Waves and energy
Here we discuss wave-like perturbations hµν of a general background metric gµν . The mathematics is similar to that of linearized theory: hµν is a tensor with respect to background coordinate transformations (as it was for Lorentz transformations in linearized theory) and it undergoes a gauge transformation when one makes an infinitesimal coordinate transformation. As in linearized theory, we will assume that the amplitude of the waves is small. Moreover, the waves must have a wavelength that is short compared to the radius of curvature of the background metric. These two assumptions allow us to visualize the waves as small ripples running through a curved and slowly changing spacetime.