Shadow of a naked singularity

We analyze the redshift suffered by photons originating from an external source, traversing a collapsing dust cloud, and finally being received by an asymptotic observer. In addition, we study the shadow that the collapsing cloud casts on the sky of the asymptotic observer. We find that the resulting redshift and properties of the shadow depend crucially on whether the final outcome of the complete gravitational collapse is a black hole or a naked singularity. In the black hole case, the shadow is due to the high redshift acquired by the photons as they approach the event horizon, implying that their energy is gradually redshifted toward zero within a few crossing times associated with the event horizon radius. In contrast to this, a naked singularity not only absorbs photons originating from the source, but it also emits infinitely redshifted photons with and without angular momenta. This emission introduces an abrupt cutoff in the frequency shift of the photons detected in directions close to the radial one, and it is responsible for the shadow masking the source in the naked singularity case. Furthermore, even though the shadow forms and begins to grow immediately after the observer crosses the Cauchy horizon, it takes many more crossing times than in the black hole case for the source to be occulted from the observer’s eyes. We discuss possible implications of our results for testing the weak cosmic censorship hypothesis. Even though at late times the image of the source perceived by the observer looks the same in both cases, the dynamical formation of the shadow and the redshift images has distinct features and time scales in the black hole versus the naked singularity case. For stellar collapse, these time scales seem to be too short to be resolved with existing technology. However, our results may be relevant for the collapse of seeds leading to supermassive black holes.

Figure 1

The phase portrait for the autonomous dynamical system defined in Eq. (29). Note that we use the variable $1/\mathrm{\Pi }$ instead of $\mathrm{\Pi }$ in order to have a better view of the phase space in the region $0<\mathrm{\Pi }\le 1$. The arrows show the direction of the vector field, the blue solid lines are the stable and unstable local manifolds associated with the critical point $({\chi }_2,{\mathrm{\Pi }}_2)=(\sqrt{2/3},\sqrt{2/3})$, and the red dashed line is an integral curve corresponding to a radial light ray. As the figure suggests, the basin of attraction for the critical point $({\chi }_1,{\mathrm{\Pi }}_1)=(1,1)$ is the region that lies below the stable manifold.

Figure 2

Conformal diagram for the spherical collapse of a dust cloud forming a naked singularity. We illustrate the world line of the distant observer and the spacetime trajectories of two light rays ${\gamma }_0$ and $\gamma$. The particular radial light ray ${\gamma }_0$ is the one originating from the distant source which penetrates the cloud at the moment of time symmetry and reaches the observer at time ${\tau }_{\text{obs}}=0$. $\gamma$ is an arbitrary trajectory of a null geodesic with angular momentum.

Figure 3

The critical angle $\widehat{\alpha }$ (thick solid lines) as well as $\overline{\alpha }$ (thick dashed lines) as functions of $2m_1/r_1^{+}$. The thin solid lines correspond to slices of constant proper time of the observer. Left panel: Black hole case with initial compactness ratio $\mathcal{C}={\mathcal{C}}_0$. Central panel: Naked singularity with $\mathcal{C}={\mathcal{C}}_0/2$. Right panel: Naked singularity with $\mathcal{C}={\mathcal{C}}_0/4$. $r_{\mathrm{Ch}}$ denotes the areal radius of the surface of the star at the moment it crosses the Cauchy horizon. Note that in the black hole case $\widehat{\alpha }=\pi$ since the asymptotic observer always lies to the past of the Cauchy horizon in this case.

Figure 4

Self-convergence test for the numerical solution of the dynamical system in Eq. (28) for a null geodesic traced back from the event $x$ at the surface of the cloud, passing very close to the singularity, and intersecting again the surface of the cloud in the past. The relative numerical error decreases (left panel) by a factor of $\sim 16$ (right panel) when the step size decreases by a factor of 2.

Figure 5

Total redshift measured by the asymptotic observer at certain proper times as a function of the normalized viewing angle $\zeta /\overline{\zeta }$ of the collapsing object. Left panel: Black hole case with initial compactness ratio $\mathcal{C}={\mathcal{C}}_0$, for which ${\tau }_{3m_1}/2m_1\approx 10.8$. Central panel: Naked singularity with $\mathcal{C}={\mathcal{C}}_0/2$. In this case ${\tau }_{\mathrm{Ch}}/2m_1\approx 18.4$ and ${\tau }_{3m_1}/2m_1\approx 41.1$. Right panel: Naked singularity with $\mathcal{C}={\mathcal{C}}_0/4$, for which ${\tau }_{\mathrm{Ch}}/2m_1\approx 47.5$ and ${\tau }_{3m_1}/2m_1\approx 121.9$.

Figure 6

Redshift snapshots at different proper times of the asymptotic observer as the dust star of initial compactness ratio ${\mathcal{C}}_0$ collapses to a black hole. Here, ${\tau }_{3m_1}/2m_1\approx 10.8$.

Figure 7

Redshift snapshots at different proper times of the asymptotic observer as the dust star of initial compactness ratio ${\mathcal{C}}_0/2$ collapses to a naked singularity. In this case, ${\tau }_{\mathrm{Ch}}/2m_1\approx 18.4$, and ${\tau }_{3m_1}/2m_1\approx 41.1$. The shadow of the naked singularity appears at ${\tau }_{\text{obs}}={\tau }_{\mathrm{Ch}}$, and then it covers the whole region at ${\tau }_{\text{obs}}={\tau }_{3m_1}$.

Figure 8

Redshift snapshots at different proper times of the asymptotic observer as the dust star of initial compactness ratio ${\mathcal{C}}_0/4$ collapses to a naked singularity. Here, ${\tau }_{\mathrm{Ch}}/2m_1\approx 47.5$, and ${\tau }_{3m_1}/2m_1\approx 121.9$. The same qualitative features as in Fig. 7 are present.

Figure 9

Normalized angular extension of the shadow of the black hole (dotted line) compared to the shadow of two naked singularities (dashed lines), as functions of the observer’s proper time. In the black hole case, the shadow becomes apparent around ${\tau }_{\text{obs}}={\tau }_{3m_1}$ ($r_1^{+}=3m_1$), and then it grows continuously and takes an infinite time to cover the object, leaving a thin bright ring around it. In the naked singularity cases, the shadow appears in ${\tau }_{\text{obs}}={\tau }_{\mathrm{Ch}}$, and then it grows continuously and covers the whole region under consideration in finite time ${\tau }_{\text{obs}}={\tau }_{3m_1}$.

Figure 10

A plot of the effective potential $W(x)$ for the parameter value $\lambda =0.6$.

Figure 11

The critical angle of incidence $\widehat{\alpha }$ versus the compactness ratio $\mathcal{C}≔2m/r$ along the world line of a free-falling observer at constant $R$. The interval shown corresponds to the interval $J_1<x<J_{AH}$. As in the previous figure, the parameter $\lambda$ is set to 0.6. We note that the behavior of $\widehat{\alpha }$ in the self-similar case resembles the behavior of $\widehat{\alpha }$ in the bound collapse case; see the central and right panels of Fig. 3.