In this paper, we present static and spherically symmetric vacuum solutions to the mass-dimension $d\le 4$ action of an effective-field theory, choosing the diffeomorphism symmetry to be broken explicitly. By using the reduced-action method with a Schwarzschild seed solution, we find static and spherically symmetric black-hole solutions to the field equations to linear order in the symmetry-breaking coefficients, which are consistent solutions to the modified Einstein equations at the same order. Using several Ansätze for the symmetry-breaking coefficients, we classify the allowed solutions, and we compute standard consequences and observables, including horizons, thermodynamics, photon geodesics, and perihelion precession. We find that the horizon structure of some of our solutions are similar to the Reissner-Nordström case and that several of them exhibit physical singularities at $r=2M$ as well as a nonzero angular deficit at $r\to \infty$. We note in particular that introducing more than one nonzero coefficient for spacetime-symmetry breaking leads to a solution with three horizons; the aim is to obtain observables that can be confronted to black-hole observational data.

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• Alternative gravity theories
• Gravitation