behaves as a blackbody due to its gravity, even though the material of the shell causes the specular dr2 reflection of or is transparent to null rays: both the null ds2 ¼ −F ðrÞdt2 þ þ r2ðdθ2 þ sin2 θdϕ2Þ; ð32Þ i i F ðrÞ ray reflected by the shell and that transmitted through the i shell suffer the large redshift or are trapped in the where neighborhood of the shell. Hence, the behavior of any physical field in this spacetime will be very similar to those 2M Λ F ðrÞ¼1 − i − i r2: ð33Þ in the maximally extended Schwarzschild spacetime with i r 3 the boundary condition under which nothing appears from the white hole: the contracting shell corresponds to the white hole horizon. In the late stage of the gravitational collapse, the image of the shell and the spectrum of the temporal quasinormal modes will be very similar to the black hole Shell 2 Shell 1 shadow and the quasinormal modes of the Schwarzschild a u(2) D2 spacetime. By contrast, the static shell will show images distinctive from the black hole shadow, if it is composed a u(1) of transparent or reflective materials, since, as shown in Appendix A, the angular frequency of a null ray perfectly D0 d D1 reflected at or transmitted through the static shell is the same as that of the incident one. Even if the inside of the a u(0) static shell is not empty but filled with the vacuum energy, the result will be unchanged as long as the vacuum energy radial is transparent to the null rays. The time necessary for traveling from the source of the null rays to the distant observer can be very long due to the strong redshift if the Shell 0 radius of the static shell is very close to the gravitational radius. However, even though its radius is very close to the FIG. 4. The schematic diagram representing the decay of shell 0 gravitational radius, as long as the shell has already been into shell 1 and shell 2 is depicted.

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For later convenience, we introduce the dyad basis We require the conservation of four-momentum at d, related to the two-sphere whose components are, in all a ¼ a þ a ð Þ domains, given as mð0Þuð0Þ mð1Þuð1Þ mð2Þuð2Þ; 44 1 where θˆμ ¼ 0; 0; ; 0 ; ð34Þ r ≔ 4π 2 σ ð Þ mðIÞ RðIÞ ðIÞ: 45 ˆ μ 1 ϕ ¼ 0; 0; 0; : ð35Þ σ r sin θ Note that mðIÞ is positive since we assume ðIÞ is positive. The derivation of the conservation law from the Bianchi By virtue of the spherical symmetry, the surface stress- identity is shown in Appendix B. energy tensor of shell I (I ¼ 0, 1, 2) is given in the form The u component of Eq. (44) leads to ab ¼ σ a b þ ab ð Þ ¼ Γ þ Γ ð Þ SðIÞ ðIÞuðIÞuðIÞ PðIÞH ; 36 mð0Þ mð1Þ ð1Þ mð2Þ ð2Þ; 46 σ a whereas the n component leads to where ðIÞ, PðIÞ, and uðIÞ are the surface energy density, the tangential pressure, and the four-velocity of shell I, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ¼ ϵ Γ2 − 1 þ ϵ Γ2 − 1 ð Þ respectively, and mð1Þ ð1Þ ð1Þ mð2Þ ð2Þ ð2Þ : 47 ab ¼ θˆaθˆb þ ϕˆ aϕˆ b ð Þ H : 37 Since mðJÞ is positive, Eq. (47) implies that ϵð1Þ ¼þ1 and ϵð2Þ ¼ −1 should hold in the situation we consider. We assume that σð Þ is positive. I From Eq. (46),wehave The radial coordinate of the decay event d is denoted by ¼ r rd. Hereafter, the time and radial coordinates of shell I 2 2 2 2 2 mð2ÞΓð2Þ ¼ mð0Þ − 2mð0Þmð1ÞΓð1Þ þ mð1ÞΓð1Þ; ð48Þ are denoted by TðIÞi and RðIÞ, where i is the index to specify ¼ 0 the time coordinate in the domain Di (i , 1, 2): as whereas, from Eq. (47), we have mentioned, the time coordinate is not continuous at the 2 2 2 2 2 shells. Then, we introduce the orthonormal basis of the mð2ÞΓð2Þ ¼ mð1ÞðΓð1Þ − 1Þþmð2Þ: ð49Þ center of mass frame at d. The components of them are given as Equations (48) and (49) lead to α _ _ m2 þ m2 − m2 uð0Þ ¼ðTð0Þi; Rð0Þ; 0; 0Þ; ð38Þ ð0Þ ð1Þ ð2Þ i Γð1Þ ¼ : ð50Þ 2mð0Þmð1Þ _ α Rð0Þ _ n ¼ ;Fðr ÞTð0Þ ; 0; 0 ; ð39Þ ð0Þi ð Þ i d i Through the similar procedure, we obtain Fi rd 1 2 þ 2 − 2 ˆα mð0Þ mð2Þ mð1Þ θ ¼ 0; 0; ; 0 ; ð40Þ Γð2Þ ¼ : ð51Þ rd 2mð0Þmð2Þ 1 ϕˆ α ¼ 0 0 0 ð Þ Equations (50) and (51) lead to ; ; ; θ ; 41 rd sin mð0Þ ¼ mð1ÞΓð1Þ þ mð2ÞΓð2Þ ≥ mð1Þ þ mð2Þ: ð52Þ where i ¼ 0 (i ¼ 1) represents the components in D0 (D1), and a dot means the derivative with respect to the proper From Eq. (43) with J ¼ 1,wehave time of shell 0. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t t 2 t Hereafter, we assume that the decay occurs before shell 0 uð1Þ1 ¼ Γð1Þuð0Þ1 þ Γð1Þ − 1nð0Þ1 forms a black hole, i.e.,   Γ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ ð1Þ rd mð0Þ 2 Rð0Þ ð Þ 0 ð Þ ¼ ðF0 − F1Þ − þ Γ − 1 ; F2 rd > : 42 2 2 ð1Þ F1 mð0Þ rd F1 a ¼ 1 The four-velocity uðJÞ (J ,2)atd is written in the ð53Þ form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where we have used Eqs. (27) and (31) for ut , and a ¼ Γ a þ ϵ Γ2 − 1 a ð Þ ð0Þ1 uðJÞ ðJÞuð0Þ ðJÞ ðJÞ nð0Þ; 43 ¼ ð Þ Fi Fi rd . On the other hand, by using Eq. (31),wehave   where ΓðJÞ is a positive number larger than one, and ϵðJÞ ¼ 1 r mð1Þ 1 is the sign factor which will be fixed by the momentum t ¼ d ð − Þ − ð Þ uð1Þ1 2 F2 F1 2 : 54 conservation. F1 mð1Þ rd

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temporal Then Eqs. (53) and (54) lead to    2 D2 mð1Þ 2mð1Þ r mð0Þ Shell 2 Shell 1 ¼ þ þ Γ d ð − Þ − de Sitter domain F2 F1 2 ð1Þ F0 F1 (neutral null) (neutral timelike) r r 2mð0Þ 2r d d d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a u u(1) 2 _ (2) þ Γð1Þ − 1Rð0Þ : ð55Þ

D0 d1 D1 By the similar procedure starting from Eq. (43) with J ¼ 2, Minkowski domain Schwarzschild domain a we have u(0)

2    radial mð2Þ 2mð2Þ mð0Þ ¼ þ − Γ rd ð − Þþ F2 F0 2 ð2Þ 2 F0 F1 2 Shell 0 rd rd mð0Þ rd (neutral timelike dust) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 _ − Γð2Þ − 1Rð0Þ : ð56Þ

