According Stephen Hawking and Leonard Mlodinow, the Universe is locally stable but globally unstable, because they consider Quantum effects. Berman and da Costa, have only considered Classical General Relativity and then, the Universe is locally and globally stable.^{[1]}^{[2]}
The work of Berman and da Costa, has been centered not only on the usual Robertson-Walker´s metric, that considers an expanding homogeneous isotropic geometry, but rather on the modified (they termed "generalized") metric that arises by considering a time-varying metric coefficient, whereby, this last term, forces the metric to represent a rotational Universal state.Both the original metric and the "generalized", are Gaussian metrics, which means that the geometrical interpretation is that the time axis lies in an orthogonal direction to the ordinary three-space.The difference lies, in that the three-space may rotate, around the orthogonal time axis, this qualifies for a Universal rotation.
The conclusion of Witten was that Minkowski's space was also stabl,because, perturbations in the form of gravitational waves should not decrease the total energy, due to the fact that it is known that gravitational waves have positive energy. We now conclude that our Universe is also stable, due to the reparametrization, that transforms Robertson-Walker´s metric, into Minkowski´s. But, first, let us deal with some conceptual issues.
We have three kinds of stability criteria:
Since a physical system shows a tendency to decay into its state of minimum energy, the criterion states that the system should not be able to collapse into a series of infinitely many possible negative levels of energy. There should be a minimum level, usually zero-valued, which is possible for the physical system.;
The matter inside the system must not be possibly created out of nothing, or else, the bodies should have positive energy.;
"Small" disturbances should not alter a state of equilibrium of the system (it tends to return to the original equilibrium state). In the case of the Universe, disturbances, of course, cannot be external.
According with our discussion, the rotating Robertson--Walker's Universe is locally and globally stable, whenever Classical Physics is concerned. Now, Berman and Trevisan (2010)^{[3]} have shown that Classical General Relativity can be used to describe the scale-factor of the Universe even inside Planck's zone; provided that we consider that the calculated scale-factor behavior reflects an average of otherwise uncertain values, due to Quantum fluctuations.
Berman and Gomide (2012, 2012)^{[4]}^{[5]} have obtained a zero-total energy proof for a rotating expanding Universe. The zero result for the spatial components of the energy-momentum-pseudotensor calculation are equivalent to the choice of a center of Mass reference system in Newtonian theory, likewise the use of co-moving observers in Cosmology. It is with this idea in mind, that we are led to the energy calculation, yielding zero total energy for the Universe, as an acceptable result. We are assured that we chose the correct reference system; this is a response to the criticism made by some scientists to argue that pseudotensor calculations depend on the reference system, and thus, those calculations are devoid of physical meaning.
Related conclusions should be consulted (see all Berman's references and references therein). As a bonus, we can assure that there was not an initial infinite energy density singularity, because attached to the zero-total energy conjecture, there is a zero-total energy-density result, as was pointed by Berman elsewhere . The so-called total energy density of the Universe, which appears in some textbooks, corresponds only to the non-gravitational portion. The zero-total energy density results when we subtract from the former, the opposite potential energy density.
As Berman showed elsewhere, we may say that the Universe is singularity-free , and was created ab-nihilo, in particular, there is no zero-time infinite energy-density singularity.
Referring to rotation, it could be argued that cosmic microwave background radiation should show evidence of quadrupole asymmetry and it does not, but one could argue that the angular speed of the present Universe is too small to be detected; also, we must remark that CMBR deals with null geodesics, while Pioneers' anomaly, for instance, deals with time-like geodesics. In favor of evidence on rotation, we remark neutrinos' spin, parity violations, the asymmetry between matter and anti-matter, left-handed DNA-helices, the fact that humans and animals alike have not symmetric bodies, the same is happening to molluscs. We predict that chaotic phenomena and fractals, rotations in galaxies and clusters, may provide clues on possible left handed preference through the Universe.
Berman and Trevisan have remarked that creation out-of-nothing seems to be supported by the zero-total energy calculations. Rotation was included in the derivation of the zero result by Berman and Gomide. We could think that the Universes are created in pairs, the first one (ours) has negative spin and positive matter; the second member of the pair, would have negative matter and positive spin: for the ensemble of the two Universes, the total mass would always be zero; the total spin, too. The total energy (twice zeros) is also zero.
Hawking and Mlodinow (2010) conclude their book with a remark on the fact that the Universe is locally stable, but globally unstable because spontaneous creation is the reason why the Universe exists, and new creations like this may still happen. Of course, this is a question of interpretation.
We now want to make a conjecture related to the stability criteria as above.
A physical system is not "chaotic," if small perturbations in its initial state do not originate "large" variations in its future behavior. According to our discussion, the Robertson--Walker's Universe, with or without rotation, is locally and globally stable under the three criteria. As its total energy is zero, we conjecture that this type of Universe is not globally chaotic, and that the three criteria for stability imply that any such system cannot be globally chaotic altogether. We remark nevertheless, that because Einstein's field equations are non-linear, chaos is not forbidden in a local sense.
We regret that the name of a basic result in General Relativity Theory, is called "positive energy theorem" instead of the "non-negative energy theorem."
References
↑Stephen Hawking and Leonard Mlodinow, The Grand Design^{[page needed]} - E Bantam Books, New York, 2010