CC BY
44
УДК 524.8, 539.42, 530.1
Invariant integral: The earliest works and most recent application
G.P. Cherepanov
The New York Academy of Sciences, New York, USA
The present paper embraces mainly the three-year period of 1966 to 1968 when the invariant integral of fracture mechanics appeared and became popular, and the last two years of 2015 to 2016 when the neoclassic cosmology based on the invariant integral came up. A mention is given to the previous works of Euler, Cauchy, Maxwell, Nother, Gunther and Eshelby who dealt with invariant integrals in mathematics, hydrodynamics, electrodynamics, and the theory of dislocations. A brief review is given of the creation of the invariant integral of fracture mechanics under static and dynamic conditions for a solid continuum including elastic, plastic and viscoelastic materials, as well as of some of its most important applications, ramifications and generalizations for other physical fields. The initial phase of the expansion and revolution of the large-scale universe is studied in the framework of the neoclassic approach, including the Big Bang and the Dark Energy; it is shown that the spheroidal shape of the universe assumed at the Big Bang retains its eccentricity constant in the initial phase. The assumption of a superphoton as a primordial universe was analyzed.
Keywords: invariant integral, continuum and quantum mechanics, cosmology, fracture mechanics, the universe shape, the Dark Energy, the Big Bang, expansion and revolution of the universe
Инвариантный интеграл: первые работы и новое приложение
Г.П. Черепанов
Нью-Йоркская академия наук, Нью-Йорк, США
Настоящая статья касается в основном работ периода 1966-1968 гг., когда появился и стал популярным инвариантный интеграл механики разрушения, а также периода 2015-2016 гг, когда возникла неоклассическая космология, основанная на инвариантном интеграле. Упоминаются подходы Эйлера, Коши, Максвелла, Нетера, Гюнтера и Эшелби, которые ввели инвариантные, т.е. независимые от пути интегрирования, интегралы в математике, гидродинамике, электродинамике и теории дислокаций. Дан краткий обзор инвариантных интегралов для упругих, пластических и вязкоупругих тел при статическом и динамическом нагружении. Кратко обсуждаются некоторые основные приложения этого интеграла, а также его обобщения для других физических полей. Рассматривается начальная фаза расширения и вращения сфероидальной Вселенной в большом масштабе порядка 100 МПк в рамках неоклассического подхода, включая Большой взрыв и Темную энергию; показано, что в этой фазе начальный эксцентриситет Вселенной сохранится постоянным. Анализируется предположение о том, что Вселенная возникла из изначального суперфотона.
Ключевые слова: инвариантный интеграл, механика сплошных сред и квантовая механика, космология, механика разрушения, форма Вселенной, Темная энергия, Большой взрыв, расширение и эволюция Вселенной
1. Introduction
The invariant or path-independent integrals first appeared in works of Euler and Cauchy who introduced them to create the theory of functions of a complex variable used in the two-dimensional hydrodynamics of ideal incompressible fluids. As a matter of fact, these integrals represented the conservation laws of physics. Later, similar invariant integrals appeared in Maxwell's electromagnetics and in mathematical works by Nöther and Günther.
In 1951, Eshelby studied the infinitesimal translation 8e of a field singularity inside a closed surface in an elastic
body and defined the force on the singularity as 8W/8e where W is the work of tractions on the surface. By this way, Eshelby derived the force on a dilatation center and on a small spherical inhomogeneity and found out the Peach-Kohler dislocation-driving force for arbitrary anisotropic elastic media. All Eshelby's singularities while moving produced no irreversible changes in the material, and because of lack of applications Eshelby's discoveries remained little known until the late 1960th. Even today, however, it is not easy to discern some of the present invariant integrals in his fundamental theoretical work [1]. It had
© Cherepanov G.P., 2017
been only since the year 1968 that Eshelby had started on working in fracture mechanics.
It is noteworthy that, except for the analytical functions of a complex variable and these referenced cases, the invariant or path-independent integrals had never been used until 1966-1967.
2. Discovering the invariant integral of fracture mechanics
In the 1950th, thanks to the pioneering works of Irwin and Williams, it became clear that to understand the fracture phenomena one should study the fracturing process occurring in the small vicinity of the crack front.
