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37
УДК 53.02+524 + 537 + 622+501 + 531 + 532
Вселенная в большом масштабе: прошлое, настоящее, будущее
Г.П. Черепанов
В настоящей статье предложенный ранее автором инвариантный интеграл физической мезомеханики используется для изучения Вселенной в евклидовом пространстве в большом масштабе порядка 100 Мпк. В этом масштабе Вселенную можно считать однородной и изотропной. На основании многолетних астрофизических измерений в рамках космических программ WMAP и PLANCK, кривизну Вселенной можно считать равной нулю вопреки предположениям общей теории относительности. В основе статьи лежит закон взаимодействия двух точечных масс в космическом гравитационном поле, выведенный из инвариантного интеграла физической мезомеханики. Доказано, что сила их взаимодействия равна сумме двух слагаемых, одно из которых представляет силу притяжения по закону Ньютона, а другое — силу отталкивания, вызванную космологической составляющей поля. Оба слагаемых образуют правую часть эволюционного уравнения Вселенной, представляющего собой уравнение движения пробной массы на краю Вселенной. Это уравнение почти совпадает с основным уравнением моделей FLRN и ACDM, основанных на общей теории относительности. Получено сингулярное решение этого уравнения, которое описало расширение Вселенной — от Большого взрыва и стадии отрицательного ускорения ранней Вселенной до стадии ее ускоренного расширения, в которой мы живем. Возраст Вселенной найден равным 12.3 миллиарда лет, что довольно близко к современной теории, основанной на предпосылках общей теории относительности о ненулевой кривизне Вселенной. Затем вводится гипотеза вращения расширяющейся Вселенной, которая позволяет определить космологическую постоянную (темную энергию) как простую функцию угловой скорости вращения. Оказалось, что общепринятому значению космологической постоянной отвечает угловая скорость вращения, согласно которой угол поворота Вселенной за 12 миллиардов лет составил лишь около 30°. Предложено дополнительное уравнение эволюции вращения расширяющейся Вселенной. Обсуждаются гипотезы, относящиеся к фрактальной размерности Вселенной и гравитонам минимальной частоты как строительным блокам Вселенной. Показано, что нейтронные звезды становятся черными дырами, если их масса более чем в 6.7 раз больше массы Солнца. Вычислена орбитальная скорость звезд Млечного Пути и других галактик, которая оказалась независимой от расстояния звезды до центра галактики и равной порядка 250 км/с в полном соответствии с астрофизическими измерениями.
Ключевые слова: космология, инвариантный интеграл, расширение Вселенной, вращение Вселенной, возраст Вселенной, космологическая постоянная, уравнения эволюции Вселенной, темная энергия, темная материя, черные дыры
The large-scale universe: The past, the present and the future
G.P. Cherepanov
The New York Academy of Sciences, New York, 10007-2157, USA
Contrary to the common approach of the general relativity, the author uses his invariant integral of physical mesomechanics to model and study the universe at the large scale of about 100 MPc in the Euclidian space. The flatness of the universe proven by numerous probes of the WMAP and PLANCK satellite missions necessitates this approach. From the invariant integral of cosmology, the interaction force of two point masses in the cosmic-gravitational field is derived. This force is proven to be a sum of two terms, the one being the Newtonian gravity and the other the repulsion force caused by the cosmological constant. Both terms make up the right-hand part of the evolution equation of the dynamic universe. Qualitatively in agreement with the FLRW and ACDM models, and WMAP and PLANCK mission data, the exact solution of this equation has provided the history of the early decelerating universe and the asymptotic description of the Big Bang, the expansion at an almost constant rate in the middle age, and the current stage of the accelerated expansion of the universe. The age of the universe is found to be equal to 12.3 billion years. It is shown that neutron stars become stable Black Holes when their masses are greater than 6.7Msun. Then, it is assumed that the universe not only expands but also revolves, and the evolution equations of the revolving and expanding universe are advanced, with the cosmological constant being defined in terms of the angular velocity of the universe. A singular solution of these evolution equations has described the history of the revolving and expanding universe, at least, up to the age of about ten billion years. Orbital velocities of stars in the Milky Way are calculated to be about 250 km/s independent of the distance of stars from the galaxy center. Using the equation of the fractal dimension of the universe as a power-law fractal, the thickness of a disk-shaped universe is found. The graviton of minimum frequency is hypothesized to be the smallest elementary particle and the building block of everything.
Keywords: cosmology, invariant integral, expansion of the universe, rotation of the universe, age of the universe, cosmological constant, evolution equations of the universe, Dark Energy, Dark Matter, Black Holes
© Cherepanov G.P., 2016
1. Introduction
Cosmology is a speculative science/philosophy about the universe/cosmos based on astrophysical observations and human imagination. It feeds the curiosity of human beings in quest of their place in the world and fate in the future. Sometime they hoped to find the answers in what could be seen in the sky. The chain of events there appeared more persistent than anything on the Earth, "eternal", which made them use the heaven for worshiping as well as for measuring the time and coordinates on the Earth.
Still the Ancient Egyptians could with the naked eye see in the sky what Aristotle (388-323 BC), an Ancient Greek polymath and a disciple of Plato, set forth later in his treatise On the Heaven. He viewed the Earth as the center of the Cosmos, with Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, and stars being at an ever increasing distance from it—everything enclosed by a prime solid sky. In his masterpiece The Sand Reckoner, Archimedes (287212 BC), another Greek genius, calculated the total number of indivisible grains in the universe to be about 1064 in modern designation. Amazingly, this figure coincides with the current estimate of the mass of the universe about 1056 g, if we assume the mass of one grain to be equal about 10-9 g! Archimedes possessed a knowledge of calculus two thousand years before Leibnitz (1646-1716), a German polymath, and Newton (1642-1727), a British genius.
