You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'The stability of transparent spherically symmetric thin shells (and wormholes) to linearized spherically symmetric perturbations about static equilibrium is examined. This work generalizes and systematizes previous studies and explores the consequences of including the cosmological constant. The approach shows how the existence (or not) of a domain wall dominates the landscape of possible equilibrium configurations.'
address: 'Department of Physics, Queen’s University, Kingston, Ontario, Canada, K7L 3N6 '
author:
- 'Mustapha Ishak[@email] and Kayll Lake[@email1]'
date:
- 
- 
title: Stability of Transparent Spherically Symmetric Thin Shells and Wormholes
---

Introduction
============

It is difficult to imagine a geometrical construction more fundamental than an embedded hypersurface across which the second fundamental form is discontinuous. Studies of such constructions within the Darmois [@Darmois] - Israel [@Israel] formalism (and others) have seen widespread application in classical general relativity (e.g. [@mus]). The importance of such a construction is of course not limited to classical general relativity but is in a real sense central to a much wider class of considerations, such as brane world cosmologies [@xxx].

In this paper we examine the stability of spherically symmetric thin shells, which are in a clearly defined sense “ transparent", to linearized spherically symmetric perturbations about static equilibrium under the assumption that the shells remain transparent under perturbation. The study of the stability of shells is not new [@chase]. Here we follow Poisson and Visser [@poissonandvisser] and parameterize the stability of equilibrium so that we do not have to specify a surface equation of state. The present work generalizes and systematizes previous studies and shows how the existence (or not) of a domain wall [@domainwall] dominates the landscape of possible equilibrium configurations.

Stability
=========

Equation of Motion
------------------

We use the notation of Musgrave and Lake [@mus] throughout and consider a 3-surface $\Sigma$ dividing spherically symmetric spacetime into two distinct parts. In the spherically symmetric case, in terms of the intrinsic coordinates $(\tau,\theta,\phi)$, the metric intrinsic to $\Sigma$ can be given as $$ds_{\Sigma}^2 = R^2(\tau)d\Omega^2+\epsilon d\tau^2 \label{intrinsic}$$ where $\Sigma$ is timelike (spacelike) for $\epsilon = -1 (+1)$ and $d\Omega^2$ is the metric of a unit sphere. In terms of the surface energy density $\sigma(\tau)$, the surface mass of $\Sigma$ is defined by $$M(\tau) = 4 \pi R^2 \sigma = \epsilon \gamma_{\theta \theta} \label{surfacemass}$$ where $\gamma_{\theta \theta} = [K_{\theta \theta}]$ and $K_{ab}$ denotes the second fundamental form of $\Sigma$. (Our notation is explained in [@notation].) We have $$K_{\theta \theta}^2 = R^2\Big(\dot{R}^2-\epsilon(1-\frac{2 m}{R})\Big) \label{kthetatheta}$$ where $m$ (in general not a constant) is the (invariantly defined [@mass]) effective gravitational mass of one of the enveloping spacetimes. The identity $$K_{\theta \theta}^{+ 2} =  \frac{1}{4M^{2}}(K_{\theta \theta}^{+ 2} - K_{\theta \theta}^{- 2} + M^2)^2 \label{identity}$$ can be written in the form [@lake] $$\label{eqn-lake}
\dot{R}^2 = \epsilon + \left( \frac{\left[ m \right]}{M} \right)^2
   - \frac{2 \epsilon \overline{m}}{R} +
   \left( \frac{M}{2R} \right)^2.$$ We refer to (\[eqn-lake\]) as “the equation of motion" for $\Sigma$. This is a quadratic identity and applies to both shells and wormholes.

Transparency Condition
----------------------

In this paper we use the transparency condition $$[G^\alpha_\beta n_\alpha u^\beta]=0, \label{transparency}$$ where $G^\alpha_\beta$ is the (4-dimensional) Einstein tensor, $u^\alpha$ is the 4-tangent to $\Sigma$ and $n^\alpha$ the 4-normal to $\Sigma$. It follows from (\[transparency\]) that for $\sigma^{'}\neq0$ $$\Big(\frac{M}{2R}\Big)^{''} =\frac{\Upsilon}{2R^3}\Big(1+2\frac{P^{'}}{\sigma^{'}}\Big) \label{transparency1}$$ where $$\Upsilon \equiv 3 M -(MR)^{'} = 8 \pi R^2(\sigma + P),
\label{upsilon}$$ $'\equiv \frac{d}{dR}$ and $P$ denotes the intrinsic surface pressure.

