Wave Metrics in the Cotton and Conformal Killing Gravity Theories

We study wave metrics in the context of Cotton Gravity and Conformal Killing Gravity. First, we consider pp-wave metrics with flat and non-flat wave surfaces and show that they are exact solutions to the field equations of these theories. More explicitly, the field equations reduce to an inhomogeneous Laplace and Helmholtz differential equations, depending on the curvature of the two-dimensional geometry of the wave surfaces. An interesting point here is that the ones with non-flat wave surfaces are not present in classical GR, which manifests a crucial distinction between these theories and GR. Moreover, we investigate Kerr-Schild-Kundt metrics in the context of these theories and show that, from among these metrics, only the AdS wave metrics solve the field equations of these theories. However, AdS spherical and dS hyperbolic wave metrics do not solve the field equations of these theories, which is in contrast to the classical GR. In the case of AdS wave metrics, the field equations of these theories reduce to an inhomogeneous Klein-Gordon equation. We give all the necessary and sufficient conditions for the metric function $V$ to solve these field equations. Lastly, we address the colliding gravitational plane waves problem only in Cotton gravity due to the complexity of the field equations in Conformal Killing Gravity.