It was almost two thousand years ago that Ptolemy (100170 AD), an Ancient Greek-Roman astronomer, created the astronomical tables and geographical maps used during next 1500 years throughout the world. With his famous treatise Elements, Euclid (circa 320-260 BC ), an Ancient Greek mathematician and one of "the giants on which shoulders Newton stood", created the geometry unshakeable for more than two thousand years until Riemann (1826-1866), a German genius, and Lobachevski (1792-1856), a Russian mathematician, advanced the None-Euclidian geometry.
The Ancient Greek community, less in number than today's Luxembourg, for a short time brought up Pythagoras, Socrates, Plato, Aristotle, Archimedes, Euclid, Ptolemy and a dozen of other brilliant brains that eclipsed the achievements of all great minds of the last four centuries bred in the pool which is 104 larger than the ancient Greece. The ancient Greek theory about the geocentric Aristotelian/Ptolemaic cosmology reigned in the world during two thousand years and was then destroyed by the self-sacrificing deeds of Copernicus (1473-1543), a German astronomer, Bruno (1568-1600), an Italian hero, and Galileo (1564-1642), an Italian great. Meanwhile, still Archimedes recognized the heliocentric system called now the Copernican one. Today we see that bitter controversy only as the difference of opinion concerning a convenient origin of the frame of reference so that both Aristotle and Copernicus are right. (Everywhere in this paper, the national identification of a person corresponds to the state of his birth).
In 1907 Minkowski (1864-1909), a Russian mathematician and Einstein's teacher, coined the term spacetime and suggected the following Minkowski metric of the flat spacetime [1]:
ds2 = -c2dt2 +dx2 + dy2 + dz2. (1)
Here c is the speed of light in vacuum, t is the time, and x, y, and z are the Cartesian coordinates. In mathematical terms this metric characterizes the special relativity theory earlier advanced by Lorentz (1853-1928) and Einstein (18791955), German physicists, and by Poincare (1854-1912), a French mathematician.
Declaring c to be an absolute constant, Einstein advanced the following dualisms of mass-energy and spacetime as the physical properties of everything
E = mc2, L = cT. (2)
Here E, m, L, and T are the energy, mass, length, and time. These dualisms have since revolutionized the science/philosophy of humankind.
In 1916 Einstein generalized the Minkowski metric and advanced the general relativity theory of the curved spacetime in order to describe the Newtonian gravity as a geometrical property of the curved spacetime, with a greater curvature being caused by a denser matter. He derived the following Einstein equations [2]:
G«p =
8nG
T«p,
(3)
GaP = Rap --Rgap, Rap = ^p, a, P ^ = 1, 2, 3, 4.
Here G is the gravitational constant, Tap is the energy-momentum tensor of matter, x1, x2, x3 and x4 are the redesignated t, x, y, and z, gap is the covariant metric tensor of the spacetime ds2 = gapdxadxp, Gap is the Einstein tensor, Rap is the Ricci curvature tensor, R = R^ is the Ricci scalar, and R^p is the Riemann curvature tensor of the spacetime.
In 1922 Friedmann (1880-1926), a Russian mathematician, applied the general relativity to the homogeneous and isotropic universe and derived the following Friedmann equations [3]:
- I = — Gp-, - = -—GI p + 3^ I. (4)
a J 3 a a 3 ^ c2 J
Here a = a(t) is the scale factor of spacetime, t is the time, p is the mass density of the universe, p is the pressure defined by a cosmological model chosen, and k is the normalized curvature index of the universe (k = 0 for the flat universe, k = 1 for the sphere, and k = -1 for the hyperboloid).
Although Einstein was a referee of the Friedmann paper, he failed to appreciate its value for cosmology. Equations (4) were later modified and called the FLRW equations after Friedmann and also after Lemaitre (1894-1966), a Belgian astronomer, Robertson (1903-1961), an American physicist, and Walker (1909-2001), a British mathematician. The FLRW equations have been also used in the
ACDM model called the standard cosmological model which is mostly recognized today.
From the second Friedmann equation, it follows that for positive p the universe contracts, which corresponds to the original view of Einstein. However, later Hubble (18891953 ), an American astronomer, measured the value of the following Hubble parameter [4]
H = ai/ a. (5)
He proved that this parameter was a positive constant (the Hubble law) so that the universe appeared to be expanding at a constant rate. The further studies supported this law, but corrected the value of the constant obtained by Hubble.
During the next 50 years Einstein and other theoreticians tried to modify the general relativity and the Friedmann equations in order to match the theory to the numerous astronomical discoveries of many new galaxies, quasars, pulsars, Black Holes, Dark Matter, supernovae and so forth. Then, as a result of the long-term WMAP satellite mission, the curvature of the universe was proven to be equal to zero with only 0.4% margin of error. Moreover, in the past when the universe was much denser the curvature of the universe was much closer to zero contrary to the general relativity, according to which the denser matter made the curvature of the spacetime greater.
The universe is, and was at any time, flat so that the general relativity is incorrect because it is built on the assumption of a non-flat universe. In the literature, this controversy is politely called the flatness problem of the general relativity. In an attempt to save the general relativity, even a new fantastic inflation theory was made up.
In the course of the Supernova Cosmology Project and the WMAP and PLANCK satellite missions, it was also established that in the cosmos, beyond the gravity, some tensile forces of an unclear nature provide the observed acceleration of the expansion of the universe. These forces were called the Dark Energy. This discovery as well contradicts to the general relativity. Many other particular facts, for example, the almost constant orbital velocity of stars in spiral galaxies, have been unexplainable by the general relativity.
In the present paper we will, in main, revive the Aristotelian/Ptolemaic view of the cosmos and study the large-scale universe using the invariant integral of cosmology introduced by this author. Our intention is, in particular, to explain and make clear the basic astronomical phenomena recently discovered but not understood in the framework of the general relativity such as the Dark Energy, the Dark Matter, the Black Hole and others. For short, we will call this new approach the NEOC (the Neoclassic Cosmology).
2. Basic assumptions: the large-scale universe
At first, we need to evaluate the reasonable time hori-
zon determined by natural biological limits of the human
race lifetime.
