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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 {it 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 ${r^l} {{(-{r^2}+{t^2})}^{frac{1-N}{2}-l}}$ 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 Greens 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 Einsteins linear equation of a small perturbation in a space-time with a homogeneous section of low dimension, is studied. For every harmonic mode of the horizon, there are two solutions which behave differently at large distance $r$. In the basi
Relativistic quantum field theory in the presence of an external electric potential in a general curved space-time geometry is considered. The Fermi coordinates adapted to the time-like geodesic are utilized to describe the low-energy physics in the
Using Relativistic Quantum Geometry we study back-reaction effects of space-time inside the causal horizon of a static de Sitter metric, in order to make a quantum thermodynamical description of space-time. We found a finite number of discrete energy
The complete analytical solutions of the geodesic equation of massive test particles in higher dimensional Schwarzschild, Schwarzschild-(anti)de Sitter, Reissner-Nordstroem and Reissner-Nordstroem-(anti)de Sitter space--times are presented. Using the
We apply the Induced Matter Model to a five-dimensional metric. For the case with null cosmological constant, we obtain a solution able to describe the radiation-dominated era of the universe. The positive $Lambda$ case yields a bounce cosmological m