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We consider a General Relativistic generalized RWs metric, and find a field of Universal rotational global centripetal acceleration, numerically coincident with the value of the Pioneers Anomalous one. Related subjects are also treated. The rotation defined here is different from older frameworks, because we propose a Gaussian metric, whose tri-space rotates relative to the time orthogonal axis, globally.

Detailed description of the subjects treated in this paper may be found in the two books recently published by Berman in 2012 [1,2]). Additional paper references are Berman in 2007 [

The subject treated in three papers by Marcelo Samuel Berman in this issue, two of them co-authored by Fernando de Mello Gomide (in 2012 [

Attempts to ascribe a rotational state to the Universe, were carefully described by Godlowski (in 2011 [

Ni [12,13], has reported observations on a possible rotation of the polarization of the cosmic background radiation, around 0.1 radians. As such radiation was originated at the inception of the Universe, we tried to estimate a possible angular speed or vorticity, by dividing 0.1 radians by the age of the Universe, obtaining about 10^{–}^{19} rad·s^{–}^{1}.

The numerical result is very close to the theoretical estimate, by Berman (in 2007 [

where c, R represent the speed of light in vacuum, and the radius of the causally related Universe.

We must remember, as Berman and Gomide [

The value of Bermans rotation, fits with the Pioneers anomaly, which consists on decelerations sufferred by Nasa space probes in non-closed curves, extending to outer space. Thermal emission was cited as resolving the Pioneers anomaly, but it does not explain the fly-bys, like Berman and Gomide [

About this same numerical value of the angular speed is predicted also in Godel’s rotational model, but it is not an expanding one (see Adler, Bazin and Schiffer [

Rotating metrics in General Relativity were first studied by Islam (in 1985 [

In his three best-sellers Hawking (in 1996 [

There are four methods, in GRT, to create rotations. Non-diagonal metrics, like Kerrs, is one. The adoption of an imperfect fluid model, with vorticities, as in Raychaudhuris equation, is second. Third, you may follow the Godlowski et al. (in 2004 [

where,

It was seen that when one introduces a metric temporal coefficient which is not constant, the new metric includes rotational effects. In fact, we have a generalized Gaussian metric, because besides the fact that the trispace is orthogonal to the time-axis, the spatial part of the metric, rotates as a whole, relative to this time axis. This is a new concept being introduced in the theory.

The present paper follows the steps of the semi-relativistic treatment by Berman (in 2007 [

In a previous paper Berman (in 2009 [

The pioneer works of Berman (in 1981 [

The zero-total-energy of the Roberston-Walker’s Universe, and of any Machian ones, have been shown by many authors (Berman in 2006 [30,31]; in 2007 [

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.

Carmeli et al. in 1990 [

We shall argue below that, for the Universe, local and global Physics blend together. The pseudo-momentum, is to be taken like the linear momentum vector of Special Relativity, i.e., as an affine vector. In a previous paper (Berman in 2009 [

Consider first a temporal metric coefficient which depends only on t. The line element becomes:

The field equations, in General Relativity Theory (GRT) become:

and,

Local inertial processes are observed through proper time, so that the four-force is given by:

Of course, when, 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 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.

From the energy-momentum conservation equation, in the case of a uniform Universe, we must have,

The above is necessary in the determination of cosmic time, for a commoving observer. We can see that the hypothesis (2)—that is only time-varying—is now validated.

In order to understand Equation (6), it is convenient to relate the rest-mass m, to an inertial mass, with:

It can be seen that represents the inertia of a particle, when observed along cosmic time, i.e., coordinate time. In this case, we observe that we have two acceleration terms, which we call,

and,

The first acceleration is linear; the second, resembles rotational motion, and depends on and its timederivative.

If we consider a centripetal acceleration, we conclude that the angular speed is given by,

By comparison between the usual—metric, and the field equations in the t—metric, we are led to conclude that the conventional energy density and cosmic pressure p are transformed into and, where:

and,

We plug back into the field equations, and find,

For a time-varying angular speed, considering an arc, so that,

we find, from (11),

Returning to (14), we find,

This completes our solution.

The case where depends also on and was considered also by Berman (in 2008 [

Even in popular Science accounts (Hawking in 1996 [

The pseudotensor, also called Einstein’s pseudotensor, is such that, when summed with the energy-tensor of matter, gives the following conservation law:

In such case, the quantity

is called the general-relativistic generalization of the energy-momentum four-vector of special relativity (Adler et al. in 1975 [

It can be proved that is conserved when:

a) only in a finite part of space; andb) when we approach infinity, where is the Minkowski metric tensor.

However, there is no reason to doubt that, even if the above conditions were not fulfilled, we might eventually get a constant, because the above conditions are sufficient, but not strictly necessary. We hint on the plausibility of other conditions, instead of a) and b) above.

