domingo, 7 de junio de 2015

From Newton to 5-Dimensionality, Hamilton-Jacobi, Black Hole Cosmology, One Robertson-Walker Universe

Part 1 - From Newton to 5-Dimensionality, Hamilton-Jacobi, Black Hole Cosmology, One Robertson-Walker Universe

by Jacques Chauveheid
Version X-7b
Contrasting with dark energy cosmology seemingly viewing the universe as a star whose gravitation exclusively acts on its own source, factors such as motivation for inflation and a two-body configuration lead to Schwarzschild’s cosmological time coupled with the source mass of a second body constituted by a primeval black hole of negative mass-energy. To the end, we follow Einstein’s geometric approach in his first (static) universehereafter referred to as Einstein´s universe (anterior to that with cosmological constant in 1917), whose 3-space metric in standard form is structurally identical to that in the spherical Robertson-Walker metric.

Key-Words: Lemaître, Einstein's Universe, 5 dimensions, Interior and exterior cases, Cosmic gravitation, Newton, Hamilton-Jacobi, Universe wave-function, Vinti, Primeval black hole, Repulsive gravity, Symmetry breaking, Double big bang, Cosmological time, No maximal symmetry.

Figure A

I-A Overview

The expansion of the universe was discovered in 1927 by Georges Lemaître [1-A], followed by Edwin Hubble in 1929. To simplify the particulars about the independent works on spacetime metric by Friedman, Robertson and Walker, what follows uses the Robertson-Walker metric, hereafter referred to as .

1. This paper attempts to uncover Einstein´s reasoning and the implications of his first static universe [1-B], according to general relativity formalized by him on November 25th, 1915. Einstein´s first universe essentially focused on the three dimensions of space, because Einstein defended the concept of a stable universe, which implied the absence of time coupling in the term c2dt2 of his metric. Admissible in Einstein´s time, this situation persisted until now, despite the present search of new force(s) pushing for expansion, possible dark energy, etc. - situation about which we foresee the possibility of repulsive gravity originating in negative black hole mass-energy.

2. However, three spatial dimensions showed insufficient for Einstein to close theoretically his model of universe, so that a 4th dimension of space was imperative (calculations in Section IV): This paper details Einstein´s footnote in which he referred to a Euclidean 4-dimensional space as an , because Einstein used it only once. Besides, Einstein, who just finished 4-dimensional general relativity, could hardly take seriously the addition of a 4th spatial dimension, to become five dimensions when adding the time, this situation causing that nobody would then swallow five dimensions in general relativity already put in doubt by the scientific community in Einstein´s time, situation lasting until 1965 in European universities (personal remembering).

3. In some contrast, the 4th space dimension x4 only figures in next equations (10) to (16), its explicit presence being eliminated in the final equations, so that Einstein´s universe looks 4-dimensional, as current general relativity. However, the 4th spatial dimension x4 (part of the 5 dimensions that include time) remains implicitly present in all possible cosmological models, according to Figure A showing all values of the universe radius R, from the big bang (R = 0) to the today R value. This universe radius R is time-dependent, as well as the variable universe 3-hypersurface, but both cannot be interpreted as kind of imprecise time measures, because the time-dependent circumference exclusively represents the universe 3-hypersuface, which did expand since the time zero until now. Importantly, this circumference also expanded over the whole Finite Euclidean 4-dimensional Space in Fig. A (5-dimensional when including time), as well as in Eq. (28), which proposes a completed R-W metric.
4. Since explaining the expansion of the universe, the R-W metric was considered incomparably superior to what Einstein did in his seemingly primitive first universe. However, the R-W metric was theoretically derived by Einstein in 1916 (eleven years before Lemaître), as evidenced in Section IV, the only change required for theorizing an expanding universe being nothing more than making Einstein´s universe radius time-dependent (one word difference). Moreover, Einstein´s procedure appears today indispensable to improve the outdated R-W metric, exemplified in Section V (Eq. 28).

5. The problem of universe expansion is mainly mechanical because the universe is essentially an electrically neutral, massive, object, hardly anything else for now. So, the use, in Section III, of Jacobi´s mechanics whose Hamiltonian (energy) covers Hamilton’s contribution. This somewhat differs from the Hamilton-Jacobi equation, basis of celestial-orbital mechanics, whose more abstract character seems less helpful when looking for something a bit novel. Using this method quickly evidenced the negative mass of the primeval black hole preceding universe apparition, negativity used to complete the R-W metric in Section V. According to this, billions years ago, this black hole formed our universe through ejection achieved by repulsive gravitation caused by the negative black hole mass acting gravitationally on our universe, just created with a positive mass. In this view, this giant black hole is still located outside our universe, at the exact center of Figure A. We will never observe it in a naive fashion, but this black hole had all reasons to exist and still does.

