I-Love-Q Relations: From Compact Stars to Black Holes Kent Yagi 6 1 Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 0 eXtreme Gravity Institute, Department of Physics, Montana State University, 2 Bozeman, Montana 59717, USA y a M Nicolás Yunes 6 eXtreme Gravity Institute, Department of Physics, Montana State University, 2 Bozeman, Montana 59717, USA ] c Abstract. q - The relations between most observables associated with a compact star, such as the r g mass and radius of a neutron star or a quark star, typically depend strongly on their [ unknown internal structure. The recently discovered I-Love-Q relations (between the moment of inertia, the tidal deformability and the quadrupole moment) are however 2 v approximately insensitive to this structure. These relations become exact for stationary 1 black holes in General Relativity as shown by the no-hair theorems, mainly because 7 black holes are vacuum solutions with event horizons. In this paper, we take the 1 2 first steps toward studying how the approximate I-Love-Q relations become exact in 0 the limit as compact stars become black holes. To do so, we consider a toy model . 1 for compact stars, i.e. incompressible stars with anisotropic pressure, which allows 0 us to model an equilibrium sequence of stars with ever increasing compactness that 6 approaches the black hole limit arbitrarily closely. We numerically construct such a 1 : sequence in the slow-rotation and in the small-tide approximations by extending the v Hartle-Thorne formalism, and then extract the I-Love-Q trio from the asymptotic i X behavior of the metric tensor at spatial infinity. We find that the I-Love-Q relations r approach the black hole limit in a nontrivial way, with the quadrupole moment and the a tidal deformability changing sign as the compactness and the amount of anisotropy are increased. Through a generalization of Maclaurin spheroids to anisotropic stars, we showthatthemultipolemomentsalsochangesignintheNewtonianlimitastheamount of anisotropy is increased because the star becomes prolate. We also prove analytically that the stellar moment of inertia reaches the black hole limit as the compactness reaches a critical black hole value in the strongly anisotropic limit. Modeling the black hole limit through a sequence of anisotropic stars, however, can fail when considering other theories of gravity. We calculate the scalar dipole charge and the moment of inertia in a particular parity-violating modified theory and find that these quantities do not tend to their black hole counterparts as the anisotropic stellar sequence approaches the black hole limit. PACS numbers: 04.30.Db,04.50Kd,04.25.Nx,97.60.Jd I-Love-Q Relations: From Compact Stars to Black Holes 2 1. Introduction A plethora of compact stars with masses between 1 M and 2 M and with radii (cid:12) (cid:12) of approximately 12 km have been discovered through a variety of astrophysical observations [1–4]. The limited accuracy of these observations, coupled to degeneracies in the observables with respect to different models for the nuclear physics at supranuclear densities encoded in the equation of state (EoS), have prevented observations from elucidating the internal structure of compact objects. For example, X-ray observations do not typically allow us to confidently state whether the compact objects observed are standard neutron stars [5–8], or hybrid stars with quark-gluon plasma cores [9,10], or perhaps even strange quark stars [11]. Future observations of compact objects could shed some light on this problem, as the accuracy of the observations increases and more observables are obtained [12,13]. The extraction of information from these future observations is aided by the use of approximately universal relations, i.e. relations between certain observables that are roughly insensitive to the EoS [13–17]. For example, the moment of inertia I, the tidal deformability λ (or tidal Love number) and the (rotation-induced) quadrupole moment 2 Q satisfy relations (the so-called I-Love-Q relations) that are EoS insensitive to a few % level [14,15]. Such relations are useful to analytically break degeneracies in the models used to extract information from X-ray and gravitational-wave observations of compact objects. This information, in turn, allows us to better probe nuclear physics [13] and gravitational physics [14,15]. Similar universal relations exist among the multipole moments of compact stars [18–22], i.e. the coefficients of a multipolar expansion of the gravitational field far from the compact object. These no-hair like