Towards extreme mass-ratio inspiral calculations in the time domain using hyperboloidal slicing and compactification

Our goal is to accurately predict gravitational wave signals by solving the Regge-Wheeler equation in the time domain. We first built foundational understanding by solving a simpler case with a simpler method: an oscillating point source on an infinite string, solved by implementing a second-order finite difference approximation to solve the 1D wave equation. We simulated waves propagating to infinity by imposing an internal boundary condition consistent with a Dirac delta source. This was solved and simulated in Mathematica. One important feature of our technique is the use of hyperboloidal slicing and compactification because it enhances accuracy for astrophysical applications. We used hyperboloidal slicing to transform the time coordinate so there are only a finite number of wavelengths within the infinite domain; this enables compactification while avoiding the infinite blue shift problem. We compared our numerical results to the exact solution, which agreed up to the numerical discretization error.