6 9 12 15 18 a 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 p 3 For the attractive case, a r > 0, the condition becomes For the repulsive case, a r < 0, the sense of the inequality is reversed. 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 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 : and where, 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 Taking the form of the equation of state (EoS) to be p = wσ , we obtain from Eqs. (14) and (15), 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 → −∞ . p σ = w = 1 4 (2 a − r$_{−}$ − r$_{+}$ )( r 2 0 − a $^{2}$) ( a − r$_{−}$ )( a − r$_{+}$ )[ a + (1 2 n ) r$_{0}$ ] . (22) Ω$_{σ}$ ≈ − 2 a. (21) d $^{2}$r dτ 2 = − Γ r tt ( dt dτ ) 2 = − a $^{r}$. (18) ( r − B 2 A ) 2 = B 2 − 4 AC 4 A 2 . C = r 2 $_{0}$( r$_{−}$ + r$_{+}$ ) + (4 nr$_{0}$ − 2 r$_{0}$ ) r$_{−}$r$_{+}$. B = 2 r 2 0 + ( r$_{−}$ + r$_{+}$ )(4 nr$_{0}$ − 2 r$_{0}$ ) + 2 r$_{−}$r$_{+}$, A = r$_{−}$ + r$_{+}$ + 4 nr$_{0}$ − 2 r$_{0}$, Ω$_{σ}$ = ∫ 2 π 0 ∫ π 0 [ $_{ρ}$√ − g ] ∣ r = $_{a}$dθ dφ = 4 πh ( a ) σ ( a ) = − 16 π [ a + (1 − 2 n ) r$_{0}$ ] D [ ( a − r$_{−}$ )( a − r$_{+}$ ) ( a − r$_{0}$ ) 2 n − $^{1}$( a + r$_{0}$ ) 1 − 2 n ] . (20) Ω$_{σ}$ = ∫ 2 π 0 ∫ π 0 ∫ ∞ −∞ [ ρ + p $_{]}$√ − g dR dθ dφ. Ω$_{σ}$ = ∫ [ ρ + p $_{]}$√ − gd $^{3}$x. (19) ( r − B 2 A ) 2 > B 2 − 4 AC 4 A 2 . IV. THE TOTAL AMOUNT OF EXOTIC MATTER FIG. 1: Plot for p versus a . The black, blue, and red colors represent n = 0 . 98, 0.5 and 0.02, respectively. For every color, thin, thick, and thicker curves, respectively, represent r$_{+}$ = 10, 8, and 6. For every combination of r$_{+}$ and n , we plot three different sets, ( r$_{−}$ = 5, r$_{0}$ = 2), ( r$_{−}$ = 5, r$_{0}$ = 3), and ( r$_{−}$ = 4, r$_{0}$ = 2), which are represented by chain and solid curves, respectively. V. AN EQUATION OF STATE