relations resemble the well-known, black hole (BH) no-hair relations of general relativity (GR) [23–29]. The latter state that all multipole moments of an uncharged, stationary BH in GR can be prescribed only in terms of the first two (the BH mass and spin). The no-hair like relations of compact stars differ from the BH ones in that the former require knowledge of the first three stellar multipole moments to prescribe all higher moments in a manner that is roughly insensitive to the underlying EoS. But how are the approximate I-Love-Q and no-hair like relations for compact stars related to those that hold for BHs exactly? One way to address this question is to carry out simulations of compact stars that gravitationally collapse into BHs, extract the I- Love-Q and multipole moments and study how the relations evolve dynamically. However, not only are such simulations computationally expensive, but the machinery employed in the past would no longer be useful, as it is valid only for stationary spacetimes, i.e. the Geroch-Hansen multipole moments [30,31] used for example in [14,15,18,20] are not well-defined for non-stationary spacetimes. One would have to employ a dynamical generalization of these moments and develop a procedure to extract them from dynamical simulations. A simpler way to gain some insight is to consider how the universal relations evolve I-Love-Q Relations: From Compact Stars to Black Holes 3 in a sequence of equilibrium stellar configurations∗ of ever increasing compactness that approaches the compactness of BHs arbitrarily closely. Such a sequence, however, cannot be constructed from neutron star solutions with isotropic pressure, as used in the original I-Love-Q [14,15] and no-hair like relations [18,20]; such stars have a maximum stellar compactness (i.e. the ratio between the stellar mass and radius) that is well below the BH limit. An alternative approach is to consider a sequence of anisotropic stars† (see e.g. [33] for a review of anisotropic stars), which, for example, in the Bowers and Liang (BL) model [34] can reach BH compactnesses for incompressible stars in the strongly anisotropic limit. Following this logic, in [35] we studied how the no-hair like relations for compact stars approach the BH limit. We first showed that the stellar shape transitions from prolate to oblate as the compactness is increased. We then showed that the multipole moments approach the BH limit with a power-law scaling and that the no-hair like relations also approach the BH limit in a very nontrivial way. In this paper we extend these investigations in a variety of ways and clarify several points that were left out of the initial analysis. First, in this paper we consider both slowly-rotating stars and tidally-deformed stars, which allow us to study how the I-Love-Q relations approach the BH relations in the BH limit. In [36], we constructed tidally-deformed or slowly-rotating, anisotropic compact stars to third order in spin for various realistic EoSs. We here follow [36] but focus on incompressible stars, as this allows us to construct an equilibrium sequence of anisotropic stars that approaches the BH limit arbitrarily closely. Second, we extend the analysis of [35] by carrying out analytic calculations in various limits: (i) the weak-field limit, (ii) beyond the weak-field limit, (iii) the strong-field limit and (iv) the strongly-anisotropic limit. In the first limit, we expand all equations in small compactness and retain only the leading terms in the expansion. This leads to anisotropic stars modeled as incompressible spheroids with arbitrary rotation that reduce to Maclaurin spheroids [37–39] in the isotropic limit. When going beyond the weak-field limit, we retain subleading terms in the small compactness expansion, which is equivalent to a post-Minkowskian (PM) expansion; we extend the work of [40] for isotropic stars to the anisotropic case and derive the moment of inertia and tidal deformability. In the third limit, we expand all equations about the maximum compactness allowed for incompressible stars, extending the analysis of [41] to anisotropic stars and deriving the tidal deformability for specific choices of the anisotropy parameter. In the fourth limit, we expand all equations about the maximum anisotropy allowed by the BL model, analytically deriving the moment of inertia for incompressible stars as a function of the compactness. Third, we study whether an equilibrium sequence of anisotropic compact stars can ∗ Another approach is to consider “BH mimickers” whose compactness can reach that of BHs, such