Causal Structure of an Inflating Magnetic Monopole

We clarify the causal structure of an inflating magnetic monopole. We also discuss general nature of inflationary spacetimes. PACS number(s): 04.70.Bw, 14.80.Hv, 98.80.Cq Electronic address: [email protected] 1 For the last decade spacetime solutions of magnetic monopoles have been intensively studied in the literature [1–5]. This originated from the rather mathematical interest in static solutions with non-Abelian hair [1]. It was shown [2] that static regular solutions are nonexistent if the vacuum expectation value of the Higgs field η is larger than a critical value ηsta, which is of the order of the Planck mass mPl. Linde and Vilenkin independently pointed out that such monopoles could expand exponentially in the context of inflationary cosmology [3]. Because this “topological inflation” model does not require fine-tuning of the initial conditions, it has been attracting attention. In particular, it is recently found that topological inflation takes place in some of the plausible models in particle physics [6]. In Ref. [4] (Paper I), dynamical solutions of magnetic monopoles for η > ηsta were numerically obtained: monopoles actually inflate if η ≫ ηsta. Recently, the causal structure of an inflating magnetic monopole is discussed in Ref. [5]. Such arguments are also important for initial conditions for successful inflation [7]. However, there are serious errors about the causal structure of an inflating monopole in Refs. [5,7], as we shall explain below, which would throw into question the entire discussions there. In this paper we clarify the causal structure of an inflating magnetic monopole, as a complement of Paper I. In order to see the spacetime structure for numerical solutions, we observe the signs of the expansion of a null geodesic congruence. For a spherically symmetric metric, ds = −dt + A(t, r)dr +B(t, r)r(dθ + sin θdφ), (1) an outgoing (+) or ingoing (−) null vector is given by 1 k ± = (1,±A , 0, 0) (2) and its expansion Θ± is written as Θ± = k θ ±;θ + k φ ±;φ = 2 B ( ∂B ∂t ± 1 Ar ∂(Br) ∂r ) , (3) which is defined as the trace of a projection of k ;ν onto a relevant 2-dimensional surface. The derivation of Eq.(3) as well as more general arguments were given by Nakamura et al. [9] Apparent horizons are defined as the surfaces with Θ = 0 or Θ = 0 . We label those 1 In Paper I and Ref. [8], the null vector was expressed as k ± = (−1,±A , 0, 0). The minus sign of kt ± was just a typo, and the expression of Θ±, which is the same as Eq.(3), was correct. 2 An apparent horizon is usually defined as the outermost surface with Θ=0 for an asymptotically flat spacetime. In this article, however, we call any marginal surface with Θ = 0 or Θ = 0 an apparent horizon.