Spontaneous Lorentz violation and asymptotic flatness

The Standard Model Extension (SME) is a generic parametrization for Lorentz violation and the phenomenological consequences of the minimal gravity sector of the SME are usually studied using a post-Newtonian expansion that requires spacetime to be asymptotically flat. However, there is a term in this sector for which these approximations are unable to make predictions; this is known as the $t$ puzzle. The present paper studies a model of spontaneous Lorentz violation in the minimal gravity sector of the SME in a static and spherically symmetric situation, when no additional matter fields are present. It is shown that, under the above mentioned assumptions, $t$ is the only term in the minimal gravity sector for which no asymptotically flat solutions exist. This stems from the fact that the $t$ term fixes the asymptotic behavior of all the pieces of the curvature tensor.