Solutions in the generalized Proca theory with the nonminimal coupling to the Einstein tensor

We investigate the static and spherically symmetric solutions in a class of the generalized Proca theory with the nonminimal coupling to the Einstein tensor. First, we show that the solutions in the scalar-tensor theory with the nonminimal derivative coupling to the Einstein tensor can be those in the generalized Proca theory with the vanishing field strength. We then show that when the field strength takes the nonzero value, the static and spherically symmetric solutions can be found only for the specific value of the nonminimal coupling constant. Second, we investigate the first-order slow-rotation corrections to the static and spherically symmetric background. We find that for the background with the vanishing electric field strength the slowly rotating solution is identical to the Kerr--(anti-)de Sitter solutions in general relativity. On the other hand, for the background with the nonvanishing electric field strength the stealth property can be realized at the first order in the slow-rotation approximation.