Light rings around five dimensional stationary black holes and naked singularities

The existence of light rings in a spacetime is closely related to the existence of black hole horizons and observables such as the ringdown and the shadow. Black holes, compared to nonvacuum ultracompact objects, have rather unique environments. To this aim, recently [P. V. P. Cunha and C. A. R. Herdeiro, Phys. Rev. Lett. 124, 181101 (2020)] topological arguments, independent of the underlying gravity theory, were developed to prove the existence of unstable light rings outside the Killing horizon of four dimensional asymptotically flat, stationary, axisymmetric, nonextremal black holes. Here we extend these arguments to five-dimensional stationary black holes. Generically in five dimensions, there are two possible conserved angular momenta, hence the four-dimensional discussion does not extend verbatim to five dimensions; nevertheless, we prove that there is a light ring for each rotation sense for a stationary black hole. We give the static and the Myers-Perry rotating black holes as examples. We also show that when the horizon of the black hole disappears and the singularity becomes naked, only one of the light rings survives; a similar phenomenon also occurs in four dimensions which might allow testing the cosmic censorship hypothesis.

Figure 1

The representation of the results found in the contour analysis. The vector field, $\overrightarrow{v}$, obtained by using the effective potential of the positive rotation sense can be seen along the contour. The full negative winding is apparent.

Figure 2

$\overrightarrow{v}=(\frac{1}{r^4\sin \theta \cos \theta }(2-r^2),-\frac{4\sqrt{r^2-1}}{r^3}\left(\frac{\cos 2\theta }{{\sin }^{2}2\theta }\right))$ obtained by using the effective potential function associated with the positive rotation sense is plotted in neighborhood of the standard light ring ($r=\sqrt{2}$, $\theta =\pi /4$) for the static spacetime with two equal angular momenta.

Figure 3

$\overrightarrow{v}=(\frac{-1}{r^4\sin \theta \cos \theta }(2-r^2),\frac{4\sqrt{r^2-1}}{r^3}\left(\frac{\cos 2\theta }{{\sin }^{2}2\theta }\right))$ obtained by using the effective potential function associated with the negative rotation sense is plotted in neighborhood of the standard light ring ($r=\sqrt{2}$, $\theta =\pi /4$) for the static spacetime with two equal angular momenta.

Figure 4

Black hole case: The vector field, $\overrightarrow{v}$, obtained by using the impact parameter $b_{+}$ as given in (75) and (76) can be seen in the neighborhood of the standard light ring for the Myers-Perry black hole with two distinct angular momenta. For this plot, we assumed that the mass term is $\mu =1$, the rotation parameters are $a=0.1$ and $b=0.4$ and the angular momenta of the photon are ${\mathrm{\Phi }}_1=1.1$ and ${\mathrm{\Phi }}_2=1.5$. The horizon is located at $r_H=0.83$.

Figure 5

Black hole case: The vector field, $\overrightarrow{v}$, obtained by using the impact parameter $b_{-}$ as given in (75) and (76) can be seen in the neighborhood of the standard light ring for the Myers-Perry black hole with two distinct angular momenta. We took $\mu =1$, $a=0.1$ and $b=0.4$ and the angular momenta of the photon are ${\mathrm{\Phi }}_1=1.1$ and ${\mathrm{\Phi }}_2=1.5$. The horizon is located at $r_H=0.83$.

Figure 6

No horizon case: The vector field, $\overrightarrow{v}$, obtained by using the impact parameter $b_{+}$ as given in (75) and (76) can be seen in the neighborhood of the single light ring for the naked singularity. So this light ring survives. For this plot, we assumed that the mass term is $\mu =1$, the rotation parameters are $a=0.3$ and $b=0.8$ and the angular momenta of the photon are ${\mathrm{\Phi }}_1=1.1$ and ${\mathrm{\Phi }}_2=1.5$.

Figure 7

No horizon case: The vector field, $\overrightarrow{v}$, obtained by using the impact parameter $b_{-}$ as given in (75) and (76) has no winding as can be seen. The light ring with $b_{-}$ disappears for the naked singularity case. We took $\mu =1$, $a=0.3$ and $b=0.8$ and the angular momenta of the photon are ${\mathrm{\Phi }}_1=1.1$ and ${\mathrm{\Phi }}_2=1.5$.