FIG. 5. The schematic diagram representing the formation of a By using Eqs. (50) and (51), we can see that Eq. (55) is gravastar triggered by decay of shell 0 into shell 1 and shell 2 is equivalent to Eq. (56). The momentum conservation (44) depicted. uniquely determines the geometry of D2 which appears after the decay event d if we fix the values of rd, F0, F1, _ so that the rhs of Eq. (58) is less that 2M1. Then, by mð0Þ, mð1Þ, and mð2Þ; Rð0Þ is determined through Eqs. (28) investigating the effective potential Uð0Þ, we can easily see and (29) except for its sign that we have to choose. that the allowed domain for the motion of shell 0 is V. GRAVASTAR FORMATION 0 ≤ ≤ M1 ð Þ Rð0Þ 2 ð1 − Þ 60 In this section, we consider the gravitational collapse of Eð0Þ Eð0Þ shell 0 accompanied by the gravastar formation. Here, we 1 2 1 will adopt the gravastar model devised by Visser and for = 0. the effective potential of shell 0 as Shell 2 shrinks to zero radius, so that the innermost domain 2 D0 disappears at some stage (see Fig. 5). By contrast, shell mð0Þ ð Þ¼1 − þ ð Þ 1 corresponds to the crust of the gravastar. The decay event Uð0Þ Rð0Þ Eð0Þ 2 ; 57 Rð0Þ of shell 0 and its areal radius are denoted by d1 and rd1, respectively. where mð0Þ is constant due to the conservation law (22), and It is observationally very important when the gravastar ¼ hence Eð0Þ M1=mð0Þ is also constant. formation starts. In Ref. [7], the gravastar formation is From Eq. (30),wehave implicitly assumed to start when the radius R of the contracting object satisfies R − 2M ≃ l ,wherel M1 pl pl ð Þ −33 Rð0Þ > 2 ; 58 (≃1.6 × 10 cm) is the Planck length. The timescale in 2E ð0Þ which the radius of the collapsing object satisfies 0 2M1. We are ð0Þ 0 interested in the case that shell 0 contracts and forms a From Eqs. (28), (29),and(31), we can see that once < R − 2M ≪ 2M is satisfied, the time evolution of the radius black hole, i.e., Rð0Þ ≤ 2M1, if the decay of shell 0 does not − t ¼ ≃ 2M occur. Hence, we assume of a dust shell (m constant) is given by R Const: × e , where t is the proper time for an asymptotic observer. Thus, 1 − 2 ≃ the timescale in which R M lpl is achieved will be Eð0Þ > ; ð59Þ 2 much less than our average lifetime if the mass M of the

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1 108 ab contracting object takes M⊙

dM π2 It is easy to see that ka is null in this limit. Furthermore, we ¼ − T4 A ; ð62Þ ð2Þ dt 30 BH H have in natural units, where, m being the Planck mass, T ¼ ab pl BH Sð2Þ Mð2Þ 2 → a b ð Þ m =8πM is the Bekenstein-Hawking temperature, and A 2 kð2Þkð2Þ: 68 pl H Eð2Þ 4πR 16π 2 4 ð2Þ is the horizon area M =mpl [12]. We assume that after a small fraction ϵ (≪ 1) of the initial mass of the As expected, shell 2 becomes the null dust in this limit. collapsing objects is released through the particles created ab Although Sð2Þ itself diverges due to the Lorentz contraction, by the semiclassical effect, the gravastar formation begins. the Misner-Sharp mass kept by shell 1 is finite by assumption Then by solving Eq. (62), we can see that the timescale tϵ [see Eq. (24)]: this divergence should be absorbed in the in which the initial mass M of the contracting object i integral measure (please see Ref. [13] for the proper stress- becomes ð1 − ϵÞM is given by i energy tensor of the null shell). In the massless limit of shell 2, 3 we have Mð2Þ at the decay event, from Eq. (56),inthe Mi tϵ ¼ 7680π½1 þ OðϵÞϵ following form: m4 pl " 3 2 − 2 67 Mi mð0Þ mð1Þ mð0Þ ¼ 3.1 × 10 ½1 þ OðϵÞϵ yr: ð63Þ Mð2Þj ¼ Eð0Þ þ d1 2 2 M⊙ mð0Þ rd1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 2 If this is true, asymptotic observers should wait to observe mð0Þ the gravastar formation for a very long time after the þ Eð0Þ − − F1 : ð69Þ 2r 1 gravitational collapse has begun: the time will be much d longer than the age of the Universe for a black hole of the The cosmological constant in D2 is determined at the mass larger than the solar mass if ϵ ≳ 10−50. decay event d1 through Anyway, the radius of shell 0 might be very close to the gravitational radius in the domain D1 when the gravastar 6 Λ2 ¼ Mð2Þj : ð70Þ formation starts. Hence, hereafter we assume so. 3 d1 rd1 A. Motion of shell 2 Then the Misner-Sharp energy of shell 2 is a function of the Let us start on the discussion about shell 2. We assume radius of shell 2, that shell 2 moves inward with the energy much larger than Λ ¼ M ≫ 1 M Rð2Þ 2 3 its proper mass, i.e., Eð2Þ ð2Þ=mð2Þ , where ð2Þ is Mð2Þ ¼ ðF0 − F2Þ¼ Rð2Þ: ð71Þ the Misner-Sharp energy of shell 2. We introduce 2 6

a As can be seen from Eq. (71), Mð2Þ vanishes when the mð2Þuð2Þ a ≔ ð Þ radius of shell 2 becomes zero, or in other words, shell 2 kð2Þ M 64 ð2Þ disappears when it shrinks to the symmetry center r ¼ 0.

044027-8 GRAVASTAR FORMATION: WHAT CAN BE THE EVIDENCE … PHYS. REV. D 99, 044027 (2019)

B. Motion of shell 1 is necessary so that Eq. (78) is satisfied, since we require σð1Þ ≥ 0, or equivalently, mð1Þ ≥ 0; the allowed domain for From Eq. (43), the of shell 1 at d1 is written in the form the motion of shell 1 is bounded from above. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ _ 2 _ 1. Dissipation through further decay Rð1Þ ¼ Γð1ÞRð0Þ þ Γð1Þ − 1F1Tð0Þ1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Shell 1 is the crust of the gravastar. It is dynamical and 2 mð0Þ ¼ −Γ − − hence should dissipate its energy so that the gravastar is ð1Þ Eð0Þ 2 F1 rd1 stable and static. Chan et al. studied the gravastar formation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by taking into account a dissipation through the emission of mð0Þ þ Γ2 − 1 − ð Þ null dust [14]. In this paper, instead of the emission of the ð1Þ Eð0Þ 2 : 72 rd1 null dust, we assume that the crust, shell 1, emits outward shell 3 at the event d2 with r ¼ r 2 and becomes static and _ d Hence, Rð1Þ is positive at d1, if and only if stable; the static crust of the gravastar is called shell 4. This process is equivalent to the decay of shell 1 into shell 3 and 2 2 1 mð0Þ shell 4 (see Fig. 6). Γ > Eð0Þ − ð Þ ð1Þ 2 73 The domain between shell 3 and shell 4 is denoted by D3. F1 rd1 Replacing shell 0, shell 1, shell 2, D0 and D2 by shell 1, D2 D3 holds. Taking into account Eqs. (50) and (52), Eq. (73) shell 3, shell 4, and in Eq. (55), the same argument as that in Sec. IV is applied, and we obtain leads to the condition on mð1Þ as " 2    mð3Þ 2mð3Þ r 2 mð1Þ mð0Þ mð0Þ ¼ þ þ Γ d ð − Þ − F3 F1 2 ð3Þ F2 F1 mð1Þj 0 implies D0 d1 D1 Minkowski domain Schwarzschild domain 1 Λ2 3 mð1Þ M1 − Rð1Þ − > 0: ð78Þ mð1Þ 6 2Rð1Þ Shell 0 It is easy to see that, irrespective of the equation of state of (neutral timelike dust) shell 1, FIG. 6. The schematic diagram representing the stabilization of 1 the gravastar due to the decay of shell 1 into shell 3 and shell 4. 6 3 ≔ M1 ð Þ Rð1Þ

044027-9 NAKAO, YOO, and HARADA PHYS. REV. D 99, 044027 (2019)

