Generic spherically symmetric dynamic thin-shell traversable wormholes in standard general relativity

We consider the construction of generic spherically symmetric thin-shell traversable wormhole spacetimes in standard general relativity. By using the cut-and-paste procedure, we comprehensively analyze the stability of arbitrary spherically symmetric thin-shell wormholes to linearized spherically symmetric perturbations around static solutions. While a number of special cases have previously been dealt with in scattered parts of the literature, herein we take considerable effort to make the analysis as general and unified as practicable. We demonstrate in full generality that stability of the wormhole is equivalent to choosing suitable properties for the exotic material residing on the wormhole throat.

Figure 1

The figure depicts an embedding diagram of a traversable thin-shell wormhole. A single manifold $\mathcal{M}$ is obtained by gluing together two distinct spacetime manifolds, ${\mathcal{M}}_{+}$ and ${\mathcal{M}}_{-}$, at their boundaries, i.e., $\mathcal{M}={\mathcal{M}}_{+}\cup {\mathcal{M}}_{-}$, with the natural identification of the boundaries $\Sigma ={\Sigma }_{+}={\Sigma }_{-}$.

Figure 2

The spacetime diagram for a thin-shell wormhole in an asymptotically flat spacetime, represented by two manifolds ${\mathcal{M}}_{+}$ and ${\mathcal{M}}_{-}$, joined at a radial coordinate $a$, at the junction surface $\Sigma$.

Figure 3

The figure depicts an embedding diagram of a traversable thin-shell wormhole. The single manifold $\mathcal{M}$ is obtained by gluing together ${\mathcal{M}}_{+}$ and ${\mathcal{M}}_{-}$, at junction surface $\Sigma$. The surface stress-energy tensor on $\Sigma$ is given in terms of the surface energy density, $\sigma$, and the surface pressure, $\mathcal{P}$. See the text for details.

Figure 4

Stability analysis for thin-shell Schwarzschild traversable wormholes. The stability region is that above the grey surface depicted in the plot. The grey surface is given by the dimensionless quantity $F(M_{\pm },a_0)$, defined by the right-hand side of inequality (83). We have considered the range $0<x=M_{+}/a_0<1$ and $0<y=M_{-}/M_{+}<1/x$, respectively. Note that a large stability region exists for low values of $x=2M_{+}/a_0$ and of $y=M_{-}/M_{+}$. For regions close to the event horizon, $x\to 1$, the stability region decreases in size and only exists for low values of $y$. See the text for details.

Figure 5

Stability analysis for thin-shell Reissner-Nordström traversable wormholes. The stability region is that above the grey surface. The grey surface is given by the dimensionless quantity $G(M_{\pm },Q_{\pm },a_0)$, defined in Eq. (86). The values of $|Q|/M_{+}=1$ and $|Q|/M_{+}=\sqrt{3}/2$ are depicted in the left and right plots, respectively. Note that for decreasing values of $|Q|/M_{+}$, the stability regions decrease substantially for low values of $M_{-}/M_{+}$ and for high values of $2M_{+}/a_0$. See the text for more details.

Figure 6

The plot depicts the stability region for the thin-shell variant of the Ellis wormhole, for which $b_{\pm }=R_{\pm }^2/r$. The grey surface is given by the dimensionless quantity $H(R_{\pm },a_0)$, given by the right-hand side of inequality (89). The stability region, represented by the function $a_0{m}_s^{\prime \prime }(a_0)$, is that above the grey surfaces depicted in the plot. We have considered the range $0<x=R_{+}/a_0<1$ and $0<y=R_{-}/R_{+}<1/x$. Note that large stability regions exist for small values of $x$ and of $y$. The stability regions decrease for large values of $y$, i.e., for $R_{-}\gg R_{+}$, and for large values of $x$. See the text for details.