Here we assume that the time horizon of the human race is about ten million years. For ten million years light will have covered the distance which is much less than 100 MPc that is much less than about 1/300 of the diameter of the universe. Based on the astrophysical data, at the scale of 100 MPc the current universe can be considered homogeneous and isotropic. In the NEOC approach, at any time the distance of about 1/300 of the universe diameter, or its time equivalent, will be called the epoch. It is the large scale we accept in this paper to treat the universe as homogeneous and isotropic at any time. This scale will play the role of an elementary cell resembling similar notions of elementary cells in continuum physics.
Based on the WMAP and PLANCK missions data, within less than 0.4% margin of error, the universe is proven to be flat in the previous and current epochs so that its FLRW metric in the spherical coordinates is reduced to the following Minkowski metric in the Cartesian coordinates
ds2 = -c2dt2 + a(t)2(dr2 + r 2d92 + r 2sin29d =
= -c2dt2 + dx2 + dy2 + dz2. (6)
At the scale of Ax ~ Ay ~ Az ~ 100 MPc and At ~ ~ 107 years the Minkowski metric is reduced to the Euclidian metric
ds2 = dx2 + dy2 + dz2 (7)
because the spatial terms in Eq. (6) hundreds times greater than the temporal term. For ten million years light runs less than 1/10 of an elementary cell of the large-scale structure of the universe.
In other words, at this large scale the universe on the average will be exactly the same even during the time, which is thousand times greater than the reasonable lifetime of the human race. Particularly, any cosmological theories of the future beyond this lifetime cannot ever be supported or refuted and, hence, all of them are speculative. Within our epoch, that is, during the Homo Sapience lifetime the large-scale structure of the universe is always one and the same— the last man on the Earth will see the same universe the first man saw.
At the large scale in the previous, current and future epochs the universe will be considered flat, homogeneous and isotropic. Within our epoch we, mainly, remain in the framework of the static Aristotelian and Ptolemaic universe described by the Euclidian metric. The latter metric is accepted in this paper for any epoch.
3. Invariant integral of cosmology
Let the physical matter interact by the field potential
x1, x2, x3) where x1, x2, and x3 are the Cartesian coordinates of the Euclidian space with the origin of the reference frame at the point of observation. We can write the law of the energy conservation in this system by means of the following invariant T-integral [5-9]
M = — pR3, (17)
where p is the average density of the gravitational matter (the baryonic matter plus the Dark Matter). Let us consider any mass m on the edge of this sphere. The field force acting upon this mass is, evidently, given by Eq. (14).
From Eqs. (14) and (17), it follows that any mass m moves away with some acceleration if A > p, that is, if the density of the Dark Energy is greater than the average density of the gravitational matter. It is the fundamental property of the expanding universe at the current epoch. Acceleration of the expansion is determined by the value of difference A - p. Though this expansion increases the distance between objects that are under shared gravitational influence, it does not increase the size of these objects, e.g. galaxies.
From Eqs. (14) and (15), it follows also that, if A < p, the universe experiences a deceleration, and if A = p, the universe expands at a constant velocity which corresponds to the principle of Galileo.
At the scale less than 100 MPc, the gravitational matter is inhomogeneous and thus, under the prevailing force of gravitation, it can collapse and form Black Holes, galaxies, quasars, pulsars, stars, planets, and various particles. At any scale, the competition of the gravity versus repulsion force is characterized by the dimensionless number Ch = = M/(aL ), where L and M are the specific linear size and gravitational mass of the system under consideration. When Ch >> 1, we can ignore the Dark Energy, and when Ch << 1, we can neglect the Newtonian gravity.
Let us provide some figures for the current epoch assuming the cosmological constant to be equal to A = = 10-26 kg/m3:
the solar system: M = 2-1011 kg, L ~ 1011m, Ch ~ 1023; the Milky Way (our galaxy): M = 1.4-1042 kg, L ~ ~ 1021m, Ch ~ 105;
a supercluster of million galaxies: M = 1048 kg, L ~ ~ 3 -1024 m ~ 100 MPc, Ch ~ 1;
the universe: M = 1053 kg, L = 101/3 -1026m, Ch = 1. In the scale of 100 MPc and greater the effect of the Dark Energy is, at least, of the same order as that of the Newtonian gravity. In the smaller scales, the effect of the Dark Energy can be ignored. In the space beyond the universe, the Dark Energy dominates because number Ch may be much less than 1.
According to the recent most accurate measurements of the PLANCK satellite mission, the ordinary matter plus the Dark Matter make up 31.7% of the mass of the universe, and the other 68.3% accounts for the Dark Energy. The ordinary matter accounts only for 4.9% of the total mass. At the current epoch, the repulsion forces can dominate at the large scale, while the gravity dominates at the smaller scales [10].
The universe represents a community of the gravitational matter and the antigravitational Dark Energy; for the scales
less than 100 MPc at the current epoch, the universe is bound by the prevailing forces of gravitation. The Dark Energy is uniformly distributed in the universe. For example, in the Earth there is about 0.01 g of the Dark Energy.
6. The dynamic universe: the evolution history
Let us study the radial motion of an arbitrary mass in the cosmic-gravitational field of the universe. The outward motion of the masses forms the expansion of the universe. During one epoch this dynamic expansion is very small as compared to 100 MPc. Hubble was first to measure its rate using the Doppler effect that describes the decrease of frequency of a receding source of waves (redshift) and the increase of frequency of an approaching source (blueshift) in the spectrum of hydrogen, helium, and other chemical elements of the source.
By means of Eqs. (14) and (17), the radial motion equation of any mass at the edge of the universe of radius R is written as follows
4n
R = — GR(A-p).
(18)
This is the evolution equation of the expanding universe.
Let us accept the following assumptions of our model.
(i) The universe is flat, homogenous and isotropic at any time of its history.
(ii) The cosmological constant A is one and the same at any time.
(iii) The gravitational mass M of the universe is constant at any time.
