From Newton to
5-Dimensionality, Black Hole Cosmology, Hamilton-Jacobi, One Robertson-Walker
Universe (Part 1)
by Jacques
Chauveheid
Version
X-8c
Figure A
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Abstract
Contrasting with dark energy
cosmology seemingly viewing the universe as a star whose gravitation
exclusively acts on its own source, a two-body configuration leads to
Schwarzschild’s cosmological time coupled with the negative source mass of a
primeval black hole. To the end, we follow Einstein’s geometric approach in his
first (static) universe, hereafter
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's primeval atom, Einstein's universe, Vacuum
energy, Higgs, Tonnelat, Hamilton-Jacobi, Vinti, Dirac, Repulsive gravity, Double big bang
I-A Overview
In 1927, Georges Lemaître
discovered the expansion of the universe [1-A], confirmed by
Edwin Hubble in 1929. To simplify the particulars about the independent works
on spacetime metrics by Friedmann, Robertson and Walker, what follows uses the
Robertson-Walker metric, hereafter referred to as R-W metric.
1. This paper intends to uncover Einstein´s
reasoning and the implications of his first static universe [1-B] according to general relativity
formalized by him in November 1915. Einstein focused on the dimensions of space
to construct a 3-dimensional metric, to which he added the source-free term c2dt2
from special relativity, because the unique source of gravitation was the mass
of the universe itself (no exterior source). Admissible for a non expanding
universe, this situation did last until now, in spite of the search of new
force(s) pushing for expansion, possible dark energy, etc.
Nevertheless, three
spatial dimensions were insufficient to close theoretically Einstein's first universe, so that a 4th dimension of
space was imperative (calculations in Section V): What follows details
Einstein´s footnote in which he referred to a Euclidean 4-dimensional space as
an artifice, because he used it only
once. Moreover, Einstein, who just finished 4-dimensional general relativity,
could hardly take seriously four dimension of space, to become five with the
time, so that nobody would swallow
five dimensions in general relativity already put in doubt by the scientific
community in Einstein´s time, situation lasting until 1965, eventually more
(?), in European universities (personal remembering).
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, this
dimension x4 remains implicit 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 expanding 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).
Since explaining the
expansion of the universe, the R-W metric was considered incomparably superior
to what Einstein did in his seemingly primitive universe. However, the R-W
metric was theoretically derived by Einstein in 1916 [1-B] (roughly eleven years before Lemaître), as evidenced in
Section V, 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 VI.
2, We foresee
the possibility of repulsive gravity (see further) originating in the negative
mass of a primeval black hole, which here constitutes the unique initial
condition. When emitting subsequently a definite quantity of positive energy to
build a spacetime with a universe, the corresponding amount of black hole
negative energy will raise of the same absolute quantity (negative number of
energy units), by virtue of strict mass-energy conservation. To produce the
universe with its spacetime of constant total mass M,
this operation can be executed once, avoiding continuous, or sequential, energy
"creation" perturbing the universe equilibrium, for example with small
big bangs, etc. Somewhat summarizing, all this is about the universe and its
spacetime originating from less than nothing, in reference to the negativity of
primeval black hole mass.
The problem of universe
expansion is mainly mechanical because the universe is essentially an
electrically neutral, massive, object, hardly anything else for now. This
justifies the use, in Section IV, 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 is
less intuitive when looking for something a bit novel. Using this method
quickly evidenced the negative mass of the primeval black hole necessarily preceding
the apparition of the universe, this negativity being used to complete the R-W
metric in Section VI. 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 formed
with a positive mass. As a result, 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
today.
