The perturbation equation of a static symmetrical homogeneous space-time

In absence of explicit solutions of the perturbation equation of a static symmetrical homogeneous space-time, the best we can do is to construct a quasi-transformation. In this framework, we solve the perturbation equation with initial data and a number of results are derived. Far from the horizon of a black hole of even space dimension N , a mass-less field decays as r(−r + t) 1−N 2 −l in space-time, where l is a harmonic number of the sphere. A relation of energy and momentum of a particle with mass in a hyper black hole is discovered and a solution to the equation of Klein-Gordon in the metric of Schwarzschild-Tangherlini with initial data on the hypersphere is proposed. Also, the Green’s function of the Klein-Gordon equation in Schwarzschild coordinates is calculated. This function is a sum on the harmonic modes of the sphere. The first term is a double integration on the spectrum of energy and the momentum of the particle. Far from the horizon, the double integration is approximated by an integration on a line defined by the relation of energy and momentum of a free particle. From here, the potential of Yukawa is derived. Finally, the linear perturbation equations are derived and solved exactly. The master equation with initial data of a small perturbation in a static symmetrical homogeneous space-time, like a (possibly higher dimensional) Schwarzschild black hole, is studied. A main statement of the article is that for each harmonic mode of the horizon there are two solutions that behave similarly at large. In the basic mode, the asymptote of a field (an eigentensor of the Lichnerowicz operator, for example) decays at infinity according to the universal law (−r + t) 1−N 2 . These solutions occur in an integral form. The analysis we present is of a small perturbation to the full static symmetrical solution. The higher order perturbations will appear in a sequel. We determine independently perturbations of the space-time in dimension 1+N ≥ 4, where the system of equations can be reduced to a master equation — a tensor differential equation. The solutions are integral transformations which in some cases reduce to explicit functions.