Gravitational Collapse and Formation of Universal Horizons

Gravitational collapse and formation of universal horizons

Miao Tian a,b, Xinwen Wang b, M.F. da Silva c, and Anzhong Wang b,c,d ∗† a School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China b GCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USA c Departamento de F´ısica Teórica, Universidade do Estado do Rio de Janeiro, Rua SãoFrancisco Xavier 524, Maracanã,CEP 20550013, Rio de Janeiro, RJ, Brazil d Institute for Advanced Physics & Mathematics, Zhejiang University of Technology, Hangzhou 310032, China (Dated: August 21, 2021) In this paper, we first generalize the definition of stationary universal horizons to dynamical ones, and then show that (dynamical) universal horizons can be formed from realistic gravitational collapse. This is done by constructing analytical models of a collapsing spherically symmetric star with finite thickness in Einstein-aether theory.

PACS numbers: 04.50.Kd, 04.70.Bw, 04.20.Jb, 97.60.-s

I. INTRODUCTION operators take over, and a healthy low-energy limit is pre- sumably resulted 1. It is remarkable to note that, despite The invariance under the Lorentz symmetry group is a of the stringent observational constraints of the violation cornerstone of modern physics, and is strongly supported of the LI [1], the nonrelativistic general covariant HL by observations. In fact, all the experiments carried out gravity constructed in [11] is consistent with all the solar so far are consistent with it [1], and no evidence to show system tests [12, 13] and cosmology [14, 15]. In addition, that such a symmetry must be broken at certain energy it has been recently embedded in string theory via the scales, although it is arguable that such constraints in nonrelativistic AdS/CFT correspondence [16]. Another the gravitational sector are much weaker than those in version of the HL gravity, the health extension [17], is the matter sector [2]. also self-consistent and passes all the solar system, astro- physical and cosmological tests 2. Nevertheless, there are various reasons to construct Another example that violates LI is the Einstein-aether gravitational theories with broken Lorentz invariance theory, in which the breaking is realized by a timelike vec- (LI) [3]. In particular, our understanding of space-times tor field, while the gravitational action is still generally at Plank scale is still highly limited, and the renomaliz- covariant [18]. This theory is consistent with all the solar ability and unitarity of gravity often lead to the violation system tests [18] and binary pulsar observations [21]. of LI [4]. One concrete example is the Hoˇrava theory of However, once the LI is broken, speeds of particles can quantum gravity [5], in which the LI is broken via the be greater than that of light. In particular, the dispersion anisotropic scaling between time and space in the ultra- relation generically becomes nonlinear [14], violet (UV), 2 4! −z i −1 i 2 2 2 p p t → b t, x → b x , (i = 1, 2, 3), (1.1) E = cpp 1 + α1 + α2 , (1.2) M∗ M∗ where z denotes the dynamical critical exponent. This is a reminiscent of Lifshitz scalars in condensed matter where E and p are the energy and momentum of the physics [6], hence the theory is often referred to as the particle considered, and cp, αi are coefficients, depend- Hoˇrava-Lifshitz (HL) quantum gravity at a Lifshitz fixed ing on the species of the particle, while M∗ denotes the arXiv:1501.04134v2 [gr-qc] 24 Jan 2015 point. The anisotropic scaling (1.1) provides a crucial suppression energy scale of the higher-dimensional oper- mechanism: The gravitational action can be constructed ators. Then, one can see that both phase and group ve- in such a way that only higher-dimensional spatial (but not time) derivative operators are included, so that the UV behavior of the theory is dramatically improved. In 1 particular, for z ≥ 3 it becomes power-counting renor- It should be emphasized that, the breaking of LI can have sig- nificant effects on the low-energy physics through the interac- malizable [5, 7]. The exclusion of high-dimensional time tions between gravity and matter, no matter how high the scale derivative operators, on the other hand, prevents the of symmetry breaking is [9]. Recently, Pospelov and Tamarit ghost instability, whereby the unitarity of the theory is proposed a mechanism of SUSY breaking by coupling a Lorentz- assured [8]. In the infrared (IR) the lower dimensional invariant supersymmetric matter sector to non-supersymmetric gravitational interactions with Lifshitz scaling, and showed that it can lead to a consistent HL gravity [10]. 