Nonlinear stability of a brane wormhole

We analytically study the nonlinear stability of a spherically symmetric wormhole supported by an infinitesimally thin brane of negative tension, which has been devised by Barcelo and Visser. We consider a situation in which a thin spherical shell composed of dust falls into an initially static wormhole; the dust shell plays the role of the nonlinear disturbance. The self-gravity of the falling dust shell is completely taken into account through Israel’s formalism of the metric junction. When the dust shell goes through the wormhole, it necessarily collides with the brane supporting the wormhole. We assume the interaction between these shells is only gravity and show the condition under which the wormhole stably persists after the dust shell goes through it.

Figure 1

The initial configuration is depicted.

Figure 2

Shell-1 forms the wormhole structure.

Figure 3

Shell-1 supporting the wormhole is initially static. Shell-2 falls into the wormhole and collides with shell-1. The interaction between these shells is assumed to be gravity only: the shells merely go through each other.

Figure 4

Adopting the units $M_{\mathrm{w}}=1$, the function $W(r)$ with $(a,m_2,E)=(1.3,0.05,1)$ is depicted. There is a maximum of $W(r)$ at $r=r_{\mathrm{A}}<0$. However, it is very large compared with extrema in $r\ge 0$, and hence we do not show it in this figure.

Figure 5

The same as Fig. 4 but with $m_2=0$.

Figure 6

The $(a,m_2)$ space with $E=1$ is depicted. The domain in which the wormhole stably persists is shown as the unshaded region.

Figure 7

A close-up of the neighborhood of the intersections of the curves $W(r_{\mathrm{B}})=0$, $W(r_{\mathrm{D}})=0$, and $4m_2=a-M_{\mathrm{w}}$.

Figure 8

The same as Fig. 6, but $E=2$.

Figure 9

We depict $(r_1-{\rho }_{+})\mu$ with $m_2=0$ as a function of $a$ in units of $M_{\mathrm{w}}=1$.

Figure 10

We depict $w_{\mathrm{lb}}^{\prime }$ as a function of $a$ in units of $M_{\mathrm{w}}=1$. It is positive in the domain of our interest.