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Versione Senza Appendice - 7 February 2014 - ore 13.50
WE COME FROM
A BLACK–HOLE
AND WE ARE STILL IN
A BLACK–HOLE
Antonino Zichichi
The last attempt to solve the problem of unifying all
Fundamental Forces of Nature is the RQST (Relativistic
Quantum String Theory). Unfortunately, despite the large
number of theoretical physicists involved in the game, RQST
still needs more time to solve the problem. Meanwhile we
would like to give answers to simple facts such as the number
of protons, neutrons and electrons (p, n, e),
𝑁(𝑝 𝑛 𝑒) ≃ 1080 ,
in our Universe.
Purpose of this note is to provide an answer based on the
three Fundamental Constants of Nature and the Schwarzschild
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solution [1] of the Einstein equation.
In his universal outlook of the world – independent of our
restricted environment – Planck in 1899 wanted that the
fundamental units of Mass, Length and Time should be
derived [2] from the values of the Fundamental Constants of
Nature: c (the speed of light), h (the Planck Constant) and GN
(the Newton Gravitational coupling).
Planck included the Boltzman’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, 3.5  1032 Kelvin.
Here are Planck’s units in what we now call the “Planck
Universe”:
LPlanck
= (G  h /c3)1/2
= 1,616  1033 cm
TPlanck
= (G  h /c5)1/2
= 5,39  1044 sec
MPlanck
= (h  c /G)1/2
= 2,18  105 gr
The exact values of c, h and GN are given below.
c = (29˙979˙245˙740 ± 120) cm / sec.
h = (6,62606876 ± 0,00000052)  1034 Joule  sec.
GN = (6,67259 ± 0,00085)  1011 (m3 
2
1 
2).
These quantities had a special meaning for Planck [2]:
«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». 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.
When Planck was expressing his ideas on the meaning of
his fundamental natural units there was neither the Big–Bang
nor the Einstein equation which describes the cosmic
evolution. And no one knew that the Einstein equation had a
solution, discovered by Schwarzschild [1], which describes
the gravitational field of a massive point particle. John
Wheeler in 1967 gave to this solution the name of “Black–
Hole”, the reason being that it corresponds to such a density of
matter that even the light cannot escape the gravitational
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attraction. Schwarzschild formula establishes the coupling
between the radius of a Black–Hole, RBH , and its mass, M BH :
BH
R
=
2G MBH
c2
≅ 1.5 ∙ 10−28 ∙ cm ∙ gr −1 ∙ M BH .
(1)
The Black–Hole Radius increases linearly with its Mass,
as shown in Figure 1. The Schwarzschild formula remains as
it is despite all developments [3] in the physics of Black–
Holes including what has been discovered by RQST.
The remarkable fact is however that if we look at the
point where the Radius is that of the world where we leave
(about 1028 cm) the mass turns to be
mU ≃ 8 ∙ 1055 grams,
which is the mass of our Universe.
Let us now assume that M BH , is not concentrated in a
point, as in the Schwarzschild solution of the Einstein
equation, but distributed inside the volume defined by the
sphere of the Black–Hole-horizon.
We assume that the Black–Hole-horizon [4] is the surface
of a sphere where M BH is distributed.
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We neglect details like [(4/3) ] in front of RBH to have
the Black–Hole volume. Since the density is given by the
mass over the volume
ρBH =
MBH
VBH
=
MBH
(K ∙ MBH )3
,
the result – following the Schwarzschild equation (1) – is that
the Black–Hole density decreases with the square of the
Black–Hole mass
r BH = K -3 × M-2
BH
with
K =
2G
−28
−1
≅
1.5
∙
10
∙
cm
∙
gr
.
c2
In Figure 2 the density of our Universe, ρU , and the Planck
density, ρPlanck , are given as function of the Radius of all
possible horizons produced by all possible masses allowed by
the Schwarzschild solution of the Einstein equation.
It is interesting to see the different values of densities
including those which have attracted the interest of John
Michell in 1783; independently of Pierre-Simon de Laplace in
1796 and in 1939 of Robert Oppenheimer, George Volkoff,
Hartland Snyder and Fritz Zwicky.
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Conclusion: the Universe where we are is the proof that a
Black–Hole can expand its radius by something like 62 orders
of magnitudes going from 1033 cm up to 1029 cm. 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.
It is interesting to see (Figure 2) the different values of
densities which can go from the minimum, ρUniverse , to the
maximum, ρPlanck .
On many occasions, during the activities of the
International School of Cosmology and Gravitation, I have
been discussing with friends and colleagues (including John
Wheeler [5], Nathan Rosen [6] and Peter Bergmann [7]) how
it happens that no one has been able so far to derive the result
that our Universe should have the number of protons,
electrons (and neutrons) which our Universe is made of; i.e.
the number quoted before which is about
𝑁(𝑝 𝑛 𝑒) ≃ 1080 .
Despite the enormous work devoted to understand the
physics of Black–Holes [3] including the study of Quantum
Gravity
[8]
and
the
Relativistic
Quantum
String
Theory (RQST) [9] with the interesting discovery of the
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“Landscape” [10], no one has been able to get even the easier
goal, which is the Universe mass. This is about 1056 grams, if
we ignore the problem of Dark-Matter and Dark-Energy.
In fact, our Universe has a number of Galaxies of about
𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒
𝑁(𝐺𝑎𝑙𝑎𝑥𝑖𝑒𝑠)
≃ 2 ∙ 1011 .
