Non-Singular Horizonless Objects

• Non-singular horizonless objects are spacetimes engineered to avoid curvature singularities and event horizons using mechanisms like extra dimensions and infinite-derivative dynamics.
• They serve as counterexamples to classical black hole formation by bypassing the standard energy conditions and modifying causal structures through regularization scales and exotic sources.
• These models predict distinctive observational features, including unique lensing patterns, ringdown signatures, and quasi-periodic oscillations, which can be used to test strong-field gravity.

Non-singular horizonless objects are spacetime configurations that are globally free of curvature singularities, lack event horizons, and can—depending on theoretical context—support extreme gravitational compactness or mimic many observational features of black holes. Their construction and physical implications span modified gravity theories, semiclassical gravity, classical general relativity with special sources, and lower-dimensional toy-models. These configurations provide powerful counterexamples to the inevitability of singularity formation in gravitational collapse as dictated by the classical Penrose singularity theorem. The paper of non-singular horizonless objects delivers insights into the role of energy conditions, the topology and causal structure of spacetime, the microphysical resolution of singularities, observational signatures in strong-field gravity scenarios, and challenges deep aspects of cosmic censorship.

1. Theoretical Motivation and Definitions

Non-singular horizonless objects arise as alternatives or generalizations to the canonical outcomes of gravitational collapse predicted in general relativity (GR). Classical GR prescribes that, for matter satisfying the null energy condition (NEC) and after the formation of a trapped surface, continued collapse unavoidably yields a singularity clothed by an event horizon—a black hole. In contrast, non-singular horizonless objects circumvent this outcome through various mechanisms:

• Modification of the gravitational field equations (e.g., torqued extra dimensions, infinite-derivative gravity, Lorentz-violating frameworks).
• Introduction of exotic sources or new regularization scales in the matter sector.
• Utilizing topology (e.g., wormhole throats) or the global violation/avoidance of certain energy and convergence conditions during dynamical transitions.

A prototypical example is the static, spherically-symmetric, singularity-free compact shell in "Torsion to Hide Extra Dimension" (THED) gravity, which possesses a flat interior, non-trapped exterior, and supports arbitrarily large compactness without horizon or singularity while still satisfying NEC for observable matter (Shankar, 2015).

These configurations must be carefully distinguished from naked singularities, which also lack event horizons but possess divergent curvature scalars, leading to pathologies at their core, and from regular black holes, which retain event horizons but have regular interiors (e.g., de Sitter cores).

2. Mechanisms of Singularity Avoidance

Several theoretical mechanisms underpin the existence of non-singular horizonless objects:

a) Modified Gravity:

THED gravity, which incorporates a five-dimensional bulk with induced torsion serving to algebraically “hide” the extra dimension, allows the 4D world to host positive mass, horizonless compact objects as exterior solutions. The field equations,

$$G_{\nu}^{\mu} - {}_{\nu}^{\mu} = \Sigma_{\nu}^{\mu},$$

where the torsion-induced stress tensor ${}_{\nu}^{\mu}$ depends on the extra-dimensional scalar component Φ, modify the usual Einstein equations such that the standard causal and energy-focusing properties leading to singularity formation can be bypassed (Shankar, 2015).

b) Nonlocal and Infinite-Derivative Dynamics:

Infinite-derivative gravity (IDG) achieves regularization at small scales by introducing nonlocal interactions that modify the graviton propagator and lead to a softened gravitational potential,

$$\Phi(r) = \frac{m}{M_p^2} \frac{1}{r} \operatorname{erf}(M_s r/2),$$

where the error function leads to constant potential and vanishing force in the ultraviolet, preventing metric singularities and precluding horizon formation given a suitable mass gap (Koshelev et al., 2017).

c) Lorentz-violating Gravity and Universal Horizons:

In Hořava/khronometric or Einstein–aether models, the preferred foliation alters causal and horizon structures. Universal horizons (defined by %%%%1%%%% where $u^a$ is the aether vector and $\chi^a$ the Killing vector) may exist inside or replace event horizons, enforcing regular causal boundaries even for arbitrarily fast signals (Mazza et al., 2023). Horizonless regular objects ("black bounces," traversable wormholes) are possible within these frameworks.

d) Regularized Geometries with New Scales:

Metric modifications that introduce a regularizing length scale ℓ (e.g., the Hayward prescription) shift the innermost regions from singularities to regular cores or throats. For instance,

$$F(r) = 1 - \frac{2 M \rho^2(r)}{\rho^3(r) + 2M \ell^2},$$

with either simply connected (inner horizon) or non-simply connected (wormhole) causal structures. These allow the configuration to smoothly transition from horizonful to horizonless states as regularization parameters vary (Carballo-Rubio et al., 2023).