I-B Introduction
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 do not look very compatible although the
Wikipedia vision seems hard to dismiss. This situation reflects the difficulty
to conciliate mathematical viewpoints with practical concerns, besides accepting
that usual metrics are problematic 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 accommodates a black hole,
according to Schwarzschild´s solutions, (gravity is static here). By virtue of (static)
centered-spherical symmetry, this central black hole remained at the exact
place of the big bang that originated our universe. Besides, 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 from 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 then reads
Tab
= (σoc2 + p)wawb + p.gab +
λ.gab (C)
(wa ≡ dxa/ds), where σo.is the rest mass density of matter, p is the usual pressure. The scalar field λ represents the positive energy density of vacuum, expected
to be R-dependent and related to the Englert-Higgs field
in vacuum.
In conformity with
current terminology, the word dimension
most often refers to spatial dimensionality. For example, a 3-dimensional
hypersurface is in reality 4-dimensional when including time. Einstein´s finite
4-dimensional Euclidean space in Fig. A is 5-dimensional, due to the presence
of time in the metric, etc. According to details above, the infinite Einstein
Euclidean 4-space might simply be replaced by nothingness since apparently playing
no role in the expansion of the universe.
The organization of
this paper is the following. Section II is about the interior and exterior
cases in cosmology. Section III details some generalities. Section IV
introduces the relation between cosmic gravitation and the Hamilton-Jacobi formalism.
Section V recalls the origin of the Robertson-Walker metric in Einstein's
universe. Section VI proposes a variant of the R-W metric.
II. Interior and Exterior Cases in
Cosmology
In unifying attempts
during the 1919-1955 period [3], the
interior case refers to the equations in matter (sources and particles of finite
size), the exterior case being that of unified field(s) outside sources and
particles. At the micro level, Tonnelat's comments evidence the dichotomy
between two distinct field structures, inside and outside a massive (charged)
elementary particle [4].
The galactic case is
not detailed here (Part 2). At an intermediate level constituted by galactic
and intergalactic configurations, both structures coexist on a same footing. In
cosmology, what follows is about the formal separation between interior and
exterior cases.
1. The Interior Case
The interior case,
without unique center since isotropic, is that viewed by an observer inside the universe in free
fall [2]. This observer differs formally from an outside observer such
as a cosmologist characterizing the distinct exterior case. One represents the interior
case by the isotropic 3-dimensional hypersurface (three space dimensions) constituting
the flat geometric background of locally vanishing cosmic gravitation [1-B], this hypersurface being 4-dimensional
when including time. Einstein´s word Euclidean
thus means A = B =
1 in the 4-dimensional metric
ds2
= B(c2dt2) - A[dr2
+ r2(dθ2 + sin2θ.dφ2)] (1)
written in spherical
coordinates in special relativity. This expresses the essence of general
relativity based on the equivalence principle (Einstein [1-B]).
2. The Exterior Case Defining Cosmology
In Fig. A, Einstein´s Euclidean
4-space appears inside and outside the circumference representing the
3-hypersurface of zero thickness, according to the words
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
evidences that Einstein underrated higher dimensionality by referring to it as
an artifice, which is understandable because he only used it once.
Nevertheless, 5-dimensions including time led to exact calculations that became
a written part of theoretical physics. In what follows, there is therefore no
basis for discarding Einstein´s original 5-dimensional vision and its corresponding
geometry.
In the exterior case, the
universe 3-hypersurface is 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 in the right member of
Einstein's equation. Accordingly, the universe 3-hypersurface is acted upon by
cosmic gravity working along radial cosmic geodesics determining Newton's gravitational
potential
-Gm/R on the 3-hypersurface, m being the negative mass of the primeval black hole. 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 reflecting Einstein's
method used for 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 Friedmann, 1922).
III. Generalities
1..The structure Y = 2Gm/rc2 in Newton´s potential -Gm/r plays a crucial role in gravitation theory because exact
in Newton´s theory, as well as in the static case of general relativity (see
below). This Newton potential seems underestimated since retrieved in a weak
field approximation from Einstein´s field equation of gravitation. However,
this approximation only applies to A in Eq. (1)
since leading to A
≈ 1 + Y,
instead of A = 1/(1 - Y) for Y considered
small [1-B]. This approximation is
corrected by the exact Schwarzschild solution maintaining B = 1 - Y in the static case, which defines black hole
solutions through B
= 1 - Y, the R-dependence of A defining the expansion.
