Exact Black Hole Solutions and Their Thermodynamics in Bumblebee Gravity with Lightlike or Spacelike VEVs

Motivated by recent developments in Lorentz-violating theories of gravity, we obtain new black hole solutions within the framework of Bumblebee gravity, where the Bumblebee vector field possesses two independent nonzero components and acquires either a lightlike or spacelike vacuum expectation value. Within this framework, we derive new Schwarzschild-like and Schwarzschild–(A)dS-like black hole solutions. By further incorporating a nonminimally coupled electromagnetic field, we generalize these to new charged black hole solutions. These solutions extend previous results by including additional Lorentz-violating parameters. A key finding is that even for lightlike vacuum expectation values, the black hole solutions exhibit distinct corrections from Lorentz violation. Furthermore, we present a preliminary analysis of their thermodynamic properties. Similar to previous studies that reported a discrepancy between the black hole entropy and the Wald entropy in Bumblebee gravity with spacelike vacuum expectation values, our solutions in the spacelike case exhibit the same behavior. In contrast, for the lightlike case considered here, the two entropies coincide.