2 In fact, in the IR the theory can be identified with the hypersurface-orthogonal Einstein-aether theory [18] in a particu- ∗The corresponding author lar gauge [3, 19, 20], whereby the consistence of the theory with †Electronic address: Anzhong˙[email protected] observations can be deduced. 2 locities of the particles are unbounded with the increase Killing horizons by apparent horizons [29, 30], and in the of energy. This suggests that black holes may not exist stationary limit, the latter reduces to the former. In Sec. at all in theories with broken LI, and makes such theo- III, we study a collapsing spherically symmetric star with ries questionable, as observations strongly indicate that a finite thickness in the framework of the Einstein-aether black holes exist in our universe [22]. theory. When the effects of the aether are negligible, the Lately, a potential breakthrough was the discovery vacuum space-time outside the star is uniquely described that there still exist absolute causal boundaries, the so- by the Schwarzschild solution and the junction condition called universal horizons, in the theories with broken LI across the surface of the star reduces to those of Israel [23]. Particles even with infinitely large velocities would [31]. Once this is done, we find that the khronon equa- just move around on these boundaries and cannot es- tion can be solved analytically when the speed of the cape to infinity [24]. The universal horizon radiates like khronon is infinitely large, for which the sound horizon a blackbody at a fixed temperature, and obeys the first of the khronon coincides with the universal horizon. It is law of black hole mechanics [25]. remarkable that this is also the case for the Schwarzschild Recently, we studied the existence of universal hori- solution [26], for which the universal horizon and surface zons in the three well-known black hole solutions, gravity are given by Eq.(1.4). The paper is ended in Sec. the Schwarzschild, Schwarzschild anti-de Sitter, and IV, in which our main conclusions are presented. Reissner-Nordström,and found that in all of them universal horizons always exist inside their Killing horizons [26]. In particular, the peeling-off behavior of the globally II. DYNAMICAL UNIVERSAL HORIZONS AND BLACK HOLES timelike khronon field uµ was found only at the universal horizons, whereby the surface gravity κpeeling is calcu- lated and found equal to [27], A necessary condition for the existence of a universal horizon of a given space-time is the existence of a globally 1 κ ≡ uαD u ζλ , (1.3) time-like foliation [3, 23, 26]. This foliation is usually UH 2 α λ characterized by a scalar field φ, dubbed khronon [23], µ and the normal vector uµ of the foliation is always time- where ζ and Dµ denote, respectively, the time transla- like, tion Killing field and covariant derivative with respect to λ the given space-time metric gµν (µ, ν = 0, 1, 2, 3). For the u uλ = −1, (2.1) Schwarzschild solution, the universal horizon and surface gravity are given, respectively, by [26], where φ ∂φ 3/2 √,µ αβ Sch. 3rs Sch. 2 1 uµ = , φ,µ ≡ µ ,X ≡ −g ∂αφ∂βφ. (2.2) RUH = , κpeeling = , (1.4) X ∂x 4 3 rs In this paper, we choose the signature of the metric as where rs denotes the Schwarzschild radius. (−1, 1, 1, 1). It is important to note that such a defined In this paper, we shall study the formation of the uni- khronon field is unique only up to the following gauge versal horizons from realistic gravitational collapse of a transformation, spherically symmetric star 3. To be more concrete, we shall consider such a