2. A massive object such as a star is
expected to expand, or contract, according to a metric including a time-dependent
radius. We might refer to that as "forced expansion, or contraction",
possibly occurring with the R-W metric not including a source of gravitation,
issue related to dark energy (similar problem with dark matter in galaxies).
3. As in the solar
system, a static gravitational field deals with various motions of massive
matter, so that the word "static" refers to a static gravitational
field, physical concept unrelated to eventual expansion of matter. Not omitting
the central black hole present in static (spherical) gravitational
configurations, as in Fig. A, the universe can expand in relation with a primeval
black hole located very close to the previous big bang place in Fig. A. Looking
at this, it seems hard to avoid the negativity of black hole mass inducing
repulsive gravitation provoking the formation of a first universe. Moreover, the
gravitational force is here typically central, meaning exclusively distance-dependent,
causing the radial expansion of the universe in accordance with a static, centered-spherical
field configuration leaving no room for magneto-gravitation
[5].
IV. Exterior Vacuum Case according to Hamilton-Jacobi
Formulation
1. Parenthesis
What follows does not
recall the Lagrangian derivation of Jacobi's equations and the Hamiltonian
formalism not really used here, except next trivial equation (2). In this view,
the Hamilton-Jacobi formalism mainly reduces to the Jacobi equations providing
a sufficient basis in the presently simple situation.
2. The
Hamilton-Jacobi Formulation
In cosmology, the flat Euclidean
4-space plus time in Fig. A corresponds to the exterior case describing the
geometry inside an expanding universe 3-hypersurface, the internal geometry of
this universe constituting the only object of interest in cosmology. In
contrast, the Hamilton-Jacobi formalism [6,
7] exclusively describes the motion of any object of mass M in a continuum, without looking inside this object.
The way general
relativity is commonly developed, its applications do not include the external motion
of a universe, particularly not in a flat continuum. Quite the opposite, the
Hamilton-Jacobi formulation visualizes the motion of this universe in a flat
continuum where all first derivatives of the metric vanish. General relativity might
therefore work in conjunction with the Hamilton-Jacobi formulation. In other
words, we go ahead with this.
The Hamilton-Jacobi theory
also includes the Hamilton-Jacobi equation 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 the Hamilton-Jacobi formulation in 4-dimensional
special relativity, the (constant) Hamiltonian H of the universe reads
H = Mc2/(1-v2/c2)1/2 - GmM/r = k (2)
where k is a
constant, M being the constant rest mass of the universe with
its vacuum energy, by virtue of energy
conservation, The mass m is the negative mass
of a primeval black hole, black holes being included in all Schwarzschild (static)
solutions, which is extended here to radial
expansion. We assume that this primeval black hole originated our universe through
repulsive gravitation implied by the negativity of black hole mass. Moreover, a
vanishing initial radius before expansion would imply infinite potential energy
opposing a necessarily finite rest mass M of the universe, according to k being
constant in Eq. (2). To obviate this difficulty, we consider next scenario detailing
four distinct phases.
The first phase is the apparition of an
anisotropic, centered-spherical, black hole of negative mass, inevitably
related to Dirac's negative energy solutions.
The second phase, introducing a first big bang, describes
the ejection, from the primeval black hole, of a centered-spherical universe with
positive mass. This first universe corresponds to Lemaître's primeval atom (see further).
The third phase presents a second big bang occurring
during the transition from the anisotropic, centered-spherical, universe to a second
isotropic universe through symmetry breaking. The fourth phase marks the start
of universe expansion at the time zero.
Summarizing, this proposed context defines the
time zero as the start of universe expansion occurring after formation of two
successive universes, according to
-GMm/r = -GMm/X (3)
(t = 0), where X is the initial finite
radius of the second (isotropic) universe.