as the gravastars considered in [32]. The latter, however, are very different from neutron stars or quark stars. † Anisotropic stars are here only used as a toy model to study an equilibrium sequence of compact stars that can reach the BH limit, and not as a realistic model for compact stars. I-Love-Q Relations: From Compact Stars to Black Holes 4 be used to study the BH limit of stellar observables in theories other than GR. As an example, we work in dynamical Chern-Simons (dCS) gravity [42–44], a parity violating modified theory of gravity that is motivated from string theory [45], loop quantum gravity [46–48] and effective theories of inflation [49]. We treat this modified theory as an effective field theory and assume that the GR deformation is small. Such a treatment ensures the well-posedness of the initial value problem [50]. Slowly-rotating, anisotropic compact stars to linear order in spin in dCS gravity were constructed in [36] using realistic EoSs within the anisotropy model proposed by Horvat et al. [51]. We now extend the treatment in [36] to the BL anisotropy model and focus on the incompressible case. 1.1. Executive Summary Let us now present a brief summary of our results. We find that the I-Love-Q relations for strongly anisotropic stars in GR indeed approach the BH limit as one increases the compactness. Figure 1 shows evidence for this by presenting the I-Love-Q relations for incompressible stars with a variety of anisotropy parameters λ in the BL model [34]. BL The isotropic case is recovered when λ = 0, while λ = −2π corresponds to the BL BL ¯ ¯ strongly anisotropic limit. The BH limit (λ = 0) corresponds to I = 4 and 2,BH BH ¯ Q = 1, shown with dashed horizontal lines. We confirm the validity of our numerical BH results by comparing them to an analytic calculation of the I-Love relations in the PM approximation (solid curves in the top panel of Fig. 1). Observe that the relations approach the BH limit as the compactness is increased (shown with arrows) in a way ¯ ¯ that depends quite strongly on λ , with λ and Q changing sign as the BH limit is BL 2 approached. We also find that the approach of the I-Love-Q relations to the BH limit appears to be continuous, as shown in Fig. 1. That is, we find no evidence of the discontinuity hypothesized in [52], based on a weak-field calculation of the quadrupole moment of strongly anisotropic, incompressible stars. We in fact prove analytically that the moment of inertia of a strongly anisotropic, incompressible compact star reaches the BH limit continuously as the compactness is increased. We do so by constructing slowly-rotating, anisotropic incompressible stars to linear order in spin in the strongly anisotropic limit ¯ (λ = −2π) and analytically deriving I as a function of the compactness C in terms of BL hypergeometric functions. Taylor expanding I¯ about C = 1/2+O(χ2), with χ the BH dimensionless spin parameter, we find that I¯(C) = I¯ +O(C −C ,χ2). BH BH The quadrupole moment changes sign as it approaches the BH limit, as shown in Fig. 1, but is this the case for all multipole moments? We find that this is not the case by constructing incompressible spheroids with anisotropic pressure and arbitrary rotation in the weak-field limit. We derive a necessary condition on the anisotropy model such that spheroidal configurations are realized and find that the BL model satisfies such a condition. We then calculate the (cid:96)th mass and current multipole moments, M and S , (cid:96) (cid:96) in the slow-rotation limit within the BL model and find that M and S are both 2(cid:96)+2 2(cid:96)+3 proportional to 1/(4π +5λ )(cid:96)+1. This means that the sign of only (M , M , M ...) BL 2 6 10 I-Love-Q Relations: From Compact Stars to Black Holes 5 102 λ =π BL λ =0 BL λ =-0.5π BL λ =-1.5π I λBL=-1.9π 101 PBML Padé I BH 101 Q| | 100 Q BH 10-1 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 |λ| 2 Figure 1. (Color online) Relations between the dimensionless moment of inertia I¯≡I/M3 and the dimensionless tidal deformability λ¯ ≡λ /M5 (top), and between ∗ 2 2 ∗ the dimensionless quadrupole moment Q¯ ≡ −Q/(M3χ2) and λ¯ (bottom) for an ∗ 2 equilibrium sequence of anisotropic, incompressible stars with varying compactness (the arrows indicate increasing compactness), given some anisotropy parameter λ , to BL leading-order in slow rotation and tidal deformation. M is the stellar mass, χ≡J/M2 ∗ ∗ is the dimensionless spin parameter, with J the magnitude of the stellar spin angular momentum. Isotropic stars correspond to λ = 0, while strongly-anisotropic stars BL