Locally Naked Null Singularity

• Locally naked null singularity is a lightlike curvature singularity on a null hypersurface from which outgoing null geodesics emerge without the shielding of an event horizon.
• The models employ precise methodologies using Bondi–Sachs and double-null coordinates and have been validated in Vaidya, scalar field, and axially symmetric collapse scenarios.
• Distinct observable signatures include divergent lensing and shadow profiles, while infinite redshift ensures that any transmitted energy is effectively undetectable, upholding cosmic censorship.

A locally naked null singularity is a curvature singularity on a null hypersurface from which outgoing null geodesics emerge and reach external observers. Unlike spacelike or timelike singularities, the null property means that the singular locus itself is lightlike in the causal structure. Local nakedness denotes that there exists a family of outgoing null generators whose past endpoint is the singular locus, and that the singularity is not hidden behind a trapped surface or event horizon within the local neighborhood. These objects arise in gravitational collapse and critical phenomena, generalizing cosmic censorship violations from the classical context to regimes where the singularity is “bare” only within some region. Their definition, local causal structure, observable characteristics, energetic and mathematical properties, and their place in scalar field, dust, and axially symmetric collapse models have been established rigorously across diverse scenarios.

1. Geometric and Causal Structure

Locally naked null singularities have been constructed in spherically symmetric, axially symmetric, and rotating spacetimes. The general metric form can be written in Bondi–Sachs or double-null coordinates, reflecting the outgoing null character. In the Bondi–Sachs scheme, the singular surface is defined by

$$\frac{dr}{du}\bigg|_{r=r_*}=0 \quad\Longrightarrow\quad \frac{V}{r_*}e^{2\beta}-r_*^2 h_{AB} U^A U^B e^{-2\beta}=0,$$

where $r=r_*$ is the null hypersurface supporting the singularity (Deshingkar, 2010). In double-null coordinates $(u,v)$ for the exterior region,

$$ds^2 = -\Omega^2(u,v) du dv + r^2(u,v) d\sigma_{S^2}^2,$$

the singular locus is typically at $(u=0, v=0)$, with outgoing null geodesics satisfying $u=\text{const}$ (Cicortas et al., 2024).

Outgoing null generators escape from the singular locus and reach future null infinity ($\mathscr{I}^+$) in finite affine parameter. This has been verified for incoming Vaidya models, generalized Vaidya stiff fluid collapse, and more, with explicit ODE criteria and global causal analysis (Wheeler, 2022, Vertogradov, 2022). Local visibility is guaranteed by the existence of a nonempty family of future-directed outgoing null geodesics $\gamma(v)=(v,r(v))$ satisfying $\lim_{v\to 0^+} r(v)=0$.

Distinct signatures include: the absence of horizons (no real root for $f=0$ in the lapse function), removal of closed trapped surfaces, the presence of diverging curvature invariants ($K \rightarrow \infty$ as $r \rightarrow 0$), and a Penrose diagram where the null singular line is locally accessible to outgoing null rays (Paul, 2020, Patel et al., 2022).

2. Physical Properties and Energy Transmission

Physical consequences hinge on the infinite redshift mechanism. In completely general spacetimes (no symmetry, arbitrary matter satisfying the weak energy condition), any energy or information carried along outgoing null geodesics emitting from the singularity is subject to infinite redshift (Deshingkar, 2010): $$1+z = \frac{\left[K_a u^a_{\text{(o)}}\right]_{\text{obs}}}{\left[K_a u^a_{\text{(s)}}\right]_{\text{src}}} \simeq \frac{X_{\text{obs}}}{0} \to \infty$$ Outgoing luminosity thus vanishes,

$$I_p = \frac{P_0}{A_0 (1+z)^2} \implies I_p \to 0\quad (z\to\infty)$$

This establishes a “perfectly absorbing boundary” effect: while locally naked null singularities are geometrically visible, they are physically sterile—no signal, energy, or causal influence can reach external observers. From the viewpoint of cosmic censorship, the existence of locally naked null singularities does not imply a breakdown of predictability or observable pathology (Deshingkar, 2010).

In certain collapse scenarios (e.g., Vaidya null dust), energy can be physically transported out along the first outgoing singular null geodesic—the Cauchy horizon—when spacetime is matched across two regions with different mass accumulation rates. The Cauchy horizon becomes a null boundary layer carrying positive surface energy (Jhingan et al., 2010): $\mu = \frac{[M]}{4\pi r^2}$ This transmitted energy is computable and, in principle, observable, furnishing an energetic signature departing from the generic “trapped energy” case.

3. Formation Criteria and Model Dependence

Formation mechanisms have been rigorously established for Einstein–scalar field systems, Vaidya dust models, generalized fluids, and axially symmetric null dust. For incoming Vaidya models, the local nakedness criterion is governed by the mass function growth rate (Wheeler, 2022):

• If $m'_+(0) \le 1/16$, $(v=0, r=0)$ is locally naked;
• If $m'_+(0) > 1/16$, no outgoing null rays reach $(0,0)$.

In Einstein-scalar field collapse, precise self-similarity (Christodoulou’s solutions) or approximate self-similar fine-tuning enables naked null singularity formation. The essential condition resides in the activation (or not) of blueshift instabilities and the behavior of the scale-invariant quantity $\Psi=\lambda^{-1}\partial_v \phi$ along the singular cone. Non-generic data perturbations with high-order vanishing are required to suppress the blueshift and maintain null local nakedness (Singh, 2022).

