Generalized JMN Naked Singularity Models

Abstract

We construct a generalized class of Joshi-Malafarina-Narayan (JMN) naked singularity spacetimes that arise as equilibrium end states of gravitational collapse with non-vanishing tangential pressure. The generalization introduces density inhomogeneity through a radially dependent mass function $F(r)=(M_0+M_n r^n)r^3$, leading to a two-parameter family of solutions matched smoothly to an exterior Schwarzschild spacetime. The observational properties of the spacetime are then examined through shadow formation and thin accretion disk emission. We find that when the photon sphere lies in the exterior Schwarzschild region, the shadow is identical to that of a Schwarzschild black hole. Accretion disk spectra show enhanced high-frequency emission compared to Schwarzschild, while deviations from the original JMN model remain small due to strong constraints on the inhomogeneity parameter. These results indicate that the generalized model effectively serves as a small perturbation of the JMN spacetime, demonstrating the robustness of JMN-type naked-singularity geometries.

Figures (5)

• Figure 1: Density profiles before and after collapse for the generalized JMN (GJMN) spacetime. In all cases, the matching radius $R_b$ is kept the same as in the corresponding JMN configuration to enable a direct comparison. The additional parameter $M_n$ introduces radial inhomogeneity in the density profile, leading to equilibrium configurations that differ from the homogeneous-density JMN toy model. We have fixed $n=2$ and $\mathbf{M}=1$
• Figure 2: Allowed parameter space in the $(M_0,M_n)$ plane for two physically relevant ranges of the matching radius $R_b$. The left panel corresponds to $2 \le R_b < 3$, where the parameters form a continuous region. The right panel corresponds to $R_b \ge 6$, where the allowed parameter space collapses into thin discrete bands due to the rapid shrinking of the admissible range of $M_n$ as $R_b$ increases. The plots are generated for $n=2$ and $\mathbf{M}=1$.
• Figure 3: The effective potential for JMN1($M_0=0.6794$) and GJMN($n=2,\, M_0=0.7,\, \&\, M_2=-0.01$) spacetime inside the boundary $R_b=2.9435$, which smoothly matches with the effective potential for the Schwarzschild spacetime($\mathbf{M}=1$).
• Figure 4: Intensity distribution and the shadow cast by GJMN spacetime. We have fixed $n=2$ and $\mathbf{M}=1$
• Figure 5: Accretion disk spectral luminosity distribution for the Schwarzschild, JMN, and generalized JMN (GJMN) spacetimes. The left panel shows the full spectrum, while the right panel zooms into the high–frequency region. The plots are generated for $n=2$, $\mathbf{M}=1$, $M_n=-0.005$, $M_0=0.3498$ and $R_b=6\mathbf{M}$. The same value of $R_b$ is used for both the JMN and GJMN models to allow a direct comparison.