Dynamical wormholes

Abstract

We numerically investigate the dynamical evolution of spherically symmetric charge free wormholes. We concentrate on two specific examples, both of which exhibit wormhole expansion and wormhole collapse: the Ellis-Bronnikov wormhole, which is sourced by a real massless ghost scalar field, and the quantum corrected Schwarzschild black hole in semiclassical gravity (which has a wormhole structure and is not a true black hole), which is sourced by a renormalized energy-momentum tensor. Despite their very different sources, we demonstrate that the dynamics of these two wormholes are remarkably similar. Our analysis focuses on diagrams for the areal radius and components of the energy-momentum tensor. This work also serves as a review, offering a detailed description of how to perform a spherically symmetric dynamical evolution using double null coordinates as well as a review of the static solutions for our two examples.

Figures (21)

• Figure 1: (a) A two dimensional discrete grid of $(u,v)$ values is used as our computational domain, here illustrated as a $7\times 7$ grid (in practice, each dimension will typically have thousands of grid points). The grid points have uniform spacing $\Delta u$ and $\Delta v$ and take values in the ranges $u_i \leq u \leq u_f$ and $v_i \leq v \leq v_f$. (b) A single grid cell, where points 1, 2, 3, and 4 are points on the grid, while point 0 is an intermediate point and is not on the grid. In general, we will know field values at points 1, 2, and 3 and will compute field values at point 4.
• Figure 2: These diagrams display static Ellis-Bronnikov wormhole solutions which are parametrized in terms of the constant $a_1$. (a) The areal radius, $r$, as a function of the radial coordinate, $x$, for (from darkest curve to lightest) $a_1 = 0$, 0.25, 0.5, 0.75, and 1. (b) The wormhole throat radius, $r_\mathrm{th}$, as a function of $a_1$. (c) The ADM mass as viewed from the positive side, $M_+$, as a function of $a_1$.
• Figure 3: Contour diagrams for the areal radius, $r$, in double null coordinates for static Ellis-Bronnikov wormholes defined by (a) $a_1 = 0$, (b) 0.25, and (c) 0.5. In each diagram, the gray lines are contours with values of $r$ as labeled and the thick black lines mark the wormhole throat with radii (a) $r_\mathrm{th}/x_0 = 1$, (b) 0.970, and (c) 0.887.
• Figure 4: Contour diagrams for the areal radius, $r$, for the dynamical evolution of the symmetric massless Ellis-Bronnikov wormhole. Each diagram displays an expanding wormhole. The black curves display the apparent horizon $r_{,u} = 0$ and the blue curves display the apparent horizon $r_{,v} = 0$. Each diagram uses the same initial data, but evolves the system with grid spacing $\Delta u = \Delta v = 1/N$ with (a) $N = 100$, (b) 200, and (c) 400. As the grid spacing decreases, the evolution maintains the static configuration longer. This is the expected behavior for a unstable static solution.
• Figure 5: Each diagram is for the same dynamical evolution shown in figure \ref{['fig:EB r symmetric no pulse']}(c). The yellow curves are the apparent horizons from figure \ref{['fig:EB r symmetric no pulse']}(c). The diagrams display contours for (a) the metric field $\sigma$, (b) the outgoing energy-momentum $T_{uu}$, (c) the ingoing energy-momentum $T_{vv}$, and (d) the net flow of energy momentum, $T_{vv} - T_{uu}$.