6 9 12 15 18 a −0.00020 −0.00015 −0.00010 −0.00005 0.00000 FIG. 2: Plot for σ versus a . The description of the curves is the same as in FIG. 1. Expanding V ( a ) around a$_{0}$ , we obtain where the prime denotes the derivative with respect to a , assuming a static solution situated at a$_{0}$ . Since we are linearizing around a = a$_{0}$ , we must have V ( a$_{0}$ ) = 0 and V $^{′}$( a$_{0}$ ) = 0. The configuration will then be in stable equilibrium if V $^{′′}$( a$_{0}$ ) > 0. where A = 4 πh ( a ) by Eq. (2). The first term on the left side corresponds to a change in the throat’s internal energy, while the second term corresponds to the work done by the throat’s internal forces. According to Ref. [19], the right side represents a flux. From Eq. (27), we get So the distribution of matter in the shell is of the dark- energy type. Now, purely mathematicall speaking, if a → 1 $_{2}$( r$_{−}$ + r$_{+}$ ), then p → 0. Since 1 $_{2}$( r$_{−}$ + r$_{+}$ ) < r$_{+}$ , however, such a dust shell is never found. Our spacetime metric implies that the surface mass of this thin shell is given by M$_{shell}$ = 4 πh ( a ) σ . (For a static solution, we have ˙ a = 0 and ¨ a = 0.) Thus M$_{shell}$ = 2 [ a + (1 − 2 n ) r$_{0}$ ] ( r 2 0 a $^{2}$) × ( a − r$_{0}$ ) 1 − $^{n}$( a + r$_{0}$ ) n √ ( a − r$_{−}$ )( a − r$_{+}$ ) . (23) 1 Now observe that for n = $_{2}$, the mass of the black hole in Eq. (10) is increasing with r$_{0}$ . At the same time, for a fixed value of the throat radius a , the mass of the thin shell is decreasing with r$_{0}$ , as long as r$_{0}$ remains much less than a . To carry out this analysis, we start with the energy conservation equation. Using Eqs. (12) and (13), one can verify that Now we will focus our attention on the stability of the configuration under small perturbations around a static solution at a$_{0}$ . The starting point is the definition of a potential, extended to our metric [Eq. (2)]. Rearranging Eq. (12), we obtain the thin shell’s equation of motion Here the potential V ( a ) is defined as To do so, observe that Next, we define a parameter β , which is interpreted as the subluminal sound speed, by the relation It is also shown in Ref. [19] that and, finally, V ( a ) = V ( a$_{0}$ ) + V $^{′}$( a$_{0}$ )( a − a$_{0}$ ) + 1 2 V $^{′′}$( a$_{0}$ )( a − a$_{0}$ ) 2 + O [ ( a − a$_{0}$ ) $^{3}$] , (26) d dτ ( σ A ) + p d A dτ = { [ h $^{′}$( a )] 2 − 2 h ( a ) h $^{′′}$( a ) } ˙ a √ f ( a ) + ˙ a 2 2 h ( a ) , (27) d da [ σh ( a )] + P d da [ h ( a )] = −{ [ h $^{′}$( a )] 2 − 2 h ( a ) h $^{′′}$( a ) } σ 2 h $^{′}$( a ) , h ( a ) σ ′ + h $^{′}$( a )( σ + p ) + { [ h $^{′}$( a )] 2 − 2 h ( a ) h $^{′′}$( a ) } σ 2 h $^{′}$( a ) = 0 . (28) V $^{′′}$( a ) = f $^{′′}$( a ) + 16 π $^{2}$× {[ h ( a ) h $^{′}$( a ) σ $^{′}$( a ) + ( 1 − h ( a ) h $^{′′}$( a ) [ h $^{′}$( a )] 2 ) σ ( a ) ] [ σ ( a ) + 2 p ( a )] + h ( a ) h $^{′}$( a ) σ ( a )[ σ $^{′}$( a ) + 2 p $^{′}$( a )] } . (29) β $^{2}$( σ ) = ∂p ∂σ ∣ σ . (30) V ( a ) = f ( a ) − [ 4 πh ( a ) σ ( a ) h $^{′}$( a ) ] 2 . (25) ˙ a 2 + V ( a ) = 0 . (24) σ $^{′}$( a ) + 2 p $^{′}$( a ) = σ $^{′}$( a )[1 + 2 p $^{′}$( a ) /σ $^{′}$( a )] = σ $^{′}$( a )(1 + β $^{2}$) . VI. STABILITY 4