In this work, we study a vector model of spontaneous spacetime symmetry breaking coupled to gravity: the bumblebee model. The primary focus is on static spherically symmetric solutions. Complementing previous work on black hole solutions, we study the effects on the solutions when the vector field does not lie at the minimum of its potential. We first investigate the flat spacetime limit, which can be viewed as a modified electrostatic model with a nonlinear interaction term. We study the stability of classical solutions generally and in the spherically symmetric case. We also find that certain potentials, based on hypergeometric functions, yield a Hamiltonian bounded from below. With gravity, we solve for the spherically symmetric metric and vector field, for a variety of choices of the potential energy functions, including ones beyond the quadratic potential like the hypergeometric potentials. Special case exact solutions are obtained showing Schwarzschild-anti–de Sitter and Reissner-Nordström spacetimes. We employ horizon and asymptotic analytical expansions along with numerical solutions to explore the general case with the vector field away from the potential minimum. We discover interesting features of these solutions including naked singularities, repulsive gravity, and rapidly varying gravitational field near the source. Finally, we discuss observational constraints on these spacetimes using the resulting orbital behavior.

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• Alternative gravity theories
• Classical black holes
• Classical solutions in field theory
• Dynamical symmetry breaking models
• Spontaneous symmetry breaking
• Symmetries
• Continuous symmetries
• Lorentz symmetry
• Spacetime symmetries