2 2 2 2 and, in this Sec. VB1, all quantities are evaluated at d2. mð1Þ þ mð4Þ mð1Þ mð1Þ − mð4Þ pffiffiffiffiffiffi þ − _ − ¼ 0 Here, as in the case of shell 2, we take the limit mð3Þ → 0 2 Eð1Þ 2 2 Rð1Þ F2 ; mð1Þmð4Þ rd2 mð1Þmð4Þ under the assumption of Pð3Þ ¼ wð3Þσð3Þ with wð3Þ fixed: ð Þ this limit is equivalent to the assumption that shell 3 is a 89 null dust. From Eq. (80),wehave and hence we have 2 2 mð1Þ − mð4Þ mð1Þ ¼ þ − þ _ ð Þ mð1Þ pffiffiffiffiffiffi F3 F1 Eð1Þ 2 Rð1Þ : 82 þ þ _ 2 − 2 mð1Þrd2 rd2 Eð1Þ Rð1Þ mð4Þ mð1Þmð4Þ F2 2r 2 d mð1Þ Then, from this result, we have þ þ − _ 2 ¼ 0 ð Þ Eð1Þ 2 Rð1Þ mð1Þ : 90 rd2 rd2 Eð4Þ ¼ ðF2 − F3Þ 2mð4Þ The above quadratic equation for mð4Þ has a degenerate root rd2 rd2 pffiffiffiffiffiffi ¼ ðF2 − F1Þþ ðF1 − F3Þ 2 2 mð1Þ F2 mð4Þ mð4Þ ¼ ð Þ mð4Þ mð1Þ ; 91 2 2 _ − Eð1Þ þ 2 þ Rð1Þ mð1Þ mð1Þ mð4Þ mð1Þ rd2 ¼ − − þ _ Eð1Þ 2 Eð1Þ 2 Rð1Þ mð4Þ mð1Þmð4Þ rd2 where we have used Eq. (86). By using Eq. (86), we can see 2 þ 2 2 − 2 _ mð1Þ mð4Þ mð1Þ mð4Þ mð1Þ that Eq. (88) is satisfied only if Rð1Þ is positive. Thus, we ¼ þ − _ 2 Eð1Þ 2 2 Rð1Þ : _ mð1Þmð4Þ mð1Þmð4Þ rd2 consider the only situation in which Rð1Þ is positive at the _ ð83Þ event d2. Rð4Þ vanishes if and only if the proper mass of shell 4 satisfies Eq. (91), and hereafter we assume so. By The crust of the gravastar is shell 4 after the event d2.As virtue of the future directed condition of the four-velocity mð1Þ mentioned, since the gravastar becomes static after the of shell 1, i.e., Eð1Þ − 2 > 0, and Eq. (86),wehave _ rd2 event d2, Rð4Þ always vanishes. Since we have mð1Þ _ Eð1Þ − − Rð1Þ > 0: ð92Þ 2 2 _ 2 mð4Þ rd2 Rð4Þ ¼ −Uð4ÞðRð4ÞÞ ≔ Eð4Þ þ − F2ðRð4ÞÞ; ð84Þ 2Rð4Þ Here note that Eq. (83) can be rewritten as _ Eq. (83) and Rð4Þj ¼ 0 lead to 2 2 Rð4Þ¼r 2 þ d mð4Þ mð1Þ mð4Þ mð1Þ _ Eð4Þ − ¼ Eð1Þ − − Rð1Þ   2 2 2 2 þ 2 2 − 2 2 rd2 mð1Þmð4Þ rd2 mð1Þ mð4Þ mð1Þ mð1Þ mð4Þ _ Eð1Þ þ − Rð1Þ − F2 ¼ 0: m2 − m2 2mð1Þmð4Þ 2r 2 2mð1Þmð4Þ ð1Þ ð4Þ mð4Þ _ d þ þ Rð1Þ: ð93Þ 2mð4Þr 2 mð1Þ ð85Þ d _ 0 Since we have Hence, if Rð1Þ > holds, the future directed condition, − 2 0 Eð4Þ mð4Þ= rd2 > , for shell 4 also holds. The decay of 2 2 _ 2 mð1Þ mð1Þ shell 1 to make the gravastar static is possible. R ¼ Eð1Þ þ − F2 ¼ Eð1Þ − − F1 ð1Þ 2 2 The effective potential of shell 4, Uð4Þ, vanishes at Rð4Þ ¼ rd2 rd2 r 2 by assumption. The first- and second-order derivatives ð86Þ d of Uð4Þ should vanish and be positive, respectively, at ¼ Rð4Þ rd2 so that the gravastar is stably static. Hereafter we at Rð1Þ ¼ r 2, the following inequality holds: d assume so; these assumptions partly determine the equation of state of shell 4 as follows. mð1Þ _ Eð1Þ þ > jRð1Þj: ð87Þ Equation (18) implies that the surface energy density of 2r 2 d shell 4 is given in the form By the same argument as that of Eq. (52), we have 1 hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σð4Þ ¼ F2ðRð4ÞÞ − Uð4ÞðRð4ÞÞ 0 ð Þ 4πRð4Þ mð1Þ >mð4Þ > : 88 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii − F3ðRð4ÞÞ − Uð4ÞðRð4ÞÞ ; ð94Þ Then, Eq. (85) leads to

044027-10 GRAVASTAR FORMATION: WHAT CAN BE THE EVIDENCE … PHYS. REV. D 99, 044027 (2019) and the tangential pressure of shell 4, Pð4Þ, is given from Eq. (22) in the form   1 1 R dU ðR Þ d 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4Þ ð4Þ ð4Þ Pð4Þ ¼ − ðσð4ÞRð4ÞÞ¼− þ σð4Þ 2Rð4Þ dRð4Þ 2 dRð4Þ 4 ½F2ðRð4ÞÞ − Uð4ÞðRð4ÞÞ½F3ðRð4ÞÞ − Uð4ÞðRð4ÞÞ

Λ2Rð4Þ M3 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð95Þ 2 24π F2ðRð4ÞÞ − Uð4ÞðRð4ÞÞ 8πRð4Þ F3ðRð4ÞÞ − Uð4ÞðRð4ÞÞ

¼ 0 ð Þ ≪ 1 ≪ At Rð4Þ rd2,wehave Let us estimate Ru. Since

and hence Ru defined as Eq. (79) is written in the form 2. Case of shell 1 expanding at d1   _ β2 First we consider the case of Rð1Þ > 0 at the first decay F1 4 2 2 R ¼ r 1 1 þ þ þ Oðβ ;β F1;F Þ ; u d 3 mð0Þ 1 event d1 with the areal radius rd1. Since we consider the 12Eð0ÞðEð0Þ − 2 Þ 2 rd1 case that rd1 is very close to M1, we have, from Eq. (74), ð103Þ mð0Þ pffiffiffiffiffiffi mð1Þj < ½1 þ OðF1Þ F1: ð98Þ d1 1 2ðEð0Þ − Þ where all quantities are evaluated at d1. Since rd2

The proper mass of shell 1 should be much smaller than mð0Þ. ¼ ½1 þ Oðβ2 Þ ð Þ We show the effective potential Uð1Þ in the case of rd1

Pð1Þ ¼ wð1Þσð1Þ ð99Þ with jwð1Þj ≤ 1 is too similar to distinguish from that of the ( r ) dust, even if it is depicted together in Fig. 7. The dominant U(1) energy condition for shell 1 is given by

σð1Þ ≥ jPð1Þj: ð100Þ

As long as the dominant energy condition is satisfied, the effective potential of shell 1 behaves as that shown in Fig. 7. r As mentioned below Eq. (86), shell 1 should decay into −1 _ 2M1 shell 3 and shell 4, when Rð1Þ > 0 so that the gravastar is static. Hence, shell 1 should decay before it bounces off FIG. 7. The effective potential Uð1Þ near Rð1Þ ¼ rd1 is depicted the potential barrier. The allowed domain for the motion in the case that shell 1 is the dust, i.e., Pð1Þ ¼ 0. We assume ¼ 1 001 2 ¼ 0 9 ¼ 10−1 of shell 1 is bounded from above as Eq. (79) and hence rd1 . × M1, Eð0Þ . , and mð1Þ mc. Shell 1 rd1