3. Energy Conditions, Collapse, and the Penrose Theorem

A key result stemming from these models is that horizonless singularity-free objects can be constructed without violating the NEC for observable matter. In THED gravity, while the effective torsion-induced stress tensor violates NEC, the physical shell matter can satisfy $\sigma + p \geq 0$ even at arbitrarily small radii, with

$$\sigma = \frac{2}{R} \left(1 - \frac{1}{\sqrt{B(R)}}\right) - \frac{f(R)}{R\sqrt{B(R)}}, \quad p = \frac{1}{R} \left( \frac{1 + r A'(R)/(2A(R))}{\sqrt{B(R)}} - 1 \right) + \frac{f(R)}{R\sqrt{B(R)}},$$

where $f(r)$ and $c$ are geometry parameters determined by the extra dimension (Shankar, 2015).

However, during dynamical transitions—such as between a singular black hole and a horizonless regular object—the null convergence condition (NCC),

$R_{\mu\nu}\ell^\mu\ell^\nu \geq 0,$

is generically violated during the transition phase even if it is globally restored in the final stationary endpoint. For instance, if the mass function in a time-dependent metric interpolates between $m_i(r)$ and $m_f(r)$, then

$$R_{\mu\nu}l^\mu l^\nu = \frac{2 \dot{m}(r,v)}{r^2} < 0$$

during any period with $\dot{m}(r,v) < 0$ (Borissova et al., 1 Feb 2025). This local NCC violation is required in order to defocus null congruences and prevent singularity formation as invoked in the Penrose theorem, demonstrating that the theorem’s global assumptions (on geodesic completeness and energy conditions) can be bypassed in time-dependent regularization scenarios.

4. Explicit Constructions: Shells, Geometries, and Field-Theoretic Models

The literature provides a rich set of explicit models for non-singular horizonless objects:

• Singularity-free Static Shells (THED Gravity):

These consist of a flat interior, a shell of matter obeying modified Israel junction conditions, and a THED vacuum exterior. Numerical analysis shows that when the geometry parameter $c \leq -1$, even as the shell's radius is taken to be arbitrarily small, NEC is satisfied and no event horizon forms (Shankar, 2015).

• Smooth Supergravity Microstate Solutions:

In string theory, horizonless solutions matching the charges and angular momentum of supersymmetric D1-D5-P black holes exist, with the singularity replaced by a smooth cap. These provide “microstate geometries” and admit a dual description in $(4,4)$ CFTs (Bena et al., 2016).

• Regular Black Hole-to-Wormhole Metrics:

Metrics of the form

$$F(r) = 1 - \frac{2 M \rho^2(r)}{\rho^3(r) + 2M \ell_1^2}, \quad \rho^2(r) = r^2 + \ell_2^2,$$

with tunable $\ell_1$, $\ell_2$, continuously interpolate between regular black holes, black bounces, and horizonless wormholes depending on parameter regime (Carballo-Rubio et al., 2023).

• Non-canonical Scalar Lump Objects in 2+1D:

Exact solutions for horizonless, nonsingular “lump-like” (centerless) static compact objects arise from k-field scalar Lagrangians $L(Y) = \lambda Y^\alpha$ with $\lambda < 0$ and $1/2 < \alpha < 1$, featuring a localized energy density away from both the center and the boundary (Maluf et al., 23 Apr 2024).

5. Stability, Light Rings, and Dynamical Features

Stability analysis reveals both possibilities and intrinsic constraints for horizonless objects:

• Light Ring Multiplicity and Stability:

Any ultracompact non-singular horizonless object sufficiently compact to support a light ring (closed null geodesic) necessarily supports at least one additional (inner) light ring (Filippo, 10 Apr 2024). A geometric/topological argument (involving the Brouwer degree of the effective photon potential) demonstrates that, when the outer ring possesses Kerr-like (observationally constrained) properties (radial maximum, angular minimum), an inner stable (local minimum) light ring is inevitable. This holds independently of the underlying gravitational dynamics or matter content, provided the outer ring matches observations. Stable inner light rings may trigger non-linear instabilities due to the accumulation of waves or perturbations at these orbits.