correspond to λ = −2π. Our numerical results are validated by analytic PM BL calculations(solidcurves). ThedashedhorizontallinescorrespondtotheBHvaluesofI¯ andQ¯,whileλ¯ =0istheBHvalueforthedimensionlesstidaldeformability. Observe 2,BH that the I-Love-Q relations of anisotropic stars approach the BH limit continuously. and (S , S , S ...) is opposite to that of the isotropic case when λ < −4π/5, which is 3 7 11 BL consistent with the results of [52] for the quadrupole moment M . In particular, the sign 2 of (M , M , M ...) and (S , S , S ...) is the same as that of the sign of the isotropic 4 8 12 5 9 13 case even in the strongly-anisotropic limit. Although the I-Love-Q relations for compact stars approach the BH limit as one increases the compactness in GR, we find that this is not always the case in other theories of gravity when the limit is modeled through an equilibrium sequence of anisotropic stars. Figure 2 presents evidence for this by showing the scalar dipole charge and the correction to the dimensionless moment of inertia in dCS gravity as a function of the stellar compactness. Once more, we validate our numerical results by comparing them to analytic PM relations for the scalar dipole charge. Observe that unlike in the GR case, these quantities do not approach the dCS BH limit (shown with black crosses) as one increases the compactness. This result suggests that modeling the BH limit through strongly anisotropic stars is not appropriate in certain modified theories of gravity. The remainder of this paper presents the details of the calculations that led to the results summarized above. In Sec. 2, we explain the formalism that we use to construct slowly-rotating and tidally-deformed anisotropic stars. We also describe the I-Love-Q Relations: From Compact Stars to Black Holes 6 λ =0 BL 12 dCS λBL=-1.2π λ =-1.5π BL λ =-1.9π BL 10 PM Padé µ BH 8 0.4 BH 0.3 0.2 Iδ 0.1 0 -0.1 -0.2 0.1 0.2 0.3 0.4 0.5 C Figure 2. (Color online) Dimensionless scalar dipole charge µ¯ [Eq. (72)] (top) and the dCS correction to the dimensionless moment of inertia δI¯ (normalized by the dimensionless dCS coupling constant and the GR value of I¯) [Eq. (73)] (bottom) for a sequence of anisotropic, incompressible stars labeled by stellar compactness C and anisotropy parameter λ . Corresponding BH values are shown by black crosses. Our BL numerical results are validated by analytic PM calculations (solid curves) [Eq. (B.12)] in the top panel. Observe that, unlike in the GR case, µ¯ and δI¯do not approach the BH limit as one decreases λ and increases C. BL BL anisotropy model and show how the maximum stellar compactness for a non-rotating configuration approaches the BH one in the strongly anisotropic limit. In Sec. 3, we present analytic calculations of the stellar moment of inertia, tidal deformability and multipole moments in certain limits. In Sec. 4, we present numerical results that show how the I-Love-Q relations approach the BH limit in GR. We also show that the scalar dipole charge and the correction to the moment of inertia in dCS gravity do not approach the BH limit. Finally, in Sec. 5, we give a short summary and discuss various avenues for future work. We use the geometric units of c = 1 = G throughout this paper. 2. Formalism and Anisotropy Model In this section, we first explain the formalism we use to construct slowly-rotating or tidally-deformed compact stars with anisotropic pressure and extract the stellar multipole moments and tidal deformability. We then explain the specific anisotropic model that we will use throughout the paper. We present the spherically-symmetric background solution and describe the maximum compactness such a solution can possess for polytropic EoSs of the form p = Kρ1+1/n. Here p and ρ are the stellar radial pressure and energy density, while K and n are constants. Henceforth, the stellar compactness is defined by C ≡ M /R , where M and R are the stellar mass and radius for a non-rotating ∗ ∗ ∗ ∗ configuration respectively. I-Love-Q Relations: From Compact Stars to Black Holes 7 2.1. Formalism Let us first explain how one can construct slowly-rotating compact stars with anisotropic pressure, by following [36,53–55] and extending the Hartle-Thorne approach [56,57] to third order in spin. Let us assume the spacetime is stationary and axisymmetric, such that the metric can be written as (cid:20) 2(cid:15)2m(r,θ)(cid:21) (cid:2) (cid:3) ds2 = −eν(r) 1+2(cid:15)2h(r,θ) dt2 +eλ(r) 1+ dr2 r−2M(r) (cid:16) (cid:17) +r2(cid:2)1+2(cid:15)2k(r,θ)(cid:3) dθ2 +sin2θ(cid:8)dφ−(cid:15)(cid:2)Ω−ω(r,θ)+(cid:15)2w(r,θ)(cid:3)dt(cid:9)2 +O((cid:15)4), (1) where ν and λ are functions of the radial coordinate r only, while ω, h, k, m and w are functions of both r and θ. The quantity (cid:15) is a book-keeping parameter that labels the order of an expression in (M Ω), where Ω is the spin angular velocity. The surface ∗ is defined as the location where the radial pressure vanishes. We transform the radial coordinate via r(R,θ) = R+(cid:15)2ξ(R,θ)+O((cid:15)4), (2) so that the spin perturbation to the radial pressure and density vanish throughout the star [56,57]. The enclosed mass function M(r) is defined via 2M(r) e−λ(r) ≡ 1− , (3) r and thus, M is the value of M(r) evaluated at the stellar surface R . We decompose ω, ∗ ∗ h, k, m, ξ and w in Legendre polynomials [36]. The stress-energy tensor for matter with anisotropic pressure can be written as [36,58,59] T = ρ u u +p k k +q Π , (4) µν µ ν µ ν µν where q is the tangential pressure and uµ is the fluid four-velocity, given by uµ = (u0,0,0,(cid:15)Ωu0), with u0 determined through the normalization condition uµu = −1. µ kµ is a unit radial vector that is spacelike (kµk = 1) and orthogonal to the four- µ velocity (kµu = 0) of the fluid, while Π ≡ g +u u −k k is a projection operator µ µν µν µ ν µ ν onto a two-surface orthogonal to uµ and kµ. We introduce the anisotropy parameter σ ≡ p−q [58,59] with σ = 0 corresponding to isotropic matter. Following the treatment of metric perturbations, we expand σ in the slow-rotation approximation and decompose each term in Legendre polynomials as (cid:110) (cid:111) σ(R,Θ) = σ(0)(R)+(cid:15)2 σ(2)(R)+σ(2)(R)P (cosθ) +O((cid:15)4). (5) 0 0 2 2 Notice that the superscript (subscript) in σ(n) corresponds to the order of the spin (cid:96) (Legendre) decomposition. The function σ(0) needs to be specified a priori, and it in fact 0 defines the anisotropy model. The function σ(2) is determined consistently by solving the 2 perturbed Einstein equations, once σ(0) is chosen [36]. The function σ(2) is irrelevant in 0 0 this paper as it only affects the stellar mass at subleading order in a small spin expansion. I-Love-Q Relations: From Compact Stars to Black Holes 8 We construct slowly-rotating compact star solutions with anisotropic pressure as follows. First, we substitute the metric ansatz and the matter stress-energy tensor mentioned above into the Einstein equations. We then expand in small spin (or equivalently in (cid:15)) and solve the perturbed Einstein equations order by order in (cid:15). In the interior region, we solve the equations numerically with a regularity condition at the center. In the exterior region, we solve the equations analytically with an asymptotic flatness condition at spatial infinity. We finally match the two solutions at the stellar surface to determine any integration constants. The latter determine the moment of inertia I, the quadrupole moment Q and the octupole moment S of the exterior solution 3 at linear, quadratic and third order in spin respectively. In this paper, we also construct non-rotating but tidally-deformed compact stars to extract the stellar tidal deformability [41,60]. We are here particularly interested in the quadrupolar, electric-type tidal deformability, λ , which is defined as the ratio 2 of the tidally-induced quadrupole moment and the external tidal field strength. We follow [41,60,61] and treat tidal deformations as small perturbations of an isolated compact star solution. Such a tidally-deformed compact star can be constructed similarly to how we construct slowly-rotating solution, except that we set Ω = ω = w = 0, as we are only interested in electric-type, even-parity perturbations. For convenience, we work with the following dimensionless quantities throughout: I Q S λ ¯ ¯ ¯ 3 ¯ 2 I ≡ , Q ≡ − , S ≡ − , λ ≡ . (6) M3 M3χ2 3 M4χ3 2 M5 ∗ ∗ ∗ ∗ Here, the dimensionless spin parameter χ is defined through the magnitude of the spin angular momentum J by χ ≡ J/M2, with J only kept to O((cid:15)). The BH value of ∗ ¯ ¯ ¯ each dimensionless quantity above is I = 4 [62], Q = 1 [31], S = 1 [31] and BH BH 3,BH ¯ λ = 0[29,41,60,63,64]. Weherechoosetoworkwiththeabovechoiceofnormalization 2,BH introduced in [14,15,18–20], but clearly this choice is not unique. In fact, one can choose other normalizations, for example involving the stellar compactness, which may improve the universality in the I-Love-Q relations and no-hair like relations for compact stars among stellar multipole moments [22]. Other choices of normalization, nonetheless, will not affect the conclusions we arrive at in this paper. 