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Sectional Curvatures in an Expanding Universe

Updated: Jul 6, 2021

In what appeared to be a ground-breaking paper, Hawking ["Perturbations of an Expanding Universe" ]intended to study "perturbations of a spatially homogeneous universe...in terms of small variations of the curvature...Density perturbations grow relatively to the background, but galaxies cannot be formed by the growth of perturbations that were initially small."


By the analysis of sectional curvature, I will show that these results are completely untenable. And to make matters worse, Hawking did not even stop to check to see if his solutions satisfy the sectional curvatures!


What seemed surpising, at first, was that the second-order equation for the conformal factor C was integrated with respect to proper time, \tau,

dt/d\tau=1/C

so that the metric

ds^2=-d\tau^2+C^2d\gamma^2,

where d\gamma^2 is "the line element of a space of zero or unit positive or negative curvature, could be written conformally as

ds^2=C^2{-d\tau^2+ d\gamma^2}.

The "equations of motion" can be expressed in terms of the conformal factor as

\mu'=-(\mu+p)3C'/C

3C''/C=-(1/2)(\mu+3p),

where \mu is the mass density, p is the hydrostatic pressure, and the prime means differentiation with respect to proper time. (On could question the relation between proper and coordinate time between the two metrics, but we won't bother ourselves which such "peculiar" queries.)


Hawking then goes on to describe two scenarios: p=0 (dusty universe) and p=\mu/3 (radiation dominant universe), not realizing that \mu is mass density and not the internal energy density. The conformal factor is treated as a volume. In the first case, \mu=M/C^3, with the total mass M as a constant. The equations of motion reduce to

(3/M)C''/C=1/2C^3, (3/M)C'^2-1/C=E,

where E supposedly stands for twice the total energy. Hawking distinguishes 3 scenarios with E>.<,=0.

But these do not correspond to the above two equations of motion. The first equation of motion is an identity, while the second equation reduces to the first of the above two equations. The second equation is its first integral! So the question then is: How can the solutions depend on the sign of E?


Supposedly, the mass M is kept constant when the first equation is integated to obtain the second. The non-Euclidean solutions that Hawking finds are hyperbolic sine (E>o) and sine (E<0), hyperbolic and elliptic cases, respectively. These have constant sectional curvatures. As far as the first of the two equations is concerned, only the E>0 is allowed meaning that the density \mu=M/C^3=constant. The solution C~cos\tau would need a negative sign in the first equation, independent of the second equation and the sign of E.


(Again their is a pecularity in the Hawking reports the solution in terms of lab time t, and not proper time \tau. But this can't be right because dC/dt=dC/d\tau C, and that won't satisfy the second equation. It suffices to consider his case E=0, C=Mt^2/12. Now C'=(M t/6)/C, but C'' does not equal (M/6)/C^2 because C must also be differentiated in the denomator, and this gives the term -M/3C^2. Adding the two together gives C''=-M/6C^2, which has the wrong sign. )


So, Hawking is really considering the case of constant density M/C^3 and not M=const. Moreover, his solution

C=(1/2E){cosh[(EM/3)^1/2 t]-1}

does not satisfy anything. Moreover, the sectional curvatures would necessarily have to satisfy

C'^2/C^2-EM/3C^2=2C''/C,

which is not

(C'/C)'= - integral of (d\tau/C^2). (*)


In the radiation case,

C''/C=-(M/3C^4)

which upon integration gives

C'^2/C^2-EM/3C^2=M/3C^4=-C''/C.

What was positive radial curvature in the dust cast has now become negative. Moreover, the sectional curvatures do not comply with (*), so they cannot be constant sectional curvatures as Hawking's results

C=sinh t/E for E>0 and

C=-sin t/E for E<0.

Observing the relation between sectional curvatures and the Euler-Lagrange equations, any contradictions to the former would imply non-geodesics paths that the latter imply.


The radial curvature,

C''/C= - 1/E^2C^4

does not coincide with Hawking's result

C''/C= - M/3C^4.

His solution is independent of the mass. In any case, the radiation case cannot be in terms of constant mass density since \mu=M/C^4.


Regarding his density perturbations, for p=0 he obtains

\mu'=-\mu\theta, \theta'=-\theta^2/3-\mu/2, (***)

where

\theta=3C'/C.

The first equation is a mere identity while the second

C''/C= -M/6C^3

has the wrong sign! The radial sectional curvature is positive. Thus, he can

conclude nothing about the evolution of the universe in terms of density perturbations, which are not really perturbations at all! Hawking calls (***) an expansion equation. Rather, I would call it the Jacobi equation. In the case of the negative sign means that geodesics are contracting.


The full expansion equation,

\theta'=-(1/3)\theta^2-(1/2)(u+3p),

has the pressure entering in the same way as the mass density. This is obviously incorrect since positive pressure creates expansion. It also explains the ludicrous nature of the inflationary scenario which implicates negative pressure as the cause of expansion. This can be traced back to the incorrect form of the mass-stress tensor for a perfect fluid in general relativity. Parenthetically, it is interesting to note that the first term in the expansion equation cancels the tangential sectional curvature component, leaving pure radial sectional curvature.


Note that the conclusions we have arrived at on radial sectional curvature is independent of the sign of the constant of integration when that equation is integrated. And being an arbitrary constant of integration, it can be chosen at will (even being set equal to zero) without affecting the radial sectional curvature. So, if we are to attach a physical significance to the constant of integration, the tangential sectional curvature must be considered as distinct from the radial sectional equation, which is its derivative.


So by changing from radiation to dust, was was contracting is expanding--even without the help of pressure! When dealing with gravitational radiation, the gravitational energy density enters with the same negative sign as the mass density so it should have the same effect of drawing neighboring geodesics nearer to one another. What then is the cause of the expansion of the universe? According to the inflationary scenario, it is due to negative pressure. But, in Hawking cases at least, the pressure is either zero or positive (radiation pressure). And, yet, he makes the claim that the universe is expanding. This is a classic example where equations can be manipulated to suit the foregone conclusions. Is there any connection between the two?


As a final point, the fact that the radial and tangential sectional curvatures sum to zero, according to (*), should be related to the fact that the vanishing of the Schwarzian

(C''/C')'=(1/2) integral (C"/C')^2 (**)

vanishes for cross-ratio. In fact, the cross-ratio is expressed in terms of hyperbolic sine and sine in the cases the geometries are hyperbolic and elliptic, respectively. It is well-known that the Schwarzian is a projective invariant, because its vanishing leaves the cross-ratio invariant, but, the sectional curvatures it involves third order derivatives in comparison with the of second-order derivatives of sectional curvatures.


The Schwarzian satisfies a Hill equation, like the Jacobi equation so it should make a statement on the divergence or contraction of neighboring geodesics. The reason for believing that the Schwarzian may, in fact, be more fundamental rests on the following: Geometric transformations involve rotations and translations, and the most important invariants are angle and distance, respectively. For conformal transformations the invariance of angles is most important. Whereas, for projective transformations, the most important invariant is the cross-ratio. Deviations from the cross-ratio can be expressed in terms of the Schwarzian.

According to Mundy, it seems likely that all invariant properties of a geometrical configuration can ultimately be interpreted in some number of cross-ratio constructions. Thus, in some way, the Schwarzian should supersede the radial and tangential sectional curvatures in providing the most satisfying, if not complete, description of geometric configurations.




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