In line with Eq. (2), Jacobi's writings
describing the motion of a massive body (1834-1843), introduced the action S whose infinitesimal variation δS reads
δS = p.δx - Wδt (4)
where W is the
energy, p being the 3-momentum. Eq. (4) is consistent with
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) reduces to
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. All this. not counting the relation between primeval
black hole mass and Dirac's negative energy solutions.
A bit surprisingly, Dirac's equation for the
electron looks like the only theoretical basis sustaining the sudden apparition
of negative black hole mass in a flat, 5-dimensional vacuum spacetime, which eliminates
spinor coupling with gravitation, commonly seen as the major difficulty.
3. Comments
1. 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 double big bang the universe kept the
general configuration it had at the time zero, the CMB (cosmic microwave
background) being distributed quite simultaneously.
2. The instability of the first universe, of positive
mass produced by the primeval black hole, is not detailed here since requiring some
elements of galactic gravitation (Part 2). It is, however, conceivable that this
first universe had the same symmetry as the primeval black hole. Afterward, the
change of geometry, from centered-spherical anisotropy to spherical isotropy,
provoked a chaos caused by this symmetry breaking, which added to the first
explosive emission of hot matter from the black hole. Both (successive)
phenomena constitute 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 ?
3. 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 (roughly 3 or 4 million solar masses), located at the center of
our galaxy. These observations led to comments such as ...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]. But black holes at centers of
spiral galaxies do not have negative mass-energy like the primeval black hole
having here originated the big bang. Moreover, non expanding galaxies seem to
emit as much mass-energy as that 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. Nevertheless, galactic
black holes look like small big bangs according to Prigogine´s comments on
French speaking TV (around 2003 ? - not reported on Internet). As a first conclusion,
we will only understand black holes and their mystery when clarifying their field structure in a (centered-spherical)
galactic configuration.
4. According to Luminet (Internet), Lemaître's
concept of a primeval atom, synonymous to primitive
atom, came out before 1930. Two years later, Lemaître recalled the old concept
of primeval nebula introduced by Kant and Laplace [11], who detailed how a diffused primeval nebula (not yet
Lemaître's atom) filled the whole space and progressively condensed into
partial nebulae, finally producing stars. Lemaître also developed some themes
related to the cosmology of his time, for example mentioning the big crunch in which he did not believe, which is
casually confirmed by the simple Hamilton-Jacobi model detailed above. According
to this model, a stabilization of the universe, after exhaustion of (positive) potential
energy in Eq. (2), will only occur if the primeval black hole mass is constant,
which works according to energy conservation (see below). Somewhat detailing
this, the primeval black hole constitutes an initial condition, apparently contradicting
any conservation principle. However, such principles only make sense within an
existing spacetime, which here appears during, or after, the formation of the
first spacetime. Lemaître also underlined the great importance of the missing
mass problem. Moreover, Dirac's work (see above) confirms LemaItre's vision
following which the immense universe is based on quantum microphysics, reason
why he replaced the Kant-Laplace word nebula
by atom, conceptually covering atomic
physics of quantum essence, independently of the reductionist concept of atomic
smallness.
5. Lemaître's concept of primeval atom looks therefore
like a key opening a door to major theoretical issues he related to quantum
physics. Confirming this, a crucial paper appeared in 1978 [12], detailing the big bang through the apparition of a quantum
fluctuation, whose energy exactly compensates the negative gravitational energy
by virtue of energy conservation, respecting therefore causality according to
the authors (important). The basic concept of this 1978 paper somewhat differs
from the present paper presenting a supposedly giant black hole as initial
condition. We then assume that positive energy was extracted from a fraction of
the primeval negative mass, slightly raising black hole mass negativity, quite simultaneously
with the emergence of spacetime with positive vacuum energy, process without
which space and time could not surface out of nothingness (spacetime vacuum
should have a cost). Summarizing, energy conservation applies from the time
zero, or the beginning of the formation of the first universe. Afterward, the
negative black hole mass remains constant forever, so that the universe will reach
a maximum size after total exhaustion of the positive potential energy.