During the last fifty years or so, energy density estimates of the universe varied between 1.4 % [2] to 2.5%, even 5 % (?), reason why some Internet comments qualify cosmology as erratic, One also reads <spacetime geometry is influenced by whatever matter and radiation are present> (General Relativity, Wikipedia). Both views, not very compatible, reflect the difficulty to conciliate mathematical viewpoints with practical concerns, accepting the possibility that usual metrics do not work too well in cosmology.

However, a legitimate mathematical perspective would mean little if limiting mass-energy to act exclusively on itself, neglecting other sources of gravitation when all sources should be accounted for. In what follows, the radial expansion of the universe corresponds to a two-body system, whose center is occupied by a black hole, according to Schwarzschild´s solutions. By virtue of centered-spherical symmetry, this central black hole has all reasons to remain at the exact place of the big bang that originated our universe. Besides this, gravitational energy, not being explicit in general relativity, impedes the intuition to work as in technology, situation solved by Newton's potential in Jacobi´s formulation depicting cosmology as a typical exterior case implying an adaptation of the R-W metric.

Physical-Mathematical Conventions and Dimensionality

This paper agrees with Einstein's choice of a closed, spherical universe [1-B].

Changing the sign of Riemann´s tensor in refs. [1-B] and [2], Einstein´s equation for gravitation reads in four dimensions

Rab - (1/2)gab.R = K.Tab (A)

where Einstein´s constant of gravitation K is defined by

K = 8πG/c4 (B)

with G being Newton´s gravitational constant. The energy tensor of a perfect cosmic fluid with pressure reads

Tab = (σoc2 + p)wawb + p.gab (C)

(wa ≡ dxa/ds), where σ the rest mass density and p is the pressure.

In conformity with current terminology, the word often refers to spatial dimensionality. For example, a 3-dimensional hypersurface is in reality 4-dimensional when referring to time. Both Einstein´s finite and infinite 4-dimensional Euclidean spaces in Fig. A are in reality 5-dimensional, due to the presence of time, etc.

The organization of this paper is the following. Section II is about the interior and exterior cases in cosmology. Section III details the connection between cosmic gravitation and the Hamilton-Jacobi formalism used in orbital-celestial mechanics. Section IV recalls the origin of the Robertson-Walker metric in Einstein's universe. Section V proposes a variant of the R-W metric.

In unifying attempts during the 1919-1955 period [3], the interior case refers to the equations in matter (sources and particles), the exterior case being that of the (unified) field outside sources and particles. At the micro level, Tonnelat's comments are about the dichotomy between two distinct field structures, inside and outside a massive (charged) elementary particle [4]. At an intermediate level constituted by galactic and intergalactic configurations (Part 2), both structures coexist on a same footing. At the cosmological level, what follows is about the formal separation between interior and exterior cases.

1) The Interior Case
The interior case, obviously with no visible center, is that viewed by an observer inside the universe in free fall [2], geometrically represented by a 3-dimensional isotropic hypersurface, with flat Euclidean geometry [1-B]. Here, Einstein´s word means A = B = 1 in the metric

ds2 = B(c2dt2) - A[dr2 + r2(dθ2 + sin2θ.dφ2)] (1)

as in special relativity, which expresses the essence of general relativity based on the equivalence principle (Einstein [1-B]).

2) The Exterior Case
In Fig. A, Einstein´s Euclidean 4-space appears inside and outside the circumference representing the 3-hypersurface of zero thickness, according to Einstein´s phrase <A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions...[1-B]. However, Einstein´s footnote <The aid of a fourth dimension has naturally no significance except that of a mathematical artifice>. A bit curiously, Einstein underrated higher dimensionality by referring to it as an , which is understandable because he only used it once. Nevertheless, 5-dimensions including time led to calculations that became a written part of theoretical physics. In accordance with this, what follows maintains Einstein´s original 5-dimensional vision and its corresponding geometry.

