Gravitational Wave Energy Attenuation in Active Gravitational Fields

This paper investigates the propagation of gravitational waves through regions of spacetime influenced by strong gravitational fields. We reformulate the wave energy expression in terms of affine length, exploring how metric perturbations and the associated energy densities deform proper geodesics. A central claim is proposed: gravitational waves experience an energy drop when traversing regions where the spacetime is expressed predominantly in affine terms. This claim is developed through variational analysis of the metric and its impact on wave amplitude.