Gravitational collapse of a massless scalar field in a periodic box

Abstract

Gravitational collapse of a massless scalar field with the periodic boundary condition in a cubic box is reported. This system can be regarded as a lattice universe model. We construct the initial data for a Gaussian-like profile of the scalar field taking the integrability condition associated with the periodic boundary condition into account. For a large initial amplitude, a black hole is formed after a certain period of time. While the scalar field spreads out in the whole region for a small initial amplitude. It is shown that the expansion law in a late time approaches that of the radiation dominated universe and the matter dominated universe for the small and large initial amplitude cases, respectively. For the large initial amplitude case, the horizon is initially a past outer trapping horizon, whose area decreases with time, and after a certain period of time, it turns to a future outer trapping horizon with the increasing area.

Constant-mean-curvature slicing of the Swiss-cheese universe

“…It is found from this figure that CMC slices do not pass through the black hole region but the white hole region. A similar slice is observed in an analysis of massless scalar field collapse in an expanding background [29]. In order to understand the spacetime structure of a numerical solution, one useful way is to find marginally trapped surfaces associate with the outgoing or ingoing null vector fields.…”

“…Actually, in Ref. [29], one of the present authors and his collaborators have revealed the spacetime structure by virtue of the results given in this paper. This is an example showing that the knowledge about the sequence of CMC slices obtained in this paper helps in understanding the spacetime structure of the numerical spacetime solution.…”

“…Non-spherical simulation of gravitational collapse in an expanding background has been recently performed in Ref. [29].…”

Primordial black hole formation with full numerical relativity

“…We choose an initially expanding spacetime with K = −3H 0 , so that the periodic integrability condition is satisfied 4 [122][123][124]. The eventual Hamiltonian constraint only depends on the radial coordinate r due to the spherical symmetry of the setup, and we solve for the conformal factor ψ numerically…”

CTTK: a new method to solve the initial data constraints in numerical relativity

“…Note that the condition from the linearisation will be different if ρ = ρ(ψ), as it is the case for the gradient energy density of a scalar field, with ρ ∼ ψ −4 (∂iϕ) 2 and one can choose a constant K that satisfies the integratibility condition[22][23][24][26][27][28].…”