9 marzo 2014 – ore 14.38
THE EVOLUTION
OF THE UNIVERSE
AND GAP
Antonino Zichichi
INFN and University of Bologna, Italy
CERN, Geneva, Switzerland
Enrico Fermi Centre, Rome, Italy
World Federation of Scientists, Beijing, Geneva, Moscow, New York
It has been recently pointed out [1] that our Universe
seems to obey the conditions posed by the Schwarzschild
solution [2] of the Einstein equation.
The Einstein equation establishes a correlation between
mass-energy and the curvature of Space-Time: high curvature
corresponds to high values for the mass-energy. Since high
curvature corresponds to small Radius, the mass-energy goes
like 1/R. On the other hand the Schwarzschild equation
establishes a correlation between the Radius of the Horizon
1
produced by the point-like mass, M, which depends only on
the fundamental constants G and C.
𝑅𝐵𝐻 =
2𝐺
𝐶2
𝑀𝐵𝐻 ≃ 1.5 × 10−28 ∙ 𝑐𝑚 ∙ 𝑔𝑟 −1 ∙ 𝑀𝐵𝐻 (1)
The mass MBH increases with the Radius RBH. The equation
in (1) is only one point in the Einstein equation which gives
the Radius versus the mass-energy, as illustrated in Figure 1.
Figure 1
2
What is needed is an equation which describes the
evolution of the Radius and the mass of the Universe as a
function of Time.
Suppose we were able to find – from basic principles –
the function which describes the Universe, 𝜓 𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒 , this
function must evolve in Time
𝐽𝜓 𝑈 (𝑅, 𝑀)
𝐽𝑡
in such a way that the correlation between Radius and mass of
the Universe obeys the Schwarzschild equation:
𝑅𝑈 (𝑡) ≅ 1.5 × 10−28 ∙ 𝑐𝑚 ∙ 𝑔𝑟 −1 ∙ 𝑀𝑈 (𝑡)
The evolution of 𝜓 𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒 must describe the change of
density of the Universe, from
𝜌𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒 (𝑡 = 0) ≅ 1093 ∙ 𝑔𝑟 ∙ 𝑐𝑚−3
up to
𝜌𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒 (𝑛𝑜𝑤) ≅ 5 × 10−30 ∙ 𝑔𝑟 ∙ 𝑐𝑚−3
3
i.e. along a change of density by 123 powers of ten. All
possible Einstein equations are represented by all functions
(𝑅 =
1
𝑀
) in Figure 1. What is needed is the function
𝜓 𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒
which evolves with Time along the line (R = M) given by the
Schwarzschild condition in (Figure 1).
The Einstein equation and the Schwarzschild solution
ignore the existence of the SU(3) × SU(2) × U(1). The
convergence of the three couplings α3 α2 α1 at EGUT is at least
two orders of magnitudes below ESU, the energy where RQST
(Relativistic Quantum String Theory) has established the
origin of the Gravitational Forces (i.e. the String Unification
Scale).
At present the existence of the gap between
EGUT and ESU
is based on the most exact study of the evolution with
Energy of the three gauge couplings α1 α2 α3 , as illustrated in
Figure 2.
4
Figure 2
5
The four Fundamental Forces of Nature should all start at
ESU. The consequences of the GAP in understanding the
evolution of our Universe is one of the most interesting
problems in front of us.
******
Purpose of this note is to call attention on the remarkable
developments in the study of Black–Holes and on the possible
connection with the high precision determination of the
convergence of the three couplings α1 (QED), α2 (QFD) and
α3 (QCD) towards a common origin αGUT .
The starting point is in fact the discovery (long time
ago) that
the
three
gauge
couplings
converge towards a unique value αGUT ≃
(α1 α2 α3 )
1
25
do
at the energy
αGUT ≃ 1.2 × 1016 GeV if the existence of supersymmetry
is introduced in the evolution equations of α1 α2 α3 [3]. The
energy evolution of the three couplings
α1 (q2)
α2 (q2)
(1)
α3 (q2)
had never before been performed with the new condition [4]
based on the energy dependence, not only of the gauge
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couplings themselves as indicated in (1), but also with the
energy dependence of the masses: i.e. the EGM (Evolution of
Gaugino Mass) effect [5].
This inclusion produced a factor (700)1 nearly three
orders of magnitudes, for the threshold of supersymmetry
breaking.
Suppose the convergence of the three couplings (α1 α2 α3 )
is computed taking into account the evolution of each
couplings with q2, neglecting the variation of the masses
associated with the physics of the given gauge–group, i.e.
U(1) for α1 , SU(2) for α2 and SU(3) for α3 . Suppose the
(≠)
“prediction” is ESUSY = 700 TeV. Using the same model the
prediction becomes 1 TeV, if the EGM effect [5] is included.
