Radiation collapse and gravitational waves in three dimensions.

Two non-static solutions for three dimensional gravity coupled to matter fields are given. One describes the collapse of radiation that results in a black hole. This is the three dimensional analog of the Vaidya metric, and is used to construct a model for mass inflation. The other describes plane gravitational waves for coupling to a massless scalar field. PACS numbers: 4.20, 4.40, 97.60.L Typeset using REVTEX 1 Lower dimensional gravity has often been used as an arena for investigating various problems that arise in four dimensions, but are not solvable there. Among those that have been substantially investigated include quantum gravity in three dimensions [1,2] and black hole evaporation in two dimensions [3]. Obtaining classical solutions in lower dimensions is often a first step in these models. In three dimensional gravity the solutions for point masses were the first to be studied [4]. More recently a black hole solution has been given by Banados et. al. [5], which also provides an arena for investigating black hole evaporation. There are two classical problems in general relativity in four dimensions that have recently attracted some attention, and a three dimensional version of them may be useful to address. The first problem has to do with the inner (Cauchy) horizon of the Kerr and ReissnerNordstrom black holes. This horizon is believed to be unstable to time dependent perturbations because it is a surface where infalling radiation is infinitely blue shifted. The question is what effect the back reaction of the blue shifted radiation has on the internal geometry. More precisely one would like to know what type of singularity develops at or before the Cauchy horizon as a result of this back reaction. This question is important for the cosmic censorship hypothesis, for if the Cauchy horizon can be crossed, the timelike curvature singularity in such spacetimes becomes naked. This question may be asked of plane wave spacetimes, which also have Cauchy horizons. Recent approaches to this problem, within spherical symmetry, take into account nonlinear perturbations at the Cauchy horizon as well as their back reaction on the geometry. The results suggest that the singularity has a null portion, where the internal mass function of the black hole diverges [6,7]. However it is not yet known what type of singularity replaces the Cauchy horizon under general perturbations. The second problem is the investigation of the collapse of matter fields to form black holes. This has been studied numerically and the results are intriguing [8,9]. It has been found that when the initial matter field is an ingoing pulse, the collapsing matter forms a 2 black hole with mass given by M = K(c − c∗), where K is a constant, c is any one of the parameters in the initial data for the matter field, c∗ is the critical value of this parameter (that gives a zero mass black hole), and γ ∼ .36. In particular, no black hole is formed when c < c∗. An important feature of this result is that it appears to be independent of spherical symmetry and the type of matter fields, with the same numerical exponent γ appearing in all cases studied to date. This seems to reflect a universal property of the Einstein equations in strong field regions. So far there is no analytical understanding of this result. It would be interesting to see if a similar result is true in lower dimensions, and whether it can be better understood there, perhaps analytically. This paper is concerned with the first problem, and two metrics in three dimensions are given that have Cauchy horizons. One has the Vaidya form and describes collapsing spherically symmetric radiation. It allows the construction of the Ori model for mass inflation [7] in three dimensions . The other describes plane gravitational waves for coupling to a massless scalar field. This metric may also be used as a starting point for studying perturbations of Cauchy horizons [10]. The Vaidya form of a three dimensional metric may be written using an advanced time coordinate v, and polar coordinates r, θ in the plane. It is ds = −f(r, v)dv + 2drdv + rdθ. (1) The total energy momentum tensor we use contains contributions Iαβ = ρ(v) 4πr ∂αv∂βv (2) for infalling radiation with luminosity ρ(v), and E α = q 4πr diag(−1,−1, 1) (3) (in the coordinates (v, r, θ)) for the external electric field due to a charge q. The Einstein equations with cosmological constant Λ 3 Gαβ + Λgαβ = 2π(Iαβ + Eαβ) (4) have, with the ansatz (1), the solution f(r, v) = −[Λr + g(v) + qlnr], (5) where g(v) is given by dg(v) dv = ρ(v). (6) This gives the three dimensional analog of the charged Vaidya metric [11]. If asymptotically (r → ∞ and v → −∞) the radiation inflow vanishes so that g(v) = 0, the metric assumes the ‘vacuum’ form ds = (Λr + qlnr)dv + 2dvdr + rdθ. (7) This shows that Λ ≡ −l−2 must be negative for the metric to be asymptotically Lorentzian. The case ρ(v) = 0, g(v) = M , a constant, gives the spherically symmetric static black hole found by Banados et. al [5]. The event horizon (for q = 0) is at rEH = l √ M. (8) When ρ(v) 6= 0, the mass of the black hole formed from the collapse depends on the parameters in the ingoing pulse. Asymptotically (v → ∞), the apparent horizon becomes null and its radius gives the black hole mass. The radial coordinate of the apparent horizon rAH is a measure of the black hole mass function m(v), and is given (again for q = 0) by m(v) := rAH(v) = l √