Shadows of spherically symmetric black holes and naked singularities (1802.08060v2)

Abstract: We compare shadows cast by Schwarzschild black holes with those produced by two classes of naked singularities that result from gravitational collapse of spherically symmetric matter. The latter models consist of an interior naked singularity spacetime restricted to radii $r\leq R_b$, matched to Schwarzschild spacetime outside the boundary radius $R_b$. While a black hole always has a photon sphere and always casts a shadow, we find that the naked singularity models have photon spheres only if a certain parameter $M_0$ that characterizes these models satisfies $M_0\geq 2/3$, or equivalently, if $R_b\leq 3M$, where $M$ is the total mass of the object. Such models do produce shadows. However, models with $M_0<2/3$ (or $R_b>3M$) have no photon sphere and do not produce a shadow. Instead, they produce an interesting `full-moon' image. These results imply that the presence of a shadow does not by itself prove that a compact object is necessarily a black hole. The object could be a naked singularity with $M_0\geq 2/3$, and we will need other observational clues to distinguish the two possibilities. On the other hand, the presence of a full-moon image would certainly rule out a black hole and might suggest a naked singularity with $M_0<2/3$. It would be worthwhile to generalize the present study, which is restricted to spherically symmetric models, to rotating black holes and naked singularities.

• The paper presents a comparative analysis of shadow formation in Schwarzschild black holes and naked singularities using detailed mathematical modeling.
• It demonstrates that the presence of a photon sphere is key, with JMN-1 singularities casting shadows only under specific conditions while JMN-2 singularities do not.
• The study emphasizes gravitational lensing effects that challenge the cosmic censorship conjecture and pave the way for future research on rotating singularity models.

Analysis of Shadows and Images in Black Holes and Naked Singularities

The paper "Shadows and Images of Naked Singularities" by R. Shaikh et al. explores the comparative analysis of shadows cast by Schwarzschild black holes and by naked singularities resulting from the gravitational collapse of spherically symmetric matter. This research identifies critical aspects of understanding the observational differences between black holes and naked singularities, highlighting significant theoretical implications for general relativity and the cosmic censorship conjecture.

The authors present detailed mathematical models to examine two classes of naked singularities, referred to as JMN-1 and JMN-2, characterized by their respective parameters $M_0$ and $\lambda$. The analysis considers these singularities under various conditions, particularly focusing on the presence or absence of a photon sphere, which critically determines whether these entities cast a shadow similar to that of a black hole.

Photon Spheres and Shadows

The shadow of a compact object is primarily governed by the existence of a photon sphere. In Schwarzschild black holes, a photon sphere is present, leading to the formation of a characteristic shadow in the observer's sky. However, the paper reveals that not all naked singularities possess a photon sphere. For instance, JMN-1 singularities have a photon sphere only under the condition $M_0 \geq 2/3$ (or $R_b \leq 3M$), resulting in a shadow. Conversely, when this condition is not met, the JMN-1 singularity does not produce a shadow, creating what is termed a "full-moon" image. In the case of JMN-2 singularities, no photon sphere is formed across the parameter space, hence they do not cast a shadow.

Gravitational Lensing and Relativistic Images

The research extends beyond the mere detection of shadows to include gravitational lensing and relativistic image formation. The trajectory of light and its bending around these compact objects offer additional observational signals to distinguish between black holes and naked singularities. For photons traversing near the object, the deflection angle provides critical insights, particularly when analyzing images such as relativistic Einstein rings in the observer's field.

Practical and Theoretical Implications

From an astrophysical standpoint, the results underscore the intricacies involved in interpreting potential black hole images captured by instruments like the Event Horizon Telescope (EHT). While the detection of a shadow suggests the existence of a photon sphere, it does not conclusively indicate the presence of an event horizon. Hence, other observational clues are necessary to differentiate between a black hole and a naked singularity possessing a photon sphere.

The research challenges the cosmic censorship conjecture by illustrating scenarios where gravitational collapse leads to naked singularities instead of black holes. Such findings call for deeper exploration into the formulation of the conjecture, considering both theoretical and cosmological aspects.

Future Directions

While the paper meticulously explores spherically symmetric models of naked singularities, expanding this research to rotating naked singularities would provide a more comprehensive understanding. The generalization to rotating solutions akin to Kerr black holes presents a promising avenue for future investigation, potentially unearthing new forms of singularities within the general relativity framework.

This paper's analysis provides crucial insights into the complex nature of black holes and naked singularities, marking significant progress in gravitation physics. Understanding the conditions under which these entities cast shadows and unraveling their observational signatures will continue to be pivotal for future advancements in the domain.