Indubitably, the deviations from geodesic motion are more tale-tale in the theory of gravitation than Newton's expression for gravitational acceleration. Einstein sought to justify the relativity of accelerated motion by claiming (locally) that an accelerated frame is equivalent to a gravitational field. However, it is precisely the tidal forces that distinguish between the their reference frames and destroy said "equivalence." Tidal forces are deviations from geodesic motion, and that is why they are, and should be, central to any theory of gravitation.

Einstein modelled his field equations on generalizing Poisson's equation. It poses an equivalence between geometry, embodied by the Ricci tensor, and an energy-stress tensor, which accounts for the source term in Poisson's equation. Confusing the density with an energy density, the energy-stress tensor invokes of an energy-momentum 4-vector, with an energy density standing in for a density of matter, and the hydrostatic pressure components playing the role of momentum components. The time component of the Ricci tensor is thus equated with an energy density, while its space components with those of a uniform hydrostatic pressure. The first indication that something is badly amiss was in its application to the Friedmann-Robertson-Walker (FRW) metric, where for constant uniform density, the energy density becomes that of a negative pressure. This ludicrous result, instead of being question, was made the basis of an "inflationary" scenario that irons out the inhomogeneities in the early universe by claiming that a "negative" pressure causes a rapid expansion of the universe. But, before coming to that, let us begin with the tried-and-true Schwarzschild metric.

In the plane theta=0, the Schwarzschild metric can be written simply as

-1=-F^2(r) (dt/ds)^2+G^2(r) (d phi/ds)^2,

where the metric coefficients F^2(r)=1-2M/G and G^(' 2)=1-2M/G are functions only of the radial coordinate, r, M is a constant mass, and the prime indicates the derivative with respect to r. Local time is indicated by "s" while coordinate time by "t". The tidal force components are calculated in terms of the second derivatives of F and G, F"/F=-2M/G and G"/G=M/G, and the cross term, F'G'/FG=F'/G=M/G^3, since F=G'. This corresponds to the distortion of a sphere into a prolate ellipsoid. The tidal force has zero-divergence since the z-component of the tidal force is twice that of the tidal components in the x-and y-directions. It turns out to be the same as the vanishing of the Ricci tensor, which was Einstein's condition of "emptiness". Hardly, it could be considered as a condition of emptiness, but one of a stress equilibrium cause by the presence of a finite M, which, certainly contradicts there being an empty universe! In other words, the equation for geodesic deviation involves the Riemann tensor, while its divergence reduces it to the Ricci tensor. The vanishing of the latter is certainly not a condition of emptiness!

Now, instead of a constant mass, M, we consider a constant density rho. The components of the tidal stress are all constant---and equal! This is the case of a constant curvature metric, and becomes the model of Schwarzschild's so-called inner solution. According to Einstein's field equations this calls for setting the Ricci tensor components equal to the mass density and the hydrostatic pressure. But with a constant mass density there results

rho=-p (mass density=-pressure)

which is an absurdity anyway you look at it.

Before continuing with the FRW metric, which is even more telling, let us derive the Schwarzschild metric for the outer solution by r to obtain:

Omega^2=(d phi/ds/d t/ds)^2=M/G^3,

a geodesic equation, which at constant radius, will trace out a circular orbit with frequency Omega.

We will now show that the FRW metric has an entirely different meaning than the conventional one given to it. In particular, the geodesics will turn out to be half circles, and in the limit straight lines, in the radius of the universe phi plane, analogous to geodesics of the Poincare half-plane model. The conditions placed on the tidal stresses do not allow for different scenarios of "open", "closed", and "flat" universes for which there is no "critical" density separating them.

The FRW metric tries to describe a homogenous, spherically symmetrical expanding universe. According to Gauss, the curvature of a surface defined by the element

ds^2=dR^2+G^2 d phi^2

is k=-G"/G, where the prime indicates the derivative with respect to R. Now define r=G(R) so that

dr=G'(R)dR.

Since k is constant, we may integrate Gauss's relation to obtain

G^(' 2)(R)=1-kG^2,

where we have set the constant of integration equal to unity. Consequently, the metric of the surface can also be written as

ds^2=dr^2/(1-kr^2)+r^2d phi^2,

where 1/sqrt(k)=R_0 is the constant radius of the sphere.

To take into account the expansion of the sphere we multiply the metric by a dimensionless, time-dependent factor a^2(t), which has often, incorrectly, been defined as the so-called radius of the universe. Combining this with the hyperbolic time factor dt, the metric becomes

ds^2=dt^2-a^2(t)G^2(R)d phi^2.