The search for the lightest supersymmetric particle would
become possible for LHC.
The reason for the study of the evolution with energy
of α1 α2 α3 was two–fold.
The first reason was in order to work out the energy level
where the supersymmetry could be “predicted” to break.
These “predictions” were – and are – model dependent. It
was found that – whatever the model is – the EGM effect [5]
lowers, by nearly three orders of magnitude, the energy level
where the lightest supersymmetry particle should be produced.
7
The second reason for the study of the evolution with q2
of α1 α2 α3 was in order to see what was needed for the three
couplings to converge towards the same point.
A detailed study of this convergence, when all
experimental uncertainties are taken into account [5], gives as
result that, if we take all “best” values, the convergence level
where they become equal
α1 = α2 = α3 ≃
1
25
is at
EGUT ≃ 1.2 × 1016 GeV
which is somewhat three orders of magnitude below the
Planck Energy level.
Let us take for granted that this is indeed the case; i.e. the
three couplings converge at EGUT .
For the Planck Energy level, taking into account the
RQST (Relativistic Quantum String Theory) approach to
include Gravitational Forces in the game we have to multiply,
EPlanck by √𝛼𝑢 and the result goes down to
ERQST ≃ 1018 GeV.
This means that the Gravitational Forces start to operate
at
EGrav. ≃ 1018 .
8
Suppose that this is correct, and suppose that there are
no other forces in this energy range, from ≃ 1018 Gev to ≃
1016 Gev.
Problem 1. What happens in the energy range
EGrav. and EGUT
where the only forces are the Gravitational.
In this Energy interval the Universe consists only of what
the Gravitational Forces can do. They can only produce
masses without any charge and without other properties,
which are generated by the three other Fundamental Forces
and fundamental particles. All that the Universe can consist of
are masses as Black–Holes.
At present Galaxies and Stars are supposed to have their
origin in Space-Time quantum–fluctuations.
If at this time only Gravitational Forces were present,
they could generate only Black–Holes. The origin of the
Galaxies are these Black–Holes, now at the centre of each
Galaxy. Their masses are in the range of 106 ÷ 108 solar
masses; the total mass of each Galaxy being of the order of
1011 solar masses:
MGalaxy. ≃ 2 × 1011 𝓂⊙ ,
the “primordial” Black–Hole acted as seeds for each Galaxy
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formation.
At that time only the Gravitational Forces were present.
No SU(3) × SU(2) × U(1). The world we come from knew
nothing of SU(3) × SU(2) × U(1).
In this primitive Universe the Einstein equation was
operative and the solution of the Einstein equation found by
Schwarzschild was at work.
The Schwarzschild solution of the Einstein equation
corresponds to such a density of matter that the light cannot
escape the gravitational attraction. John Wheeler called this
solution: “Black–Hole”.
The smallest Black–Hole is given by the Planck scale
where the unity of length is
ℓ𝑃𝑙𝑎𝑛𝑐𝑘 = 10−33 𝑐𝑚.
There is no limit for the largest length.
If we take the present radius of our Universe
ℓ𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒 ≅ 1029 𝑐𝑚
the ratio of the two lengths is a very large number
ℓ𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒
ℓ𝑃𝑙𝑎𝑛𝑐𝑘
≃ 1062 .
10
If we take the Schwarzschild formula in order to know
what is the mass which correspond to this Radius, the answer
is the mass of our Universe
𝑚𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒 ≅ 8 ∙ 1055 𝑔𝑟 ≃ 1056 gr
since 𝑚𝜌 ≃ 10−24 𝑔𝑟 the total number of nucleons in the
Universe is
𝑁𝑛𝑢𝑐𝑙𝑒𝑜𝑛𝑠 ≃ 1080 .
The conclusion is that we come from the smallest Black–Hole
which during its evolution has followed the Schwarzschild
equation producing the result of our Universe being like a very
big Black–Hole.
It is generally believed that Black–Holes phenomena are
scale-invariant, small and big Black–Holes are such that in
their inner structure the law of physics known to us are no
longer valid.
But, our Black–Hole allows the study all laws of physics
which go from the Standard Model to the extrapolation called
Beyond Standard Model (BSM). One of the possibilities for
BSM being the Superworld.
11
It is therefore established that the laws of Physics we
know remain valid well inside a Black–Hole, provided that
this Black–Hole is as large as our Universe.
This finding could be related with the properties
theoretically mentioned by Gerardus 't Hooft [6]. He says on
page 77: «If the original amount of material was big enough,
the contraction will proceed, and, in the limit of zero pressure
and purely radial, spherically symmetric motion, the equations
can easily be solved exactly. We obtain flat Space-Time
inside, and a pure Schwarzschild metric outside. As the ball
contracts, a moment will arrive when the Schwarzschild
horizon appears. From that moment on, an outside observer
will no-longer detect any radiation from the shell, but a
Black–Hole instead».
