Lorentz violation in the gravity sector: The t puzzle

Lorentz violation is a candidate quantum-gravity signal, and the Standard-Model Extension (SME) is a widely used parametrization of such a violation. In the gravitational SME sector, there is an elusive coefficient for which no effects have been found. This is known as the $t$ puzzle and, to date, it has no compelling explanation. This paper analyzes whether there is a fundamental explanation for the $t$ puzzle. To tackle this question, several approaches are followed. Mainly, redefinitions of the dynamical fields are studied, showing that other SME coefficients can be moved to nongravitational sectors. It is also found that the gravity SME sector can be consistently treated \textit{\`a la} Palatini, and that, in the presence of spacetime boundaries, it is possible to correct its action to get the desired equations of motion. Moreover, through a reformulation as a Lanczos-type tensor, some problematic features of the $t$ term, which should arise at the phenomenological level, are revealed. The most important conclusion of the paper is that there is no evidence of a fundamental explanation for the $t$ puzzle, suggesting that it may be linked to the approximations taken at the phenomenological level.