collapsing object in the Einstein- φ˜ = F(φ), (2.3) aether theory [18]. To make the problem tractable, we further assume that the effects of the aether are negligi- where F(φ) is a monotonically increasing (or decreasing) ble, so the space-time outside of the star is still described and otherwise arbitrary function of φ. Clearly, under the by the Schwarzschild solution, while inside the star we above gauge transformations, we haveu ˜µ = uµ. assume that the distribution of the matter is homoge- The khronon field φ is described by the action [18], neous and isotropic, so the internal space-time is that of Z 4 p h 2 µ 2 the Friedmann-Robertson-Walker (FRW). Although the Sφ = d x |g| V0 (Dµuν ) + V0 (Dµu ) model is very ideal, it is sufficient to serve our current µ ν µ i purposes, that is, to show explicitly that universal hori- + c3 (D u )(Dν uµ) − c4a aµ , (2.4) zons can be formed from realistic gravitational collapse. α Specifically, the paper is organized as follows: In Sec. where ci’s are arbitrary constants, and aµ ≡ u Dαuµ. II, we shall present a brief review on the definition of The operator Dµ denotes the covariant derivative with re- the stationary universal horizons, and then generalize it spect to the background metric gµν , as mentioned above. to dynamical spacetimes. This is realized by replacing Note that the above action is the most general one in the sense that the resulting differential equations in terms of uµ are second-order [18]. However, when uµ is written in the form of Eq.(2.2), the relation 3 Here “realistic” means that the collapsing object satisfies at least the weak energy condition [28]. u[ν Dαuβ] = 0, (2.5) 3 is identically satisfied. Then, it can be shown that only B. Universal Horizons in Non-Stationary three of the four coupling constants ci are independent. Spacetimes In fact, from Eq.(2.5) we find [18], To study the formation of universal horizons from grav- µ α β β α ∆Lφ ≡ a aµ + Dαuβ D u − Dαuβ D u = 0. itational collapse, we need first to generalize the above (2.6) definition of the universal horizons to non-stationary Then, one can always add the term, spacetimes. For the sake of simplicity, in the rest of Z this paper we shall restrict ourselves only to spherical p 4 ∆Sφ = c0 |g| d x∆Lφ, (2.7) space-times, and its generalization to other spacetimes is straightforward. The metric for a specially symmetric space-time can into Sφ, where c0 is an arbitrary constant. This is effec- 0 be cost in the form, tively to shift the coupling constants ci to ci, where 2 i j 2 i 2 ds = gijdx dx + R x dΩ , (i, j = 0, 1), (2.14) 0 0 0 0 c = c1 + c0, c = c2, c = c3 − c0, c = c4 − c0. (2.8) 1 2 3 4 in the spherical coordinates, xµ = x0, x1, θ, $ , (µ = 2 2 2 2 Thus, by properly choosing c0, one can always set one of 0, 1, 2, 3), where dΩ = dθ + sin θd$ . ci (i = 1, 3, 4) to zero. However, in the following we shall The normal vector nµ to the hypersurface R = C0 is leave this possibility open. given by, The variation of Sφ with respect to φ yields the ∂(R − C0) 0 1 khronon equation, nµ ≡ = δ R0 + δ R1, (2.15) ∂xµ µ µ µ i DµA = 0, (2.9) where C0 is a constant and Ri ≡ ∂R/∂x . Setting ζµ = δ0 R − δ1 R , (2.16) where [20], µ 1 µ 0 µ we can see that ζ is always orthogonal to nµ, µ µ µ (δν + u uν ) ν λ A ≡ √ Æ , ζ nλ = 0. (2.17) X ν γν ν γ For spacetimes that are asymptotically flat there al- Æ ≡ Dγ J + c4aγ D u , α αβ α β α β ways exists a region, say, R > R∞, in which nµ J µ ≡ V0g gµν + V0δµ δν + c3δν δµ and ζµ are, respectively, space- and time-like, that is, − c uαuβg D uν . (2.10) n nµ| > 0 and ζµζ | < 0. An apparent hori- 4 µν β µ R>R∞ µ R>R∞ zon may form at RAH , at which nµ becomes null, λ nλn = 0, (2.18) A. Universal Horizons in Stationary and R=RAH Asymptotically flat Spacetimes where RAH < R∞. Then, in the internal region R < RAH , nµ becomes timelike. Therefore, we have In stationary and asymptotically flat spacetimes, there ( µ > 0, R > RAH , always exists a time translation Killing vector, ζ , which n nλ = = 0,R = R , (2.19) is timelike asymptotically, λ AH < 0, R < RAH . λ ζ ζλ < 0, (2.11) Since Eq.(2.17) always holds, we must have

