Black holes with a cosmological constant in bumblebee gravity

In this work, we present black hole solutions with a cosmological constant in bumblebee gravity, which provides a mechanism for the Lorentz symmetry violation by assuming a nonzero vacuum expectation value for the bumblebee field. From the gravitational point of view, such solutions are spherically symmetric black holes with an effective cosmological constant and are supported by an anisotropic energy-momentum tensor, conceived of as the manifestation of the bumblebee field in the spacetime geometry. Then we calculate the shadow angular radius for the proposed black hole solution with a positive effective cosmological constant. In particular, our results are the very first relation between the bumblebee field and the shadow angular size.

Figure 1

Relations between the metric parameters and the event horizon radius, $r_{+}$, for the Schwarzschild-anti–de Sitter-like case. ${\mathrm{\Lambda }}_e$ and $\ell$ stand for the effective cosmological constant and the Lorentz-violating parameter, respectively. As we can see, that parameter can decrease the event horizon radius. In this graphic, we adopted $M=1$.

Figure 2

Relations between the metric parameters and the event horizon radius, $r_{+}$, and the cosmological horizon radius, $r_c$, for the Schwarzschild-de Sitter-like case. ${\mathrm{\Lambda }}_e$ and $\ell$ stand for the effective cosmological constant and the Lorentz-violating parameter, respectively. The parameter $\ell$ may increase the event horizon and decrease the cosmological horizon. In this graphic, we adopted $M=1$, thus white regions are forbidden for that value.

Figure 3

Gravitational potential (33) and the influence of the Lorentz-violating parameter $\ell$ on it. As we can see, the parameter $\ell$ does not modify the photon sphere radius, indicated by $r_{ph}=3M$. In this graphic one adopts $M=1$ and $|{\mathrm{\Lambda }}_e|=0.01$.

Figure 4

Definition of the shadow angular radius ${\mathrm{\Theta }}_{\mathrm{stat}}$ for a static observer at rest ($r=r_o$).

Figure 5

Shadow angular radius ${\mathrm{\Theta }}_{\mathrm{stat}}$ as seen by a static observer at different locations ($r=r_o$). The dashed line indicates the photon sphere, $r_{ph}=3M$, which is the surface that causes the black hole shadow. As we can see, the larger the Lorentz-violating parameter $\ell$ is, less the shadow angular radius. If the angular radius is zero, our observer will be at the cosmological horizon. In this graphic, we adopted $M=1$ and ${\mathrm{\Lambda }}_e=0.01$.