  The Einstein's 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 basic 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 Einstein's
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+$n\ge$ 4 where the system of
equations is simplified to the Einstein's 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.