( < 0, R > RAH , for r → ∞.A Killing horizon is defined as the existence λ ζλζ = = 0,R = RAH , (2.20) of a hypersurface on which the time translation Killing > 0, R < RAH , vector ζµ becomes null, that is, ζµ becomes null on the apparent horizon, and λ spacelike inside it. ζ ζλ = 0. (2.12) We define a dynamical universal horizon as the hyper- On the other hand, a universal horizon is defined as the surface at which existence of a hypersurface on which ζµ becomes orthog- λ uλζ = 0. (2.21) R=RUH onal to uµ, Since uµ is globally timelike, Eq.(2.21) is possible only λ uλζ = 0. (2.13) when ζµ is spacelike. Clearly, this is possible only inside the apparent horizons, that is, RUH < RAH . Since uµ is timelike globally, Eq.(2.13) is possible only In the static case, the apparent horizons defined above when ζµ becomes spacelike. This can happen only inside reduce to the Killing horizons, and the dynamical univer- the apparent horizons, because only in that region ζµ sal horizons defined by Eq.(2.21) are identical to those becomes spacelike. given by Eq.(2.13). 4

III. GRAVITATIONAL COLLAPSE AND of the observers that are comoving with the collapsing FORMATION OF UNIVERSAL HORIZONS surface of the star. In the current case, we have τ = t. Then, we find Let us consider a collapsing star with a finite radius R (τ) = a(τ)r , RΣ(τ), where τ denotes the proper time of the surface of Σ Σ the star. To our current purpose, we simply assume that 2M 2 1 − v˙ − 2R˙ Σv˙ − 1 = 0, (3.8) the space-time inside the star is described by the FRW RΣ flat metric, where a dot denotes the derivative with respect to τ. 2 2 2 2 2 2 ds− = −dt +a (t) dr + r dΩ , (R ≤ RΣ(τ)), (3.1) However, since in the present case we have τ = t, so we shall not distinguish it from that with respect to t, used where R = a(t)r is the geometric radius inside the col- above. The extrinsic curvature tensor on the two sides of lapsing star. From Eq.(2.15), the normal vector nµ to the surface defined by, the hypersurface R = C0 takes the form, 2 α β δ 0 1 ± h ∂ x± ± α ∂x± ∂x± i nµ =a ˙(t)rδµ + a(t)δµ, (3.2) K = −n± + Γ , (3.9) ab α ∂ξa∂ξb βδ ∂ξa ∂ξb wherea ˙ ≡ da/dt. Then, the corresponding vector ζµ reads has the following non-vanishing components [32] 4 ζµ = a(t)δ0 − a˙(t)rδ1 . (3.3) v¨ Mv˙ µ µ K+ = − − , ττ v˙ R 2 According to Eq.(2.18) the apparent horizon locates at, Σ − −2 − Kθθ = sin θKϕϕ = a(τ)rΣ, (3.10) 1 r = − . (3.4) K+ = sin−2θK+ = (R − 2M)v ˙ − R R˙ , AH a˙(t) θθ ϕϕ Σ Σ Σ ± Note that in the collapsing case we havea ˙ < 0. where nα are the normal vectors defined in the two faces The spacetime outside the collapsing star is vacuum. of the surface (3.6), In the Einstein-aether theory [18], if we consider the case ∂(r − r ) where the effects of the aether field is negligible, then − Σ r nα ≡ α = δα, the vacuum space-time will be that of the Schwarzschild, ∂x− and in the ingoing Eddington-Finkelstein coordinates ∂(R − R (v)) dR (v) n+ ≡ Σ = δR − Σ δv . (3.11) (v, R, θ, $) the metric takes the form, α α α α ∂x+ dv 2M ds2 = − 1 − dv2+2dvdR+R2dΩ2, (R ≥ R (τ)), From the Israel junction conditions [31], + R Σ − − (3.5) [Kab] − gab [K] = −8πτab, (3.12) where M denotes the total mass of the collapsing system, including that of the star surface, and has the dimension we can get the surface energy-momentum tensor τab, − + − − ab − of length L, that is, [M] = L. The surface Σ of the star where [Kab] ≡ Kab − Kab, [K] ≡ g [Kab] , and gab can be parameterized as, can be read off from Eq.(3.7), where a, b = τ, θ, $. In-

