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We argue that the Robertson-Walker’s Universe is a zero-energy stable one, even though it may possess a rotational state besides expansion.

The first pseudo-tensorial calculation of the energy of the Universe, has been made by Berman, in 1981 [

In a previous paper Berman (2009 [

The zero-total-energy of the Roberston-Walker’s Universe, and of any Machian ones, have been shown by many authors. It may be that the Universe might have originated from a vacuum quantum fluctuation. By “vacuum”, we mean the spacetime of Minkowski. In support of this view, we shall show that the pseudotensor theory (Adler et al., in 1975 [

The reason for the failure of non-Cartesian curvilinear coordinate energy calculations through pseudotensors, resides in that curvilinear coordinates carry non-null Christoffel symbols, even in Minkowski spacetime, thus introducing inertial or fictitious fields that are interpreted falsely as gravitational energy-carrying (false) fields.

Consider first Robertson-Walker’s metric, added by a temporal metric coefficient which depends only on t. The line element (Gomide and Uehara, 1981 [

Of course, when constant, the above equations reproduce conventional Robertson-Walker’s field equations.

We must mention that the idea behind RobertsonWalker’s metric is the Gaussian coordinate system. Though the condition constant, is usually adopted, we must remember that, the resulting time-coordinate is meant as representing proper time. If we want to use another coordinate time, we still keep the Gaussian coordinate properties. Berman (2008 [

Consider the following reparametrization:

In the new coordinates, the generalized RWs metric becomes:

This is Minkowski’s metric.

Even in popular Science accounts (Hawking, 1996 [

Berman and Gomide (2012 [41-43]) has recently shown that the generalized Robertson-Walker’s metric yielded a zero-energy pseudotensorial result. The same authors showed that the result applied in case of a rotating and expanding Universe.

The equivalence principle, says that at any location, spacetime is (locally) flat, and a geodesic coordinate system may be constructed, where the Christoffel symbols are null. The pseudotensors are, then, at each point, null. But now remember that our old Cosmology requires a co-moving observer at each point. It is this co-motion that is associated with the geodesic system, and, as RWs metric is homogeneous and isotropic, for the co-moving observer, the zero-total energy density result, is repeated from point to point, all over spacetime. Cartesian coordinates are needed, too, because curvilinear coordinates are associated with fictitious or inertial forces, which would introduce inexistent accelerations that can be mistaken additional gravitational fields (i.e., that add to the real energy). Choosing Cartesian coordinates is not analogous to the use of center of mass frame in New-tonian theory, but the null results for the spatial components of the pseudo-quadrimomentum show compatibility.

Witten in 1981 [

The conclusion of Witten was that Minkowski’s space was also stable, because perturbations in the form of gravitational waves should not decrease the total energy, because it is known that gravitational waves have positive energy. We now conclude that our Universe is also stable, due to the reparametrization above. But, first, let us deal with some conceptual issues.

We have three kinds of stability criteria: 1) 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; 2) The matter inside the system must not be possibly created out of nothing,or else, the bodies should have positive energy; 3) “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 RobertsonWalkers Universe is locally and globally stable, whenever Classical Physics is concerned. Now, Berman and Trevisan (in 2010 [

Berman and Gomide (2012 [41-43]) and Berman (2012 [38,39]) 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 comoving 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 which 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 (see, for instance, Berman, in 2009 [48,49]). The so-called total energy density of the Universe, which appears in some textbooks, corresponds only to the non-gravitational portion, and the zero-total energy density results when we subtract from the former, the opposite potential energy density (Berman, 2012 [38,39]).

As Berman (2009 [49,50]) shows, 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.

Rotation of the Universe and zero-total energy were verified for Sciama’s linear theory, which has been expanded, through the analysis of radiating processes, by one of the present authors (Berman, 2008 [

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 antimatter, left-handed DNA-helices, the fact that humans and animals alike have not symmetric bodies, the same 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 (2010 [

Hawking and Mlodinow (2010 [

We now want to make a conjecture related to the stability criteria of last Section.

A physical system is not “chaotic”, if small perturbations in its initial state do not originate “large” variations in its future behaviour. According to our discussion, the Robertson-Walkers 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 Einsteins 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”. Experimental observational evidence on the rotation of the Universe is dealt with, in the books by Berman (2012 [38, 39]), and references therein. Seminal papers on rotation evidence were due to Paul Birch in 1982, in the wellknown Nature .

One of the authors (MSB) thanks Marcelo Fermann Guimarães, Nelson Suga, Mauro Tonasse, Antonio F. da F. Teixeira, and for the important incentive offered by Miss Solange Lima Kaczyk,now a brand new advocate, continued during the last five years of his research in Cosmology.