Gravitational Field of a Particle Falling in a Schwarzschild Geometry Analyzed in Tensor Harmonics
Abstract
We are concerned with the pulse of gravitational radiation given off when a star falls into a "black hole" near the center of our galaxy. We look at the problem of a small particle falling in a Schwarzschild background ("black hole") and examine its spectrum in the high-frequency limit. In formulating the problem it is essential to pose the correct boundary condition: gravitational radiation not only escaping to infinity but also disappearing down the hole. We have examined the problem in the approximation of linear perturbations from a Schwarzschild background geometry, utilizing the decomposition into the tensor spherical harmonics given by Regge and Wheeler (1957) and by Mathews (1962). The falling particle contributes a $\ensuremath{\delta}$-function source term (geodesic motion in the background Schwarzschild geometry) which is also decomposed into tensor harmonics, each of which "drives" the corresponding perturbation harmonic. The power spectrum radiated in infinity is given in the high-frequency approximation in terms of the traceless transverse tensor harmonics called "electric" and "magnetic" by Mathews.