This work studies wave propagation in the most general covariant scalar-tensor theories with second-order field equations, particularly focusing on the causal structure realized in these theories and also the shock formation process induced by nonlinear effects. For these studies we use the Horndeski theory and its generalization to the two scalar field case. We show that propagation speeds of the gravitational wave and scalar field wave in these theories may differ from the light speed depending on background field configuration, and find that a Killing horizon becomes a boundary of causal domain if the scalar fields share the symmetry of the background spacetime. With regard to the shock formation, we focus on transport of discontinuity in second derivatives of the metric and scalar field in the shift-symmetric Horndeski theory. We find that amplitude of the discontinuity generically diverges within finite time, which corresponds to shock formation. It turns out that the canonical scalar field and the scalar DBI model, among other theories described by the Horndeski theory, are free from such shock formation even when the background geometry and scalar field configuration are nontrivial. We also observe that the gravitational wave is protected against shock formation when the background has some symmetries at least. This fact may indicate that the gravitational wave in this theory is more well-behaved compared to the scalar field, which typically suffers from shock formation.
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