In the exterior case, the universe 3-hypersurface is mathematically conceived as an idealized, although useful, approximation of isotropy and homogeneity by virtue of two parameters, the mass density and pressure of a perfect cosmic fluid with pressure in the right member of Einstein's equation. Accordingly, in the exterior case the universe 3-hypersurface is acted upon by cosmic gravity working along cosmic geodesics determining the gravitational potential -Gm/R of the 3-hypersurface, m being here a negative gravitational mass detailed subsequently. R is the increasing radius of the universe 3-hypersurface. Acted upon by cosmic gravitation, the universe 3-hypersurface expands accordingly. Astronomers and cosmologists imagine therefore the universe as if they were located at some distance from it, which is the essence of the exterior case, described by the R-W metric portraying Einstein's conceptual method he used to imagine his first universe [1-B].

Although based on equations providing acceptable average values at high scale, the image of a continuous perfect fluid with pressure by definition does not detail discontinuous and complex objects such as systems of stars and planets. In relation with the big bang image, still with us in the present universe, obtaining more than rough average values is hardly viable for galaxy clusters contrasting with organized systems of stars such as anisotropic centered-spherical spiral galaxies. Moreover, Einstein's embedding a 3-dimensional universe-hypersurface in a 4-dimensional Euclidean universe (5-dimensional when including time) may look like a thought experiment, when not realizing that Einstein´s vision became highly physical when his universe radius became time-dependent a few years later (Alexander Friedman, 1922).


1. What follows does not recall the Lagrangian derivation of equations and the Hamiltonian formalism not really used here, except for next trivial equation (2). In this view, the Hamilton-Jacobi formalism reduces here to Jacobi´s equations providing a sufficient basis.

2. From a physical standpoint, the structure Y = 2Gm/rc2 in Newton´s potential
-Gm/r plays a crucial role in gravitation theory because being exact in Newton´s theory, as well as in general relativity (see below). This Newton potential appears somewhat underestimated in the static case because retrieved in a approximation from Einstein´s field equation of gravitation. However, this approximation only applies to A in Eq. (1) since giving A ≈ 1 + Y, instead of
A = 1/(1 - Y) for Y considered small [1-B]. This approximation is corrected by the exact Schwarzschild solution that maintains B = 1 - Y as exact solution in the static case, independently of any approximation.

Consequently, if wishing a universe expanding in relation with a central black hole located very close to the big bang place on Fig. A, it seems a bit hard to avoid the negativity of black hole mass, which would then induce repulsive gravitation acting on the universe. Moreover, as in the static case, Newton´s gravitational force will be maintained as a typical central force (distance-dependent - gravitation is static here) during the radial expansion of the universe within a centered-spherical configuration leaving no room for  [5].

Jacobi´s Equations
Einstein's Euclidean 4-space corresponds to the exterior case describing an expanding universe, which therefore also works with the Hamilton-Jacobi formalism [6, 7], base of celestial mechanics detailed in the Preface of Vinti’s book [8]: <...The Hamilton-Jacobi equation, which in modern physics provided the transition to wave mechanics, is now seen as the starting point for the Vinti spheroidal method for satellite orbits and ballistic trajectories...>.

Visualizing this in special relativity, the constant Hamiltonian (energy) of the universe reads

H = Mc2/(1-v2/c2)1/2 - GmM/r = k (2)

where k is a constant, M being the rest mass of the universe. The second mass m is the exterior source mass corresponding to the negative mass-energy of the central black hole, present in Schwarzschild´s exact solutions. Respecting mass-energy conservation, this central black hole originated our universe and caused its expansion through the repulsive gravitation implied by its negative mass in Newton´s old gravitational force. Moreover, the variable r in the positive potential energy
-GmM/r is not exactly the initial universe radius in the exterior case (see below). According to Eq. (2) and the Schwarzschild solution of a static gravitational field, the central black hole is located outside our universe, at the exact place of the big bang characterizing its original formation.

The variable r only constitutes an approximation of the universe radius R because a vanishing universe radius r (r = 0) at the origin of the expansion would oppose a necessary finite rest mass M of a beginning universe, which works in accordance with spherical symmetry that would then imply infinite gravitational (potential) energy, unacceptable because k is constant. The universe rest mass M and its radius R, therefore necessarily finite at the big bang, define the time zero as the exact beginning of universe expansion, which implies r ≠ 0 in the coordinate change

r = X + R (3)

where the condition R = 0 defines R as the vanishing initial radius of an expanding universe, X not being detailed here. For R = 0, the positive potential energy
-GMm/r would therefore read -GMm/X at the time zero t = 0.