Restrictions from the Potential
-------------------------------

In (\[eqn-lake\]) $m$ is defined in terms of coordinates extrinsic to $\Sigma$. In order to view (\[eqn-lake\]) as defining the potential $V(R)$, where $$\dot{R}^2 = -V(R), \label{potential}$$ $R$ must define $m$ uniquely. This imposes a restriction on the enveloping spacetimes. Consider, for example, Bondi coordinates: $$ds^2 = 2c(v,r)dvdr-c^2(v,r)\Big(1-\frac{2m(v,r)}{r}\Big)dv^2+r^2d\Omega^2. \label{bondi}$$ The continuity of the intrinsic metric gives $$r^+ = r^- = R \label{intriniccont}$$ so that for (\[potential\]) to hold at equilibrium we require $$m=m(r)=m(R) \label{mass}$$ which, we note, is precisely the condition that guarantees (\[transparency\]) at equilibrium. For (\[transparency\]) to hold away from equilibrium it follows that $\frac{\partial
c}{\partial r }=0$. In the remainder of this work we assume (\[mass\]) and $\frac{\partial c}{\partial r }=0$ so that the shells remain transparent under perturbation. For timelike $\Sigma$ then our assumptions require static embeddings in the neighborhood of $\Sigma$.

$\Upsilon=0$
------------

In this paper we consider linearized spherically symmetric perturbations about static equilibrium. Following Poisson and Visser [@poissonandvisser], we use $\frac{P^{'}}{\sigma^{'}}(R)$ as a parameterization of the stability of equilibrium so that we do not have to specify an equation of state on $\Sigma$. This parameterization singles out the case $\Upsilon=0$, equivalently $$(MR)^{'}=3M,     \qquad      (\frac{M}{2R})^{''}=0, \qquad \sigma
= -P, \label{posupsilonzero}$$ where $\sigma^{'}=0$. This case (a “domain wall" [@domainwall]) signals a fundamental change in the surfaces that distinguish stable equilibria parameterized by $\frac{P^{'}}{\sigma^{'}}(R)$ as it enters as an asymptote ($\frac{P^{'}}{\sigma^{'}} \rightarrow \pm \infty$). Whereas this limit has well known concrete manifestations, for example Casimir and false vacua (*e.g.* [@poissonandvisser]), it circumvents the usual phenomenology of $\Sigma$ (*e.g.* [@mus]) since the surface stress-energy tensor of $\Sigma$ is simply proportional to the metric intrinsic to $\Sigma$. In what follows we consider $\Upsilon\neq0$. Whether or not this asymptote enters into a given configuration depends simply on the nature of the roots to the polynomial $\sigma + P=0$. Phenomenologically, for timelike $\Sigma$, $\frac{P^{'}}{\sigma^{'}}(R)$ represents the square of the speed of sound on $\Sigma$ and so one would normally expect $0\leq\frac{P^{'}}{\sigma^{'}}(R)<1$. However, one could also construct $\Sigma$ out of (say) fields for which $\frac{P^{'}}{\sigma^{'}}(R)$ need not even be positive. Since we are concerned with stability, not phenomenology, the plots we give generally extend beyond $0\leq\frac{P^{'}}{\sigma^{'}}(R)<1$.

Equilibrium
-----------

From (\[eqn-lake\]) and (\[potential\]) it follows that the equilibrium condition $V^{'}=0$ is $$\Big(\frac{M}{2R}\Big)^{'} = \frac{2R}{M}\Big\{\epsilon\Big(\frac{\overline{m}}{R}\Big)^{'}-\frac{[m]}{M}\Big(\frac{[m]}{M}\Big)^{'} \Big\} \equiv \Gamma, \label{vprime}$$ and the stability condition $V^{''}>0$ is $$\frac{M}{2R}\Big(\frac{M}{2R}\Big)^{''} < \Psi-\Gamma^2, \label{vprimeprime}$$ where $$\Psi \equiv \epsilon\Big(\frac{\overline{m}}{R}\Big)^{''}-\frac{[m]}{M}\Big(\frac{[m]}{M}\Big)^{''}-\Big(\frac{[m]}{M}\Big)^{' 2}. \label{Psi}$$ With the aide of the transparency condition (\[transparency1\]) then the conditions for stable equilibria are $$1+2\frac{P^{'}}{\sigma^{'}}<\Phi,     \qquad      M\Upsilon>0,
\label{posupsilon}$$ and $$1+2\frac{P^{'}}{\sigma^{'}}>\Phi,     \qquad       M\Upsilon<0,
\label{negupsilon1}$$ where $$\Phi \equiv \frac{4R^4}{M \Upsilon}\Big(\Psi-\Gamma^2\Big).
\label{Phidef}$$ It is perhaps worthy of note that not only is there no need for the specification of an equation of state on $\Sigma$, it is not necessary to even calculate $P$. It is only necessary to calculate $P$ for a background check on the existence of the asymptote $\Upsilon=0$ by way of a search for allowed roots to the polynomial $\sigma +P=0$.

