Shellworld: Gravity on a Thin Shell

• Shellworlds are defined as hypersurfaces where lower-dimensional gravity and matter localize via precise matching conditions, offering a unified framework in GR and string theory.
• The D-dimensional spherical construction employs cut-and-paste techniques with explicit Israel junction conditions to match intrinsic and extrinsic curvature, ensuring stability and energy conservation.
• String theory embeddings, including T̄T deformations, utilize brane dynamics to resolve bulk singularities and generate effective worldvolume gravity with implications for cosmological bounces.

A shellworld is a spacetime construction in which lower-dimensional "worlds" and their gravity are localized on a thin shell—typically a co-dimension-1 hypersurface—embedded in a higher-dimensional bulk. Such shellworlds arise in classical general relativity, string theory, and braneworld cosmology, and are characterized by the junction between distinct bulk solutions across the shell, leading to both matter and dynamical gravity localized at the hypersurface. The shell typically separates regions with different geometrical or topological properties, and its dynamics and stability are governed by generalized Israel junction conditions and associated conservation laws.

1. Geometric Foundations and Field Equations

Consider a hypersurface $\Sigma$ of co-dimension 1 embedded in a Lorentzian manifold, separating two spacetimes $(M^+,g^+)$ and $(M^-,g^-)$. The induced metric (first fundamental form) $h_{ab}$ on $\Sigma$ is taken continuous, ensuring the metric matches across the shell. The extrinsic curvature $K_{ab}^\pm$ quantifies the embedding on either side. The jump in extrinsic curvature across $\Sigma$ encodes the shell's localized stress-energy.

For non-null (timelike/spacelike) shells, the generalized Israel junction condition holds: $$\bigl[K_{ab}\bigr] - h_{ab}\, \bigl[K\bigr] = -\,\kappa\,\tau_{ab}$$ where $\tau_{ab}$ is the surface stress-energy tensor and $K\equiv h^{ab}K_{ab}$ (Senovilla, 2018). This is supplemented by differential and algebraic constraints, including local conservation: $$\nabla_a\,\tau^{a}{}_{b} = [G_{\mu\nu}]\,n^{\mu}e_b^{\nu}$$ where $n^\mu$ is the normal to $\Sigma$.

For null shells, novel equations for the Weyl tensor degrees of freedom come into play. In particular, the traceless part of the shell tensor $y_{ab}$, called $Z_{ab}$, is identified with the singular part of the Weyl tensor, $W_{ab}$, which encodes impulsive gravitational wave degrees of freedom. The dynamical evolution of $W_{ab}$ is governed by a transport equation along the shell's null generators.

The formalism unifies all shell types (non-null, null, or signature-changing) and applies in arbitrary dimensions. These geometric and field-theoretic relations are foundational for all shellworld scenarios, providing a consistent prescription for both classical and quantum gravity models involving thin shells (Senovilla, 2018).

2. D-dimensional Spherical Shellworld Construction

The canonical shellworld scenario, as formulated by Eiroa and Simeone, constructs a thin spherical shell in a D-dimensional spherically symmetric background by "cut and paste" techniques. The basic method involves:

• Taking two D-dimensional manifolds $\mathcal{M}_-$ and $\mathcal{M}_+$ described by metrics

$$ds^2_\pm = -f_\pm(r)\,dt_\pm^2 + f_\pm(r)^{-1}\,dr^2 + r^2 d\Omega_{n}^2\,, \qquad n=D-2$$

• Defining the shell $\Sigma$ at $r = a(\tau)$, where $a(\tau)$ is the shell radius as a function of proper time, and identifying the induced metric as

$ds^2_\Sigma = -d\tau^2 + a(\tau)^2\,d\Omega_n^2$

• Matching the first fundamental form and enforcing the junction via the Lanczos condition:

$[K_i^{\;j}] - \delta_i^j [K] = -8\pi S_i^{\;j}$

with $S_i^{\;j} = \mathrm{diag}(-\sigma,\,p,\ldots,p)$ for a perfect-fluid shell.

The explicit surface stress-energy components are: $$8\pi\,\sigma = \frac{n}{a}\left(\sqrt{\dot a^2 + f_-(a)} - \sqrt{\dot a^2 + f_+(a)}\right)$$

$$8\pi\,p = -\,\frac{n-1}{n}\,8\pi\,\sigma - \left( \frac{\ddot a + \frac12 f'_-(a)}{\sqrt{\dot a^2 + f_-(a)}} - \frac{\ddot a + \frac12 f'_+(a)}{\sqrt{\dot a^2 + f_+(a)}} \right)$$

with the conservation law: $$\frac{d}{d\tau}(\sigma\,a^n) + p\,\frac{d}{d\tau}(a^n) = 0$$

Imposing the weak energy condition (WEC) yields

$\sigma \geq 0\,, \qquad \sigma + p \geq 0$

which, in the static case, implies $f_-(a)\geq f_+(a)$ for non-exotic matter. The effective potential formulation,

$\dot a^2 + V(a) = 0$

with

$$V(a) = \frac{f_-(a)+f_+(a)}{2} - \left[ \frac{n(f_-(a)-f_+(a))}{8\pi a \sigma(a)} \right]^2 - \left[ \frac{4\pi a \sigma(a)}{n} \right]^2$$

enables a systematic investigation of the existence and stability of static shellworlds. The condition $V(a_0)=0$, $V'(a_0)=0$, and $V''(a_0)>0$ identifies stable configurations (Eiroa et al., 2012).