− serting Eq.(3.10) into the above equation, we find that r − rΣ = 0 (in V ), τab can be written in the form + R − RΣ(v) = 0 (in V ), (3.6) τab = σwawb + η(θaθb + $a$b), (3.13) where rΣ is a constant, when we choose the internal coordinates (t, r, θ, $) are comoving with the fluid of the where wa, θa and $a are unit vectors defined on the sur- τ collapsing star. On the surface of the collapsing star, the face of the star, given, respectively, by wa = δa , θa = θ $ interior and exterior metrics reduce to RΣδa, $a = RΣ sin θδa , and 2 2 2 2 2 ds |r=r = −dt + a (t)r dΩ 1 a˙ Mv˙ v˙ − Σ Σ σ = + + − , (3.14) 2 2 2 4πrΣa 4πa 2πr a 4πrΣa = ds+|R=RΣ(v) Σ v˙ v¨ 1 a˙ Mv˙ 2M dRΣ(v) 2 η = + − − − , = − 1 − − 2 dv 8πr a 8πv˙ 8πr a 8πa 8πr2 a2 RΣ(v) dv Σ Σ Σ 2 2 +RΣ(v) dΩ 2 2 2 ≡ −dτ + RΣ(τ) dΩ , (3.7) 4 + Note the sign difference of the first term of Kττ between the where RΣ(τ) is the geometric radius of the collapsing one obtained here and that obtained in [32], because of the sign star, v is a function of τ, where τ denotes the proper time difference of the cross term dvdR in the external metric (3.5). 5 here σ is the surface energy density of the collapsing star, coupling constants ci. However, when c14 = 0, we obtain and η its tangential pressure. Physically, they are of- a particular solution, ten required to satisfy certain energy conditions, such as V0r −Ht weak, strong and dominant [28], although in cosmology V (t, r) = , a(t) = a0e , (3.21) none of them seems necessarily to be satisfied [33]. a(t)

On the other hand, inside the collapsing star, the where V0,H and a0 are integration constants with khronon can be parametrized as, H > 0, a0 > 0. It is remarkable to note that c14 = 0 cor- responds to the case in which the speed of the khronon µ p 2 2 µ µ 2 u = 1 + a V δ0 + V δ1 , becomes infinitely large cφ = c123/c14 → ∞, a case that p u = − 1 + a2V 2δµ + a2V δµ, (3.15) was also studied in [3, 23, 26]. µ 0 1 From the definition of the dynamical universal horizon where V = V (t, r) is determined by the khronon equation Eq.(2.21) and considering Eqs.(3.3), (3.15) and (3.21), we (2.9), which now reduces to find that the collapse always forms a universal horizons inside the collapsing star, and its location is given by, r t r 2A t 3˙a(t)A s A,r + + A,t + = 0, (3.16) 2 r a(t) rUH (t) = . (3.22) p 4 2 2 2 −2Ht 2 V0 + 4V0 a0H e − V0 where From Eq.(3.8), on the other hand, we find that, t t t t A = c123A1 + c14A2 + c13A3 2 p 2 2 2 2 2 √ −rΣaa˙ + rΣa + rΣa a˙ − 2aMrΣ 1 + a2V 2 v˙ = . (3.23) Ar = At, Aθ = Aφ = 0, (3.17) 2M − arΣ a2V −Hτ Substituting it together with a(τ) = a0e back into with cab ≡ ca + cb, cabc ≡ ca + cb + cc, and Eqs.(3.14), we obtain V 1 At = (2a4V 5(−1 + 2r2aa¨) σ = (1 + G) , 1 r2(1 + a2V 2) 4πRΣ p 1 M − R − 2H2R3 +r(rV 00 + V 0(2 + ra2 1 + a2V 2V˙ )) η = Σ Σ − 1 , (3.24) p 8πRΣ RΣG +V (−2 − 3r2a˙ 2 + 3r2aa¨ + 5r2a 1 + a2V 2aV˙ 0 p where +r2a4V˙ 2 + 2ra2 1 + a2V 2(V˙ + rV˙ 0)) r 2M 2 3 2 p 2 2 0 −Hτ 2 2 +a V (−4 + r(−2ra˙ + 4ra 1 + a V aV˙ RΣ = a0rΣe , G ≡ 1 + H RΣ − . (3.25) RΣ p 2 2 ˙ ˙ 0 +a(7ra¨ + 2a 1 + a V (V + rV )))) Obviously, to have both σ and η real, we must assume 4 4 0 00 ˙ ¨ Min Min +ra V (2V + r(V + a(5˙aV + aV ))) that RΣ ≥ RΣ , where RΣ is a root of the equation, +ra2V 2(4V 0 + r(2V 00 + a(7˙aV˙ + aV¨ )))), (3.18) 2 2 2M 1 + H RΣ − = 0. (3.26) RΣ V Min At = − (4rV a˙ 2 + ra5V 4(2V a¨ + 5˙aV˙ ) When the star collapses to the point RΣ (τMin) = RΣ , 2 r(1 + a2V 2) the tangential pressure diverges, whereby a space-time p singularity (with a finite radius) is developed. This rep- +2a3V 2(2V 2 1 + a2V 2a˙ + 2rV (ä resents the end of the collapse, as the space-time beyond p p + 1 + a2V 2aV˙ 0) + 5ra˙V˙ ) + a(4V 2 1 + a2V 2a˙ this moment is not extendable. p It can be shown that the weak energy condition σ ≥ +rV (2ä + 5 1 + a2V 2aV˙ 0) + 5ra˙V˙ ) 0, σ +η ≥ 0 can always be satisfied by properly choosing +ra6V 4V¨ + a4V 2(4rV 3a˙ 2 + V 2(2V 0 + rV 00) the free parameters involved in the solution, before the p +2V 1 + a2V 2(V˙ + rV˙ 0) + 2rV¨ ) formation of the universal horizon. In particular, from Eq.(3.24) we evan see that σ is always non-negative, and +a2(8rV 3a˙ 2 + V 2(2V 0 + rV 00) + V (rV 2 p 1 3M +2 1 + a2V 2(V˙ + rV˙ 0)) σ + η = G − − 1 . (3.27) p 8πRΣG RΣ +r( 1 + a2V 2V 0V˙ + V¨ ))), (3.19) Thus, for RΣ ≥ 3M, the weak energy condition is always satisfied. When RΣ < 3M, it is also satisfied, provided t 2 2 2 2 A3 = 2V (1 + a V )(˙a − aa¨). (3.20) that G ≥ 3M/RΣ − 1, or equivalently 2 Here a prime denotes the derivative with respect to r. 2 2 4M 9M H RΣ + ≥ 2 . (3.28) It is found very difficult to solve Eq.(3.4) for any given RΣ RΣ 6

