Gravastar: A Non-Singular Ultra-Compact Object

• Gravastar is a hypothetical compact object featuring a de Sitter interior with negative pressure, replacing singularities found in black holes.
• Its structure consists of a thin, ultra-relativistic shell of stiff matter matched smoothly via Israel-Darmois conditions, ensuring stability.
• Unique observational signatures, such as modified photon orbits and gravitational wave echoes, may help distinguish gravastars from black holes.

A gravastar (gravitational vacuum star) is a theoretically proposed ultra-compact object originally conceived as an alternative endpoint to black hole formation. In its canonical form, a gravastar consists of a de Sitter-like interior (with negative pressure acting as a repulsive force), an ultra-relativistic thin shell of stiff matter, and an exterior described by classical vacuum solutions (typically Schwarzschild or a closely related geometry). Because gravastars possess no event horizon and lack a central singularity, they have attracted significant attention as models that potentially resolve deep theoretical difficulties associated with black holes, such as the information loss paradox and the inevitability of singularities in classical general relativity.

1. Internal Structure and Governing Equations

The prototypical gravastar architecture, as established in models by Mazur and Mottola, features three regions:

• Interior region (0 ≤ r < r₁): The matter sector obeys $p = -\rho$, corresponding to a constant positive energy density and negative pressure. This “dark energy” phase mimics a de Sitter vacuum and yields an interior metric, for example,

$$ds^2 = -A_{-} dt_-^2 + \frac{dr_-^2}{A_{-}} + r_-^2 (d\theta^2 + \sin^2\theta d\phi^2),$$

with $A_{-}(r_-) = 1 - H^2 r_-^2$ for the spherically symmetric case (Sakai et al., 2014).

• Thin shell (junction at r = r₁, $r_2 = r_1 + \epsilon$): The shell is composed of stiff matter ($p = \rho$), often modeled as an ultra-relativistic plasma. The shell width $\epsilon$ is taken to be small relative to the object’s overall radius. Israel junction conditions (or their generalizations for modified gravity scenarios) are imposed at this interface, typically yielding constraints like the surface energy density $\sigma$ and pressure $p$, e.g., $\sigma = p$ for 2+1 stiff matter shells.
• Exterior region ($r > r_2$): The exterior is vacuum, described by the Schwarzschild solution,

$$ds^2 = -A_+(r_+) dt_+^2 + \frac{dr_+^2}{A_+(r_+)} + r_+^2 (d\theta^2 + \sin^2\theta d\phi^2),$$

with $A_+(r_+) = 1 - (r_g/r_+)$ and $r_g = 2M$.

In alternative geometries (e.g., cylindrically symmetric models (Bhattacharjee et al., 2023, Sinha et al., 12 Feb 2025)), the interior and shell equations are modified by the specific line elements and potential structures (such as the Kuchowicz metric potential).

The key feature of this construction is the enforcement of smooth matching of the metric functions and their derivatives across the shell via the Darmois-Israel formalism, resulting in a physically well-defined global spacetime.

2. Physical Properties of the Thin Shell

The thin shell serves as a quantum critical interface, bridging the de Sitter-like repulsive interior and the vacuum exterior. Key properties include:

• Equation of state: $p = \rho$ (ultra-relativistic, stiff matter) (Sakai et al., 2014, Bhattacharjee et al., 2023, Sinha et al., 12 Feb 2025).
• Proper shell length: Computed via the metric function as

$l = \int_{r_1}^{r_2} \sqrt{e^{b(r)}} dr,$

with $e^{b(r)}$ specified by the thin shell solution (e.g., approximate logs or algebraic expressions in $r$ with parameters fixed by matching and regularity conditions).

• Shell energy: The energy content

$E = 4\pi \int_{r_1}^{r_2} \rho(r) r^2 dr$

increases monotonically with shell thickness and is dominated by the shell for realistic parameter choices (Bhattacharjee et al., 2023).

• Shell entropy: The shell carries the system’s total entropy (the interior is assigned zero entropy as a pure condensate). This is computed as

$$S = 4\pi \int_{r_1}^{r_2} s(r) r^2 \sqrt{e^{b(r)}} dr,\quad s(r) = \frac{a k_B}{\hbar} \sqrt{\frac{p}{2\pi}},$$

where $a$ is a dimensionless parameter. Entropy increases with shell thickness, vanishing in the pure de Sitter limit (single condensate).

3. Regularity, Absence of Singularities, and Event Horizon Replacement

In the gravastar model, the combination of the repulsive de Sitter core and the stiff, finite shell eliminates the central singularity and avoids the appearance of a classical event horizon:

• Regularity at the center: The interior potentials (e.g., using the Kuchowicz metric) are constructed such that $g_{tt}$ and $g_{rr}$ remain finite as $r \to 0$ (Bhattacharjee et al., 2023, Sinha et al., 12 Feb 2025).
• No event horizon: The matching between regions and the stiff shell replace the mathematically idealized event horizon by a physical (albeit extremely thin) surface. Global analyticity and continuity are preserved, and the gravitational field outside still mimics that of a black hole over a wide parameter range.
• Information paradox avoidance: Since there is no region from which signals cannot escape in principle, the gravastar sidesteps the information loss paradox inherent to black holes (Bhattacharjee et al., 2023, Sinha et al., 12 Feb 2025).

