Wormholes supported by phantom energy

By a combination of analytical and numerical techniques, we demonstrate the existence of spherical, asymptotically flat traversable wormholes supported by exotic matter whose stress tensor relative to the orthonormal frame of Killing observers takes the form of a perfect fluid possessing anisotropic pressures and subject to linear equations of state: $\tau =\lambda \rho c^2$, $P=\mu \rho c^2$. We show that there exists a four parameter family of asymptotically flat spherical wormholes parametrized by the area of the throat $A(0)$, the gradient $\Lambda (0)$ of the red shift factor evaluated on the throat as well as the values of $(\lambda ,\mu )$. The latter are subject to restrictions: $\lambda >1$ and $2\mu >\lambda$ or $\lambda <0$ and $2\mu <-|\lambda |$. For particular values of $(\lambda ,\mu )$, the stress tensor may be interpreted as representing a phantom configuration, while for other values represents exotic matter. All solutions have the property that the two asymptotically flat ends possess finite Arnowitt-Deser-Misner mass.

Figure 1

Reflectionally symmetric solutions generated choosing: $\Lambda (0)=0$ and $(\lambda ,\mu )$ labeled as case (1, 2, 3) in Table .

Figure 2

Reflectionally symmetric solutions generated by choosing: $\Lambda (0)=0$ and $(\lambda ,\mu )$ labeled as case (4, 5, 6) in Table .)

Figure 3

In these graphs we have taken $\lambda =2$, $\mu =0.5$ so that $1-\lambda +2\mu =0$, and we have kept them constant throughout. We gradually change $\Lambda (0)$ starting from $\Lambda (0)=0$ and increasing (decreasing) towards positive (negative) values. Notice that after a characteristic value of $\Lambda (0)$ the solutions become unbounded at one end.

Figure 4

In the graphs we have taken $\lambda =1.5$, $\mu =1$. Like in Fig. 3, we vary $\Lambda (0)$ from $\Lambda (0)=0$ up to $\Lambda (0)=0.8$ and $\Lambda (0)=-0.8$. Again after some characteristic value of $\Lambda (0)$ the solutions become diverging at one end.

Figure 5

We repeat the same analysis as in Figs. 3, 4 except that we have chosen $\lambda =7$, $\mu =4$. Again, the solutions diverge after some characteristic value of $\Lambda (0)$.

Figure 6

Sketch of the parameter space $(\lambda ,\mu )$. The regions labeled by (3, 4) do not generate solutions that decay to zero asymptotically. Regions labeled (1, 6) generate solutions decaying to zero at infinity but the decay is slow. Finally regions (2, 5) generate solutions that decay sufficiently fast so that they have finite ADM mass.