044027-11 NAKAO, YOO, and HARADA PHYS. REV. D 99, 044027 (2019)

ð Þ¼−αð − ÞþOðð − ÞnÞ ð Þ Since we consider the situation that rd1 is larger than but Uð1Þ r r Rb r Rb ; 107 2 2 very close to M1, rd2 should be very close to M1 from Eq. (104). Furthermore, from Eq. (101) and the inequality where α and n are positive constant and natural number 0 − ≤ Oðβ2 Þ

Λ 2Mð2Þj 2 2 ð Þ 2 2 d1 2 M1 2 M1

ð Þ ð Þ m2 − m2 Here again note that F2 rd2 >F3 rd2 holds because of ð0Þ ð1Þ Mð2Þj ¼ M1 þ OðF1Þ: ð109Þ 0 d1 2 Eð4Þ > [see Eq. (83) and the discussion below Eq. (91)]. mð0Þ From Eqs. (96), (97), and (106), Eqs. (96) and (97) imply σ ≪ ð4Þ Pð4Þ, and so the violation of the dominant energy Hence, Eq. (79) implies condition (100). This result is basically equivalent to that obtained by Visser and Wiltshire for their gravastar ¼ rd1 ½1 þ Oð Þ ð Þ Ru 2 1 F1 ; 110 model [11]. ð1 − β Þ3

β 3. Case of shell 1 contracting at d1 where has been defined as Eq. (102) and is less than but can be very close to unity, and hence R may be much larger We consider the case that shell 1 begins contracting at the u than rd1.Asaresult,rd2 can also be much larger than rd1, first decay event d1; the proper mass mð1Þ satisfies Eq. (76). and hence, as we will discuss later, the dominant energy As in the expanding case, we show the effective potential condition (100) can be satisfied by shell 4. However, as Uð1Þ in the case of Pð1Þ ¼ 0 in Fig. 8.Asinthecaseof shown below, the dominant energy condition is not satisfied expansion at d1, Uð1Þ of the equation of state (99) with ¼ at Rð1Þ Rb by shell 1. jwð1Þj ≤ 1 is too similar to that of the dust to distinguish Through the same prescription to derive Eqs. (94) and between them, even if they are depicted together in Fig. 8. (95), we have In this case, shell 1 does not bounce off the potential barrier  but directly forms a black hole through its contraction. Thus, 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ ¼ ð Þ − ð Þ in the contracting case, the equation of state of shell 1 cannot ð1Þ 4π F2 Rð1Þ Uð1Þ Rð1Þ j j ≤ 1 Rð1Þ be Eq. (99) with wð1Þ so that the gravastar forms. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Shell 1 should bounce off the potential barrier at some − ð Þ − ð Þ ð Þ 2 F1 Rð1Þ Uð1Þ Rð1Þ ; 111 radius Rb larger than M1 so that the black hole formation is halted. The effective potential should take the following form near d1: U(1) (r)

0.006 0.004 0.002 0 -0.002 U(1)( r ) R b -0.004 0 r r 2M1 d1 -0.006 -0.008 -0.01 -0.012 0 0.002 0.004 0.006 0.008 0.01 r −1 2M1

¼ 5 FIG. 8. The same as Fig. 7, but mð1Þ mc. In this case, shell 1 FIG. 9. The assumed effective potential Uð1Þ of shell 1 near ¼ begins contracting at d1 and then a black hole forms. R Rb is schematically depicted.

044027-12 GRAVASTAR FORMATION: WHAT CAN BE THE EVIDENCE … PHYS. REV. D 99, 044027 (2019) and   1 1 R dU ðR Þ d 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1Þ ð1Þ ð1Þ Pð1Þ ¼ − ðσð1ÞRð1ÞÞ¼− þ σð1Þ 2Rð1Þ dRð1Þ 2 dRð1Þ 4 ½F2ðRð1ÞÞ − Uð1ÞðRð1ÞÞ½F1ðRð1ÞÞ − Uð1ÞðRð1ÞÞ

Λ2Rð1Þ M1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð112Þ 2 24π F2ðRð1ÞÞ − Uð1ÞðRð1ÞÞ 8πRð1Þ F1ðRð1ÞÞ − Uð1ÞðRð1ÞÞ

rd2 Since we have Eð1Þj ¼ ½F2ðr 2Þ − F1ðr 2Þ: ð118Þ d2 d d 2mð1Þ

dUð1ÞðRð1ÞÞ ¼ −α 0 ð Þ σ 2 − 1 < ; 113 We depict ð4Þ and Pð4Þ as a function of rd2= M3 in dRð1Þ ¼ ¼ 0 9 ¼ 0 99 ¼ Rð1Þ Rb the case of Eð0Þ . , mð1Þ . mð0Þ at d1, and rd1 1.00001 × 2M1 for three values of v in Figs. 10–12. ¼ ¼ 1 00001 we obtain, from Eq. (112), We also show M3 in the case of rd2 rd1 . × 2M1 as a function of v in Fig. 13; the larger v is, the smaller 1 Λ M3 is. This behavior implies that if shell 1 has the larger j ≥− σ j þ p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiRb Pð1Þ ¼ ð1Þ ¼ outward velocity v>0, the larger energy should be Rð1Þ Rb 2 Rð1Þ Rb 24π ð Þ F2 Rb released through the emission of shell 3 so that shell 4 þ pMffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ð Þ is at rest. Furthermore, we depict M3 as a function of 2 : 114 8π ð Þ r 2=2M1 − 1 in Fig. 14 for three values of v; here note that Rb F1 Rb d rd2 is normalized by not M3 but M1. The mass parameter 2 M3 is a decreasing function of rd2. Due to Eq. (108) and since rd1 is very close to M1, ð Þ ≪ 1 We can see from Figs. 10 and 11 that the dominant F1 Rb holds. Hence, we have energy condition σð4Þ ≥ jPð4Þj is satisfied in the case of ≳ 1 04 2 2 rd2 . × M3, it is not so for rd2 very close to M3; the M1 ¼ 1 04 2 j ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≫ σ j ð Þ domain in Fig. 12 does not include rd2 . × M3 due Pð1Þ ¼ 2 ð1Þ ¼ : 115 Rð1Þ Rb 8π ð Þ Rð1Þ Rb Rb F1 Rb to the behavior of M3 shown in Figs. 13 and 14. Since rd2 is ≳ 1 04 the radius of the gravastar in its final state, if rd2 . × The dominant energy condition cannot be satisfied by shell 2M3 holds, the formed gravastar satisfies the dominant ¼ energy condition, even though the crust of the gravastar 1 at and around Rð1Þ Rb by continuity. Now we see the equation of state of shell 4 which is the does not in its formation process. The quantum gravita- crust of the gravastar after the second decay event d2. The tional effects should play an important role so that the surface energy density and the tangential pressure of shell 0.25 4, σð4Þ, and Pð4Þ are evaluated by using Eqs. (96) and (97). Although we have determined the effective potential Uð1Þ ¼ 0.2 of shell 1 in the only vicinity of Rð1Þ Rb as Eq. (107),we ¼ have not yet in the vicinity of Rð1Þ rd2. Thus, the value of _ 0.15 Uð1Þ, or equivalently, Rð1Þ at d2 is regarded as a free 2M1P(4) parameter. Once we assume the values of rd2 and 0.1 _ v ≔ Rð1Þj ; ð116Þ d2 0.05 2M1σ(4) we have 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 r 2 d2 −1 mð1Þj ¼ 4πσð1ÞRð1Þj 2M3 d2 d2 hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 2 2 FIG. 10. The surface energy density and tangential pressure of ¼ r 2 v þ F2ðr 2Þ − v þ F1ðr 2Þ ð117Þ d d d shell 4 are depicted as a function of the final radius of the gravastar ¼ ¼ 0 9 ¼ 0 99 Rð4Þ rd2. We assume Eð0Þ p. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, mð1Þ . mð0Þ at d1 and ¼ 1 00001 2 ¼ ð Þ ¼ 3 16 10−3 and rd1 . × M1,andv F1 rd1 . × .