• Ergoregion Instabilities:

Ultracompact horizonless objects constructed as Kerr exteriors truncated at a surface (with, e.g., Dirichlet boundary conditions) are linearly mode-unstable when sufficiently compact to contain ergoregions, with the instability tied to the existence of zero-modes precisely at the equatorial ergosurface. If the reflecting surface is outside the ergoregion, the configurations are stable at the linear level—even if light rings are present (Zhong et al., 2022).

• Quasinormal Modes and Ringdown:

In the membrane paradigm extended to spinning horizonless objects, the quasinormal mode spectrum generically breaks isospectrality between parity sectors and displays enhancement of the deviation from black hole QNM frequencies as the object’s compactness departs from the black hole limit. For reflective ultracompact objects, increasing spin may drive some modes toward instability (Saketh et al., 14 Jun 2024).

• Very-high-frequency QPOs:

Non-singular horizonless objects in static, spherically-symmetric backgrounds with regulator scale $L \gtrsim GM$ admit an “L-induced” innermost stable circular orbit at radii $r \ll r_{\rm ISCO}$, leading to very-high-frequency quasi-periodic oscillations (VHFQPOs) with frequencies $\sim 1$–$25$ kHz for stellar-mass objects. The absence of such VHFQPOs in X-ray binary data would imply the presence of a horizon (Boos et al., 1 Oct 2025).

6. Observational Signatures and Astrophysical Implications

Non-singular horizonless objects possess potentially distinctive observational signatures:

• Lensing and Shadows:

The absence of an event horizon allows novel gravitational lensing signatures. For example, hot spots orbiting boson or Proca stars generate a "plunge-through" image resulting from photons crossing the central region and escaping, yielding characteristic flux peaks and centroid shifts not present for black holes (Rosa et al., 2022). Nonlinear electrodynamics (NED) models, when probed via the effective (photon) metric, predict larger shadow diameters and often only a single unstable photon ring in horizonless ultracompact object backgrounds, which may distinguish them from both traditional black holes and other compact objects (Walia, 20 Sep 2024).

• Ringdown and Gravitational Waves:

Membrane-based horizonless models predict QNM spectra that depart from the Kerr templates, particularly as spin increases or membrane reflectivity departs from the black hole value. This can imply that spinning horizonless compact objects may be detectable via deviations in ringdown signals (Saketh et al., 14 Jun 2024).

• Lack of VHFQPOs as a Horizon Signature:

The generic existence of an inner, high-frequency stable orbit in non-singular horizonless objects implies that the absence of VHFQPOs in X-ray binary timing may be interpreted as indirect evidence for a central horizon (Boos et al., 1 Oct 2025).

• Phenomenological Prospects:

Horizonless compact object mergers, accretion signatures, echoes in gravitational waves, and multi-wavelength imaging all represent potential venues for testing the existence and properties of these objects. The universality of certain signatures (e.g., plunge-through images or QNM deviations) across broad model classes provides robust criteria for their empirical falsification.

7. Future Directions and Open Questions

Active areas of research include:

• Dynamical formation: The realization of non-singular horizonless objects as endpoints of dynamical collapse, especially when including realistic matter and dissipative processes, remains a major unresolved issue.
• Nonstationary and realistic geometries: Many analyses are performed for stationary/idealized configurations; extension to generic time-dependent backgrounds with radiative and nonlinear effects is needed.
• Quantum Gravity and Beyond: The embedding of these regularities within ultraviolet-complete theories remains open. Lorentz-violating and string-inspired formulations are under investigation.
• Astrophysical constraints: Increasingly precise measurements of shadows, gravitational wave signals, and accretion flows are expected to constrain parameter spaces and possibly rule out specific models.

---

In sum, non-singular horizonless objects constitute a diverse and theoretically fertile class of spacetimes that offer alternatives to classical black hole endpoints, elucidate the roles of energy and convergence conditions, and suggest concrete, model-independent observational diagnostics in the strong-gravity regime. Their further paper bridges gravitational theory, astrophysics, and quantum gravity phenomenology.