Scalar waves in the exterior of a Schwarzschild black hole

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Abstract
Using the method of separation of variables, we study rigorously scalar waves due to a point source in the exterior of a Schwarzschild black hole. First, a Fourier analysis gives general formulas for the interior and exterior radial wave functions and their relations to solutions of special cases, the Green's function, and the frequency spectrum. Three special cases are examined. Second, Laplace transforms of the field are obtained and their properties are studied. Using the results of the Laplace transformation and some general properties of timelike curves, we prove the following theorem: The time‐dependent scalar field of a point source goes to zero outside the horizon as the source falls into the black hole.