Self-Gravitating Matter Thin Shell

• Self-gravitating matter thin shells are singular hypersurfaces that serve as zero-thickness interfaces carrying energy, pressure, and charges in spacetime.
• The junction conditions rooted in the Israel formalism match interior and exterior metrics, enabling accurate modeling of collapse, bounce, and oscillatory behaviors.
• Analytical studies of these shells illuminate black hole thermodynamics, multiverse transitions, and semiclassical quantum effects in gravitational systems.

A self-gravitating matter thin shell is a singular hypersurface of zero thickness carrying localized energy, pressure, and possibly additional charges, embedded in a spacetime whose geometry it nontrivially influences. It serves as an exact model for idealized interfaces in general relativity, quantum gravity, and modified gravity. Its implementation—via the junction (matching) conditions—connects interior and exterior metric regions, thereby capturing the nonlinear interplay between matter distributions, geometrical structure, and global causal properties. Thin shells play a pivotal role in the dynamical and thermodynamic modeling of black holes, gravastars, early-universe phase transitions, and semiclassical gravity.

1. Junction Conditions and Shell Dynamics

The evolution of a spherically symmetric, self-gravitating thin shell separating two regions (e.g., Minkowski/Minkowski, Schwarzschild/Schwarzschild, or Reissner–Nordström/Reissner–Nordström) is governed by the discontinuities in the extrinsic curvature across the shell, as encoded in the Israel formalism. For a shell with radius $\rho(\tau)$, proper time $\tau$, and surface energy-momentum tensor modeled by a perfect fluid, the key equation is

$$\sigma_\text{in} \sqrt{\dot{\rho}^2 + 1 - \frac{2m_\text{in}}{\rho} + \frac{Q^2}{\rho^2}} - \sigma_\text{out} \sqrt{\dot{\rho}^2 + 1 - \frac{2m_\text{out}}{\rho} + \frac{Q^2}{\rho^2}} = 4\pi\rho\, \mu(\rho),$$

where $m_\text{in},\,m_\text{out}$ are the interior/exterior ADM masses, $Q$ is the (shared) charge (for an electrically neutral shell), and $\mu(\rho)$ is the shell’s energy density, possibly with pressure $p$ obeying a polytropic equation of state. The sign functions $\sigma_\text{in},\,\sigma_\text{out}$ encode the orientation of the normal vector.

The equation reduces to a conservation-law form: $\dot{\rho}^2 + U(\rho) = 0$ with the effective potential $U(\rho)$ dependent on the shell parametrization and spacetime parameters. For dust shells ($p=0,\,\mu(\rho) = n = A/\rho^2$), $U(\rho)$ admits closed-form analytical characterization, allowing classification of collapse, bounce, and oscillatory behaviors.

2. Global Causal Structure: Carter–Penrose Diagrams

The shell’s worldtube is embedded in a spacetime whose causal structure is best visualized via Carter–Penrose diagrams, especially for multi-horizon geometries (e.g., Reissner–Nordström). Here, regions labeled $R_+$, $R_-$, $T_+$, $T_-$ encode exterior, interior, and trapped domains. Shell evolution traces distinct paths across these, crossing horizons (where the $g_{tt}$ component vanishes) and possibly traversing from an external universe through internal black hole regions (potentially passing into successive "universes").

Oscillatory shell motion is especially nontrivial: for a suitable parameter regime, the shell undergoes repeated expansion and contraction between two bounce points, cycling through an infinite sequence of causally disconnected regions. The diagrams reveal the possibility of transition between distinct asymptotic infinities and the prospect of "baby universe" formation in quantum cosmological contexts.

3. Equations of State and Stability Criteria

The shell matter can obey general polytropic equations of state: $p = k n^\gamma$ with number density $n = A/\rho^2$, yielding total energy density

$$\mu(\rho) = \frac{A}{\rho^2} + \frac{k A^\gamma}{\gamma-1} \frac{1}{\rho^{2\gamma}}.$$

For such matter, the qualitative features of the effective potential $U(\rho)$—and hence the allowed shell trajectories—are largely preserved, contingent on the parameters $(A,\,k,\,\gamma)$. Oscillating (periodic) solutions arise when the potential admits two real roots with $U(\rho) < 0$ between them. The parameter range

$$4\pi^2 A^2 < Q^2, \qquad (m_\text{out} - m_\text{in})^2 < 16 \pi^2 A^2$$

ensures the existence of these stable, nontrivial solutions.