Integrating equations (17) and (18) under these assumptions, we come to the following basic equation:
= ^G(p + Ia) -*4, p = ^. (19) 3^2' R 4nR
H2 =
( R I2
R
v y
Here H is the Hubble parameter, and K is a constant. At the current epoch, we have t = t0, R = R0, R/R = H„. (20)
Using Eqs. (19) and (20), we get K
Kc2 = R2
4n
—GA + 3
2MG R03
- H2
(21)
It is amazing that Eq. (19) almost coincides with the basic equation of the modern FLRN and ACDM models derived from the general relativity based on the assumption of the curved universe (the difference is in the meaning of K).
The integration of Eq. (19) provides the following evolution of the universe at any epoch from the birth when t = 0 to the infinite future when t ^ <»
t =J
4n gar 2 + 2MG - kc2
3 R
12
dR.
(22)
The analysis of basic equations (18) and (19) shows that in the life of the universe there were three different
stages characterizing the early universe, the middle age universe and the old age universe.
The earliest universe had a small size R << Rm. Its rate of expansion, infinite at the birth, was decreasing with time growing. At the earliest age, p>>A so that the density of the gravitational matter was much greater than the density of the Dark Energy.
At the middle age, the universe was growing at the close-to-constant rate. At this age, p ~ A and the rate of expansion was minimal at R = Rm.
At the old age, A > p and the rate and acceleration of expansion are increasing when time grows. We live in the beginning of this stage which is characterized by the following exponential expansion when t ^ <»
V/2 i ^- ^
dR dt
4n
G A I R, R = R0 exp
y GA (t -10)
. (23)
Here t = t0 and R = R0 at the current epoch.
All these predictions of the present NEOC model have been supported by the well-known astronomical discoveries for the last 30 years. Let us notice some of them:
(i) According to Eq. (22), R ^ 0 and R ^^ when t ^ 0, that is the universe was born from nothing and grew at an infinite rate (the Big Bang).
(ii) The early universe was decelerating, and at the current epoch it is accelerating. This phenomenon was discovered by a large team of outstanding astronomers who for about 20 years have collected a tremendous amount of the supporting evidence; three of them were awarded the Nobel Prize (A.G. Riess, S. Perlmutter, and B.P. Schmidt, American astrophysicists).
At the time when t ^ the universe will disintegrate into a great number of independent communities running one from another by the Dark Energy but keeping their own constituents owing to the gravity.
7. The age of the universe
Let us estimate the current age of the universe using Eq. (22). At the present epoch, R = R0 so that we have
Tj =| r * GAR2 + M - Kc- fdR. (24)
Here TJ is the current age of the universe.
Let us transform Eq. (24) to the following shape
i r . r 1 ^ VV2
tu = t j
Here T =
+ 2a| —1 I+ß
dx.
3 a=Po ß= 3Ho
4nGA ' A ' 4nGA
-1.
(25)
(26)
Parameters a and P are dimensionless. Let us call them the
Poincare number and the Hubble number to honor these
great contributors into cosmology. The Poincare number is
directly proportional to the Ch number calculated for the
current epoch at the scale of the universe. The current age
(28)
of the universe is determined by the Hubble and Poincare numbers, and by one constant of the time dimension depending only on GA.
To determine the age of the universe let us use the results of observations collected by the PLANCK satellite mission for many years. According to these data, at the current epoch the universe contains the Dark Energy 68.3/4.9 times more than the ordinary (baryonic) matter, and the average density of the baryonic matter is equal to p0 = 4.5 • 10-28 kg/m3. From here, it follows that the density of the Dark Energy is equal to
A = 0.63 • 10-26 kg/m3. (27)
Using Eqs. (26) and (27) and other data obtained by the PLANCK and WMAP satellite missions, we get the Poin-care number and other parameters T = 0.756 • 1018 s = 2.4 • 1010 years, a = 31.7/68.3 = 0.646, H0 = 68.3 km/s per MPc.' From here and from Eqs. (26) and (27), we find the Hubble number
P = 1.8. (29)
Using Eqs. (25), (28) and (29), we calculate the current age of the universe
Tj = 12.26 -109 years ~ 12.3 billion years. (30) According to the ACDM model, the age of the universe is about 13.8 billion years. It is striking that the ACDM theory based on the assumption of the general relativity about the nonflat universe could achieve the result that is so close to the correct one!
8. The neutron universe, the Dark Matter and Black Holes
Let us study the early universe. From Eq. (19), it follows that
2MG R
, when t ^ 0,
(31)
dR dt so that
R = (2MG)1/3(3/21)2/3, when t ^ 0. (32)
Using Eq. (32), we can find the density of the early universe
P=-
1
6nGt2
-, when t ^ 0.
(33)
The density of the early universe was the function of only G and t. Evidently, in the early universe p>>A so that in this model it consisted mostly of the gravitational matter.
Neutrons and protons make up nuclei of atoms of all elements. They have mass about 1.67-10-27 kg and radius about 10-15 m. Hence, their density pB is equal to
pB = 4-1017 kg/m3. (34)
From Eq. (33), it follows that in about 45 ^s after the universe was born, it had the density which was about the density of atom nuclei.
It is reasonable to assume that at some times around t ~45 ^s the universe represented a dense gluon "soup" of down quarks and of a double amount of up quarks, and it was covered by a dense "atmosphere" of photons and neutrinos trapped by the gravitation forces of the universe. It was a gigantic Black Hole we call the neutron universe. Other elementary particles like electrons, positrons, muons and others, both boson and fermions, were also trapped by the gravity, although being either unstable or insignificant in amounts. Because the universe is electroneutral in average, proton charges were annihilated by electrons so that the ratio of the number of down quarks and up quarks was typical for neutrons.
If the gravitational mass of the universe was about 1054 kg at any time, then according to Eqs (32) and (34), the radius of the neutron universe at that time was equal to
Rn = 3.8 -107 km. (35)
It is about the distance between the Sun and Mercury, a very small part of the solar system.
All gravitating particles near a big-mass object, including photons and neutrinos, are trapped by the gravity of the big object, if
2GM > c2R. (36)
Here M and R are the mass and radius of the big object like a neutron star or the neutron universe.