In line with Eq. (2), Jacobi's writings, describing the motion of a massive body (period 1834-1843), introduced the action whose infinitesimal variation δS reads

δS = p.δx - Wδt (4)

where W is the energy, p being the 3-momentum. Eq. (4) implies

px = ∂S/∂x ; W = -∂S/∂t (5)

according to

S = p.x - Wt (6)

with the particular wave-solution [6]

ψ = exp[iS/ħ) (7) (ħ ≡ h/2π , i = √-1)

defining de Broglie's wave ψ, interpreted here as universe wave-function. In the exterior case, the universe is essentially a massive body whose radial motion implies its vanishing angular momentum, so that (7) also reads

ψ = exp[i/ħ(pR - Wt)] (8)

where R is the universe radius. In addition, the action S is an invariant in

S ≡ p.x - Wt = pμxμ + (iW/c)(ict) (9)

(x4 = ict, Greek indices refer to space), The concept of universe wave-function is not new [9], as theorists felt free to enlarge Bohr's correspondence beyond microphysics, according to quantum reality coexisting with classical physics everywhere, which is simpler than limiting the application of Bohr's correspondence principle.

According to this, probability densities follow the radial lines of cosmic gravitation causing the expansion of the universe. Separations between centers of mass of systems of stars such as galaxies maintain their radial alignment along the lines of cosmic gravity related to fixed stars. Voids between galaxy clusters only enlarge according to the expansion. In other words, after the big bang the universe kept the general configuration it had at the time zero.

Furthermore, a first unstable universe with positive mass, produced by the central black hole (inseparable part of Schwarzschild´s solutions) seems conceivable as having the same symmetry as the black hole. Thereafter, the subsequent change of geometry, from centered-spherical anisotropy to spherical isotropy, provoked a chaos caused by this symmetry breaking, adding to the first explosive emission of hot matter from the black hole, both successive phenomena constituting a double big bang whose image, enlarged through radial expansion, should correspond to that of the large-scale structure of the present universe. As introduced above, this hypothesis seems consistent with observed voids and other irregularities in astronomical pictures, the North-South blue haze in a picture of the 2MASS Project giving the impression of a possible footprint of this symmetry breaking?

Regarding black holes, the reported story (lost references) was that Gold, Bondi and Hoyle founded their Steady State Theory (1948) on observations of violent explosions from the black hole Sagittarius A (about 4 million solar masses), located at the center of our galaxy, in <...violent events do seem to be occurring in the nuclei of many galaxies, so galactic nuclei seem like natural candidates for the location of continuous creation[2]. However, at first sight black holes at centers of spiral galaxies do not seem very similar to the negative mass-energy black hole having here originated the big bang. Moreover, non expanding galaxies (Part 2) seem to emit as much mass-energy as what they absorb (approximation), in possible relation with the hypothesis of gravitational radiations of positive and negative energies [10], on which more work needs to be done. However, galactic black holes look like small big bangs, according to Prigogine´s comment on French speaking TV (around 2003?, not reported on Internet). As a conclusion, we will only understand black holes and their mystery when knowing their field structure.

Consider a spherical 3-dimensional hypersurface defining the universe embedded in a 4-dimensional Euclidean continuum [1-B] according to

xβxβ + x4.x4 = a2 (10)


dl= dxβ.dxβ + dx4.dx4 (11)

(β = 1, 2, 3 - Greek indices refer to space), where a is the constant curvature radius, dl2 being the squared infinitesimal 4-dimensional distance between two neighboring points. Differentiating (10) gives

xβ.dxβ + x4.dx4 = 0 (12)


dx4 = - (1/x4)xβdxβ (13)


(dx4)2 = (1/x4)2.xαxβdxαdxβ (14)

Since Eq. (10) gives

(x4)= a2 - xβxβ (15)

Eq. (14) becomes

(dx4)2 = [1/(a2 - xγxγ)] xαxβdxαdxβ (16)

so that Eq. (11) reads

(dl)2 = {δαβ + [xαxβ /(a2 - xγxγ)]}dxαdxβ (17)

where δαβ is the Kronecker symbol.