Dynamical stability further correlates with turning points of $U(\rho)$, and linear perturbations about static configurations yield stability when the second derivative $U''(\rho)$ at the equilibrium point is positive.

4. Analytical Structures and Classification

Explicit structural formulas underpin both the dynamic and thermodynamic analysis. Key expressions include:

• Junction condition for the moving shell:

$$\sigma_\text{in} \sqrt{\dot{\rho}^2 + 1 - \frac{2m_\text{in}}{\rho} + \frac{Q^2}{\rho^2}} - \sigma_\text{out} \sqrt{\dot{\rho}^2 + 1 - \frac{2m_\text{out}}{\rho} + \frac{Q^2}{\rho^2}} = 4\pi\rho\, \mu(\rho)$$

• Effective potential (dust):

$$U(\rho) = 1+\frac{Q^2}{\rho^2}-\frac{m_\text{out}+m_\text{in}}{\rho} - 4\pi^2 \rho^2 \mu^2(\rho) - \frac{(m_\text{out}-m_\text{in})^2}{16\pi^2 \rho^4 \mu^2(\rho)}$$

• Sign functions for orientation:

$$\sigma_\text{out} = \operatorname{sign}\left[m_\text{out} - m_\text{in} - \frac{8\pi^2 A^2}{\rho}\right], \qquad \sigma_\text{in} = \operatorname{sign}\left[m_\text{out} - m_\text{in} + \frac{8\pi^2 A^2}{\rho}\right]$$

Explicit calculation of bounce points, event horizons, maximal/minimal radii, and transition conditions enables a full regime classification.

5. Implications: Multiverse Dynamics and Quantum Effects

Oscillatory or "bouncing" shell solutions, wherein the shell worldline repeatedly transfers between $R_+$ regions, realize a mechanism for navigation between distinct universes in the maximally extended Reissner–Nordström background. This approach provides an exact classical model for studying cosmological "bubble nucleation," baby universe creation, and semiclassical quantum tunneling phenomena. In particular, thin shell instanton methods model quantum transitions between vacua and connect to the landscape picture in quantum cosmology. These scenarios underscore connections between shell stability properties and the global topology of spacetime.

Further, the analogy between centrifugal barriers in the Kerr (rotating) geometry and electromagnetic repulsion in Reissner–Nordström suggests that similar oscillatory shell solutions can exist around rotating black holes, enabling possible generalizations to axisymmetric configurations.

6. Applications and Extensions

Self-gravitating thin shells serve as models for:

• Dynamical surfaces in black hole and compact-object physics
• Phase interfaces in early-universe cosmology (e.g., false vacuum decay, bubble walls)
• Constructing exact solutions in semiclassical gravity and the paper of black hole microstates
• Investigating stability and transition mechanisms in generalized backgrounds (e.g., F(R) gravity, AdS/CFT, higher dimensions)
• Probing the plausibility and physical realization of multiverse scenarios via explicit analytic models

A comprehensive analytical framework such as that provided by the fully explicit potential and junction conditions enables systematic mapping of all possible shell evolutions under the specified energy-momentum composition and asymptotic structure.

7. Summary

A self-gravitating matter thin shell in the Reissner–Nordström (and more general) spacetime context is characterized by exact matching conditions and analytic dynamic reducibility, yielding a rich solution space including collapsing, bouncing, expanding, and periodic oscillating trajectories. The spectrum and nature of allowed shell motions depend sensitively on the shell's equation of state and the global charges/masses of the embedding geometry. Oscillatory solutions support the possibility of shell-mediated transitions between entire spacetime regions (universes), with significant implications for quantum gravity and cosmology. The technical apparatus—strict junction conditions, effective potential analysis, Carter–Penrose diagrams, and explicit state equations—establishes the self-gravitating thin shell as a uniquely tractable and informative tool in the paper of gravitating systems with localized degrees of freedom (Dokuchaev et al., 2010).