Evidently, the gravitating object meeting the condition equation (36) is a Black Hole. In terms of the density pB of the Black Hole, Eq. (36) can be written as
8nGpBR2 > 3c2. (37)
For the neutron universe, the condition equation (36) is, certainly, satisfied so that all photons and neutrinos were trapped in the "atmosphere" of the earliest universe; it was a gigantic Black Hole.
The gravitational stresses oik inside a Black Hole or a dead neutron star are distributed as follows 2n
Qik =
p + y pBg(r2 - r2)
ik •
(38)
Here r is the distance from the center of a spherical Black Hole or dead neutron star, p is the pressure of the neutrinophoton atmosphere (p ~ 0 for dead neutron stars); and 8ft is the Kronecker delta.
Equation (38) is valid both for heavy ideal fluids and for heavy elastic solids which Poisson's ratio is equal to 0.5 due to the big pressure.
According to Eq. (37), the neutron stars that are less in size than R* are losing energy and fast dying
R =-
= 20 km.
(39)
2 \ 2nGpB
Hence, the neutron stars of radius R > 20 km turn out to be stable Black Holes keeping all their energy.
From Eq. (36), it follows that the critical mass of stable Black Holes is equal to
M* = 1.34 -1031 kg = 6.7Msun. (40)
Here, Msun is the mass of the Sun.
The mass 6.7Msun is the maximum value of the mass of neutron stars and the minimum value of the mass of stable Black Holes. It is useful to remember another important critical value, the Chandrasekhar limit 1.4Msun which separates the stars of lesser mass, turning into dead white dwarfs in the long run, from the neutron stars of higher mass-energy that originally rotate extremely fast and create hot rotating clouds of the gravitational matter but fast lose energy and die [11].
As a reminder, common stars usually go through a long evolution of more than 10 billion years synthesizing hydrogen from quarks, burning hydrogen into helium and then, in Red Giant Stars, helium into carbon, oxygen and heavier elements. Finally they explode as supernovae which emit a tremendous amount of photons, neutrinos and electromagnetic radiation in pulsars. After that, they get cool, rotate slower and die fast [12].
Dead neutron stars which don't revolve can't be easily detected. However, despite its size may be about that of a big meteorite, a neutron star will disturb the solar system if moves within its reach. If this happens, the humankind will inevitably perish. The smallest dead neutron stars are most probable "candidates" to meet with the solar system.
The Black Holes which mass is greater than 6.7Msun are much more powerful. They carry a tremendous energy, and they keep trapped any photons, neutrinos and other particles and adsorb any external emission received. It is hard to detect them. They can be discovered only by the nebulae of clouds of the revolving gravitational matter in their gravitational field. Gigantic Black Holes can have the mass of many million Msun and form galaxies like the Milky Way. Some hope for the direct monitoring of the Black Holes provides the so far undetected Hawking effect that follows from the quantum mechanics [13].
At the large cosmological scale, dead neutron stars and Black Holes form the Dark Matter that reveals itself only by the gravitational effect. It is mostly from this dangerous matter that our gravitational universe is made. The ordinary matter which can be, in principle, observed makes up only 4.9% of the content of the universe.
9. The Planck epoch and the Big Bang
The neutron stars and Black Holes are the densest cos-mological objects of nature. We can only guess about the state of the universe at times less than 45 ^s after it was born. More than a hundred years ago, Max Planck (18581947), a German physicist and the reluctant originator of quantum mechanics, offered a guess about some absolute units of time and space.
The point is that the only dimensionless combination of time t, the gravitational constant G, the speed of light c and the Planck constant h, is the following one
isL
hG
(41)
From here, the specific time tP and the specific length lP that can be called the Planck time and length, are equal to
[14]
tP = = 1.3 • 10-43s, lP = ctP = 4 •10-35m. (42)
These values are of great interest because they are made from the fundamental constants characterizing the main physical phenomena of nature, namely, the gravitation, the relativity and the quantum property of the microworld. If these constants are absolute, then the Planck time and length may be the quanta of the spacetime in the future Unified Theory.
According to Eq. (32), we have R = 3 • 10-14m, when t = tP (the Planck epoch). At that time of the Big Bang the prouniverse was a little bit greater in size than the nucleus of the helium atom at our epoch.
As seen, qualitatively the NEOC approach supports all substantial points of the Big Bang theory, the FLRW model and ACDM model which are based on the general relativity of a curved universe, while the current theory uses only well-measured data within the framework of the classical mechanics of the common flat space. However, both fail to explain the nature of the Big Bang.
10. Revolution of the universe and the Dark Energy
In planetary systems of stars, as well as in galaxies, clusters and superclusters, the gravitating masses revolve around a center of gravity of the corresponding system. As "a birth defect", this revolution is caused by an asymmetry of an original system arisen, for example, when a cloud of dust and gas collapsed. Due to the law of the angular momentum conservation, the denser collapsed system has the greater angular velocity of its rotation. The greatest angular velocities are characteristic for neutron stars and Black Holes, the densest objects of nature.
The axis of rotation always goes through the center of gravity of the original cloud, and the gravitation force acting upon each body is directed towards this center. The revolution creates the centrifugal force acting upon each revolving body. However, this force can balance only the component of the gravity force which is perpendicular to the axis of rotation. The component of the gravity force directed along the axis of rotation is unbalanced; this component moves all revolving bodies onto one and same plane which is perpendicular to the axis of rotation and goes through the center of gravity.
As a result, all revolving systems of gravitating bodies become flat or close to flat depending on the age of the system. The solar system is practically flat, and all planets lie in one and same plane. The fractal dimension of the Milky Way is about 2.2 so that this galaxy is an area-like fractal
close to a pancake by its general shape. The decrease of the angular momentum of a revolving system can occur only due to an emission of energy from the system.
This general property of gravitating systems makes us suggest that the universe as well revolves in the space around a certain axis going through the center of gravity of the universe. This hypothesis is especially alluring because it allows us to give a very simple explanation of the Dark Energy, the most obscure and subject of cosmology at the present time.