Clearly, by properly choosing the free parameters in- 0.15 volved in the solution, this condition can hold until the moment where the whole collapsing star is inside the uni- 0.10 versal horizon. However, at the end τ = τMin of the 0.05 collapse this condition is necessarily violated, as can be t%0.624554 t seen from Eqs.(3.26) and (3.28). Using the geometric ra- !2 !1 1 2 −Ht dius R = ra(t) = ra0e inside the collapsing star, we !0.05 find that the apparent horizon given by Eq.(3.4) and the Σ !0.10 universal horizon given by Eq.(3.22) can be expressed as, !0.15 Σ#Η RUH = rUH a(t) !0.20 s −Ht 2 = a0e , p 4 2 2 2 −2Ht 2 V0 + 4V0 a0H e − V0 1 FIG. 1: The solid (black) line represents the surface energy R = r a(t) = . (3.29) AH AH H density σ, while the dashed (red) line denotes the quantity σ + η. When plotting these curves, we set rΣ = 1,M = In Fig. 1 we show one of the cases, in which the free 1, a0 = 3,V0 = 0.6,H = 1.5. At the moment t = 0.624554, parameters are chosen as rΣ = 1,M = 1, a0 = 3,V0 = we have σ + η = 0, RΣ = 1.1756, and RUH = 1.24387. 0.6,H = 1.5. Then, we find that the weak energy condition holds until the moment t = 0.624554, at which we have R = 1.24387 > R = 1.1756. That is, the weak RUH R" R UH Σ 5 energy condition holds all the way down to the moment when the whole star collapses inside the universal hori- 4 zon, as it is illustrated clearly in Fig.2. In this figure, Min 3 three horizontal lines, R = 2M, 3M/2,RΣ are also plotted, where RSch. = 3M/2 is the universal horizon in UH 2 M 2 the Schwarzschild space-time [cf. Eq.(1.4)]. For the cur- 3M 2 rent choice of the free parameters, the universal horizon 1 Min is not continuous. In fact, when the star collapses to the R" moment t = t , where t is given by R (t ) = 3M/2, t o o Σ o !2 !1 1 2 the universal horizon jumps from RUH (to) to 3M/2, as ! shown more clearly in Fig. 3. Physically, this is because that the surface shell of the collapsing star has Min non-zero mass. The collapse ends at t = tΣ , where Min Min FIG. 2: Universal horizon RUH , the surface radius RΣ of the RΣ(tΣ ) = RΣ , at which the surface pressure of the collapsing star becomes infinitely large. It is remark- collapsing star vs t for rΣ = 1,M = 1, a0 = 3,V0 = 0.6,H = able that RMin generically is different from zero, that 1.5. The horizontal lines R = 2M and R = 3M/2 are, respec- Σ tively, the Killing and universal horizons of the Schwarzschild is, the collapse generically forms a space-time singular- vacuum solution. The collapse ends at the moment when ity that has finite radius. The corresponding Penrose Min RΣ = RΣ , at which point the pressure of the surface of the diagram is given in Fig. 4. In this figure the location collapsing star becomes infinitely large, whereby a space-time of the event horizon denoted by the straight line EH is singularity with finite thickness is developed. also marked, although it can be penetrated by particles with sufficiently large velocities, and propagate to infinity, even they are initially trapped inside it. However, by simply replacing Killing horizons by apparent ones. this is no longer the case when across the universal hori- Then, we have constructed an analytical model that rep- zon. As explained above, once they are trapped inside resents the gravitational collapse of a spherical symmet- the universal horizon, they cannot penetrate it and prop- ric star with finite thickness, and shown explicitly that agate to infinities, even they are moving with infinitely dynamical universal horizons can be formed from such a large velocities. As a result, an absolutely black region “realistic” gravitational collapse. Here “realistic” is re- is (classically) formed from the gravitational collapse of ferred to as a gravitational collapse of a star with a finite a massive star, and this region is black, even in theories thickness that satisfies at least the weak energy condition that allow instantaneous propagations! [28]. To have the problem tractable, we have assumed that the star consists of an anisotropic and homogeneous per- IV. CONCLUSIONS fect fluid and that outside the star the space-time is vacuum in the framework of the Einstein-aether the- In this paper, we have first generalized the definition ory [18]. When the effects of the aether field is negli- of a stationary universal horizon to a dynamical one, gible, the vacuum space-time is uniquely described by 7

