How to tell a gravastar from a black hole (also when it is rotating...) Cecilia Chirenti Albert Einstein Institute, Potsdam, Germany Physics Institute, University of São Paulo, Brazil in collaboration with Luciano Rezzolla, AEI Work supported by FAPESP, DAAD, MPG and SFB. Also thanks to Shin Yoshida for many useful discussions... Plan of the talk • The challenge to the idea of black holes • The basic gravastar model: infinitesimal shells • Modelling fluids: finite shells • Perturbative analysis and results • Rotating gravastars and ergoregion instability • Conclusions and future work How to tell a gravastar from a black hole • In 2001 Mazur and Mottola (MM) proposed “gravastars” as an alternative to explain the phenomenology of the most cherished foundation of modern astrophysics: black holes. • Gravastars are non-vacuum, spherically symmetric and static solutions of the Einstein equations (hence the name stars). • However, they can be built to be arbitrarily compact, with an external surface which is arbitrarily close to the horizon of a Schwarzschild black hole with the same mass. The basic idea Take a spherically symmetric, static line element: 2 dr ds2 = −f (r)dt2 + + r2 dΩ2 , h(r) and split the spacetime into 3 regions I. Interior: II. Shell: III. Exterior: 0 ≤ r ≤ r1 , ρ = −p , (deSitter) r 1 ≤ r ≤ r2 , ρ = +p , (TOV) ρ = p = 0, (Schwarzschild) r2 ≤ r , Because the gravity and pressure of the shell compensate the expansion of the interior, an equilibrium solution of the Einstein equations can be found. The challenge In the “thin-shell” limit, i.e. δ=r2 - r1 → 0, where δ is the thickness of the shell, a gravastar (g*) solution can be found even analytically. Because a g* can be made arbitrarily compact, i.e. , with an external radius r2=2M + ε, with ε → 0, it may be impossible to distinguish it from a black hole via EM radiation: • in both cases the gravitational redshift would be essentially divergent • the heat capacity of g*’s is expected to be very large (hard to heat them up and hence no radiation from cooling) Two basic questions A large literature followed the first MM idea and nowadays g* have been built with the most exotic interior. However, basic questions remain unanswered: • Are g*’s stable to generic perturbations? • If so, can an external observer distinguish a g* from a black hole of the same mass? To answer these questions and perform a stability analysis, we have considered a general class of g*’s having a shell of finite thickness but with tangential pressures, to have fluid variables continuous everywhere and avoid junction conditions on the metric. A general class of gravastars In essence, we solve the generalized TOV equations for the metric and pressure (both radial pr and tangential pt) ! r e −λ 2m(r) =1− , r m(r) ≡ 3 m(r) + 4πr pr 2(pt − pr ) ! pr = −(ρ + pr ) + , r(r − 2m(r)) r with a prescribed polynomial density and equation of state  0 ≤ r ≤ r1  ρ0 , ar3 + br2 + cr + d , r1 < r < r2 ρ(r) =  0, r2 ≤ r 4πr2 ρdr . 0 A general class of gravastars In practice we have computed a large class of solutions by varying the compactness µ= M/r2 of the g* and its thickness δ=r2 - r1 . For many of these models we have then performed a stability analysis against axial perturbations determining the real and imaginary part of the eigenfrequencies Axial-Perturbations analysis For this we solve the Regge-Wheeler equation for axial perturbations of compact stars spacetimes, ∂2ψ ∂2ψ − 2 = V! (r)ψ , 2 ∂r∗ ∂t where r∗ ≡ ! 0 r e (λ−ν)/2 dr , # eν " 3 V# (r) ≡ 3 !(! + 1)r + 4πr (ρ − pr ) − 6m . r The numerical calculations are effectively done using the null coordinates u ≡ t − r∗ , v ≡ t + r∗ , and which lead to the more compact form ∂2ψ −4 (u, v) = V! (r)ψ(u, v) ∂u∂v Axial-Perturbations analysis We use a “triangular grid”, i.e a purely outgoing null slicing, and evolve the perturbation equation after introducing an initial Gaussian pulse . The solution ψ is extracted at a large distance and after the initial transient has died off. This shows a typical example of the QNM evolution for a g* with M = 1, r1 = 1.85, r2 = 2.2 Some eigenfrequencies are reported below for the fundamental mode and its first two overtones model δ = 0.30 δ = 0.35 δ = 0.40 Schwarzchild black hole Schwarzchild star ωR 0.3281 0.2943 0.2575 0.3737 0.1090 n=0 −ωI 2.481e-3 7.081e-4 1.543e-4 8.896e-2 1.239e-9 ωR 0.4865 0.4459 0.4011 0.3467 0.1484 n=1 −ωI 6.264e-2 3.202e-2 1.227e-2 2.739e-1 3.950e-8 ωR 0.6534 0.5922 0.5384 0.3011 0.1876 n=2 −ωI 1.590e-1 1.093e-1 5.814e-2 4.783e-1 5.470e-7 Telling them apart... While δ and µ can be chosen so that the