Indeed, in the revolving universe every mass experiences the centrifugal force which is equal to the product of the mass, the square of its angular velocity and its distance to the axis of rotation. Let us assume that this centrifugal force of the revolving universe is the repulsion force of the Dark Energy acting upon any gravitating mass.
From here, using Eqs. (14) and (15), we get
2 4n ro2 =— GA.
(43)
Here ro is the angular velocity of the universe. It should be emphasized that it is an average quantity for all gravitating objects of the universe; the values of ro for particular objects can differ very much.
Substituting A in Eq. (43) by Eq. (27), we have
ro = 1.33 • 10-18 s-1. (44)
This small angular velocity of the universe cannot be ever detected by human beings. At such an angular velocity, for 12.3 billion years the universe would turn only by angle 30°.
The same and even more difficult problem is to detect the center of gravity of the universe and the axis of its rotation. It should be mentioned that the notions of the gravity center and rotation axis of the universe contradict to the Copernican principle accepted in the general relativity. These are most essential objections against the present neoclassical theory. However, they don't matter, since in the NEOC approach all observers within the 100 MPc distance around the Earth and about one million years apart are equivalent because they would observe one and same picture of the large-scale universe.
It is, probably, impossible to prove or disprove the current simple approach to the Dark Energy using direct measurements of the angular velocity or angular displacement of the universe.
The hypothesis of the revolving universe leads us as well to the conclusion that the fractal dimension of the universe should be less than 3, and in the long run it should approach to 2 similarly to clusters, galaxies, planetary systems and any revolving systems of gravitating masses.
11. Modified evolution equations of the universe
The revolution can drastically change the dynamics of the evolution of the expanding universe, particularly, because it makes the cosmological constant vary with time.
Let us derive the basic equations of the revolving and expanding universe. Because the symmetric expansion is unstable, any asymmetry in the process of the expansion of the universe from the Big Bang could produce a moment of force and revolution.
The moment of force causing the revolution is equal to the product of an eccentric force and its distance from the axis of rotation. From the dimensional analysis, it follows that the perturbation force is directly proportional to the gravity of eccentric masses, and that their distance from the axis is proportional to the radius of the universe. Based on these assumptions, the equation of the rotational dynamics of the universe can be written as follows:
d , .Wx GMme n
—(ciMRIm) =-ce R.
dt' R2 e
(45)
Here M, R and ro are the mass, maximum radius and average angular velocity of the universe, me is some eccentric masses, and ct and ce are some dimensionless coefficients depending on the shape of the universe and on the position of eccentric masses.
In particular, the value of ct is equal to 0.4 for solid spheres, 0.2 for thin circular plates, and 0.2 + 0.2(b/R)2 for oblate solid spheroids of the maximum thickness 2b (0 < < b < R). The latter two are some possible shapes the universe can get due to the revolution.
The left-hand part of Eq. (45) represents the rate of the angular momentum, and the right-hand part the moment of eccentric forces. Let us rewrite Eq. (45) as follows:
d2
r—(ar ) = PE, where a = a(T). (46)
dT
Here r, t, a, and PE are the following dimensionless parameters:
R t Gm c T* ,.n .
r —, T=^ a = roT*. Pe =—e-e^, (47) R* T* cR
where R* and T* are the radius and age of the expanding
and revolving universe at some specific epoch.
Parameter PE called the eccentricity parameter characterizes the dynamics of the revolving universe. Its value is determined by some unknown disturbances of the symmetric distribution of masses in the early universe at the time of the Big Bang, with this "birth defect" asymmetry being remained forever. In the current NEOC model of the expanding and revolving universe, the value of PE is an empirical constant like the gravitational constant.
Based on Eqs. (17), (43) and (47), the main equation (18) of the universe expansion takes the following shape
G* (48)
d2r
dT2 r2
Here G* is the dimensionless gravitational number of the universe at some specific epoch. It is equal to
G =
GMT*2
3 K* *
Here p* is the average density of the universe at the specific epoch.
The equation system (46) and (48) determines the evolution of the revolving and expanding universe in this NEOC model. The solution of this system for 0 < t < <» should meet the following boundary conditions:
r = 0, a ^ <», when t = 0, (50)
r = 1, when t = 1. (51)
It is easy to find the following singular solution to Eqs. (46) and (48) satisfying the boundary conditions, Eqs. (50) and (51),
r = t2/3,
a = 3PE T
Pe = KlG* --.
E 3 V * 9
(52)
From here, it follows that the Hubble parameter corresponding to this singular solution is equal to
H = —. (53)
3t
According to Eqs. (49), (52) and (53), this singular solution is valid when G* > 2/9 so that the density of the universe at the specific epoch is greater than a certain critical value pc
P* >PC =-
3H2
1
(54)
(49)
8nG 6nGT*2
It is interesting that this critical density of the revolving and expanding universe coincides with the density of the early universe in the model of the expanding, but not revolving universe at this specific epoch, see Eq. (33).
For comparison, if we assume the specific epoch to be the current one, then from Eqs. (53) and (54), using the known Hubble constant, we get
pc = 0.88 -10-26 kg/m2, TU = 10 billion years. (55)
This is pretty close to the results of both the general relativity and the present NEOC theory for the expanding universe.
The singular solution of the equation system of this model describes, at least, a considerable part of the prehistory of the universe, up to the age of about ten billion years. However, the complete solution of Eqs. (46) and (48) is still an open problem.
12. Orbital velocities of stars in spiral galaxies
The average orbital velocity V of a planet of mass m in the solar system is determined by the balance equation of the inertia force mV2/R and the force GmM/R2 of the attraction to the Sun of mass M where R is the average distance between the planet and the Sun. V= (gm/r)1/2 so that the orbital velocity decreases and tends to zero when this distance increases. Since Ch >> 1 for the Milky Way and all other galaxies, the orbital velocity of stars in galaxies seems to be described by the same law. However, based on the astrophysical data, this velocity practically does not depend on the distance between a star and the galaxy center, and it is usually equal to about 220-260 km/s.