Because important, we recall Einstein´s footnote referring to Eqs. (10) and (11) [1-B]: he aid of a fourth space dimension has naturally no significance except that of a mathematical artifice>
. Although agreeing with Einstein's approach, we do not go as far as ratifying his prudence undervaluing a correct mathematical operation, calling it an . Quite the opposite, embedding the 3-dimensional universe in an Euclidean 4-space refers to the work of a theorist describing the exterior case by drawing and calculating a universe on a piece of paper. Doing this, the researcher imagines he is in a 4-dimensional Euclidean space, which will not modify the equations he writes. This context was that of Einstein's mind, enabling him to visualize the 3-dimensional isotropic curved hypersurface of his first static universe, from which the Robertson-Walker metric derives, detailed now. Differentiating the definition

xβxβ ≡ r2 (18)


xβdxβ = rdr (19)

so that writing dxβdxβ in spherical coordinates

dxβdxβ = dr2 + r2(dθ2 + sin2θ.dφ2) (20)

puts Eq. (17) in the form

(dl)2 = dr2 + r2dr2/(a2 - r2) + r2(dθ2 + sin2θ.dφ2) (21), or

(dl)2 = a2dr2/(a2 - r2) + r2(dθ2 + sin2θ.dφ2) = dr2/(1 - r2/a2) + r2dΩ2 (22)

where 2 is the usual notation for 2 + sin2θ.dφ2.

Through the change of variables r*= r/a, (22) leads to

(dl)2 = a2[(dr*2/(1 - r*2) + r*2.dΩ2] (23)

where r* is dimensionless. The standard form [11] of Eq. (23) then reads

(dl)2 = a2dr*2/(1 - r*2) + r*22 (24)

where a is Einstein’s universe radius. The right member of Eq. (24) is the 3-dimensional part of the R-W metric. Since not multiplied by a2, the dimensionless term r*22 in the standard form is somewhat devoid of physical meaning, so that the null-value of the R-W metric in standard form is not generally applicable to the calculation of the speed of light. However, the conversion of (23) to (24) corresponds to a coordinate transformation [2] not affecting the field equations by virtue of general covariance, the motive being therefore a simpler derivation of the field equations.

Discarding maximal symmetry [2], the Schwarzschild expression of B in Eq. (1) is

B = (1- 2Gm/rc2) (25)

(m < 0), where m comes from the potential -Gm/r in Eq. (2). Due to the coordinate transformation r = a.r*, where a is Einstein´s universe radius of his first static universe described in Eqs, (10) to (24). Replacing now a by the usual symbol R for a time-dependent universe radius according to r = R.r*, Eq. (25) turns into

B = (1- 2Gm/Rr*c2) (26)

The standard form of the completed R-W metric thus reads

ds2 = c2[1- 2Gm/Rr*c2]dt2 - R2[dr*2/(1- r*2)] - r*22 (27)

Suppressing the unnecessary asterisk then gives

ds2 = c2[1- 2Gm/Rrc2]dt2 - R2[dr2/(1- r2)] - r22 (28)

where r is dimensionless.

[1-A] G. Lemaître: Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques Comptes Rendus Soc. Sci. Bruxelles, 49, April 25, 1927), p.11.

[1-B] A. Einstein: The Meaning of Relativity (Princeton University Press, 5th ed., NJ 1955), pp. 86, 90, 100, 102, 104, 111.

[2] S. Weinberg: Gravitation and Cosmology (Wiley, NY 1972), pp. 87, 177, 300, 302, 412, 413, 460, 471, 491, 613.

[3] M.-A. Tonnelat: Les Théories Unitaires de l'Électromagnétisme et de la Gravitation (Gauthier-Villars, Paris 1965), pp. viii, ix, xix.

[4] J. Chauveheid: A Proposal for a Quadratic Electromagnetic Coupling Based on the Underlying Philosophy of Einstein-Maxwell Theory, Phys. Essays, vol.10, Nr 3, p. 474 (1997).

[5] R.M. Wald: General Relativity (Chicago University Press, 1984), p. 78.

[6] W. Yourgrau and S. Mandelstam: Variational Principles in Dynamics and Quantum Theory (Dover, NY 1979), p. 123.

[7] J. Chauveheid and F.X. Vacanti: Schrödinger and the Hamilton-Jacobi Equation. Phys. Essays, Vol. 15, Nr 1, p. 5 (2002).

{8] John Vinti, in

[9] J. Hartle and S. Hawking: Wave function of the Universe, Phys. Review, D28, (12): 2960 (1983).
[10] F.X. Vacanti and J. Chauveheid: Is Radiation of Negative Energy Possible?. Physics Essays, Vol. 15, Nr 4, p. 387 (2002).

[11] Y. Choquet-Bruhat: General Relativity and the Einstein Equations (Oxford University Press, NY 2009), chapter 5, section 4, Eq. 4.26.