Null geodesics and shadow of a rotating black hole in extended Chern-Simons modified gravity

The Chern-Simons modification to general relativity in four dimensions consists of adding to the Einstein-Hilbert term a scalar field that couples to the first-class Pontryagin density. In this theory, which has attracted considerable attention recently, the Schwarzschild metric persists as an exact solution, and this is why this model resists several observational constraints. In contrast, the spinning black hole solution of the theory is not given by the Kerr metric but by a modification of it, so far only known for slow rotation and small coupling constant. In the present paper, we show that, in this approximation, the null geodesic equation can be integrated, and this allows us to investigate the shadow cast by a black hole. We discuss how, in addition to the angular momentum of the solution, the coupling to the Chern-Simons term deforms the shape of the shadow.

Figure 1

Plot of equatorial photon orbits with impact parameters $b=4.5$ (left panel) and $b=-4.5$ (right panel), for a rotating black hole situated at the origin of coordinates with spin parameter $a=0.2$. The different curves correspond to CS parameters $\gamma =0.3$ (dashed-dotted line), $\gamma =0.15$ (dashed line), and $\gamma =0$ (solid line). All quantities were adimensionalized with the mass of the black hole.

Figure 2

Boundary of the shadow of a black hole situated at the origin of coordinates with inclination angle ${\theta }_0=\pi /2$, and spin parameters $a=0.2$ (left panel) and $a=0.4$ (right panel). In both cases, the CS coupling parameters are $\gamma =0$ (Kerr case, solid curve), $\gamma =0.3$ (dashed-dotted curve), and $\gamma =0.4$ (dashed curve). All quantities were adimensionalized with the mass of the black hole (see text).

Figure 3

Observables $R_s$ and ${\delta }_s$ as functions of the CS coupling parameter $\gamma$, corresponding to the shadow of a black hole situated at the origin of coordinates with inclination angle ${\theta }_0=\pi /2$, and spin parameters $a=0.2$ (dashed-dotted curves) and $a=0.4$ (dashed curves). For comparison, the values corresponding to the Schwarzschild black hole are $R_s=3\sqrt{3}\approx 5.19615$ and ${\delta }_s=0$ (solid curves). All quantities were adimensionalized with the mass of the black hole (see text).