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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 1 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 1033 cm TPlanck = (G h /c5)1/2 = 5,39 1044 sec MPlanck = (h c /G)1/2 = 2,18 105 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) 1034 Joule sec. GN = (6,67259 ± 0,00085) 1011 (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 3 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. 4 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. 5 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 1033 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 6 “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. 7 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 33 210 gr = m RU 27 mT = 610 gr 25 mL = 7.410 gr 3 1 Kg = 10 gr 0 10 = 1 5 -5 210 gr mPL 1.610 -33 10 -25 5 cm 10 210 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. 10 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). 11