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We investigate the quasinormal modes of a test massless, minimally coupledscalar field on a static and spherically symmetric black hole in thescalar-tensor theory with field derivative coupling to the Einstein tensor,which is a part of the Horndeski theory with the shift symmetry. In oursolution, the spacetime is asymptotically AdS (anti-de Sitter), where theeffective AdS curvature scale is determined solely by the derivative couplingconstant. The metric approaches the AdS spacetime in the asymptotic infinitylimit and precisely recovers the Schwarzschild-AdS solution in generalrelativity if the coupling constant is tuned to the inverse of the cosmologicalconstant. We numerically find the lowest lying quasinormal frequency for theperturbation about a test massless, minimally coupled scalar field. Thequasinormal frequency agrees with that of the Schwarzschild-AdS solution forthe tuned case. For other parameters, in the large black hole limit, as themetric coincides with that of the Schwarzschild-AdS black hole, the quasinormalfrequency almost agrees with that of the Schwarzschild-AdS black hole and isinsensitive to the value of the cosmological constant. On the other hand, for asmall back hole the real part of the quasinormal frequency decreases as theabsolute value of the cosmological constant increases.

Black hole quasinormal modes in a scalar-tensor theory with field derivative coupling to the Einstein tensor