Figure 7

The plot depicts the stability regions for the thin-shell variant of the Ellis wormhole, with $b_{\pm }=R_{\pm }^2/r$, but now in the presence of nonzero external forces. The latter nonzero momentum flux arises from the functions given by ${\Phi }_{\pm }=-R_{\pm }/r$. The grey surface depicted in the left plot is given by $H_2(R_{\pm },a_0)$, i.e., the right-hand side of inequality (91), and the stability regions are given below this surface. We have considered, as before, the range $0<x=R_{+}/a_0<1$ and $0<y=R_{-}/R_{+}<1/x$. Now, collecting the results, note that the stability regions are given from the regions above the surface in Fig. 6, and below the surface provided by the left plot. These stability regions are depicted in the right plot in between the grey surfaces, represented by the functions $H(R_{\pm },a_0)$ and $H_2(R_{\pm },a_0)$, respectively. Note the absence of stability regions for high values of $x$ and $y$. See the text for more details.

Figure 8

The stability regions for the toy model thin-shell wormhole with $b_{\pm }=\sqrt{r{R}_{\pm }}$, ${\Phi }_{\pm }=0$ are given above the grey surfaces depicted in the plot. The grey surface is given by the dimensionless quantity $I(R_{\pm },a_0)$, given by the right-hand side of inequality (94). We have considered the range $0<x=R_{+}/a_0<1$ and $0<y=R_{-}/R_{+}<1/x$. Large stability regions exist for small values of $x$ and of $y$. Note that the stability regions decrease for large values of $y$, i.e., for $R_{-}\gg R_{+}$, and for large values of $x$.

Figure 9

The plot depicts the stability regions for the toy model thin shell, with $b_{\pm }=\sqrt{R_{\pm }r}$, but now in the presence of nonzero external forces. The latter nonzero momentum flux arises from the functions given by ${\Phi }_{\pm }=-R_{\pm }/r$. The grey surface depicted in the left plot is given by $I_2(R_{\pm },a_0)$, i.e., the right-hand side of inequality (96), and the stability regions are given below this surface. We have considered, as before, the range $0<x=R_{+}/a_0<1$ and $0<y=R_{-}/R_{+}<1/x$. Now, collecting the results, note that the stability regions are given from the regions above the surface in Fig. 8, and below the surface provided by the left plot. These stability regions are depicted in the right plot in between the grey surfaces, represented by the functions $I(R_{\pm },a_0)$ and $I_2(R_{\pm },a_0)$, respectively. Note the absence of stability regions for high values of $x$ and $y$. See the text for more details.

Figure 10

Thin-shell dilaton wormhole for $\beta =1/2$. The left plot describes the stability regions above the grey surface, given by (70), while the right plot describes the stability regions below the grey surface, i.e., inequality (74). From the left plot, one verifies shows that the stability regions decrease significantly for increasing values of $x$, within the considered range of $0<y<1$. In counterpart, the right plot shows that the stability regions decrease significantly for decreasing values of $y$. These latter stability regions further decrease for increasing values of $x$. See the text for more details.

Figure 11

Thin-shell dilaton wormhole for $\beta =1/8$. The left plot describes the stability regions above the grey surface, given by (72), while the right plot describes the stability regions below the grey surface, i.e., inequality (74). The left plot shows qualitatively that the stability regions decrease significantly for increasing values of $x$, in the relevant range of $0<y<1$. From the right plot, one verifies that the stability regions decrease significantly for decreasing values of $y$. See the text for more details.

Figure 12

The plots depict the final stability regions for the thin-shell dilaton wormhole. The left plot describes the stability regions given by $\beta =1/2$, while the right plot describes stability regions given by $\beta =1/8$. The final stability regions are depicted in between the surfaces of different shades of grey. Note that the stability regions decrease for decreasing values of $\beta$. More specifically, for decreasing values of $\beta$, stability regions exist for practically very low values of $y$, i.e., $M_{-}\le M_{+}$. In this context, it is interesting to note that in the vicinity of the event horizon, the stability regions increase for increasing values of $\beta$, provided one has low values of $y$. See the text for more details.