Each Galaxy has, on the average, a number of Stars of
about the same order of magnitude, 2 ∙ 1011, i.e.
𝐺𝑎𝑙𝑎𝑥𝑦
𝑁(𝑆𝑡𝑎𝑟𝑠)
≃ 2 ∙ 1011 .
Each Star has a mass in the range of the mass of our Sun
mass of the Sun = 𝓂⊙ ≃ 2 ∙ 1033 grams.
The total mass of the Universe turns out to be about
mUniverse ≃ 8 ∙ 1055 grams ≃ 1056 grams.
Taking into account the advocated presence of DarkMatter and Dark-Energy the total mass of the Universe could
go up to about
𝑚𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑒 ≃ 1058 grams,
with the Radius increasing accordingly.
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The Dark-Matter and Dark-Energy do not contribute to
increase the number of 𝑁(𝑝 𝑛 𝑒) .
Since the mass of an elementary particle (p, n) is
m (p, n) ≃ 10−24 grams,
the electron being 2000 times lighter, the total number of
(p n e) turns out to be
𝑁(𝑝 𝑛 𝑒) ≅ 𝟏𝟎𝟖𝟎 ,
as mentioned before. Sooner or later this number must be
given by RQST; meanwhile its origin is in the values of the
three Fundamental Constants and in the Schwarzschild
solution of the Einstein equation.
It cannot be a casual coincidence the fact that the
Schwarzschild equation (1) gives the correct value for the
mass and the density of the Universe when its Radius
increases by 62 powers of ten.
There is a series of experimental consequences being
studied of what can be observed when the Universe Radius
goes from 10−33 cm to 1029 cm with its mass increasing by
the same order of magnitude and the density changing by the
product, i.e. by as much as 10124 .
8
mU
8  1055gr
33
210 gr = m
RU
27
mT = 610 gr
25
mL = 7.410 gr
3
1 Kg = 10 gr
0
10 = 1
5
-5
210 gr
mPL
1.610
-33
10
-25
5
cm
10
210 cm
-2
0
10 = 1
Figure 1
9
10
28
cm
Log R(cm)
(BLACK–HOLE) DENSITY VERSUS RADIUS
1093
Oppenheimer,
Volkoff, Snyder,
Zwicky
R  1013 cm  AU  Astron. Unit
R  10 cm = 10 km
6
33
1.6  10
U
cm
Figure 2: 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]
Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie
K. Schwarzschild, Sitzungsberichte der Königlich Preussischen Akademie der
Wissenschaften 7: 189–196 (1916).
[2]
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).
[3]
Latest News from Black-Holes Physics
G. 't Hooft, in Proceedings Subnuclear Physics Erice School–2010, Vol. 48, World
Scientific (2013); Quantum Gravity without Space-Time Singularities or Horizons, G. 't
Hooft, in Proceedings Subnuclear Physics Erice School–2009, Vol. 47, World Scientific
(2011).
[4]
Horizons
G. 't Hooft, in Proceedings Subnuclear Physics Erice School–2003 “From Quarks to
Black Holes: Progress in Understanding the Logic of Nature”, Vol. 41, World
Scientific, 179–192 (2005); The Black Hole Information problem, G. 't Hooft, in
Proceedings Subnuclear Physics Erice School–2004 “How and Where to go Beyond the
Standard Model”, Vol. 42, World Scientific, 226–236 (2007).
[5]
INTERNATIONAL SCHOOL OF COSMOLOGY AND GRAVITATION
J. Wheeler, 1972–High Energy Astrophysics and its Relation to Elementary Particle
Physics; and 1992–String Quantum Gravity and Physics at the Planck Energy Scale.
[6]
INTERNATIONAL SCHOOL OF COSMOLOGY AND GRAVITATION
N. Rosen, 1977–Theories of Gravitation.
[7]
INTERNATIONAL SCHOOL OF COSMOLOGY AND GRAVITATION
P.G. Bergmann, 1979–Spin, Torsion, Rotation and Supergravity; 1982–Unified Field
Theories of More Than 4 Dimensions Including Exact Solutions; 1985–Topological
Properties and Global Structure of Space-Time; 1987–Gravitation Measurements,
Fundamental Metrology and Constants;
1990–Symposium on the Problem of the Cosmological Constant in Honor of Peter
Gabriel Bergmann's 75th Birthday;
P.G. Bergmann and Zheniju Zhang, 1991–Black Hole Physics;
P.G. Bergmann, V. De Sabbata and T.-H. Ho, 1993–Cosmology and Particle Physics;
P.G. Bergmann, V. De Sabbata and H.-J. Treder, 1995–Quantum Gravity;
P.G. Bergmann, G. 't Hooft and G. Veneziano, 1998–From the Planck Length to the
Hubble Radius.
[8]
Using Black Holes to Understand Quantum Gravity
G. 't Hooft, in Proceedings Subnuclear Physics Erice School–2011, Vol. 49, World
Scientific (2013).
[9]
Beyond Relativistic Quantum String Theory
G. 't Hooft, in Proceedings Subnuclear Physics Erice School–2012, Vol. 50, World
Scientific (2014).
[10] The Landscape and its Physics Foundations. How String Theory Generates the
Landscape
L. Susskind, in Proceedings Subnuclear Physics Erice School–2006, Vol. 44, World
Scientific, 161–252 (2008).
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