044027-13 NAKAO, YOO, and HARADA PHYS. REV. D 99, 044027 (2019)

0.16 1.05

0.14 1 F1 ( d1 )

F1 ( d1 ) 0.12 0.95

2M1 0.1 (4) M3 0.9 M1 0.08 0.85

0.06 0.8

2M1 (4) =100 F1 ( d1 ) 0.04 0.75

0.02 0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 d2 −1 0.1 0.2 0.3 0.4 0.5 2M1 d2 −1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2M3 FIG. 14. The mass parameter M3 in the case of v ¼ F1ðr 1Þ, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 10 F1ðr 1Þ, and 100 F1ðr 1Þ is depicted as a function of r 2. FIG. 11. The same as Fig. 10 but v ¼ 10 F1ðrd1Þ ¼ d d d 3.16 × 10−2. As in Figs. 10–12, we assume Eð0Þ ¼ 0.9, mð1Þ ¼ 0.99mð0Þ at d1, ¼ 1 00001 2 and rd1 . × M1.

0.045

0.04

0.035

0.03 2M1 (4) 0.025

0.02 2M1 (4) 0.015

0.01

0.005 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d2 −1 2M3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FIG. 12. The same as Fig. 10 but v ¼ 100 F1ðrd1Þ ¼ −1 3.16 × 10 . From Fig. 14, we can see 0.74M1

1.05

1

0.95 FIG. 15. This is the case in which the observer λ will wrongly conclude that a black hole will form if the areal radius rd at M3 0.9 the beginning of the gravastar formation is sufficiently close to M1 the gravitational radius 2M1 in D1, but no event horizon forms. 0.85 In the domain D2 corresponds to the gravastar. Here shell 0 is 1 2 1 0.8 assumed to gravitationally contract, i.e., =

0.75 process accompanied by the violation of the dominant 0.7 0 0.05 0.1 0.15 0.2 0.25 0.3 energy condition is realized. Hence the gravastar formation should rely on the quantum gravitational effects, if it begins ¼ ¼ FIG. 13. The mass parameter M3 in the case of rd2 rd1 at the very late stage of the gravitational collapse, i.e., 0 − 2 ≪ 2 1.00001 × 2M1 is depicted as a function of v.

044027-14 GRAVASTAR FORMATION: WHAT CAN BE THE EVIDENCE … PHYS. REV. D 99, 044027 (2019)

As mentioned, it is observationally very important when observers, it may be impossible for the observers with finite the gravastar formation begins. If the gravastar formation lifetime to see the gravastar formation and hence such starts after the backreaction of the observers believe that the shell will form a black hole, even begins sufficiently affecting the evolution of the contracting if there is no event horizon. On the other hand, our analysis object, the gravitationally contracting object of the mass on a simplified formation scenario suggests that the larger than that of the solar mass will form a gravastar formation of gravastar with the radius extremely close to completely outside the causal past of observers with the that of the would-be horizon may be possible only with finite lifetime like us. In such a case, the observers will large violation of the dominant energy condition by the wrongly conclude that a black hole will form (see Fig. 15). crust of the gravastar.

VI. SUMMARY VII. SOME REMARKS Any observer outside black holes cannot detect any Here we should note that the Hawking radiation can also physical signal caused by the black holes themselves but not be the observable that is evidence of the event horizon see the gravitationally contracting objects and phenomena formation. As shown by Paranjape and Padmanabhan, caused by them; for observers outside black holes, the almost Planckian distribution of particles created through contracting objects will form black holes after infinite time the semiclassical effect will appear in the contracting shell lapses. In order to see why a contracting object seems to be model [12]. Hence, the gravastar formation model cannot a black hole even if there is not an event horizon but the be distinguished from the black hole spacetime through contracting object in our view, we have studied a very the particle creation due to the semiclassical effects if the simple model which describes the gravitational contraction gravastar formation starts at the too late stage of the of an infinitesimally thin spherical massive shell and gravitational collapse to be observed by the distant observ- studied null rays in such a situation. Even in the case that ers with finite lifetimes. This issue will also be discussed by the shell made of materials which causes specular reflection Harada et al. [15]. It might be interesting that the Planckian of or is transparent to the null rays, it behaves as a distribution is consistent to the approximate blackbody blackbody due to its gravity if its radius is very close to behavior of the shell at a very late stage of its gravitational its gravitational radius; incident null rays do not return collapse. from the shell or suffer indefinitely large redshift even if As mentioned, the gravastar formation might start after the they return. Hence, the shell at the very late stage of its effect of the Hawking radiation causes significant back- gravitational collapse is well approximated by the max- reaction effects on the gravitational collapse of a massive imally extended Schwarzschild spacetime with the boun- object. If it is really so, the gravitational collapse of the dary condition under which nothing comes from massive object with the mass larger than the solar mass will the white hole. Signals of the quasinormal ringing and not cause the gravastar formation within the age of the shadow images obtained in the spacetime with the shell will Universe. By contrast, the formation of the primordial black be, in practice, indistinguishable to those of the maximally hole with the mass much smaller than the solar mass should extended Schwarzschild spacetime for any distant observer be replaced by the primordial gravastar formation that is, in the very late stage of the gravitational collapse. In this in principle, observable for us [16,17].Althoughitisvery sense, the black hole shadow is not the appropriate name in difficult to observe compact objects with very small mass, the case of the observed black hole candidates, since it is they might be very important in order to find the unexpected not a shadow of a black hole but the image of the highly events. darkened of a gravitationally contracting Rigorously speaking, it is impossible for us to conclude, object. Even in the case of the black hole spacetime, the through any observation, that it is a black hole. It is a black hole shadow is not the appropriate name, since it is an profound fact that general relativity has predicted the advent image of the white hole. of domains of which the existence cannot be confirmed However, as we have shown, even though the observers through any observation. By contrast, if it is not a black hole, detect the quasinormal ringings and take photos of shadow we can, in principle, know that it is the case. It is necessary to images, those observables do not necessarily imply that the keep observing black hole candidates. event horizon will form by the contracting object. There always remains the possibility that the formation of the ACKNOWLEDGMENTS event horizon is prevented by some unexpected event. We have given a scenario in which such a situation is realized: a K. N. is grateful to Hideki Ishihara, Hirotaka Yoshino, and gravitational contraction of the dust shell suddenly stops colleagues at the elementary particle physics and gravity due to its decay into two daughter shells concentric with the group in Osaka City University for useful suggestions and parent shell, and then a gravastar forms. If the decay occurs discussions. T. H. is grateful to V. Cardoso for fruitful at the radius so close to that of the corresponding event discussion on . This work was supported by horizon that the decay event is outside of the causal past of JSPS KAKENHI Grant No. JP16K17688 (C. Y.).

