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Figure from:Black holes with a cosmological constant in bumblebee gravity

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Figure 1: Relations between the metric parameters and the event horizon radius, r+, for the Schwarzschild-anti-de Sitter- like case. Ae and @ stand for the effective cosmological con- stant 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 1 Relations between the metric parameters and the event horizon radius, r+, for the Schwarzschild-anti-de Sitter- like case. Ae and @ stand for the effective cosmological con- stant 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.

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as modified gravitational field equations, in which G',,, is the Einstein tensor and the operator ' means derivative with respect to the potential argument. In the general case, T’ Dy and TM are the energy-momentum tensors of the bumblebee field and of the matter field, respectively. n order to solve Eq. (2), it is necessary to choose a bumblebee potential V and a metric Ansatz with some symmetry, like the spherical or the axial symmetry. By doing that, one obtains a set of equations and, solving them, a full metric. Here we focus on the spherical sym- metry and comment some potentials as options to get a full black hole metric, whether with or without a cosmo- ogical constant.

As we can see, our system shows three independent equa- tions and two unknown functions (y and p). A third would come from the matter energy-momentum tensor with a suitable equation of state. In our case, without a matter field, that function is zero. where B,, is assumed radial-like (4). With such a po- tential and the Ansatz (5), one has three independent equations from Eq. (2) because Gg = sin? 0Gg9. That is to say, our system of equations reads

The metric (18) is richer than the metric (7) that ex- cludes a cosmological constant. As will see, the bumble- bee field influence is found even on the horizons, contrary to the Schwarzschild-like solution in which the event hori- zon radius is the same of the Schwarzschild black hole. In particular, the last part of this article will show the influence of the bumblebee field on the shadow angular radius. Those influences are a consequence of the Lorentz symmetry violation, which is translated into geometry or into the general relativity language from a privileged spacetime direction or an anisotropic fluid. This is note- worthy from the total energy-momentum tensor of the metric (18) given by

The Schwarzschild-like singularity is present only at r = 0 (considering 0 < £< 1). The mentioned “fictitious” singular points are, indeed, a bad choice of the coordinate system. As we said, zeros of g’” give us not a “physical singularity”, but special surfaces, horizons, which depend on the sign of A, in the metric (18).

with a = cos 1(8M,/(1+2)Ac). As we can see in Fig. 2, the Lorentz-violation parameter, ¢, modifies the space- time structure, it decreases the cosmological horizon ra- dius and, at the same time, increases the event hori- zon radius for considerable values of Ae. The region ry <r < re is the so-called domain of outer commu- nication. In that region, observers may communicate with each other, thus there is no horizon in that region. But an observer beyond the cosmological horizon may be invisible according to someone inside the domain of outer communication. In the domain of outer commu- nication, observers can be static ones, and the Killing vector field 0/0t is timelike in that region (consequently, having both the spherical symmetry and the mentioned With € = 0 one has the well-known Schwarzschild-de Sitter condition to generate two horizons.2 And for 9M?(1+£)A. = 1, our metric shows an extreme case, in which ry = r,. Analytic expressions for the event and the cosmological horizons are, respectively,

Figure 2: Relations between the metric parameters and the event horizon radius, r+, and the cosmological horizon radius, Te, for the Schwarzschild-de Sitter-like case. Ae and ¢ stand for the effective cosmological constant and the Lorentz-violating parameter, respectively. The parameter ¢ 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.

in which the gravitational potential V(r) and the energy E are defined as The condition of unstable orbits (dV(r)/dr = 0 and d?V(r)/dr? < 0) that compose the black hole shadow give us the following radius: rp, = 3M. This is the photon sphere radius, and such a sphere is the surface

Figure 3: Gravitational potential (33) and the influence of the Lorentz-violating parameter ¢ on it. As we can see, the parameter £ does not modify the photon sphere radius, indi- cated by rp, = 3M. In this graphic one adopts M = 1 and |Ac|=0.01.

Figure 4: Definition of the shadow angular radius Ogtaz for a static observer at rest (r = To).

Figure 5: Shadow angular radius Ogtat as seen by a static observer at different locations (r = ro). The dashed line in- dicates the photon sphere, rp, = 3M, which is the surface that causes the black hole shadow. As we can see, the larger the Lorentz-violating parameter / 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 A. = 0.01.

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