  In this paper we first construct a mathematical model for the Universe
expansion that started up with the original Big Bang. Next, we discuss the
problematic of the mechanical and physical laws invariance regarding the
spatial frame exchanges. We then prove the (theoretical) existence of a
variable metric $g_t$, depending on time and satisfying to a simplified
Einstein equation, so that all free ordinary trajectories are geodesics. This
is done by considering the classical Galileo-Newtonian space and time relation,
by using generalized Newtonian principles and adding the approved physical new
ones (as covariance principle, Mach principle, the Einstein equivalence
principle ...) in order to establish a new cosmological model of the dynamical
Universe as being $(U(t))_{t\geq 0}=(B_e(O,t),g_t)_{t\geq 0}$, where $B_e(O,t)$
is the Euclidean ball of radius $t$ in $\mathbb R^3$. The cosmological metric
$g_t$ is totally determined, at time $t$, by the mass energy distribution
$E_t(X)$ on $B_e(O,t)$. We study also the black holes phenomenon and we prove
that the total and global cosmological energy distribution $E_t(X)$ satisfies
to the wave equation whose solutions are characterized by pseudo-frequencies
depending on time and related to the spectrum of the Dirichlet problem on the
unit ball $B_e(O,1)$ for the Laplace-Beltrami operator -$\Delta$. Our model is
consistent in the sense that all Newtonian and classical physical laws are
valid as particular cases in classical situations. Then, we show that all basic
results of Modern Physics are still valid without using the second postulate of
special relativity nor the uncertainty principle.