This paradox evoked a number of theories. The Modified Newton's Dynamics (MOND) theory rejects Newton's law of inertia and replaces it by another law, according to which the inertia force equals £(mV2/R)2, where £ is a mass coefficient. From here and the balance equation, it follows that the orbital velocity does not depend on the distance. According to another theory, this paradox is caused by the effect of the Dark Energy; however, it is not true because Ch >> 1 in this case.
Meanwhile, the flat, spiral structure of the Milky Way and other galaxies allows us to provide a simple explanation and prediction of the paradox within the framework of the classical mechanics and Newton's law of gravity. Indeed, it is reasonable to assume that the gravitational matter of the Milky Way is uniformly distributed along some logarithmic spirals with the common pole at the center of the disk of the Milky Way. These spirals are well-documented.
The length of an arc of a logarithmic spiral is equal to (R2 - Rj)/cos P, where P is the spiral constant equal to the angle between the radius-vector of a point on the spiral and the tangent to the spiral at the point, and R2 and R1 are the distance between the pole and the ends of the arc. When R2 >> R1, the length of the arc is directly proportional to the distance R = R2 between this point and the center of the galactic disk which is the pole of the logarithmic spirals.
Mass M of the galactic disk inside radius R is directly proportional to R for any number of spirals
M = kR. (56)
Here k is a galactic constant.
For the Milky Way M = 1.4 • 1042 kg, when R ~ ~ 1.5 • 1021 m so that
k = 1021 kg/m. (57)
From Eq. (56), it follows that the force attracting a star of mass m to the center of the galaxy is equal to (kmG)jR. This force is balanced by the centrifugal force of inertia of the star which is equal to mV2 /R. From here, we get the orbital velocity of stars in spiral galaxies [5-9]
V = 4kO. (58)
According to Eqs. (57) and (58), the average orbital velocity of stars in the Milky Way is equal to about 250 km/s. This result of calculation is confirmed by astrophysical data.
From Eq. (56), it follows that the distribution of gravitating masses in spiral galaxies obeys the following law [59]:
k
P =-. (59)
2nR v ^
Here p is the gravitating mass per unit area of the galactic
disk, and R is the distance from the galactic center.
The NEOC law expressed by Eq. (5 8) is, evidently, valid
also for any galaxies which gravitational matter, including
the Dark Matter, is distributed in the galactic disk similar
to Eq. (59). The present NEOC approach to this problem
may support the viewpoint of [15] Rubin (1928), an American astronomer, who explained this anomaly by the effect of the Dark Matter [15].
13. The fractal universe
As it was first shown by Mandelbrot (1924-2010), a French scientist, all geometrical objects of nature including the universe represent some fractals which length, area and volume depend on the scale of measurement. For example, the length of the coast line of the continents on the Earth substantially depends on whether we measure it using maps in the scale of 1 mile, or 100 miles (per inch of a map), or in the scale of 1 ft by a walker.
The results of the measurements of the coast line length can be approximated as follows [16]:
f a \
1-df
or = (1 - df) log
f A^
(60)
Here L is the length in the scale of A, L0 is the length in the scale of A0, and df is the fractal dimension. Fractals which are characterized by the power-law function of Eq. (60) are called the power-law fractals.
For the line-like fractals like a coast line, the fractal dimension varies in the range
1<df <1 + 9, where 0 < 9 < 1. (61)
The greater is the fractal dimension, the longer is the length L. For example, the fractal dimension of the Australian coast equals df = 1.14 while that of the Norway coast equals df = 1.49. If 9 = 0, then df = 1; this is the common metric dimension of length that does not depend on the scale of measurement. Fractal dimension is a measure of the geometrical complexity of a system.
Let us apply this approach to the universe which is surely the most complex fractal object. Suppose we can measure mass M of the universe using some astrophysical devices of various precision allowing us to study the universe within some device-dependent radius R which is the scale of measurement in this case. Evidently, the result of measurement depends on this scale.
As it was shown in the previous section, any rotating system of gravitating masses acquires a shape close to a disk which metric dimension equals 2. Hence, in the long run the fractal dimension of the universe should be close to 2. Based on this assumption, let us interpolate the measurements of the fractal universe by the following power-law function
M M0
f R f-df
R0
where df = 2 + 8 (S<< 1).
(62)
Here Mis the mass of the universe in the scale of R, M0 is the mass of the universe in the scale of R0, and df is the fractal dimension of the universe.
According to Eq. (62), the universe has the shape of a thin circular disk of radius R and thickness h, which fractal
dimension is close to 2. Although some astrophysical observations support this assumption, the results are still inconclusive and contradictory because the methods of measurement differ very much. Therefore, Eq. (62) can only serve as a hypothesis.
Since M = nphR2 in metric dimensions, from here and from Eq. (62), we can derive
h = R-f where z = Md
np
R0
(63)
This equation allows one to estimate the thickness of the universe using the empirical constant Z and the fractal dimension of the universe.
14. Gravitons and the Unified Theory
Gravitons and gravitational waves are still subject to even more speculations because of their very low intensity. We mention here only one hypothesis, according to which gravitons are the gravitational waves of the maximum possible length X which has the value of the order of the radius of the universe [5-9]
X = 1026 m, v = c/X = 3 - 10-18 s-1
(64)
mg = hv/c2 = 10-68 kg.
Here v is the wave frequency, and mg is the mass-energy of gravitons.
According to Eqs. (64), this hypothetical graviton is the smallest elementary particle of nature. No physical particle can be less. Its frequency corresponds to the period of the wave of the order of ten billion years which is about the age of the universe. From this hypothesis, it follows that our universe consists of about 10120 gravitons.
