Emission of spacetime waves from the partial collapse of a compact object

In this work we describe the partial collapse of a compact object and the emission of spacetime waves as a result of back-reaction effects. As a source mass term we propose a non-smooth continuous function that describes a mass-loss, and we then obtain the solution of such setting. We present three distinct examples of the evolution of the norm $|R_{nl}(t,r_{*})|$ in terms of $t$, and four different results are shown for the parameter $l=1,2,5,10$; here $r_{*}$ is the fixed radius of an observer outside the compact object. In all cases, the decay behaviour is actually present at $t\gg 1$ and becomes more evident for larger $l$. In addition, for the results that have smaller $l$'s their amplitudes are larger when the asymptotic character of $|R_{nl}(t,r_{*})|$ clearly appears. Finally, the farther away an observer is set, the fewer oscillations are perceived; however, from our particular fixed set of parameters, the best spot to observe the wiggles of the emitted spacetime waves is close to $r_{*}\simeq \alpha$.