Asymptotically flat wormhole solutions with variable equation-of-state parameter

This paper discusses exact wormhole solutions in the framework of general relativity with a general equation of state that reduced to a linear equation of state asymptotically. By considering a special shape function, we find classes of solutions which are asymptotically flat. We study the violation of the null energy condition as the main ingredient in wormhole physics. The possibility of finding wormhole solutions with asymptotically different state parameter is investigated. We show that in principle, a wormhole with a vanishing redshift function and the selected shape function, cannot satisfy the null energy condition at a large distance from the wormhole. Our solutions have the positive total amount of matter in the “volume integral quantifier” method. For this class of solutions, fluid near the wormhole throat is in the phantom regime which at some $r=r_2$, the phantom regime is connected to a distribution of dark energy regime. Thus, we need a small amount of exotic matter to construct wormhole solutions.

Figure 1

Three possible cases for ${\omega }_{\mathrm{eff}}$. In case (a) (dotted line), ${\omega }_{\mathrm{eff}}$ is always smaller than the line ${\omega }_{\mathrm{eff}}=-1$ (solid line) so NEC is violated throughout the entire range of $r$. In case (b) (dotted-dashed line), ${\omega }_{\mathrm{eff}}$ is bigger than the line ${\omega }_{\mathrm{eff}}=-1$ (solid line) in the range $r_1<r<r_3$ so NEC is not violated throughout this range. In case (c) (dashed line), ${\omega }_{\mathrm{eff}}$ exceeds the line ${\omega }_{\mathrm{eff}}=-1$ (solid line) at $r=r_2$ and tends to a constant so NEC is not violated throughout the range $r_2<r$. See the text for details.

Figure 2

The plot depicts the function $H(x,\alpha )$ for $A=\frac{1}{2}$ against $x$ and $\alpha$. It is clear that $H(x,\alpha )$ is positive throughout the range $0<x<1$ and $0<\alpha <1$, which implies $b(r)<r$ is satisfied everywhere. See the text for details.

Figure 3

The plot depicts the function $F_1(x,\alpha )$, for $A=\frac{1}{2}$. It is transparent that $F_1(x,\alpha )$ is positive throughout some range of $x$ and $\alpha$, which implies the NEC is satisfied through this range. See the text for details.

Figure 4

The plot depicts the function $F_2(x,\alpha )$, for $A=\frac{1}{2}$. It is transparent that $F_2(x,\alpha )$ is positive throughout the entire range of $x$ and $\alpha$, which implies the NEC is satisfied. See the text for details.

Figure 5

The plot depicts the function $F_1(x,A)$, for $\alpha =\frac{2}{3}$. It is clear that $F_1(x,A)$ is positive throughout some range of $x$ and $A$, which implies the NEC is satisfied through this range. See the text for details.

Figure 6

The plot depicts the function $F_2(x,A)$, for $\alpha =\frac{2}{3}$. It is clear that $F_2(x,A)$ is positive throughout entire range of $x$ and $A$ which implies NEC is satisfied. See the text for details.

Figure 7

Plot of $\omega (x)$ for $\alpha =2/3$ and $A=1/2$. It shows that $\omega (x)$ exceeds $\omega =-1$ throughout the region $0\le x\le 0.071$. It corresponds to case (c) in Fig. 1. See the text for details.

Figure 8

The plot depicts the function $F_1(x)$ and $F_2(x)$ for $\alpha =2/3$. The function $F_2(x)$ (dashed line) is positive in the entire spacetime and it is transparent that $F_1(x)$ (dotted line) is negative throughout the region $x>x_0$ when $x_0\simeq 0.071$, so the exotic matter is confined in this region. See the text for details.

Figure 9

The plot depicts the function $\rho (x)$ (dot-dashed line), $p(x)$ (dotted line), and $p_t(x)$ (solid line) for $A=-{\alpha }_3=1$, ${\alpha }_2=2/3$, and $A_1=-\frac{3^{1/5}\times (5\times 3^{2/5}-6)}{2\times (3^{3/5}-5\times 3^{2/5}+6)}\simeq -0.8136$ (note that vertical axe is scaled in $\frac{1}{8\pi r_0^2}$). The function $\rho (x)$ is positive in the range $x>x^{*}$ where $x^{*}\simeq 0.517$. See the text for details.

Figure 10

The plot depicts the function $F_1(x)$ (dotted line) and $F_2(x)$ (solid line) for $A=-{\alpha }_3=1$, ${\alpha }_2=2/3$, and $A_1=-\frac{3^{1/5}\times (5\times 3^{2/5}-6)}{2\times (3^{3/5}-5\times 3^{2/5}+6)}\simeq -0.8136$. The function $F_1(x)$ and $F_2(x)$ are positive in the range $x>x^{*}$ where $x^{*}\simeq 0.517$. So, NEC is satisfied in this region. See the text for details.