Examples
========

In the following examples the explicit form of $\Phi$ can become lengthy. We find it instructive to demonstrate the possible equilibrium configurations graphically. In the approach we have used, the appearance (or not) of the asymptote $\Upsilon=0$ dominates the landscape of possible equilibrium configurations. The polynomial $\sigma+P=0$ is given in Appendix A.

Schwarzschild wormholes
-----------------------

Poisson and Visser [@poissonandvisser] have considered the stability of Schwarzschild wormholes ($\gamma_{\theta \theta} = 2
K_{\theta \theta}^{+}$). In our notation they consider the case $$\overline{m} = m, \qquad [m]=0, \qquad \epsilon=-1, \qquad M=-2R\sqrt{1-2\frac{m}{R}} \label{wormhole}$$ where $m$ is a constant. It follows that $\Phi$ takes a particularly simple form: $$\Phi=\frac{m(3m-2R)}{(R-2m)(R-3m)}.\label{wormhole1}$$ Equations (\[posupsilon\]), (\[negupsilon1\]) and (\[wormhole1\]) reproduce their calculation. Physical units associated with this case are given in Appendix B. The situation is generalized in Figure 1 where we have not assumed the continuity of $m$. For this example $M<0$ and $M\Upsilon =
4R(R-3m)$ so that the region *above* the surface *(a)* and *below* the surface *(b)* in Figure 1 correspond to stable equilibrium.

Schwarzschild black holes
-------------------------

Brady, Louko and Poisson [@brady] have considered the stability of a shell surrounding a classical Schwarzschild black hole and have shown that stability conditions supersede the energy conditions for this configuration as considered previously by Frauendiener, Hoenselaers and Konrad [@frauen]. In this case $$m^{+}>m^{-}, \qquad \epsilon=-1, \qquad
M=R\sqrt{1-2\frac{m^{-}}{R}}-R\sqrt{1-2\frac{m^{+}}{R}}
\label{blackhole}$$ where we have labelled $``+"$ *exterior* to the shell in the sense that the event horizon is in $``-"$. Since $M$ and $\Upsilon$ are $>0$ it follows that the region *below* the surface shown in Figure 2 corresponds to stable equilibrium. We find that $\frac{R}{m^{+}}\sim 2.37$ as $m^{-}\rightarrow 0$ and that $\frac{R}{m^{+}}\sim 3$ as $m^{-}\rightarrow m^{+}$ so that $R>3m^{-}$ in agreement with previous results. There is no asymptote $\Upsilon=0$ in this case.

Cosmological Constant
---------------------

In what follows we explore the physical consequences of including a cosmological constant ($\Lambda$) in the background. The only spacetimes we consider here are the Schwarzschild -de Sitter and Schwarzschild - anti de Sitter spaces so that the effective gravitational mass $m(r)$ is given by $$m(r) = \texttt{m} + \frac{\Lambda r^3}{6},\label{mlambda}$$ where $\texttt{m}$ is a constant.

### Schwarzschild -de Sitter $|$ Schwarzschild

Qualitatively, the inclusion of a positive cosmological constant inside a timelike shell should weaken the binding of the shell in the sense that $P$ will become negative (a surface *tension*) to stablize the shell at sufficiently large $R$. The result is quite unlike the case $\Lambda=0$ (above) were surface tensions are not required to stablize the shell at any radius. This effect is demonstrated in Figure 3. In this case $$m^{+}>\texttt{m}^{-}, \qquad \epsilon=-1, \qquad
M=R\sqrt{1-2\frac{\texttt{m}^{-}}{R}-\frac{\Lambda R^2}{3}
}-R\sqrt{1-2\frac{m^{+}}{R}} \label{des1}$$ where again we have labelled $``+"$ *exterior* to the shell in the sense that the innermost event horizon is in $``-"$. Since $M$ and $\Upsilon$ are $>0$ in this case, it follows that the region *below* the surface shown in Figure 3 corresponds to stable equilibrium. There is no asymptote $\Upsilon=0$ in this case.

