The semi classical cosmology approximation for a Friedman Robertson Walker geometry coupled to a field is considered. A power series of the field with coefficients that depend on the radius of the geometry is proposed, and the equations for the coefficients are solved.
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
c mode, the behavior of one of the solutions is ${{(-{r^2}+{t^2})}^{frac{1-n}{2}}}$ where $n$ is dimension of space. These solutions occur in an integral form. In addition, a main statement of the article is that a field in a black hole decays at infinity according to a universal law. An example of such a field is an eigentensor of the Einsteins linear operator that corresponds to an eigenvalue different from Zero. Possible applications to the stability of black holes of high dimension are discussed. The analysis we present is of a small perturbation of space-time. The perturbation analysis of higher order will appear in a sequel. We determine perturbations of space-time in dimension 1+$nge$ 4 where the system of equations is simplified to the Einsteins linear equation, a tensor differential equation. The solutions are some integral transformations which in some cases reduce to explicit functions. We perform some perturbation analysis and we show that there exists no perturbation regular everywhere outside the event horizon which is well behaved at the spatial infinity. This confirms the uniqueness of vacuum space-time within the perturbation theory framework. Our strategy for treating the stability problem is applicable to other space-times of high dimension with a cosmological constant different from Zero.
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 fundamental solution of the Schrodinger equation for a free particle is a distribution. This distribution can be approximated by a sequence of smooth functions. It is defined for each one of these functions, a complex measure on the space of path
s. For certain test functions, the limit of the integrals of a test function with respect to the complex measures, exists. We define the Feynman integral of one such function by this limit.