Each "elementary" particle represents a cluster of a huge number of gravitons. In particular, we get the following figures: neutrino mass equals
0.32 eV/s2 = 0.58-10-36 kg = 0.26-1032mg, up quark mass equals
2.3 MeV/s2 = 4-10-30 kg = 7-1037mg , down quark mass equals
4.8 MeVs2 = 8.6-10-30 kg = 2-1038mg , electron and positron mass equals 9.1-10-31 kg = 2-1038mg, and gamma photon effective mass equals 1.8 - 10-30 kg for 1 MeV photons. These gravitons as the smallest elementary particles can pretend to the role of building blocks in the future unified theory.
However, this role imposes some strict constrains on the topology of the developing universe because, according to Eq. (64), the value of the maximum radius of the universe becomes an absolute constant like the speed of light or the gravitational constant. It means that in the earli-
(66)
est stage the universe represented a rotating whip-like set of gravitating masses of the fractal dimension close to 1 which length was always equal to one and same constant. In the course of time the shape of the universe gradually changed its geometry from an almost 1D whip to the current almost 2D disk of the maximum radius that is equal to the same constant. To a certain extent, this hypothesis is similar to the Aristotelian concept of "solid sky".
15. Some remarks
The Planck dimensional analysis that led him to the Planck length and time can be supplemented by the Planck mass, force, energy, stress, momentum and so on. For example, in terms of the absolute constants we get:
mP =Jch[G, FP = c4/G, EP = c2mP,
ctp = c1 ¡(hG2), IP = c^/ch/G. (65)
Here mP, FP, EP, ctp, and IP are the Planck mass, force, energy, stress and momentum respectively. Any physical quantity which depends on time, length and mass can be represented by a similar equation.
From Eqs (65), we find:
mP = 5.4 - 10-16 kg, FP = 1.2 - 1044 N,
EP = 48.6 J, IP = 16.3 N - s.
The Planck energy and momentum have some earthly values as distinct from time, length, mass and force.
If we assume, following the Planck way of reasoning, that the cosmological constant has to be determined by the absolute constants, then we come to the following result
AP = c5 ¡(hG2) = 0.82-1096 kg/m3. (67)
This value of the Planck cosmological constant 10120 times greater than its value according to all astrophysical data!
From this paradox, it may follow that the cosmological constant can also be an absolute constant independent of other absolute constants, or there exist some new undiscovered independent variables beyond mass, length and time.
16. Conclusion
As an alternative to the general relativity, the NEOC approach to the large-scale universe gives a way to plainly describe many obscure astrophysical phenomena, which are hard to explain in the framework of the general relativity or other cosmological theories of the curved universe.
The terminology like the Bing Bang, the Black Hole, the Dark Matter, the Dark Energy and others is a good illustration of the current understanding of many astronomical phenomena, some of which are far beyond the human mind. No one can ever know where the tremendous energy of the universe appeared from, what is beyond the observable universe, how what is beyond can influence our universe, and so on.
Hopefully, the present approach to cosmology can close this gap.
References
1. Minkowski H. Raum und Zeit // Physikalische Zeitschrift. - 1907. -V. 10. - P. 75-88.
2. Einstein A. Die Grundlage der allgemeinen Relativitätstheory // Annalen
der Physik. - 1918.-V. 49. - P. 769-822. - doi 10.1002/andp. 19163540702.
3. Friedmann A.A. Über die Krümmung des Raumes // Zeitschrift für Physik. - 1922. - V. 10. - No. 1. - P. 377-386. - doi 10.1007/ BF01332580.
4. Hubble E.P. A relation between distance and radial velocity among extra-galactic nebulae // Proc. Nat. Acad. Sci. - 1929. - V. 15. - No. 3. -P. 168-173. - doi 10.1073/pnas.15.3.168.
5. Cherepanov G.P. The Dark Energy: An Example of Application of the Invariant Integral // Fracture Mechanics. - Moscow-Izhevsk: Institute of Computer Science. - P. 845-848. Some new applications of the invariant integral of mechanics // J. Appl. Math. Mech. - 2012. -V. 76(5). - P. 519-536.
6. Cherepanov G.P. The invariant integral: some news // Proc. ICF-13. Sir Alan Howard Cottrell Symp., Bejing, China. - 2013. - V. 6. -P. 4953-4962.
7. Cherepanov G.P. The mechanics and physics of fracturing: application to thermal aspects of crack propagation and to fracking // Proc. Roy. Soc. Phil. Trans. A. - 2014. - V. 373. - P. 20140119. - doi 10.1098/rsta.2014.0119.
8. Cherepanov G.P. The invariant integral of physical mesomechanics as the foundation of mathematical physics: some applications to cosmology, electrodynamics, mechanics and geophysics // Phys. Meso-mech. - 2015. - V. 18. - No. 3. - P. 203-212.
9. Cherepanov G.P. A Neoclassic Approach to Cosmology Based on the Invariant Integral. Chapter 1 // Horizons in World Physics. V. 288. -New York: Nova Science Publishers, 2016 (in print).
10. Bennet C.L. Wilkinson Microwave Anisotropy Probe (WMAP) // Questions of Modern Cosmology: Galileo's Legacy / Ed. by M. D'Ono-frio, C. Burigana. - Berlin: Springer, 2009. - 525 p.
11. Camenzind M. Compact Objects in Astrophysics: White Dwarfs, Neutron Stars and Black Holes. - Berlin: Springer, 2007. - 588 p.
12. Weinberg S. Cosmology. Oxford: Oxford University Press, 2008. -616 p.
13. Hawking S.W. Black holes in general relativity // Commun. Math. Phys. - 1972. - V. 25. - No. 2. - P. 152-166.
14. Planck M. Über irreversible Strahlungsvergänge // Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin. -1899. - V. 5. - P. 440-480.
15. Rubin V.C. Bright Galaxies, Dark Matter. - New York: Woodbury Publ., 1997.
16. Mandelbrot B. The Fractal Geometry of Nature. - New York: W.H. Freeman and Co., 1982.
Поступила в редакцию 15.07.2016 г.
Сведения об авторе
Genady P. Cherepanov, Prof., Hon. Life Member, The New York Academy of Sciences, USA, genacherepanov@hotmail.com