4. Observational Signatures and Distinguishing Features

Although externally the spacetime can closely resemble that of a black hole, gravastars exhibit unique observational properties:

• Photon orbits and shadows: Unstable photon orbits are present in both gravastar and black hole exteriors, leading to the formation of quasi-shadows (Sakai et al., 2014). However, the gravastar’s transparent shell can produce a bright disk surrounded by a dark ring, distinct from the pure black hole shadow (where the central region is entirely dark).
• Gravitational lensing and echoes: The pattern of bright disks, arcs, and rings produced in lensing by gravastars (for either background planes or companion stars) can, in principle, reveal the mixed geometry at radii inside the photon sphere, accessible to high-resolution VLBI observations (Sakai et al., 2014).
• Thermal and quantum emission: For certain dynamical formation models, thermal emission can be associated with the de Sitter core’s Gibbons–Hawking temperature, redshifted by the shell properties (Nakao et al., 2022). In merger scenarios, gravitational wave echoes might appear due to the shell’s reflective boundary conditions (Wang et al., 2018).

5. Extensions, Stability, and Theoretical Implications

Gravastar models have been generalized to a variety of spacetime symmetries, matter content, and underlying gravitational theories:

• Modified gravity settings: Gravastar constructs have been implemented in alternative frameworks, including $f(\mathbb{T}, \mathcal{T})$ gravity (Ghosh et al., 2020), $f(\mathcal{G}, T)$ gravity (Shamir et al., 2018), braneworld gravity (Sengupta et al., 2020, 2203.12027), energy-momentum squared gravity (Sharif et al., 2022), Rastall gravity (Ghosh et al., 2021, Bhattacharjee et al., 25 Jan 2024), Loop Quantum Cosmology (Ghosh et al., 2023), and $f(R, L_m, T)$ gravity (Sinha et al., 12 Jul 2024). In each, the matching conditions and some properties of the shell are altered, but the core structure persists.
• Cylindrical and non-spherical solutions: The gravastar scenario has been extended to cylindrical spacetime, again using a non-singular interior, a stiff-matter thin shell, and a vacuum exterior (Bhattacharjee et al., 2023, Sinha et al., 12 Feb 2025).
• Stability criteria: Stability is probed via surface redshift bounds (typically $Z_s < 2$ outside the shell), sound speed constraints, energy condition verification, and entropy maximization techniques (Bhattacharjee et al., 2023, Sinha et al., 12 Feb 2025). The equilibrium between negative core pressure and stiff shell matter is robust over a range of physically reasonable parameters.
• Comparison with black holes: The presence of the thin shell as a quantum-critical interface uniquely distinguishes gravastars from black holes, both in the avoidance of event horizons/singularities and in the expected observational signatures.

6. Mathematical Expressions for Core Gravastar Features

Region	Equation of State	Key Metric Potential	Energy Density/Pressure
Interior	$p = -\rho$	$e^{a(r)} = e^{\alpha r^2 + 2\ln\beta}$	$\rho = \text{const}$
Thin shell	$p = \rho$ (stiff)	$e^{b(r)} = \frac{\alpha^2 r^2 + 3\alpha}{2\Lambda}$	$p = \rho = A e^{-\alpha r^2}$
Exterior	$p = \rho = 0$	E.g., $$ds^2 = -[\,\cdot\, ] dt^2 + [\,\cdot\, ]^{-1} dr^2 + \cdots$$	—

The entropy of the shell:

$$S = \int_{r_1}^{r_2} 4\pi r^2 s(r) \sqrt{e^{b(r)}} dr,\quad \text{with}\ s(r) = \frac{a k_B}{\hbar} \sqrt{\frac{p}{2\pi}}$$

The surface energy density and pressure (Darmois-Israel matching):

$$\sigma = -\frac{1}{4\pi R}\left[\sqrt{f}^+ - \sqrt{f}^-\right],\quad \Xi = -\frac{\sigma}{2} + \frac{1}{16\pi} \left[ \frac{f'}{\sqrt{f}} \right]^+_-$$

7. Theoretical and Observational Implications

Gravastar models provide a non-singular, horizonless endpoint for gravitational collapse that is consistent with general relativity (or its extensions) and is stabilized by quantum effects or exotic matter at high density. The replacement of the event horizon with a stiff shell invites new interpretations of gravitational-wave events, compact-object stability, and the spectral signatures of ultra-compact, dark astronomical sources.

A plausible implication is that future observations—from high-resolution radio interferometry, gravitational-wave detectors, or other probe techniques capable of resolving the compact-object shadow or post-merger ringdown structure—might be able to distinguish gravastars from classical black holes. The physical criteria for stability, the absence of singularities, and the unique matter–geometry interplay of the stiff shell remain central themes in current research (Sakai et al., 2014, Bhattacharjee et al., 2023, Sinha et al., 12 Feb 2025).