If somebody from outside our Universe would like to see
what there is in a sphere whose radius is the radius of our
Universe ( 𝑅𝑈 ≃ 1029 𝑐𝑚 ) he will find our Black–Hole,
which is the Universe were we have life and knowledge.
Planck discovered [7] what we now call “the Planck
Universe”. In his universal outlook of the world – independent
of our restricted environment – Planck wanted that the
fundamental units of Mass, Length and Time must depend
12
only on the values of the Fundamental Constants of Nature:
the speed of light c, the constant of action h, and the
gravitational constant G.
In this Universe the units of Length, Mass, Time and
Temperature must be independent of special bodies or
substances such as it is the case for our units of Length
(centimetre), Mass (gram), Time (second) and Temperature
(degree Celsius or Kelvin).
Planck included also the Boltzmann’s constant k which
converts the units of Energy into units of Temperature. This
allowed Planck to have a fundamental value also for the
Temperature. Here are Planck’s units:
Length
= (G  h /c3)1/2
= 4.13  1033 cm
Time
= (G  h /c5)1/2
= 1.38  1044 sec
Mass
= (h  c /G)1/2
= 5.56  105 gr
Temperature = K1  (hc5/G)1/2 = 3.5  1032 Kelvin.
It is remarkable the way Planck considered these
quantities: «In the new system of measurement each of the
four preceding constants of Nature (G, h, c, k) has the value
one». This is the meaning of measuring Lengths, Times,
Masses and Temperatures in Planck’s units.
13
These quantities had a special meaning for Planck [7]:
«These quantities retain their natural significance as long as
the Law of Gravitation and that of the propagation of light in a
vacuum
and
the
two
principles
of
thermodynamics
remain valid; they therefore must be found always to be
the same,
when
measured
by
the
most
widely
differing intelligence according to the most widely differing
methods».
When
Planck
was
expressing
his
ideas
on
the
meaning of his fundamental natural units there was neither
the Big-Bang nor the cosmic evolution. The very instant of the
cosmic expansion is the Planck-Time and the corresponding
density of the Universe is the Planck density.
In our world the density is dictated by the fact that we are
made by atoms and therefore the basic quantities are the mass
of the nucleon (mN  1024 gr) and the radius of the atom
(108 cm) which is dictated by the electric charge and the
mass of the electron.
Let us call the value of this “density” the “atomic
density”:
M nucleon
10-24 gr
10-24 gr
gr
ratomic @ 3
@
=
=
1
3
-24
3
3 .
-8
Ratom
10
cm
cm
10
cm
(
)
14
The density of water is typical of the atomic density. In
the above formula we neglect details like (4/3 ) in front of
R3atom to estimate the “atomic volume”.
When we go from water to lead the “atomic density”
increases by an order of magnitude.
This is due to the increase in the mass of the nucleus by
two orders of magnitude
m nucleus
 102 Mnucleon
Pb
and a correspondent increase by an order of magnitude in the
atomic volume.
The next possible density – many orders of magnitudes
higher – is the “nuclear” density: 1015 times greater than the
“atomic” density.
The reason being the value of the nuclear radius, which is
of the order of one Fermi-unit (1013 cm), i.e. five orders of
magnitude smaller than the atomic radius.
Atomic and nuclear bindings are specific for a given
Element of the Mendeleev Table and do not change when the
amount of the given Element changes.
For both forms of matter, atomic and nuclear, the density
does not change when the amount of matter increases: one ton
of lead has the same density as one kilogram of lead.
15
For matter where the binding force is gravitational
(without any other forces being involved), the density
decreases when the amount of mass increases.
More precisely the density decreases with the square of
the mass. This is the great discovery of Schwarzschild and is
coming from the relation which exists between the Radius of a
Black–Hole,
RBH ,
and the mass of the same Black–Hole,
MBH .
Here is the Schwarzschild formula:
R BH =
2G MBH
c2
= K MBH @ 1.5 ´ 10 -28 × cm × gr -1 × MBH
with G being the Gravitational Constant, c the speed of light
and
K=
2G
c2
@ 1.5 ´ 10 -28 × cm × gr -1 .
16
In the Table below we show the values of the matter density
in our world today and in the world we come from, i.e. the
Planck Universe.
Every
day’s
World
The World
we
come from
Human Body Density
Planck Density
1gr/cm3
1037 Universes/cm3
Our World
The Planck Universe
1.6  1033 cm
2.2  105 gr
5.4 x 1044 sec
cm
gr
sec
When the mass is the value of the Planck unit given in the
previous Table. i.e.