2.2. Anisotropy Model Let us now describe the specific anisotropy model that we use in this paper. Following BL [34], we choose λ (cid:18) 2M(cid:19)−1 σ(0)(R) = BL(ρ+3p)(ρ+p) 1− R2. (7) 0 3 R Here, λ is a constant parameter that characterizes the amount of anisotropy. Isotropic BL pressure corresponds to λ = 0, since then both σ(0) and σ(2) vanish. The particular BL 0 2 form of σ(0) in Eq. (7) was proposed such that the Tolman-Oppenheimer-Volkoff equation 0 I-Love-Q Relations: From Compact Stars to Black Holes 9 could be solved analytically for a spherically-symmetric, incompressible (polytropic index n = 0, i.e. ρ = const.) anisotropic star to yield [34] C M(R) = R3, (8) R2 ∗ 3 C ρ(R) = , (9) 4πR2 ∗ 3 C (1−2C)γ −(1−2CR2/R2)γ p(R) = − ∗ , 4πR23(1−2C)γ −(1−2CR2/R2)γ ∗ ∗ (10) 1 (cid:20)3(1−2C)γ −(1−2CR2/R2)γ(cid:21) ν(R) = ln ∗ , (11) γ 2 where γ is defined by (cid:18) (cid:19) 1 λ γ ≡ 1+ BL . (12) 2 2π How do the maximum compactness C of anisotropic stars differ from the isotropic max case? One can find C for anisotropic stars by finding the value of C for which the max central radial pressure p(R = 0) diverges [34]: 1 (cid:0) (cid:1) C = 1−3−1/γ . (13) max 2 Clearly, the solution in Eqs. (10) and (11) is only well-defined for λ ≥ −2π, such that BL C ≥ 0. Such a condition on λ also ensures that p ≥ 0 and the solution does not max BL diverge when C ≤ C . The red solid curve in Fig. 3 presents C as a function of λ max max BL for incompressible stars [Eq. (13)]. Observe that in the isotropic incompressible case, C = 4/9 ≈ 0.444..., while C approaches 1/2, the compactness of a non-rotating BH, max max in the λ → −2π limit. This is exactly why we consider anisotropic stars in this paper, BL as they allow us to construct a sequence of equilibrium stars that approaches the BH limit arbitrarily closely. Henceforth, C ≡ 1/2, the compactness of a non-rotating BH, since BH we work to leading order in the slow rotation approximation and C = 1/2+O(χ2). Kerr For reference, Fig. 3 also shows the maximum compactness for anisotropic stars with an n = 1 polytropic EoS (blue dashed curve), which is always smaller than that of incompressible stars. Although the maximum compactness of non-rotating, anisotropic stars can reach the compactness of non-rotating BHs in the strongly anisotropic limit (λ → −2π), the BL causal structure inside such a star is quite different from that of a BH. The left, middle and right panels of Fig. 4 show the causal structure of non-rotating anisotropic compact stars with C = 0.3 and C = 0.5, and that of a BH respectively. To construct these panels, we introduce a new (retarded) time coordinate T = v −R [65], where v = t+r ∗ is a null coordinate with r the tortoise coordinate in the exterior and interior regions, ∗ given by Eqs. (A.1) and (A.2) respectively. The ingoing null geodesics (blue lines in the figure) are given by v = const., while the outgoing null geodesics (red curves in the I-Love-Q Relations: From Compact Stars to Black Holes 10 0.5 BH n=0 n=1 0.4 max C 0.3 0.2 -2 -1 0 1 2 λ /π BL Figure 3. (Color online) Maximum compactness realized for the n = 0 (solid) and n=1(dashed)polytopesasafunctionoftheanisotropyparameterλ . Thehorizontal BL dashed line corresponds to the compactness of a non-rotating BH. �=��� �=��� �� � � � � � � � � � �* �* �* /� /� /� � � � � � � � � � � � � � � � � � � � � � � � � � � � �/�* �/�* �/�* Figure 4. (Color online) Causal structure of a non-rotating compact anisotropic star with C = 0.3 (left) and C = 0.5 (middle) with λ = −2π and a non-rotating BL BH (right). Blue and red curves correspond to ingoing and outgoing null geodesics respectively. The opening angle between these curves at each crossing point shows that of a light cone at each point. Observe that the surface of an anisotropic star with C = 0.5 is a trapped surface and radiation inside the star cannot escape to outside. Observe also how the causal structure of the interior region for such a star is different from that of a BH. figure) are given by t−r = const. The opening angle between blue and red curves at ∗ each point represents that of the light cone, while the stellar surface or the event horizon are denoted by a black dashed vertical line. Observe that a photon emitted inside a star with C = 0.3 can escape out to spatial infinity, while that from a star with C = 0.5 cannot. This is because the surface for the latter acts as a trapped surface, just like the event horizon of a BH. However, notice that the causal structure in the interior region between an anisotropic compact star with C = 0.5 and a BH is different. In particular,