Discrete self-similar and continuous self-similar solutions have been constructed, including discretely periodic radiation fields on the singular cone, leading to bounded but infinitely oscillatory scalar profile and mass aspect (Cicortas et al., 2024). The genericity of slow mass accumulation leading to naked singularities holds under open conditions in the $C^1$ topology for the mass profile (Wheeler, 2022).

4. Observational Signatures and Gravitational Lensing

Locally naked null singularities produce distinct observable phenomena, especially in lensing and shadow formation. In static spherically symmetric cases, the absence of a photon sphere fundamentally alters the lensing pattern (Paul, 2020): $V_{\text{eff}}(r) = \frac{L^2}{(r+M)^2}$ No photon sphere exists, and the bending angle diverges polynomially ($\propto (b/b_{\rm cr}-1)^{-3/2}$), not logarithmically. This divergence produces relativistic Einstein rings that are widely separated, as opposed to the exponentially spaced rings seen for black holes. Tangential magnification diverges at each ring, but radial magnification remains finite.

For rotating null naked singularities, the shadow cast is arc-shaped rather than closed (contour-shaped) as in Kerr black holes. Only the retrograde photon orbit admits a stable photon sphere, and outgoing null rays escape freely from arbitrarily small radii (Patel et al., 2022). Such lensing features and shadow profiles, potentially resolvable by next-generation VLBI, provide concrete observational “fingerprints” distinguishing null local nakedness from black hole or spacelike/timelike singularity scenarios.

5. Curvature Structure and Tipler/Królak Strength

Curvature invariants (e.g., Kretschmann scalar) universally diverge as $r\to 0$ for all locally naked null singularity models: $$K = R_{abcd} R^{abcd} \sim \frac{[C(v)-X/r]^2}{r^6} \qquad \text{or} \qquad K \sim r^{-4}$$ In Vaidya collapse, K diverges along all outgoing null generators except at most one exceptional case; thus the singularity is generically Tipler-strong, unavoidably causing divergent tidal forces along incomplete null geodesics (Wheeler, 2022).

In axially symmetric dust collapse, the singularity is gravitationally weak (Tipler/Królak conditions not satisfied), but Petrov type II structure with nonzero Weyl scalars ($\Psi_2$, $\Psi_4$) governs local tidal fields (Ahmed, 2017). In rotating black hole contexts, the parallel-propagated (p.p.) curvature may diverge even when scalar invariants remain finite—a "truly naked rotating black hole"—manifesting as locally naked null singularity on the Killing horizon (Ovcharenko et al., 23 Jan 2025). The Petrov type may change from II/D off-horizon to III/N in-falling observer frames, marking local non-scalar singularity.

6. Stability, Regularity, and Open Research Problems

Null naked singularity solutions with continuous or discrete self-similarity are structurally unstable under generic perturbations (e.g., Christodoulou instability); generic initial data activate blueshift instability, leading to trapped surfaces and black hole formation (Singh, 2022, Cicortas et al., 2024). Fine-tuned data profiles vanishing to high order, or specific discrete self-similar seed functions, are required for persistent local nakedness. The challenge of constructing smooth “interior fill-in” regions for discretely self-similar naked singularities remains open, with active research focused on gluing interior regions that transition from small-mass oscillatory behavior to large-mass accumulation (Cicortas et al., 2024).

Regularity of all fields up to, but not through, the singular cone is typical for constructed solutions. In axially symmetric collapse, the singularity is weak and extension through it may be possible; in scalar field and Vaidya models, the singularity is generic and robust under small $C^1$ perturbations, provided the mass accumulation rate does not exceed critical bounds (Wheeler, 2022).

7. Cosmic Censorship and Predictability

The principal implication is that locally naked null singularities, though geometrically visible, are “physically censored” by infinite redshift, trapping all energy and rendering them harmless with respect to breakdowns of predictability (Deshingkar, 2010). This supports the cosmic censorship hypothesis in its physical content: even if such null singularities form, their global effects remain undetectable. Exceptions occur for energy-carrying outgoing null layers (Cauchy horizons matched between collapse regions), providing tangible fluxes observable in principle (Jhingan et al., 2010).

For rotating and axially symmetric models, non-scalar parallel-propagated curvature singularities on null surfaces imply extensions to cosmic censorship that must account for frame-dependent tidal pathologies and Petrov-type transitions (Ovcharenko et al., 23 Jan 2025). These phenomena highlight the necessity for refined conjectures in the presence of rotation and higher-order gravity corrections.

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In summary, locally naked null singularities are a rigorously-defined class of causal curvature singularities that appear in a wide variety of gravitational collapse, critical phenomena, and rotating spacetimes. Their geometric visibility does not translate into physical effect for generic observers due to infinite redshift, with scalar or non-scalar curvature divergence arising depending on the model. Their distinctive lensing, tidal, and shadow profiles, as well as their energy transmission properties in specific matched-collapse scenarios, provide the observational and theoretical diagnostics for their existence and mathematical nature. Their standing within cosmic censorship remains as physically innocuous entities, but their formation criteria, energetic signatures, and observer-dependent tidal phenomena continue to drive research into the fundamental properties of gravitational collapse and the ultimate limits of causal predictability.