Bumblebee gravity: spherically-symmetric solutions away from the potential minimum

TL;DR

The paper analyzes the bumblebee gravity model, a vector field-driven spontaneous Lorentz-violating extension of general relativity, focusing on static spherically symmetric solutions with the vector away from its potential minimum. It develops flat-spacetime and fully covariant formulations, derives and solves the coupled gravity-vector equations in EF coordinates, and uncovers a spectrum of spacetimes including RN-like and AdS-like solutions, as well as horizons, naked singularities, and strong-field features. Stability considerations are tied to the choice of vector potential, with quadratic and hypergeometric potentials showing distinct asymptotic behavior and bounded or unbounded Hamiltonians in different regimes; hypergeometric forms can yield bounded energy for fixed vector magnitude but not universally. The work further discusses observational implications, proposing orbital dynamics as a route to constrain model parameters and outlining future extensions to more general configurations and nonzero nonminimal couplings.

Abstract

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 for the case of fixed spatial vector magnitude. 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 Schwarszchild-Anti de Sitter and Reissner-Nordstrom 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.

Figures (11)

• Figure 1: The potential portion of the Hamiltonian density ${\cal H}_V$ plotted for $5$ different potential functions, versus $f=B_0/\Lambda$. The black curve is the Proca potential (massive vector), the dashed and dotted curves are the quadratic potential and an $X^8$ potential as polynomial samples. Finally the blue and green curves are the hypergeometric potentials for $n=3$ and $n=5$ respectively.
• Figure 2: Plot of a series of numerical solutions for the quadratic potential $V=\frac{\lambda}{2} X^2$ with respect to $u=\frac{1}{x}$ with each solution showing dampening oscillations at $\pm1$ as $u$ approaches 1, or $x$ approaches infinity. We set $\lambda=1$ for this plot. The colors represent distinct initial values for $f$ and $df/du$ at the point $u_0=1$.
• Figure 3: Plot of a series of numerical solutions for the hypergeometric potential $n=3$, with each solution showing dampening oscillations around $\pm\sqrt{3}$. We set $g=1$ for this plot. The colors represent distinct initial values for $f$ and $df/du$ at the point $u_0=1$.
• Figure 4: Plot of a series of numerical solutions for the hypergeometric potential when $n=4$, with some solutions showing dampening oscillations around $\pm\sqrt{2}$, but solutions that move a distance of $\sqrt{6}$ from the origin escape completely as $u$ approaches $0$, or as $x$ approaches infinity. We set $g=1$ for this plot. The colors represent distinct initial values for $f$ and $df/du$ at the point $u_0=1$.
• Figure 5: Flowchart showing the possible conditions and the resulting solutions of Eq. (\ref{['firstorder']}), all under the assumption that $(AB)'=0$.