t Acknowledgements

3M/2 This work was done partly when A.W. was visiting the State University of Rio de Janeiro (UERJ). He would like Min R = R > 0 V+ A B to A UH R 0 B V EH V + RUH V

FIG. 3: Gravitational collapse of a spherically symmetric star

with its radius RΣ(t), which divides the space-time into two R = 0 ± regions, V . The curved line RUH denotes the universal horizon formed inside the star, while the vertical straight line R = 3M/2 is the universal horizon of the Schwarzschild vacuum solution. When the star collapses inside the Schwarzschild universal horizon R = 3M/2, the universal horizon suddenly jumps from RUH (to) to 3M/2, because of the non-zero mass of the collapsing surface of the star. the Schwarzschild solution [18]. Even in this case, solv- ing the khronon equation (2.9) inside the star is still very complicated. Instead, we have further assumed that the velocity of the the khronon is infinitely large, so the sound FIG. 4: Penrose diagram of a collapsing star described in the horizon of the khronon coincides with the universal hori- content for the case where the free parameters of the solution zon. Then, we have found that an analytical solution are chosen as rΣ = 1,M = 1, a0 = 3,V0 = 0.6,H = 1.5. The surface Σ of the collapsing star divides the whole space-time exists for the star made of the de Sitter universe, and + shown explicitly how a dynamical universal horizon can into two regions, the external region V and the internal region V −. The curved line UH denotes the universal horizon, be formed, as can be seen from Figs. 2 - 4. while the straight line EH the event horizon. The star col- Although such a constructed model serves our current lapses to form generically a space-time singularity at a finite purpose very well, that is, to show that universal hori- Min non-zero radius R = RΣ > 0. zons can be indeed formed from realistic gravitational collapse, it would be very interesting to consider cases without (some of) the above assumptions, specially the case in which the space-time outside of the star is not vacuum, so that the star may radiate, when it is collaps- to thank UERJ for hospitality. It was done also during ing. the visit of M.T. to Baylor University. M.T. would like In addition, although in this paper we have consid- to express his gratitude to Baylor. This work is sup- ered the formation of the universal horizons only in the ported in part by CiênciaSem Fronteiras, Grant No. framework of the Einstein-aether theory, it is expected A045/2013 CAPES, Brazil (A.W.); NSFC Grant No. that our main conclusions should be true in other theo- 11375153, China (A.W.); CAPES, CNPq and FAPERJ ries of gravity with broken LI, including the HL gravity Brazil (M.F.A.S.); CSC Grant No. 201208625022, China [11, 17]. (M.T.); and Baylor Graduate Scholarship (X.W.).

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