content_image/411053.jpg content_image/411060.jpg content_image/411075.jpg content_image/411076.jpg content_image/411069.jpg content_image/411070.jpg content_image/411071.jpg content_image/411072.jpg content_image/411073.jpg content_image/411077.jpg content_image/411074.jpg
<img src='content_image/411053.jpg'>

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 

<img src='content_image/411060.jpg'>

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 . 

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 

## VI. STABILITY 

<img src='content_image/411075.jpg'>

<img src='content_image/411076.jpg'>

Expanding V ( a ) around a$_{0}$ , we obtain 

<img src='content_image/411069.jpg'>

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. 

To carry out this analysis, we start with the energy conservation equation. Using Eqs. (12) and (13), one can verify that 

<img src='content_image/411070.jpg'>

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 

<img src='content_image/411071.jpg'>

and, finally, 

<img src='content_image/411072.jpg'>

It is also shown in Ref. [19] that 

<img src='content_image/411073.jpg'>

Next, we define a parameter β , which is interpreted as the subluminal sound speed, by the relation 

To do so, observe that 

<img src='content_image/411077.jpg'>

<img src='content_image/411074.jpg'>

