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 ∗y 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 S~aoFrancisco Xavier 524, Maracan~a,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 yElectronic address: [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 p,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 jgj 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.

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