3. Shellworld Cosmologies and Braneworld Realizations

In higher-dimensional braneworld models, the "shellworld" or "dark-bubble" replaces the standard $\mathbb{Z}_2$-symmetric Randall–Sundrum brane with a single time-like spherical shell $\Sigma$. This shellworld divides the five-dimensional bulk, typically an AdS–Schwarzschild (or general string-cloud) spacetime, into two regions with AdS radii $\ell_\pm$ and mass parameters $m_\pm$, possibly with string-cloud contributions $b_\pm$.

The induced dynamics on the shell generate a modified Friedmann equation: $$( \dot a / a )^2 = - \frac{1}{a^2} + \frac{1}{a^4} \frac{(m_+ \ell_+ - m_- \ell_-)}{(\ell_+ - \ell_-)} + \frac{2}{3} \frac{1}{a^3} \frac{(b_+ \ell_+ - b_- \ell_-)}{(\ell_+ - \ell_-)} + \frac{8\pi G_4}{3}\Lambda_4$$ with $G_4$ and $\Lambda_4$ determined by the shell tension and bulk Planck scale. The terms correspond to shell curvature ($a^{-2}$), dark radiation ($a^{-4}$), string cloud (dust, $a^{-3}$), and effective cosmological constant. Under specified inequalities (such as $m_+ \ell_+ < m_- \ell_-$ and $b_+ \ell_+ > b_- \ell_-$), there exists a value $a_*$ at which a regular, nonsingular cosmological bounce occurs—distinct from big-bang/crunch singularities seen in standard cosmology. Stability is ensured if $a_*$ lies outside any bulk Cauchy horizons (Sherpa et al., 15 Jan 2026).

Physically, a cloud of bulk strings stretches to the shell, with their endpoints providing effective four-dimensional "dust" matter. Holographically, this can be interpreted as a gluonic field on the shell, significantly affecting the shellworld's global evolution.

4. String Theory Embedding and $T\bar{T}$-Driven Shells

Shellworlds also arise in string theoretic constructions, particularly through exactly solvable deformations such as single-trace $T\bar{T}$ flows in AdS$_3$ backgrounds. In the three-dimensional effective action (after compactification on $S^3 \times \mathcal{M}_4$), relevant bulk metrics exhibit a curvature singularity (the "repulson") for one sign of the deformation parameter $\lambda < 0$. To resolve this, one excises the singular region and glues a linear-dilaton vacuum exterior, producing a shell of $D5$-branes at the junction.

The Israel matching conditions at the shell combine metric and dilaton continuity with a jump in radial derivatives, generating a thin shell characterized by an effective tension. S-duality maps the F1–NS5 system to a D1–D5 configuration, where "enhançon" physics ensures physical consistency by restricting the brane configuration to a shell at a certain radius $r_e$: $r_e^2 = g_s \alpha' \frac{Q_5 - Q_1}{v-1}$ obliging the D5's to expand onto the shell. The shell hosts localized lower-dimensional gravity and matter sampled from fluctuations. This provides a top-down, string-theoretic realization of shellworlds, with the worldsheet viewpoint identifying the setup as a current–current deformation of the corresponding WZW model, exactly encoding the $T\bar{T}$ flow (Aguilera-Damia et al., 2020).

5. Stability Analysis and Physical Criteria

The dynamical stability of shellworlds is assessed by analyzing small perturbations of the shell radius and examining the behavior of the effective potential $V(a)$. Stability requires the existence of a local minimum of $V(a)$ supporting oscillatory or static shell configurations. The explicit stability condition involves second derivatives of $V(a)$ and, in higher dimensions, the sound-speed parameter $\eta$ on the shell. Stable parameter regimes expand with increasing spacetime dimension and depend sensitively on shell charge and cosmological constant.

In scenarios with string clouds or gauge fluxes, the interplay of different energy components (shell tension, dark radiation, string cloud matter) can be tuned to obtain a nonsingular bounce, with stability determined by checking that the bounce radius avoids pathological regions (e.g., Cauchy horizons or singularities) in the higher-dimensional geometry. For the T$\bar{T}$ string backgrounds, stability derives from the consistency of the matching conditions and the absence of CTC regions in the resultant glued spacetime.

6. Applications, Extensions, and Theoretical Significance

Shellworlds provide a unifying framework for several physical phenomena:

• Realizations of braneworld cosmologies with localized gravity and matter.
• Construction of domain walls mediating between vacua of different cosmological constant, gauge structure, or even metric signature.
• String theory applications, where shellworlds resolve bulk singularities (e.g., repulson, enhançon) while supporting localized worldvolume gravity.
• Exact treatment of impulsive gravitational wave layers in the null-shell limit, where the full Weyl tensor jump equations track the shell's radiative DOF.

The full geometric shell equations generalize the Israel–Lanczos formalism, supplying new algebraic and differential constraints between the jumps in ambient curvature and the shell's intrinsic-extrinsic geometry. This ensures local conservation of shell matter and gravitational energy, further linking the bulk field content with induced gravity and matter on the shell. Shellworlds thus serve as robust platforms for modeling cosmology, holography, and quantum gravity phenomena in both effective and string-theoretic settings (Senovilla, 2018, Eiroa et al., 2012, Aguilera-Damia et al., 2020, Sherpa et al., 15 Jan 2026).