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Density fluctuations of fluids with negative pressure exhibit decreasing time behaviour in the long wavelength limit, but are strongly unstable in the small wavelength limit when a hydrodynamical approach is used. On the other hand, the corresponding gravitational waves are well behaved. We verify that the instabilities present in density fluctuations are due essentially to the hydrodynamical representation; if we turn to a field representation that lead to the same background behaviour, the instabilities are no more present. In the long wavelength limit, both approachs give the same results. We show also that this inequivalence between background and perturbative level is a feature of negative pressure fluid. When the fluid has positive pressure, the hydrodynamical representation leads to the same behaviour as the field representation both at the background and perturbative levels.

Perfect fluid with kinematic self-similarity is studied in 2 + 1 dimensional spacetimes with circular symmetry, and various exact solutions to the Einstein field equations are given. In particular, these include all the solutions of dust and stiff perfect fluid with self-similarity of the first kind, and all the solutions of perfect fluid with a linear equation of state and self-similarity of the zeroth or second kind. It is found that some of these solutions represent gravitational collapse, and the final state of the collapse can be either black holes or naked singularities.One of the most outstanding problems in gravitation theory is the final state of a collapsing massive star, after it has exhausted its nuclear fuel. In spite of numerous efforts over the last three decades or so, because of the (mathematical) complexity of the problem our understanding is still limited to several conjectures, such as, the cosmic censorship conjecture [1] and the hoop conjecture [2]. To the former, many counter-examples have been found [3], although it is still not clear whether those particular examples are generic. To the latter, no counter-example has been found so far in four-dimensional spacetimes, but it has been shown recently that this is no longer true in five dimensions [4].Lately, Brandt et al. have studied gravitational collapse of perfect fluid with kinematic self-similarities in fourdimensional spacetimes [5], a subject that has been recently studied intensively (for example, see [6] and references therein.). In this paper, we shall investigate the same problem but in 2+1 gravity [7]. The main motivation of such a study comes from recent investigation of critical collapse of a scalar field in 3D gravity [8][9][10][11]. It was found that in the 3D case the corresponding problem is considerably simplied and can be studied analytically. In particular, Garfinkle first found a class, say, S[n], of exact solutions to Einstein-massless-scalar field equations, and then Garfinkle and Gundlach studied their linear perturbations and found that the solution with n = 2 has only one unstable mode [9]. By definition this is a critical solution, and the corresponding exponent, γ, of the black hole mass,is γ = |k 1 | −1 = 4/3, where k 1 denotes the unstable mode. Although the exponent γ is close to the one found numerically by Pretorius and Choptuik, which is γ ∼ 1.2 ± 0.02 (but not the one of Husain and Olivier, γ ∼ 0.81), this solution is quite different from the numerical one [8]. Using different boundary conditions 1 , Hirschmann, Wang and Wu found that the solution with n = 4 has only one unstable mode [11]. As first noted by Garfinkle [9], this n = 4 solution matches extremely well with the numerical critical solution found by Pretorius and Choptuik [8]. However, the corresponding exponent γ now is given by γ = |k 1 | −1 = 4, which is significantly different from the numerical ones. In this paper we do not intend to solve these problems, but study some analytical solutions that represent gravitational collapse of perfect f...

Summary. -We present a 3-dimensional model for massive gravity with masses induced by topological (Chern-Simons) and Proca-like mass terms. Causality and unitarity are discussed at tree-level. Power-counting renormalizability is also contemplated.