### Schwarzschild - anti de Sitter $|$ Schwarzschild

This case is the same as the previous one except that $\Lambda <
0$. Qualitatively, the inclusion of a negative cosmological constant inside a timelike shell should not *require* a surface tension to stablize the shell at sufficiently large $R$. Whereas this agrees with what we find (see Figure 4), in this case we find an asymptote $\Upsilon=0$ with the consequence that the stability surfaces resemble the wormhole case. However, since $M>0$ here, the regions of stability are the reverse of those for wormholes in agreement with our intuition.

### Inversions of 1 and 2

By the inversion of $1$ we mean $$\texttt{m}^{+}>m^{-}, \qquad \epsilon=-1, \qquad
M=R\sqrt{1-2\frac{m^{-}}{R}
}-R\sqrt{1-2\frac{\texttt{m}^{+}}{R}-\frac{\Lambda R^2}{3}}
\label{des1inv}$$ with $\Lambda > 0$ and where again we have labelled $``+"$ *exterior* to the shell in the sense that the innermost event horizon is in $``-"$. In this case we find that the stability regions are qualitatively similar to Figure 4. The inverse of $2$ is the same situation with $\Lambda < 0$ and in this case we find that the stability regions are qualitatively similar to Figure 3 in agreement with intuition.

### Schwarzschild - (anti) de Sitter $|$ Schwarzschild - (anti) de Sitter

One would not expect that the inclusion of $\Lambda$ *continuous* across $\Sigma$ would change, in a significant way, possible equilibrium configurations. To test this consider $$\texttt{m}^{+}>\texttt{m}^{-}, \qquad \epsilon=-1, \qquad
M=R\sqrt{1-2\frac{\texttt{m}^{-}}{R}-\frac{\Lambda R^2}{3}
}-R\sqrt{1-2\frac{\texttt{m}^{+}}{R}-\frac{\Lambda R^2}{3}}
\label{des2}$$ where again we have labelled $``+"$ *exterior* to the shell in the sense that the innermost event horizon is in $``-"$. Our findings are summarized in Figure 5 which, when compared to Figure 2, agrees with what we would expect.

Conclusions
===========

In this paper we have examined the stability of transparent spherically symmetric thin shells to linearized spherically symmetric perturbations about static equilibrium under the assumption that the shells remain transparent under perturbation. Our approach has followed that due to Poisson and Visser [@poissonandvisser] wherein $\frac{P^{'}}{\sigma^{'}}(R)$ is used as a parameterization of the stability of equilibrium. An equation of state on $\Sigma$ is not specified. The present work generalizes and systematizes previous studies. We have found that the single feature which dominates the landscape of equilibrium configurations is the appearance or not of the asymptote $\Upsilon=0$. The cosmological constant has been included in our calculations with results in agreement with intuition. We find that the closest radius for a stable shell is $R = 3
\texttt{m}^{-}$ irrespective of $\Lambda$ and at that radius $\frac{P^{'}}{\sigma^{'}}=1$.

Acknowledgments {#acknowledgments .unnumbered}
===============

This work was supported by a grant (to KL) from the Natural Sciences and Engineering Research Council of Canada and by an Ontario Graduate Scholarship (to MI). Portions of this work were made possible by use of *GRTensorII*[@grt].

G. Darmois, Memorial de Sciences Mathematiques, Fasticule XXV ch V (1927). W. Israel, Nuovo Cimento [**44B**]{} 1 (1966) and [**48B**]{} 463 (1966). P. Musgrave and K. Lake, Class. Quantum Grav. **13**, 1885 (1996) (gr-qc/9510052). Indeed, a simple search on “domain AND wall" in “astro-ph + gr-qc + hep-th" at `arXiv.org` returns over 100 results for the “Last year" and over 500 results over “All years". For an overview of early studies on the gravitational stability of fluid shells see J. E. Chase, Nuovo Cimento [**67B**]{} 136 (1970). E. Poisson and M. Visser, Phys. Rev. D **52** 7318 (1995) ( gr-qc/9506083). By “domain wall" here we mean a hypersurface with an intrinsics equation of state of the form $\sigma+P=0$, where $\sigma$ is the surface density and $P$ is the surface pressure. See *e.g.* [@mus]. We write $\left[ X \right] \equiv X^+\mid_{\Sigma} - X^-\mid_{\Sigma}$ and $\overline{X} \equiv (X^+ \mid_\Sigma + X^- \mid_\Sigma)/2$ with $X^\pm\mid_{\Sigma}$ denoting the limiting values of $X$ on $\Sigma$, and use geometrical units except in Appendix B. We use the letter “m" as follows: $M$ is the surface mass of $\Sigma$ defined by (\[surfacemass\]), $m$ is the effective gravitational mass of the enveloping spacetimes “+" and “-" (see [@mass] and (\[bondi\])), $[m] \equiv m^{+}-m^{-}$ and $\overline{m}
\equiv (m^{+}+m^{-})/2$. $\texttt{m}$ is not the effective gravitational mass but is related to it for the cases considered here by (\[mlambda\]). See, for example, S. Hayward, Phys. Rev. D [**53**]{} 1938 (1996) (gr-qc/9408002) and E. Poisson and W. Israel, Phys. Rev. D. **41**, 1796 (1990). K. Lake, Phys. Rev. D **19**, 2847 (1979). P. R. Brady, J. Louko and E. Poisson, Phys. Rev. D **44** 1891 (1991). J. Frauendiener, C. Hoenselaers and W. Konrad, Class. Quantum Grav.**7**, 585 (1990). This is a package which runs within Maple. It is entirely distinct from packages distributed with Maple and must be obtained independently. The GRTensorII software and documentation is distributed freely on the World-Wide-Web from the address [http://grtensor.org]{}