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APPENDIX A: REDSHIFT OF NULL RAY DUE 0.6 TO MASSIVE SPHERICAL SHELL 0.4 6 b + 5 M+ Here we consider the redshift of a null ray due to a 0.2 spherical massive shell considered in Sec. II. Notation U+ (r) 0 adopted in this section is the same as that in Sec. II. The null -0.2 b+ M+ ray goes along a null geodesic. The components of the null a -0.4 geodesic tangent l are written in the form 4 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b + 5 M+ 1 2 -0.6 μ ¼ ω ϵ 1 − b ð Þ 0 b ð Þ l ; 2 F r ; ; 2 ; A1 -0.8 FðrÞ r r -1 where ω and b are constants corresponding to the 1 1.5 2 2.5 3 3.5 4 r angular frequency and the impact parameter, respectively, 2M+ and ϵ ¼1: ϵ ¼þ1 for the outgoing null, whereas ϵ ¼ −1 for the ingoing one. Without loss of generality, we consider FIG. 16. The effective potential of the null ray in the domain the only case of the non-negative impact parameter b ≥ 0. Dþ whose geometrypffiffiffiffiffi is Schwarzschildianpffiffiffiffiffi is depictedpffiffiffiffiffi for three 4 6 In this section, for simplicity, we focus on the shell with cases, bþ ¼ 5 27Mþ, bþ ¼ 27Mþ, and bþ ¼ 5 27Mþ. no in the spacetime without the cosmologi- cal constant and the domain D− is Minkowskian, frequency of the reflected null ray is the same as that before 2Mþ reflection in the rest frame of the shell, we have Fþ ¼ 1 − and F− ¼ 1: ðA2Þ r ðinÞ a ðoutÞ a la u ¼ la u : ðA5Þ We also focus on the case that the spherical massive shell is contracting R<_ 0. The parameter ϵ of ingoing null ray should be equal to −1, We obtain the energy equation from the radial compo- whereas it is nontrivial which sign of ϵ is chosen after the a ϵ ϵ nent of l as reflection; after the reflection is denoted by o. Then Eq. (A5) leads to 1 dr 2 þ WðrÞ¼0; ðA3Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ω2 γ 2 d Fþ b ω 1 þ − 1 − i i 2 2 Fþ where γ is the affine parameter and V R rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 2 b Fþ bo WðrÞ¼ FðrÞ − 1: ðA4Þ ¼ ω 1 þ þ ϵ 1 − Fþ ; ðA6Þ r2 o V2 o R2

The null ray can move only in the domain of WðrÞ ≤ 0. where Fþ ¼ FþðRÞ, and The geometry of Dþ is Schwarzschildian, and as well known, the effective potential WþðrÞ has one maximum at pffiffiffiffiffi V ≔−R>_ 0: ðA7Þ r ¼ 3Mþ (see Fig. 16). If bþ is larger than 27Mþ, the maximum of Wþ is positive. The null ray going inward in In the rest frame, the component of la vertical to the shell the region of r>3Mþ bounces off the potential barrier and changes its sign at the reflection event then goes away to infinity, whereas one going outward in ðinÞ a ðoutÞ a the domain of r<3Mþ also bounces off the potential la n ¼ −la n : ðA8Þ barrier and then turns to the center. On the other hand, Equation (A8) leads to the maximumpffiffiffiffiffi of WþðrÞ is nonpositive in the case of " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# bþ ≤ 27Mþ; in this case, the null ray does not bounce off F b2 the potential barrier. The fact we should remember here is ω 1 − 1 þ þ 1 − i i 2 2 Fþ that if the null ray is ingoing, or equivalently, ϵ ¼ −1, in the V R sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi region of r<3Mþ within Dþ, it does not bounce off but " # 2 continues to go inward. Fþ bo ¼ −ω 1 þ ϵ 1 þ 1 − Fþ : ðA9Þ o o V2 R2 1. Reflected case ðinÞ On the other hand, the component of la tangential to the Let us consider the case that an ingoing null ray la from ω ¼ ω ¼ shell does not change; the equation infinity in Dþ with þ i and bþ bi is reflected at the Dþ shell and then goes away to infinity in as an outgoing ð Þ ð Þ ð Þ in ˆ a out ˆ a out ω ¼ ω ¼ la ϕ ¼ la ϕ ðA10Þ null ray la with þ o and bþ bo. Since the angular

044027-16 GRAVASTAR FORMATION: WHAT CAN BE THE EVIDENCE … PHYS. REV. D 99, 044027 (2019) leads to the conservation of the , we obtain ω ¼ ω ≕ ð Þ 2ω ibi obo L: A11 ω ¼ i 2 ð Þ o V AL: A20 ðinÞ a Fþ Since the null ray is assumed to hit the shell, la n < 0 should be satisfied, and hence ω ω Then, by regarding o as a function of L, i, R, and V,we have R2 b2 < ðA12Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 þ u V Fþ 2 u Fþ ∂ω 2ω ∂A 2b V 1 þ 2 o ¼ i V2 L ¼ i t V > 0: ðA21Þ 2 b2 should hold. ∂L Fþ ∂L R i 1 − 2 Fþ From Eqs. (A6) and (A9),wehave R qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω 2 This result implies that o of the reflected null ray that can Fþ bi 1 þ 2 − 1 − 2 Fþ ¼ ω qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV R go away is bounded from the value with L ibcr. When b2 bi is equal to bcr, AR vanishes, and hence 1 − ð1 þ FþÞð1 − i Þ V2 R2 Fþ qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 bo Fþ bo 1 − ¼ 0 ð Þ 1 þ 2 þ ϵ 1 − 2 Fþ 2 Fþ A22 V o R R ¼ − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðA13Þ 2 1 þ ϵ ð1 þ FþÞð1 − bo Þ o V2 R2 Fþ holds; the reflected null ray has a vanishing radial compo- nent of la. In this case, Eq. (A6) leads to We rewrite Eq. (A13) in the form ω rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω j ¼ iFþ ð Þ 2 o b ¼b 2 : A23 bo i cr 2V þ Fþ A 1 − Fþ ¼ ϵ A ; ðA14Þ L R2 o R Hence, the angular frequency of the reflected null ray is where bounded from the above as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Fþ Fþ b ω Fþ ¼ 1 þ − 1 þ 1 − i ð Þ ω i ð Þ AL 2 2 2 Fþ ; A15 o < 2 : A24 2V V R 2V þ Fþ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 For 0 whereas the sign of redshift of the reflected null ray is caused by the contraction AR depends on bi, R, and V. Since the lhs of Eq. (A14) is of the shell. ϵ positive, o should be chosen so that the rhs is also positive, and hence we have 2. Transmitted case  þ1 2 2 for b bcr; in Dþ initially, enter D−, and then return to Dþ. We are where interested in the case that when the null ray returns from D− 2 to Dþ, the radius R of the shell is very close to the 2 R Fþ ≔ ð Þ 2Mþ bcr 2 2 : A18 gravitational radius ; here our attention is concentrated ð2V þ FþÞ at the moment of this return. The angular frequency and the ϵ ¼ −1 impact parameter of the null ray in D− are denoted by ω− The null ray with o goes inward although it is the and b−, respectively, whereas those of the null ray after null ray reflected by the shell. Since we consider the case ω that the reflection occurs when the radius of the shell is very returning to Dþ are denoted by þ and bþ, respectively. The inequality l na > 0, or equivalently, close to the gravitational radius 2Mþ, the reflected null ray a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ ¼ −1 Dþ with o continues to move inward in ; in other 2 1 b− words, the distant observers recognize the shell as an 1 þ ϵ 1 þ 1 − > 0 ðA25Þ absorber of all null rays hitting the shell. V2 R2 By taking the square of Eq. (A14) and using the relation should hold just before the null ray hits the shell in D−, 2 3 b Fþ A2 − A2 ¼ i ; ðA19Þ where V has been defined as Eq. (A7). Equation (A25) is L R 4V4R2 necessarily satisfied if ϵ is equal to unity. On the other hand,

