The proper way to spatially decompose the gravitational-wave origin in stellar collapse simulations

Gravitational waves (GWs) hold great potential for an unobscured view of protoneutron stars (PNSs) formed as a result of stellar collapses. While waiting for discovery, deepening the understanding of GW emission in theory is beneficial for both optimizing searching strategies and deciphering the eventual data. One significant aspect is the spatially dependent contribution to the overall GW signal extracted from sophisticated hydrodynamic simulations. I present the proper way to perform the spatial decomposition of GW strain with the quadrupole formula in the slow-motion and weak-field approximation. Then I demonstrate the approach using the results of a 2D axisymmetric pseudo-Newtonian hydrodynamic simulation of core-collapse supernova. I discuss the possible misleading interpretation based on the incorrect method in the literature that favors the dominant contribution by the PNS convective layer. Moreover, with the correct approach, the GW spatial profiles agree well with those calculated from a consistent perturbative method. This work re-emphasizes the global emission picture of GWs from PNS and motivates future prudent analyses with 3D simulations.