content_image/414594.jpg content_image/414614.jpg content_image/414615.jpg content_image/414613.jpg content_image/414611.jpg content_image/414612.jpg content_image/414616.jpg content_image/414609.jpg content_image/414617.jpg content_image/414619.jpg content_image/414618.jpg content_image/414610.jpg
<img src='content_image/414594.jpg'>

where, 

and 

A radially moving test particle initially at rest obeys the equation of motion 

If a r = 0, we obtain the geodesic equation. Moreover, a wormhole is attractive if a r > 0 and repulsive if a r < 0. These characteristics depend on the parameters r$_{0}$ , r$_{−}$ , r$_{+}$ , and n , the conditions on which can be conveniently expressed in terms of the coefficients A , B , and C . To avoid negative values for r , let us consider only the root r = ( B + √$_{B}$ 2 − 4 AC ) / (2 A ) of the quadratic equation Ar 2 − Br + C = 0. It now follows from Eq. (17) that a r = 0 whenever 

<img src='content_image/414614.jpg'>

<img src='content_image/414615.jpg'>

<img src='content_image/414613.jpg'>

<img src='content_image/414611.jpg'>

<img src='content_image/414612.jpg'>

For the attractive case, a r > 0, the condition becomes 

For the repulsive case, a r < 0, the sense of the inequality is reversed. 

## IV. THE TOTAL AMOUNT OF EXOTIC MATTER 

In this section we determine the total amount of exotic matter for the thin-shell wormhole. This total can be quantified by the integral [7, 8, 9, 10, 11, 12, 13] 

By introducing the radial coordinate R = r − a , w get 

Since the shell is infinitely thin, it does not exert any radial pressure. Moreover, ρ = δ ( R ) σ ( a ). So 

<img src='content_image/414616.jpg'>

Here D is given in Eq. (16). 

This NEC violating matter can be reduced by taking the value of a closer to r$_{+}$ , the location of the outer event horizon. The closer a is to r$_{+}$ , however, the closer the wormhole is to a black hole: incoming microwave background radiation would get blueshifted to an extremely high temperature [22]. On the other hand, it follows from Eq. (20) that for a ≫ r$_{+}$ , Ω$_{σ}$ will depend linearly on a : 

Taking the form of the equation of state (EoS) to be p = wσ , we obtain from Eqs. (14) and (15), 

<img src='content_image/414609.jpg'>

Observe that if the location of the wormhole throat is very large, i.e., if a → + ∞ , then w → − 1 $_{2}$. On the other hand, if a → r$_{+}$ (from the right), then ω → −∞ . 

<img src='content_image/414617.jpg'>

## V. AN EQUATION OF STATE 

<img src='content_image/414619.jpg'>

<img src='content_image/414618.jpg'>

<img src='content_image/414610.jpg'>

