Static phantom wormholes of finite size

In this paper we derive new static phantom traversable wormholes by assuming a shape function with a quadratic dependence on the radial coordinate $r$. We mainly focus our study on wormholes sustained by exotic matter with positive energy density (as seen by any static observer) and a variable equation of state $p_r/\rho <-1$, dubbed “phantom matter.” Among phantom wormhole spacetimes extending to infinity, we show that a quadratic shape function allows us to construct static spacetimes of finite size, composed of a phantom wormhole connected to an anisotropic spherically symmetric distribution of dark energy. The wormhole part of the full spacetime does not fulfill the dominant energy condition, while the dark energy part does.

Figure 1

Qualitative behavior of energy density for $a_1<0$ and $a_3>0$. For all plots the throat is located at $r_0$. The dashed line describes the case for which $\rho (r_0)\ge 0$. The solid and dotted lines represent the cases where $\rho (r_0)=0$ and $\rho (r_0)\le 0$, respectively. We can see that in the cases of the solid and dotted lines the energy density is everywhere negative, while for the dashed line the energy density is positive for $r_0\le r<r_3$ and becomes negative for $r>r_3>r_0$.

Figure 2

Qualitative behavior of energy density for $a_1<0$ and $a_3<0$. The throat is located at $r_0$. The dotted line describes the case $|a_3|=r_0^2|a_1|-r_0$ for which $\rho (r_0)=0$. The solid and dashed lines represent the cases $|a_3|\le r_0^2|a_1|-r_0$ and $|a_3|\ge r_0^2|a_1|-r_0$, for which we have at the throat $\rho (r_0)<0$ and $\rho (r_0)>0$, respectively. We can see that in the case of the solid line the energy density is everywhere negative, while for the dashed line the energy density is positive for $r_0\le r<r_1$ and becomes negative for $r>r_1$.

Figure 3

Qualitative behavior of energy density for $a_1>0$ and $a_3>0$. The throat is located at $r_0$. The dotted line represents the case $a_3=r_0^2{a}_1+r_0$ for which $\rho (r_0)=0$. The solid and dashed lines represent the cases $a_3\le r_0^2{a}_1+r_0$ and $a_3\ge r_0^2{a}_1+r_0$, for which we have at the throat $\rho (r_0)>0$ and $\rho (r_0)<0$, respectively. We can see that in the case of the solid line the energy density is everywhere positive, while for the dashed line the energy density is negative for $r_0\le r<r_2$ and becomes positive for $r>r_2$.

Figure 4

Embedding of the spacetime (38). The embedding function is given by $z(r)=\int \sqrt{\frac{r^2+4}{5r-r^2-4}}dr$, and its second derivative vanishes at $r=2$. The whole spacetime is finite and extends from $r_0=1$ to $r=4$. The phantom wormhole configuration extends from the throat to $r=2$, where an inflection point is present. From $r=2$ to $r=4$ a dark energy distribution is located, to which is connected the phantom wormhole. For $r<2$ DEC is not satisfied, while for $r\ge 2$ it is.

Figure 5

Behavior of the energy density (dashed-dotted line), radial (solid line) and lateral (dotted line) pressures and the equation of state $p_r/\rho$ (dashed line) for $r_0=1$, $a_1=1/5$ and $a_3=4/5$. Everywhere the energy density is positive and radial pressure negative; and $p_l>0$ for $1\le r<2$, and $p_l<0$ for $2<r\le 4$. On the other hand, for $1\le r\le 2$ the equation of state has a phantom character defined by $-2.5\le p_r/\rho \le -1$, while at $2\le r\le 4$ the equation of state behaves as dark energy, since $-1\le p_r/\rho \le -0.625$. In such a way, the wormhole is connected to a distribution of dark energy.

Figure 6

Three-dimensional embedding diagram of the wormhole of finite size (38) with the energy distribution.

Figure 7

Behavior of the energy density (solid line), radial (dotted line) and lateral (dashed line) pressures, $\rho +p_r$ (dashed-dotted line) and $\rho +p_l$ (long-dashed line) for $r_0=1$, $a_1=1$ and $a_3=4$. It becomes clear that the energy density is negative for $1\le r<2$ and positive for $2<r\le 4$, while $p_r\le 0$ for $1\le r\le 4$ and $p_l>0$ for $1\le r<2$, and $p_l<0$ for $2<r\le 4$. On the other hand, $\rho +p_r$ and $\rho +p_l$ are negative for $1\le r<2$ and positive for $2<r\le 4$.

Figure 8

Plots of energy density (dotted line), radial pressure (dashed line) and the radial equation of state (solid line) for the case $r_0=1$, $a_1=1$ and $a_3=4$. The energy density is negative at the wormhole part, while out of it $\rho$ is positive. In the range $2\le r<4$ we have that $0\le p_r/\rho \le -1/4$, so DEC is fulfilled.