\label{space_mass_balance} A flat universe is consistent with the cosmological solution of general relativity that admits a fair balance between kinetic energy (expansion) and potential energy (mass). We note that even in the universe with accretion of space-mass it is possible to associate to the creation of space the kinetic energy because each galaxy moves away by others.
Also, if the increase in mass is correlated to an increase in potential energy (more gravity) then the fair balance, for the general relativity, corresponds to a fair balance between mass and space in a flat universe.
There are two distinct possibilities on balancing Space-Gravity:
the balancing Space-Gravity is an invariant of the universe, from its origins (see relativistic cosmology);
the balancing Space-Gravity is reached only at a particular time of cosmic evolution (see inflationary cosmology).
We remember that in general relativity the mass-energy tensor of the universe (defined as everything that is physically observable) is constant, in this way it defines a constant metric in time.
Instead we observe that a universe with increasing mass-energy implies a variation of the mass-energy tensor and then a variation of the metric.
In a theory where the mass is the source of space and gravity, we highlight the following possibilities: the increase in the number of massive particles in the universe generates more gravity (increasing curvature implies metric variation), but also the space is increasing with a consequent weakening of gravity (decreasing curvature).
We note also that Hubble’s law has a paradoxical consequence: an increase of space (volume) implies an increase in kinetic energy because the expansion it is equivalent to movement away.
In this way the presence of the scale factor \(a(t)\) (expansion) in the metric of the relativistic cosmological equations must be a consequence of the expanding universe and thus of the increasing kinetic energy in the universe and not be placed ad hoc to admit a solution with expansion.
In case of variable metric it can believe, then, that during the expansion of the universe, there is possible to have a particular moment or phase in which the universe achieves a fair balance between gravity and space: have so much gravity as much as space.
A condition of equal balance between gravity (mass) and space will thus be expressed by the relation \(K_{exp} = K_{escape} = U_g\). From this condition of equality, in any gravitational system, we obtain the well-known escape velocity: \[v^2 = G \frac{2 M}{R}\] If we combine this relationship with the Hubble law (with the condition that the speed of expansion of a sphere with radius \(R\) is equivalent to a speed of escape from a distant center \(R\)), we obtain: \[H^2 R^2 = G \frac{2 M}{R}\] from which we get easily \[\label{critical_density}
\rho = \frac{M}{V} = \frac{M}{\frac{4}{3} \pi R^3} = \frac{3 H^2}{8 \pi G}\] for all distance \(R\) and for any time \(t\) of any local observer.
The (\ref{critical_density}) equation is the well-known critical density \cite{friedmann1922,friedmann1924,nemiroff2008}. This condition is in the Newtonian and relativistic cosmology \cite{zeldovic1983} for a flat universe.
However, if the geometry of the universe in the relativistic cosmology remains unchanged over time (from Big Bang to the present days), in our model the metric may change over time because dependent on the growth rate of space and gravity. Only if it reaches a fair balance of increasing between space and gravity the universe reaches the condition of flat universe (Critical Universe).