Such a case will occur, for instance, when we have the integral in (19) is equal to zero.

For our generalised metric, we get exactly this result, because, from Freud’s (1939) formulae, there exists a super-potential, (Papapetrou in 1974 [

where the bars over the metric coefficients imply that they are multiplied by, and such that,

thus finding, after a brief calculation, for the rotating Robertson-Walker’s metric,

The above result, with von Freud’s superpotential, which yields Einstein’s pseudotensorial results, points to a zero-total energy Universe, even when the metric is endowed with a varying metric temporal coefficient .

A similar result would be obtained from LandauLifshitz pseudotensor (Papapetrou in 1974 [

where,

and,

A short calculation shows that, for the rotating metric, too, we keep valid the result,

Other superpotentials would also yield the same zero results. A useful source for the main superpotentials in the market, is the paper by Aguirregabiria et al. in 1996 [

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.

Though so many researchers have dealt with the energy of the Universe, our present original solution involves rotation. We may paraphrase a previous calculation, provided that we work with proper time instead of coordinate time t (Berman in 2009 [

If, the energy is zero, too. But when we write Schwarzschilds metric, and make the mass become zero, we obtain Minkowski metric, so that we got the zeroenergy result. Any flat RWs metric, can be reparametrized as Minkowskis; or, for closed and open Universes, a superposition of such cases (Cooperstock and Faraoni in 2003 [

Now, the energy of the Universe, can be calculated at constant time coordinate. In particular, the result would be the same as when, or, even when. Arguments for initial null energy come from Tryon (in 1973 [

Consider the possible solution for the rotating case. We work with the -metric, so that we keep formally the RWs metric in an accelerating Universe. The scalefactor assumes a power-law, as in constant deceleration parameter models (Berman in 1983 [

where, m, D = constants, and,

where q is the deceleration parameter.

For a perfect fluid energy tensor, and a perfect gas equation of state, cosmic pressure and energy density obey the following energy-momentum conservation law, (Berman in 2007 [10,32]),

where, only in this Section, overdots stand for -derivatives. Let us have,

(constant larger than)(25)

On solving the differential equation, we find, for any, 1, , that,

(constant)(26)

When, from (26) we see that the energy density becomes zero, and we retrieve an “empty” Universe, or, say, again, the energy is zero. However, this energy density is for the matter portion, but nevertheless, as in this case, , all masses are infinitely far from each others, so that the gravitational inverse-square interaction is also null. The total energy density is null, and, so, the total energy. Notice that the energy-momentum conservation equation does not change even if we add a cosmological constant density, because we may subtract an equivalent amount in pressure, and Equation (24) remains the same. The constancy of the energy, leads us to consider the zero result at infinite time, also valid at any other instant.

We refer to Berman (in 2006 [30,31]) for another alternative proof of the zero-energy Universe. If we took instead of t, these references would provide the zero result also for the rotational case.

Einstein’s field Equations (4) and (5) above, can be obtained, when constant, through the mere assumptions of conservation of energy (Equation (4)) and thermodynamical balance of energy (Equation (5)), as was pointed out by Barrow in 1988 [

.

Now, let us consider a time-varying. We may write the energy (in fact, the “energy-density”)—equation, as follows:

The r.h.s. stands for a constant. We can regard the l.h.s. as the a sum of constant terms, thus finding a possible solution of the field equations, such that each term in the l.h.s. of (27) remains constant. For example, let us consider,

where, , and are non-zero constants. Relation (28) makes this solution practically of the Machiantype, similar to the semi-relativistic treatment by Berman (in 2007 [

When we plug the above solution to the cosmic pressure Equation (5), we find that it is automatically satisfied provided that the following conditions hold,

(constant(32)

and,

As we found a general-relativistic solution, so far, we are entitled to the our previous general relativistic angular speed Formula (11), to which we plug our solution (30), to wit,

For the power-law solution of the last Section,

so that,

where we roughly estimated the present deceleration paramenter as, while, the centripetal acceleration,

Notice that the same result would follow from a scalefactor varying linearly with time. This is the sort of scalefactor associated with the Machian Universe. In fact,the field equations that we had (Equations (4) and (5)), were not enough in order to determine the exact form of the scale-factor, because we had an extra-unknown term, the temporal metric coefficient. When we advance a given equation of state, the original RWs field equations, with constant, may determine the scale-factors formula. Just to remember, our solution is a particular one.

This is a general relativistic result. It matches Pioneers anomalous deceleration.

