Naked singularities in three-dimensions

  • G. Oliveira-Neto
  • Published 2002


We study an analytical solution to the Einstein’s equations in 2 + 1dimensions, representing the self-similar collapse of a circularly symmetric, minimally coupled, massless, scalar field. Depending on the value of certain parameters, this solution represents the formation of naked singularities. Since our solution is asymptotically flat, these naked singularities may be relevant for the weak cosmic censorship conjecture in 2 + 1-dimensions. 04.20.Dw,04.20.Jb,04.60.Kz Typeset using REVTEX ∗Email: 1 Since the work of M. W. Choptuik on the gravitational collapse of a massless scalar field [1], many physicists have focused their attentions on the issue of gravitational collapse. An important arena where one can study the gravitational collapse is general relativity in 2 + 1-dimensions. The great appeal of this theory comes from the fact that it retains many of the properties of general relativity in 3+1-dimensions, but the field equations are greatly simplified [2]. Presently, several black hole solutions in 2 + 1-dimensional general relativity are known [3]. Including the first one to be discovered, the so-called BTZ black hole [4]. All of them have an important property in common: the presence of a negative cosmological constant, which makes them asymptotically anti-de Sitter. Indeed, in a recent work it was demonstrated that a three-dimensional solution to the Einstein’s equations, with a positive cosmological constant (Λ), such that the stress-energy tensor satisfies the dominant energy condition, contains no apparent horizons [5]. The same result applies to the case Λ = 0 in the presence of matter fields. Therefore, this result explains the necessity of a negative cosmological constant in order to a black hole to form, in three-dimensional general relativity. Based on [5], we can say that the gravitational collapse of ordinary matter, without a negative cosmological constant, will never form a black hole in 2+1-dimensional general relativity. On the other hand, one can not exclude the possible formation of naked singularities as the result of the gravitational collapse, without a cosmological constant. In fact, it has already been shown that the collapse of a disk of pressureless dust in 2+1dimensions, without a cosmological constant, has as one of its possible end states a naked singularity [6]. The singularity is space-like and it is a scalar polynomial singularity [7], in other words, scalar polynomials constructed from the Riemann tensor are unbounded there. For an observer far from the collapse region the space-time is flat and conical. Since the naked singularity defines a region of non-zero measure in the parameter space of solutions and is asymptotically flat, it may be considered as a counter-example to the weak cosmic censorship conjecture [8] in 2 + 1-dimensions. 2 In the present paper we would like to present a solution to the Einstein’s equation, without a cosmological constant, representing the self-similar, circularly symmetric, collapse of a minimally coupled, massless, scalar field, in 2 + 1-dimensions. As we shall see this solution, depending on the value of certain parameters, represents the formation of naked singularities as the result of the collapse process. We shall start by writing down the ansatz for the space-time metric. As we have mentioned before, we would like to consider the circularly symmetric, self-similar, collapse of a massless scalar field in 2 + 1-dimensions. Therefore, we shall write our metric ansatz as, ds = − 2edudv + r(u, v)dθ , (1) where σ(u, v) and r(u, v) are two arbitrary functions to be determined by the field equations, (u, v) is a pair of null coordinates varying in the range (−∞,∞), and θ is an angular coordinate taking values in the usual domain [0, 2π]. The scalar field Φ will be a function only of the two null coordinates and the expression for its stress-energy tensor Tαβ is given by [9], Tαβ = Φ,αΦ,β − 1 2 gαβΦ,λ Φ ,λ . (2) where , denotes partial differentiation. Now, combining Eqs. (1) and (2) we may obtain the Einstein’s equations which in the units of Ref. [9] and after re-scaling the scalar field, so that it absorbs the appropriate numerical factor, take the following form, 2σ,u r,u − r,uu = r(Φ,u ) , (3) 2σ,v r,v − r,vv = r(Φ,v ) , (4) 2rσ,uv + r,uv = − r(Φ,uΦ,v ) , (5)

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@inproceedings{OliveiraNeto2002NakedSI, title={Naked singularities in three-dimensions}, author={G. Oliveira-Neto}, year={2002} }