044027-17 NAKAO, YOO, and HARADA PHYS. REV. D 99, 044027 (2019) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 b− > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA26Þ bþ 2 B 1 − Fþ ¼ ϵþB ; ðA30Þ V þ 1 L R R should hold in the case of ϵ ¼ −1. We will study these where ϕˆ a cases separately. But in both cases, the continuity of la rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi! leads to 1 2 ¼ 1 þ 1 þ Fþ þ 1 − b− BL 2 2 2 ω ¼ ω ð Þ V V R −b− þbþ: A27 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ! ! u 2 Since it is nontrivial whether ϵ is equal to þ1 after entering t Fþ b− − 1 þ 1 − − 1; ðA31Þ Dþ, ϵ in Dþ is denoted by ϵþ. V2 R2 First, we consider the case of ϵ ¼ −1 in D−. In the rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi transmitted case, all components of la should be every- 2 Fþ b− 1 where continuous; the continuity of l ua leads to B ¼ 1 þ þ 1 − − 1 þ a R V2 R2 V2 "rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi# 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 ! ! 1 b− u ω 1 þ − 1 − 6u 2 7 − 2 2 t 1 þ Fþ 1 − b− þ 1 ð Þ V R × 4 2 2 5: A32 "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# V R ω 2 ¼ þ 1 þ Fþ þ ϵ 1 − bþ ð Þ 2 þ 2 Fþ ; A28 Fþ V R Since we have rffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 whereas the continuity of l na leads to 1 2 a 1 þ 1 þ Fþ þ 1 − b− " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 V V R 1 b− 2 3 ω 1 − 1 þ 1 − vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 − 2 2 u ! ! V R 6u 2 7 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − t 1 þ Fþ 1 − b− þ 1 " # 4 2 2 5 ω 2 V R ¼ þ 1 þ ϵ 1 þ Fþ 1 − bþ ð Þ þ 2 2 Fþ ; A29 ! Fþ V R rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 ¼ 1 þ Fþ þ 1 − b− þ b−Fþ 0 ð Þ 2 2 2 2 2 > ; A33 where Fþ ¼ FþðRÞ. As in the reflected case, by dividing V V R V R each side of Eq. (A28) by each side of Eq. (A29) and 0 further by a few manipulations, we have BL > holds. We also have

2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi! ! u ! ! 2 2 2 u 2 Fþ b− 1 6 Fþ b− 7 1 þ þ 1 − − 1 þ 4t 1 þ 1 − þ 15 V2 R2 V2 V2 R2 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u ! ! 2 u 2 2 1 6 Fþ b− 7 b−Fþ ¼ − 4t 1 þ 1 − þ 15 þ V2 V2 R2 V2R2 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 32vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u ! ! pffiffiffiffiffiffiffi u ! ! pffiffiffiffiffiffiffi u 2 u 2 1 6 Fþ b− b− Fþ76 Fþ b− b− Fþ7 ¼ − 4t 1 þ 1 − þ 1 þ 54t 1 þ 1 − þ 1 − 5 V2 V2 R2 R V2 R2 R 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u ! ! pffiffiffiffiffiffiffi u 2 1 6 Fþ b− b− Fþ7 ¼ − 4t 1 þ 1 − þ 1 þ 5 V2 V2 R2 R 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u ! ! u 2 pffiffiffiffiffiffiffi 6t Fþ b− b− b− 7 × 4 1 þ 1 − þ 1 − þ ð1 − FþÞ5 < 0; ðA34Þ V2 R2 R R

044027-18 GRAVASTAR FORMATION: WHAT CAN BE THE EVIDENCE … PHYS. REV. D 99, 044027 (2019) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #2 where, in the last inequality, we have used the fact that 2 pffiffiffiffiffiffiffi Fþ 1 b− ≤ 1 1 0 C ≔ 1 þ 1 þ 1 − þ 1 b−=R and Fþ < , and hence BR < holds. R 2 2 2 ϵ V V R Through Eq. (A28), this result implies that þ is equal rffiffiffiffiffiffiffiffiffiffiffiffiffi! −1 ϵ ¼ −1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 to . This result implies that in the case of − , the 1 2 − 1 þ þ 1 − b− null ray keeps going inward even after returning to Dþ and 2 2 is effectively absorbed by the contracting shell. V R sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ ¼þ1 " #2 Next, we consider the case of in D−. By the 2 2 Fþ 1 b− b− similar argument to the case of ϵ ¼ −1 in D−, the ¼ 1 þ 1 − þ 1 − a 2 2 2 2 2 continuity of lau leads to V V R V R "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # " rffiffiffiffiffiffiffiffiffiffiffiffiffi# 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fþ 1 b− b− 2 ¼ 1 þ 1 − þ 1 þ pffiffiffiffiffiffiffi 1 b− 2 2 2 ω− 1 þ þ 1 − V V R R Fþ V2 R2 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1 2 ω 2 1 þ 1 − b− þ 1 − pb−ffiffiffiffiffiffiffi ð Þ ¼ þ 1 þ Fþ þ ϵ 1 − bþ ð Þ × 2 2 : A40 2 þ 2 Fþ ; A35 V R R Fþ Fþ V R C CR is positive, if and only if R is positive. We can see that whereas the continuity of l na leads to C a R is positive if and only if 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 6 1 b− 7 1 þ 1 − b− þ 1 − pb−ffiffiffiffiffiffiffi 0 ð Þ ω−41 þ 1 þ 1 − 5 2 2 > A41 V2 R2 V R R Fþ 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi holds. If ω 6 2 7 ¼ þ 1 þ ϵ 1 þ Fþ 1 − bþ ð Þ 4 þ 2 2 Fþ 5: A36 Fþ V R b− pffiffiffiffiffiffiffi ≤ 1 ðA42Þ R Fþ

By dividing each side of Eq. (A35) by each side of is satisfied, Eq. (A41) holds. By contrast, in the case of Eq. (A36) and by a few simple manipulations, we have

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b− 2 pffiffiffiffiffiffiffi > 1; ðA43Þ bþ R Fþ C 1 − Fþ ¼ ϵþC ; ðA37Þ L R2 R we rewrite Eq. (A41) in the form where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 þ 1 − b− pb−ffiffiffiffiffiffiffi − 1 ð Þ 1 2 2 2 > A44 Fþ b− V R R Fþ C ¼ 1 þ 1 þ þ 1 − L V2 V2 R2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and take the square of its both sides and, as a result, obtain 2 1 b− − 1 þ 1 − − 1; ðA38Þ 2 2 2 pffiffiffiffiffiffiffi V R 2 2 b− 2 b− ½ðV þ 1ÞFþ þ V − 2V Fþ − Fþ < 0: ðA45Þ R2 R rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 b− Fþ C ¼ − 1 þ − 1 − þ 1 þ Then, we have R V2 R2 V2 "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 − ð 2 þ 1Þð 2 þ Þ 1 b− V Fþ V V Fþ Fþ b− × 1 þ 1 − þ 1 : ðA39Þ 2 2 < 2 2 ðV þ 1ÞFþ þ V R V R pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ ð 2 þ 1Þð 2 þ Þ V Fþ V V Fþ Fþ ð Þ < ð 2 þ 1Þ þ 2 : A46 By the similar prescription to that in the case of BL, we can V Fþ V 0 ϵ see CL > . Hence, CR should be positive so that þ is þ1 equal to , although the sign of CR is nontrivial. In order In order that the intersection between Eqs. (A43) and (A46) to know it, we study the following quantity: is not empty,

044027-19 NAKAO, YOO, and HARADA PHYS. REV. D 99, 044027 (2019) pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 þ ð 2 þ 1Þð 2 þ Þ APPENDIX B: CONSERVATION OF V Fþ V V Fþ Fþ ð Þ Fþ < 2 2 A47 THE FOUR-MOMENTUM ðV þ 1ÞFþ þ V We show that the “four-momentum conservation” (44) is ≪ 1 ab should hold. We are interested in the case of Fþ in consistent with the Bianchi identity ∇aT ¼ 0. The stress- which this inequality is satisfied. Hence, in the case of energy tensor of shell I (I ¼ 0, 2, 3) is written in the form ≪ 1 Fþ , CR is positive if and only if ab ab Tð Þ ¼ Sð ÞδðχðIÞÞ; ðB1Þ ð Þ I I b− 0: T ¼ðσð1Þu u þ Pð1ÞH Þδðχ − Xð1ÞðτÞÞ ; ∂ ∂ 2 2 ð1Þ ð1Þ ð1Þ ∂χ L L R 1 − b− R2 ðB3Þ ðA52Þ

