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We calculate the odd-parity, radiative ($\ell \ge 2$) parts of the metricperturbation in Lorenz gauge caused by a small compact object in eccentricorbit about a Schwarzschild black hole. The Lorenz gauge solution is found viagauge transformation from a corresponding one in Regge-Wheeler gauge. Like theRegge-Wheeler gauge solution itself, the gauge generator is computed in thefrequency domain and transferred to the time domain. The wave equation for thegauge generator has a source with a compact, moving delta-function term and adiscontinuous non-compact term. The former term allows the method of extendedhomogeneous solutions to be applied (which circumvents the Gibbs phenomenon).The latter has required the development of new means to use frequency domainmethods and yet be able to transfer to the time domain while avoiding Gibbsproblems. Two new methods are developed to achieve this: a partial annihilatormethod and a method of extended particular solutions. We detail these methodsand show their application in calculating the odd-parity gauge generator andLorenz gauge metric perturbations. A subsequent paper will apply these methodsto the harder task of computing the even-parity parts of the gauge generator.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Metric perturbations from eccentric orbits on a Schwarzschild black  hole: I. Odd-parity Regge-Wheeler to Lorenz gauge transformation and two new  methods to circumvent the Gibbs phenomenon