Evidently, all this does not minimize Einstein's
equation, quite far from having said its last word.
V. Origin of the R-W Metric
Consider a spherical 3-dimensional
hypersurface defining the universe embedded in a 4-dimensional Euclidean
continuum [1-B] according to
xβ.xβ
+ x4.x4 = a2 (10)
with
dl2 =
dxβ.dxβ + dx4.dx4 (11)
(β = 1, 2, 3 - Greek indices refer to space), where a is Einstein's constant
curvature radius, dl2 being the squared infinitesimal 4-dimensional distance between two
neighboring points. Differentiating (10) gives
xβ.dxβ
+ x4.dx4 = 0
(12)
or,
dx4 = -
(1/x4)xβdxβ (13)
implying
(dx4)2
= (1/x4)2.xαxβdxαdxβ (14)
Eq. (10) gives
(x4)2
= a2 - xβxβ (15)
So that Eq. (14) becomes
(dx4)2
= [1/(a2 - xγxγ)] xαxβdxαdxβ (16)
Eliminating x4 in Eq, (11) then
gives
(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]: The 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 by calling it an
artifice. Quite the opposite, embedding the 3-dimensional universe in a
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 become
physical if his equations prove to be valid. 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, which is now detailed. Differentiating the definition
xβxβ
≡ r2 (18)
gives the key relation
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 dΩ2 is the usual
notation for dθ2 +
sin2θ.dφ2.
Through the change of variables r*= r/a, (22) leads to
(dl)2
= a2dr*2/(1 - r*2)
+ r*2.dΩ2
(23)
here written in standard form [13], where r* is dimensionless and a is Einstein’s
universe radius. The right member of Eq. (23) is the 3-dimensional part of the usual
R-W metric. Since momentarily not
multiplied by a2, the dimensionless term r*2dΩ2 in this space
metric in standard form is somewhat devoid of physical meaning, because the vanishing
of any metric in standard form is not generally applicable to the calculation
of the speed of light. However, the conversion of (22) to (23) corresponds to a
coordinate transformation [2] not affecting the field equations by
virtue of general covariance, the only motive being a simpler derivation of the
field equations.
VI. A Variant of the R-W metric
Discarding inapplicable
maximal symmetry [2], the following expression
of B from Eq. (1) includes the negative source-mass m of the primeval black hole, so that
B =
(1- 2Gm/Rc2) (24)
which is the usual Schwarzschild
black hole solution in the static case, based on Newton's potential -Gm/r, with r = R on the equipotential
of the universe-3-hypersurface. According to the above coordinate transformation r = a.r*, where a was Einstein´s constant universe radius of his first
static universe, we replace r by R.r*, in
conformity with what was done above
to retrieve the 3-dimensional part of the R-W metric in the form of Eq. (23). Not
using the standard form and treating the universe radius R as a time-dependent variable, the
complete R-W metric reads
ds2
= c2[1- 2Gm/Rc2]dt2 - R2[dr*2/(1- r*2) + r*2dΩ2]
(25)
Suppressing the
unnecessary asterisks then gives
ds2
= c2[1- 2Gm/Rc2]dt2 - R2[dr2/(1- r2) + r2dΩ2]
(26)
where
r is
dimensionless.
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Note:
Significant Lemaître's papers, starting with the rare original 1927 paper, are
mailed free of charge by Mrs. Moens of the Lemaître
Foundation (Louvain, Belgium - email in this web page).
[12] R. Brout, F. Englert, E. Gunzig :The Causal Universe, Faculté des
Sciences, Brussels University 1978/
[13] Y. Choquet-Bruhat: General
Relativity and the Einstein Equations (Oxford University Press, NY 2009),
chapter 5, section 4, Eq. 4.26.
Edition: MSc. Jorge Poveda.
Edition: MSc. Jorge Poveda.