−1 ∂χð2Þðτ;χÞ Hence, ωþ of the null ray with ϵþ ¼þ1 that can escape to ab ¼ðσ a b þ abÞδðχ − ðτÞÞ Tð2Þ ð2Þuð2Þuð2Þ Pð2ÞH Xð2Þ ∂χ ; infinity is bounded from above by the value of ωþj . b−¼bcr Since C j ¼ 0 holds, we have, from Eq. (A50), ðB4Þ R b−¼bcr pffiffiffiffiffiffiffi bcr Fþ where χ ¼ XðJÞðτÞ with XðJÞð0Þ¼0 represents the world C j ¼ ¼ : ðA53Þ L b− bcr V2R hypersurface of shell J (J ¼ 1, 2). We have Substituting Eq. (A53) into Eq. (A51), we have ∂χð1Þðτ; χÞ ∂ a ¼ð χ Þ ð Þ pffiffiffiffiffiffiffi ∂χ d ð1Þ a ∂χ ; B5 bcr Fþ ωþj ¼ ¼ ω− ; ðA54Þ b− bcr R ∂χð2Þðτ; χÞ ∂ a ¼ðdχð2ÞÞ ; ðB6Þ ∂χ a ∂χ and hence, in the case of V>0, where, at the decay event d, ω−Fþ ωþ < ωþj ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 þ OðFþÞ: ðA55Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b− bcr 2 Vð V þ 1 − VÞ ð χ Þ ¼ ϵ Γ2 − 1 ð0Þ þ Γ ð0Þ ð Þ d ð1Þ a ð1Þ ð1Þ ua ð1Þna ; B7 This result implies that the transmitted null ray going away qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð χ Þ ¼ ϵ Γ2 − 1 ð0Þ þ Γ ð0Þ ð Þ to infinity suffers indefinitely large redshift in the limit of d ð2Þ a ð2Þ ð2Þ ua ð2Þna ; B8 Fþ → 0. Note that in the case of V ¼ 0, i.e., the static shell, the angular frequency ωþ of the transmitted null ray is the and same as that of the incident null ray, in Dþ. The redshift of ∂ a a the transmitted null ray is caused by the contraction of ¼ nð0Þ: ðB9Þ the shell. ∂χ

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ab Hence, we have We integrate the Bianchi identity ∇bT ¼ 0 over the small neighborhood of the decay event d shown in Fig. 17: ∂χð1Þðτ; χÞ ¼ Γ ð Þ ∂χ ð1Þ; B10 the domain of integration is chosen so that the shells do not intersect the boundaries χ ¼ε=2. Only shell 0 intersects the boundary τ ¼ −δ=2, whereas only shell 1 and shell 2 ∂χð2Þðτ; χÞ ¼ Γ ð Þ intersect the boundary τ ¼þδ=2. Then, we have ∂χ ð2Þ: B11

Z Z Z Z þδ=2 þε=2 þε=2 þε=2 pffiffiffiffiffiffi 0 ¼ τ χ θˆ ϕˆ − ð0Þ∇ ab d d d d g ub aT Z−δ=2 Z−ε=2 Z−ε=2 Z−ε=2 þδ=2 þε=2 þε=2 þε=2 pffiffiffiffiffiffi ¼ τ χ θˆ ϕˆ − ½∇ ð ab ð0ÞÞ − ab∇ ð0Þ d d d d g a T ub T aub Z−δ=2 Z−ε=2 Z−ε=2 Z−ε=2 þδ=2 þε=2 þε=2 þε=2 pffiffiffiffiffiffi ¼ τ χ θˆ ϕˆ ð − ab ð0ÞÞþOðε2δÞ d d d d gT ub Z−δ=2 Z−ε=2 Z−ε=2 −ε=2 þε 2 þε 2 þε 2 = = = pffiffiffiffiffiffi ð0Þ ð0Þ pffiffiffiffiffiffi ð0Þ ¼ dχ dθˆ dϕˆ ½ −gðTτb u þ Tτb u Þj − −gTτb u j ð1Þ b ð2Þ b τ¼þδ=2 ð0Þ b τ¼−δ=2 −Zε=2 −Zε=2 −Zε=2 þδ=2 þε=2 þε=2 pffiffiffiffiffiffi pffiffiffiffiffiffi þ τ θˆ ϕˆ ð − χb ð0Þj − − χb ð0Þj Þ d d d gT ub χ¼þε=2 gT ub χ¼−ε=2 Z−δ=2 Z−ε=2 Z−ε=2 þδ=2 þε=2 þε=2 pffiffiffiffiffiffi pffiffiffiffiffiffi þ τ χ ϕˆ ð − θˆb ð0Þj − − θˆb ð0Þj Þ d d d gT ub θˆ¼þε=2 gT ub θˆ¼−ε=2 Z−δ=2 Z−ε=2 Z−ε=2 þδ=2 þε=2 þε=2 pffiffiffiffiffiffi pffiffiffiffiffiffi þ τ χ θˆð − ϕˆ b ð0Þj − − φˆ b ð0Þj ÞþOðε2δÞ d d d gT ub ϕˆ ¼þε=2 gT ub ϕˆ ¼−ε=2 −δ 2 −ε 2 −ε 2 Z = Z = Z = pffiffiffiffiffiffi pffiffiffiffiffiffi  þε=2 þε=2 þε=2 −g ð0Þ ð0Þ ð0Þ −g ð0Þ ð0Þ ¼ χ θˆ ϕˆ ð ab þ ab Þ − ab j þ Oðε2δÞ d d d ð0Þ Tð1Þub Tð2Þub ua ð0Þ Tð0Þua ub −ε=2 −ε=2 −ε=2 uτ τ¼þδ 2 uτ τ¼−δ 2 pffiffiffiffiffiffi = = −g ¼ ðσ Γ þ σ Γ − σ Þε2 þ Oðε2δÞ ð Þ ð0Þ ð1Þ ð1Þ ð2Þ ð2Þ ð0Þ ; B12 uτ where we have used the finiteness of ∇ u in the third abϕˆ j ¼ abϕˆ j a b T aub ϕˆ ¼þε=2 T aub ϕˆ ¼−ε=2 due to the spherical abj ¼ 0 equality, T χ¼ε=2 due to the situation we consider ð0Þ symmetry in the fourth equality, and ua jτ¼δ 2 ¼ abθˆ j ¼ abθˆ j = (see Fig. 17), and T aub θˆ¼þε=2 T aub θˆ¼−ε=2 and ð0Þ ua jτ¼0½1 þ OðδÞ and Eqs. (B2)–(B4) in the final equality. Hence, in the limit of δ → 0, by multiplying Eq. (B12) by u a 4π 2 τ rd, we obtain Eq. (46). Shell 2 Shell 1 By similar manipulations to those in Eq. (B12),wehave Z Z þδ=2 þε=2 pffiffiffiffiffiffi 0 ¼ τ χ θˆ ϕˆ − ð0Þ∇ ab (0) d d d d g nb aT u a −δ 2 −ε 2 Z = = pffiffiffiffiffiffi þε=2 −g ¼ χ θˆ ϕˆ ð ab ð0Þ þ ab ð0ÞÞ ð0Þ χ d d d ð0Þ Tð1Þnb Tð2Þnb ua −ε=2 uτ τ¼þδ 2 δ d (0) = n a pffiffiffiffiffiffi  na −g − ab ð0Þ ð0Þj þ Oðε2δÞ ð0Þ Tð0Þnb ua uτ τ¼−δ=2 pffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε −g ¼ − ϵ σ Γ2 − 1 þ ϵ σ Γ2 − 1 ε2 ð0Þ ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ uτ 2 Shell 0 þ Oðε δÞ: ðB13Þ

FIG. 17. The domain of integration is schematically depicted by Hence, in the limit of δ → 0, by multiplying Eq. (B12) by 4π 2 a dashed square. rd, we obtain Eq. (47).

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