In an Appendix to this Section, we go ahead with the alternative calculation with a simple naive Special Relativistic-Machian analysis, as had been made in Berman (in 2007 [

Someone has made very important criticisms on our work. First, he says why do not the planets in the solar system show the calculated deceleration on the Pioneers? The reason is that elliptical orbits are closed, and localized. You do not feel the expansion of the universe in the sizes of the orbits either. In General Relativity books, authors make this explicit. You do not include Hubbles expansion in Schwarzschilds metric. But, those space probes that undergo hyperbolic motion, which orbits extend towards infinity, they acquire cosmological characteristics, like, the given P.A. deceleration. Second objection, there are important papers which resolve the P.A. with non-gravitational Physics. The answer—that is OK, we have now alternative explanations. This does not preclude ours. Third, cosmological reasons were discarded, including rotation of the Universe. The problem is that those discarded cosmologies, did not employ the correct metric. For instance, they discarded rotation by examining Godel model, which is non expanding, and with a strange metric. The kind of metric we employ now, or the one that we employed in the rotational case, were not discarded or discussed by the authors cited by this objecter. Then, the final question, is how come that a well respected author dismissed planetary Coriolis forces induced by rotation of distant masses, by means of the constraints in the solar system. Our answer is that, beside what we answered above, he needs to consider Machs Principle on one side, and the theoretical meaning of vorticities, because one is not speaking in a center or an axis of rotation or so. When we say, in Cosmology, that the Universe rotates, we mean that there is a field of vorticities,just that. The whole idea is that Cosmology does not enter the Solar System except for non-closed orbits that extend to outer space. We ask the reader to check Machs Principle, because in some formulations of this principle, rotation is in fact a forbidden affaire.

Another one pointed out a different “problem”. He objects, that the angular speed formula of ours, is coordinate dependent. Now, when you choose a specific metric, you do it thinking about the kind of problem you have to tackle. After you choose the convenient metric, you forget tensor calculus, and you work with coordinate-dependent relations. They work only for the given metric, of course.

We 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 by Berman should be consulted (see all Berman’s references at the end of this article). 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 energydensity result, as was pointed first by Berman elsewhere (Berman, for instance, see in 2012 [1,2]). 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.

As Berman( in 2009 [67,68]) shows, we may say that the Universe is singularity-free, and was created abnihilo, nor there is zero-time infinite energy-density singularity.

Paraphrasing Dicke (in 1964 [69,70]), it has been shown the many faces of Dirac’s LNH, as many as there are about Mach’s Principle. In face of modern Cosmology, the naif theory of Dirac is a foil for theoretical discussion on the foundations of this branch of Physical theory. The angular speed found by us, (Berman, in 2010 [

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 in 2008 [

Referring to rotation, it could be argued that cosmic microwave background radiation 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 happening to molluscs. And, of course, the results of the rotation of the polarization of CMBR.

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 (in 2010 [

For more details on the subjects treated here, the general recomendation is to refer the reader to both books published recently by Berman (in 2012 [

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.

As we now have the pseudo-tensorial zero-total energy result, for rotation plus expansion, we might write in terms of elementary Physics, a possible energy of the Universe equation, composed of the inertial term of Special Relativity, , the potential self-energy

, and the cosmological “constant” energy,

, and not forgetting rotational energy, where I stands for the moment of inertia of a “sphere” of radius R and mass M. The energy equation is equated to zero, i.e.,

It must be remembered that R is a time-increasing function, while the total-zero energy result must be timeinvariant, so that the principle of energy conservation be valid. A close analysis shows that the above conditions can be met by solutions (28) and (29), which were derived or induced from the general relativistic equations. When we plug the inertia moment,

we need also to consider the following Brans-Dicke generalised relations,

and,

If we calculate the centripetal acceleration corresponding to the above angular speed, we find, for the present Universe, with cm and cm·s^{–}^{2}

This value matches the observed experimentally deceleration of the NASA Pioneers’ space-probes.

We observe that the Machian picture above is understood to be valid for any observer in the Universe, i.e., the center of the “ball” coincides with any observer; the “Machian” centripetal acceleration should be felt by any observed point in the Universe subject to observation from any other location.

We solve also other mystery concerning Pioneers anomaly. It has been verified experimentally, that those space-probes in closed (elliptical) orbits do not decelerate anomalously, but only those in hyperbolic flight. The solution of this other enigma is easy, according to our view. The elliptical orbiting trajectories are restricted to our local neighborhood, and do not acquire cosmological features, which are necessary to qualify for our Machian analysis, which centers on cosmological ground. But hyperbolic motion is not bound by the Solar system, and in fact those orbits extend to infinity, thus qualifying themselves to suffer the cosmological Machian deceleration. Thermal emission may solve the first Pioneer anomaly, but it does not solve the spin-down, nor the fly-bys in gravity assists. It is not clear why, thermal emission did not cause decelerations in elliptical orbiters. Rotation of the Universe solves all the three (Berman and Gomide in 2012 [