Gravitational Radiation Coupled to a real massive scalar Field
We study the asymptotic behavior of small-data global solutions to the Einstein--Klein--Gordon system with a positive-mass real scalar field in wave gauge. We show that the linearized trace component of the metric perturbation admits a $\hat t^{-1}\simeq t^{-1}$ optimal decreasing rate. Moreover, a quantitative one-point observability estimate is established: the initial Klein-Gordon energy is controlled by this leading order coefficient.
gr-qc↗