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We investigate the static and spherically symmetric solutions in a class ofthe generalized Proca theory with the nonminimal coupling to the Einsteintensor. First, we show that the solutions in the scalar-tensor theory with thenonminimal derivative coupling to the Einstein tensor can be those in thegeneralized Proca theory with the vanishing field strength. We then show thatwhen the field strength takes the nonzero value the static and sphericallysymmetric solutions can be found only for the specific value of the nonminimalcoupling constant. Second, we investigate the first-order slow-rotationcorrections to the static and spherically symmetric background. We find thatfor the background with the vanishing electric field strength the slowlyrotating solution is identical to the Kerr- (anti-) de Sitter solutions ingeneral relativity. On the other hand, for the background with the nonvanishingelectric field strength the stealth property can realized at the first order inthe slow-rotation approximation.

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