g* and the bh have the same oscillation freq., the decaying times will be very different. A g* and a black hole of the same mass cannot have the same complex eigenfrequencies: an observer can tell them apart beyond dispute Rotating gravastars In a recent work (Cardoso et al, 2007), a generalization of the original static, spherically symmetric gravastar model was considered, in order to describe rotating gravastars. A slow rotation approximation was used to describe the axially symmetric spacetime of a rotating gravastar. It was argued that gravastars might be unstable due to the ergoregion instability. The preliminary results obtained by Cardoso et al indicate that the time scale of the instability could be very short when compared to the Hubble time. Further investigation is still needed... Rotating gravastars Slow rotation approximation (to first order in Ω ) ds = −e 2 ν(r) dt + e 2 λ(r) 2 dr + r dθ + r sin θ (dφ − ω(r)dt) 2 2 2 2 2 ω(r) gives the dragging of the inertial frame Anisotropic energy momentum tensor T µν = (ρ + pt )uµ uν + pt g µν + (pr − pt )sµ sν uµ uµ = −1 , sµ sµ = 1 , uµ sµ = 0 , ur = uθ = 0 , uφ = Ωut , ! "−1/2 t 2 u = −(gtt + 2Ωgtφ + Ω gφφ ) Ω is the angular velocity of the gravastar “Ergoregion instability” vs “superradiance” Superradiance: waves can be amplified when reflected by a rotating black hole. Waves coming from infinity with σ < mΩ are reflected back with greater (but finite) amplitudes. The ergoregion instability appears in any system with ergoregions and no horizons, e.g. models of dense, rotating fluids. In an ergoregion, the dragging of inertial frames is so strong that all trajectories must rotate in the prograde direction. A star without a horizon but with an ergoregion is unstable to the emission of scalar, electromagnetic and gravitational radiation: any initially small perturbation will grow exponentially with time. Ergoregion instability Rotating, very compact relativistic stars can develop an ergoregion. It is natural to think that this might also be the case for gravastars, which can be made almost as compact as black holes. The ergoregion is limited by (topologically toroids) The field equation for the dragging is In vacuum 0 = ξt ξt = g00 = −eν + r2 ω 2 sin2 θ !!! + ! 4 4πr (ρ + pr ) − r r − 2m 2 !(r) = Ω − ω(r) 2J !(r) = Ω − 3 r " !! = 16πr(ρ + pt ) ! r − 2m Ergoregion instability: effective potentials for a scalar field in the background metric, there are the two rotationally split “effective potentials” V+ and V− . ψ,rr + m2 T (r, Σ)ψ = 0 , T = eλ−ν (Σ − V+ )(Σ − V− ) , ν e2 V± = −ω ± r σ Σ= m Size of the ergoregion The size of the ergoregion increases with both the compactness and the thickness of the shell. But there are constrains... Where is an ergoregion possible? We have already discussed the space of possible solutions in the (µ, δ) plane for spherical g*’s This space is further divided if rotation is taken into account no ergoregion Where is an ergoregion possible? For a given point in the (µ, δ) plane it is possible to determine the critical angular velocity above which an ergoregion is present. G*’s with angular velocities smaller than the critical one do not have an ergoregion and are therefore expected to be stable. WKB analysis or calculation of eigenfrequencies will then establish whether unstable g*’s are so in a Hubble time... Conclusions o g*’s are an ingenious solution of the Einstein eqs: can be arbitrarily compact, with outer surface just outside the horizon of BH of same mass o Still unclear processes leading to formation of g*’s and if they exist at all. If exist, heat capacity of the surface high enough to avoid heating and black-body emission o We have constructed a simple but general class of g*’s with the basic features of the thin-shell model but allows for a stability analysis. o Two basic questions have two basic answers: g*’s are stable (axial perturbations): can be discerned from a BH through the emission of GWs via QNM oscillations o Rotating g*’s could be subject to an “ergoregion instability” (Cardoso et al. 2007). So far, found the space of parameters that allows the existence of an ergoregion. o Interesting new result: an ergoregion is not present for all g*’s and hence some rotating g*’s could be ergoregion-stable. o Future work: compute eigenfrequencies of unstable g*’s and assess if instability occurs on timescales shorter than Hubble time. Where is an ergoregion possible? We have already discussed the space of possible solutions in the (µ, δ) plane for spherical g*’s This space is further divided if rotation is taken into account no ergoregion