MPlanck = 2.2  105 gr,
the correspondent Radius of the Black–Hole is
R Planck
@ 1.5 ´ 10-28 ´ 2.2 ´ 10-5 ´ cm @ 3.3 ´ 10-33 cm.
BH
The Black–Hole Radius increases linearly with its mass
value, as shown in Figure 3.
17
The density is given by the mass over the volume
r BH =
MBH
=
VBH
MBH
( K × MBH ) 3
The result is that the Black–Hole density decreases with
the square of the Black–Hole mass
r BH = K -3 × M-2
BH
The remarkable fact is however that the extensions of
the world we leave in is now (1029 cm) and – as shown in
Figure 4 – its density satisfies the same relation of the world
we come from, whose radius was 1033 cm and its density
ρPlanck ≅
5
4
× 1093 gr × cm−3 .
Conclusion the Planck density and radius satisfy the
Black–Hole condition exactly as the present day density of the
Universe and
its dimensions
satisfy
the
Black–Hole
conditions: we come from a Black–Hole and we are still in a
Black–Hole.
18
Appendix 1
The Universe where we are is the proof that a Black–Hole
can expand its radius by something like 61 orders of
magnitudes going from 1033 cm up to 1028 cm. The basic
quantity in this expansion is the density.
It is interesting to see the different values of densities
which can go from the minimum to the maximum radius of
the Black–Hole, i.e. our Universe.
Note 1
It should be pointed out that the Boltzmann’s constant k,
represents the “quantum” of Entropy, or the minimum amount
of “caos”.
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THE EVOLUTON OF THE UNIVERSE
FOLLOWING THE SCHWARZSCHILD EQUATION
Figure 3
20
Figure 4: The Figure shows the relation which exists between the value of the
Black–Hole radius (RBH) and the corresponding density (BH), from the Planck
scale to the Universe scale now.
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REFERENCES
[1] It is as if we come from a Black–Hole and we are in a Black–Hole
A. Zichichi, to be published (2014).
[2] Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen
Theorie
K. Schwarzschild, Sitzungsberichte der Königlich Preussischen Akademie
der Wissenschaften 7: 189–196 (1916).
[3] A. Petermann and A. Zichichi (Geneva, Conference Note).
New Developments in Elementary Particle Physics
A. Zichichi, Rivista del Nuovo Cimento 2, n. 14, 1 (1979). The statement
on page 2 of this paper, «Unification of all forces needs first a
Supersymmetry. This can be broken later, thus generating the sequence of
the various forces of nature as we observe them», was based on a work by
A. Petermann and A. Zichichi in which the renormalization group running
of the couplings using supersymmetry was studied with the result that the
convergence of the three couplings improved. This work was not
published, but perhaps known to a few. The statement quoted is the first
instance in which it was pointed out that supersymmetry might play an
important role in the convergence of the gauge couplings. In fact, the
convergence of three straight lines (𝛼1−1 , 𝛼2−1 , 𝛼3−1 ) with a change in slope
is guaranteed by the Euclidean geometry, as long as the point where the
slope changes is tuned appropriately. What is non trivial about the
convergence of the couplings is that, with the initial conditions given by
the LEP results, the change in the slope at MSUSY ~ 1 TeV is not correct, as
claimed by some authors not aware in 1991 of what was known in 1979 to
A. Petermann and A. Zichichi.
[4] The Effective Experimental Constraints on MSUSY and MGUT
F. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi, Il Nuovo Cimento
104A, 1817 (1991).
The Simultaneous Evolution of Masses and Couplings: Consequences on
Supersymmetry Spectra and Thresholds
F. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi, Il Nuovo Cimento
105A, 1179 (1992).
22
“The Evolution of Gaugino Masses and the SUSY Threshold
A. Zichichi, A. Petermann, L. Cifarelli and F. Anselmo, Il Nuovo Cimento
106A, 581 (April 1992).
[5] A Study of the Various Approaches to MGUT and GUT
F. Anselmo, L. Cifarelli and A. Zichichi, Nuovo Cimento 105A, 1335
(1992).
[6] The Holographic Principle
G. 't Hooft, in Proccedings of the Erice 1999 Subnuclear Physics School
“Basics and Highlights in Fundamental Physics”, page 77, World
Scientific (2001).
[7] Über Irreversible Strathlungsvorgänge
M. Planck, S.B. Preuss. Akad. Wiss. 5, 440–480 (1899).
The problem of the Fundamental Units of Nature was also presented by
M. Planck in a series of lectures he delivered in Berlin (1906) and
published as “Theorie der Wärmestrahlung”, Barth, Leipzig, 1906, the
English translation is “The Theory of Heat Radiation”, (trans. M. Masius)
Dover, New York (1959).
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