**Appendix A: The Asymptote $\Upsilon=0$**

The existence of the asymptote $\Upsilon=0$ depends on the existence of allowed roots to the equation $\sigma + P=0$. Writing $$\texttt{f}(R) \equiv 1-\frac{2m(R)}{R} \label{fdef}$$ these roots are determined by the polynomial $$\delta \sqrt{F}(2f-Rf^{'})=\sqrt{f}(2F-RF^{'}) \label{poly}$$ where $\delta=+1$ for shells, $\delta=-1$ for wormholes and we have distinguished the sides of $\Sigma$ by $\texttt{f}=f$ and $\texttt{f}=F$. As an example, the solution to equation (\[upsilon\]) for the Schwarzschild wormhole case (see Figure 1) is shown in Figure 6.

**Appendix B: Physical Units and Schwarzschild wormholes**

  ------- ------- ------------- ---------------------- -- ------- ------- ------------- ----------------------
   R(km)   $R/M$   $M/M_\odot$   $(P'/\sigma')_{min}$      R(km)   $R/M$   $M/M_\odot$   $(P'/\sigma')_{max}$
     1     -1.0     -0.67721             4.75                1     -0.55    -1.23130            -0.62
     1     -1.5     -0.45148             3.50                1     -0.60    -1.12869            -0.81
     1     -2.0     -0.33861             4.98                1     -0.65    -1.04187            -1.12
     1     -2.5     -0.27089             7.14                1     -0.70    -0.96745            -1.64
     1     -3.0     -0.22574             9.84                1     -0.75    -0.90295            -2.69
     1     -10.0    -0.06772            100.76               1     -0.85    -0.79672            -25.80
     5     -1.0     -3.38607             4.75                5     -0.55    -6.15648            -0.62
     5     -1.5     -2.25738             3.50                5     -0.60    -5.64344            -0.81
     5     -2.0     -1.69303             4.98                5     -0.65    -5.20933            -1.12
     5     -2.5     -1.35443             7.14                5     -0.70    -4.83724            -1.64
     5     -3.0     -1.12869             9.84                5     -0.75    -4.51475            -2.69
     5     -10.0    -0.33861            100.76               5     -0.85    -3.98361            -25.80
    10     -1.0     -6.77213             4.75               10     -0.55    -12.31297           -0.62
    10     -1.5     -4.51475             3.50               10     -0.60    -11.28689           -0.81
    10     -2.0     -3.38607             4.98               10     -0.65    -10.41866           -1.12
    10     -2.5     -2.70885             7.14               10     -0.70    -9.67447            -1.64
    10     -3.0     -2.25738             9.84               10     -0.75    -9.02951            -2.69
    10     -10.0    -0.67721            100.76              10     -0.85    -7.96721            -25.80
    100    -1.0     -67.72131            4.75               100    -0.55   -123.12966           -0.62
    100    -1.5     -45.14754            3.50               100    -0.60   -112.86886           -0.81
    100    -2.0     -33.86066            4.98               100    -0.65   -104.18664           -1.12
    100    -2.5     -27.08853            7.14               100    -0.70    -96.74473           -1.64
    100    -3.0     -22.57377            9.84               100    -0.75    -90.29508           -2.69
    100    -10.0    -6.77213            100.76              100    -0.85    -79.67213           -25.80
  ------- ------- ------------- ---------------------- -- ------- ------- ------------- ----------------------

  : Physical units associated with the wormhole case with continuous mass. Values for the upper branch in Figure 7 are shown on the left.

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