We can now relate the two time factors by setting T=a(t) so that

dT=a* dt,

where the star * denotes differentiation with respect to t. Introducing this into the metric gives

ds^2=da^2/a*^2-a^2(t)G^2(R) d phi^2.

Now we have to introduce some physics. Since the sectional curvature, k, is constant, so too must be a**/a=-rho, which is proportional to the inverse square of Newtonian free-fall time, rho being the constant density. Integrating the Gaussian curvature we get

a*^2=C-rho a^2,

where C is a constant of integration. It places a bound on the magnitude of the dimensionless amplitude. Consider Laplace's nebular hypothesis in which an extremely hot gas began to cool and shrink. As the rotating gas began to shrink, it rotated faster and faster, causing a flattening at the poles, much like the flattening of the earth as explained by Newton.

As the angular speed, w, increases, the radius of the nebula, R_0, decreases such that the linear velocity remains constant, v=wR_0. Hence, we can identify the constant of integration, C, as w^2, and, consequently,

a*^2=w^2-rho a^2.

Stability of the nebulae depends only on the ratio w^2/rho, which also sets the limit of the growth of the dimensionless amplitude w/sqrt rho=a_0. Introducing this value of the amplitude velocity into the metric and dividing through by ds^2 result in

1= (da/ds)^2/(w^2-rho a^2)-r^2a^2(d phi/ds)^2.

The geodesic equation,

da/d phi=wr a_0(1-a^2/a_0^2)

is obtained by differentiating with respect to a. A simple integration leads to

a=a_0 sin(wr phi),

with constant linear velocity wr. This is the equation of a circle of radius a_0, and coincides with the geodesics in the Poincare metric which are semi-circles that become straight lines when phi=0. These semi-circles cut the disc orthogonally, and represent paths of shortest distance.

This is an example of Lobachevsky's axiom that there exist for some line and some point not on it for which at least two lines through the point that do not intersect the line. The "line" here is meant to be a set of interior points of the disc that lie on a circle which intersects the disc orthogonally. Included in this are circular arcs of infinite radius, or straight lines, phi=0.

Now the two radii of curvature, k and rho, are not independent of one another but are related through

a*2=k- rho a^2.

It is necessary that k (=w^2)>0 so that the three types of universes, open, closed, and flat, are a myth. The Ricci tensor components R_{00} and R_{rr} are negatives of one another so that the pressure, as identified from Einstein's field equation is negative! One necessary condition of a nebula is that the pressure be positive throughout. The pressure cannot change sign at any point of the interior, because its vanishing determines the boundary. Its directional derivative normal to the surface also vanishes which means a neutralization of gravitational attraction. Ultimately, the latter becomes outbalanced at which point centrifugal and tidal forces cause a fission of the nebula in which matter is ejected in the form of rings. This Laplace used as a mechanism for the formation of the planets and the sun. Actually, he added mathematical substance to an earlier idea of Kant, but that's how progress is made. The rings which separated from the main body of the nebula condensed into planets and their satellites with the remainder forming the sun. (see images below).

Whether or not Laplace's nebular hypothesis for the formation of the solar system, or whether it applies to the universe as a whole, is valid is irrelevant. What was supposedly a model of the early universe--the Friedmann model, together with a non-existent critical density which occurs when w=0, has nothing whatsoever to do with the putative expansion of the universe coming in three different varieties. The sectional curvatures are not arbitrary, but, rather, well-defined. The sectional curvature, w^2, does not even enter in determining the components of the Ricci tensor, let alone the three scenarios for the evolution of the universe. Only rho, the inverse of the square of Newton's free fall time is involved; however, a constraint is necessary that involves the product of the angular speed and the free fall time.

Consequently, the physical mechanisms of rotation and gravitational attraction are responsible for the sectional curvatures and determine the evolution of the nebulae. Only the latter enters into the expression for the components of the Ricci tensor which don't--and shouldn't--be set equal to the density and pressure. Einstein's field equations are inaccurate. Setting the "wooden shack" (the energy-stress tensor) equal to the marble edifice (Einstein's modification of the Ricci tensor so that its divergence vanishes), as Einstein termed, it is neither necessary nor correct. The proof lies in the fact that negative pressures don't exist, and this is what is implied by the Einstein field equations. Their justification was based on the fact that their divergences vanish. But vanishing divergences don't determine the tensor, and the vanishing of the Einstein tensor is a consequence of one of the Bianchi identities which always holds, but the vanishing of the energy-stress tensor depends on a specific model. The two are not on the same level. There is no need of the latter since all the physics is contained in the sectional curvatures that depend on the full Riemann tensor, and not its contraction, the Ricci tensor.

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