In 1957, Irwin coined the notion of stress intensity factor describing the stress-strain field at the open-mode crack tip for brittle or quasi-brittle fractures. Also, he advanced the criterion of brittle failure in terms of the stress intensity factor, based on the earlier Griffith' understanding of the crack propagation as a process of the transfer of the elastic stress energy into the surface energy. Thus, the IrwinGriffith theory of fracturing was created. Soon after, Paris experimentally found his law of the fatigue crack growth rate in terms of the stress intensity factor. These works started on fracture mechanics as applied to only brittle or quasi-brittle fracturing.
In 1966-1967, this author published the theory of crack propagation in any solid continuum [2-5], which generalized the Griffith-Irwin theory of brittle fracturing. His theory was based on the invariant integral derived from the energy conservation law for an arbitrary solid continuum.
In Ref. [3], the author considered the neighborhood of an arbitrary point O of the open-mode crack front, which is small as compared to characteristic linear dimensions of the crack and body. In this scale, the material of this neighborhood is in the state of local plane, or antiplane, strain so that all functions depend only on Cartesian coordinates x and y, or polar coordinates r and 6, with center at O in the corresponding cross section normal to the crack front. Then, for the case of local plane strain, "denote by C a simple closed curve in the xy plane encircling point O. Curve C located in this small neighborhood may be considered to be a circle of radius R without any loss of generality" [3]. In this small scale, the crack is represented by a traction-free cut along y = 0, x < 0.
The energy conservation law for the material inside contour C was under consideration when the crack tip advanced a distance small as compared to R along the x axis. The direct calculation of the energy flow into the crack tip at O led to the following equation, see Eq. (1.12) in [3]
r = j
du
(U + K) cos 6 - (ax cos 6 + txy sin 6)--
dx
- (t cos 6 + ay sin 6) dv + — (qx cos 6 + gy sin 6) dx V
r = 2y, ds = R d6.
ds, (1)
Here y is the material surface energy per unit area, U and K are the internal and kinetic energy of the material per unit volume, ax, ay, and t are stresses, u and v are the components of the displacement vector, qx and qy are the components of the heat flux vector, and V is the crack tip velocity.
The integral in Eq. (1) is taken over contour C. In the calculation, it was assumed that, in the small neighborhood under study, the field is steady-state. Only thermal and mechanical processes were taken into account.
In papers [2, 3], Eq. (1) was advanced as the general criterion of the onset of the crack propagation in any solid continuum, with 2y = r being the basic material constant. By 1973, the constant JIC which is equal to r in Eq. (1) had been accepted by ASTM for use as a material constant which characterizes the crack propagation onset in elastic-plastic metals. Currently, this constant is used in practical engineering as one of the basic material properties.
Using the tensor notation, Eq. (1) can be written as follows
r = j[(U + K)nx-aiknkUii + V xciini]ds, i, k = 1, 2,
n1 = cos 6, n2 = sin 6, x1 = x, x2 = y, u = u, u2 = v, an = ax, a12 = t,
(2)
12
a22 =ay, q1 = qx, q2 = qy.
Due to the energy conservation law, the integration contour C in Eq. (2) can be arbitrarily modified into any other contour being in the region where the field and the boundary geometry are steady-state, i.e. stationary in coordinates x- Vt and y. The nj and n2 remain their meaning of the components of the external outer ort to the arbitrary contour C closed around the point O in the local normal plane under consideration. The positive sign of the thermal term in Eq. (2) means a heat inflow from some external sources while the negative sign means a heat outflow produced by the irreversible deformation of the material inside contour C.
In terms of the temperature field, the last member in the integrand of integrals in Eqs. (1) and (2) can be written as follows [6, 7]
V-1 (qx cos 6+ qy sin 6) =
= PmcH-Tnj + V~lkTT^ni, i = 1,2. (3)
Here T is the temperature, pm is the mass density, cH is the specific heat, and kT is the thermal conductivity of the material.
In Ref. [3], the static reversible case of an elastic material was, in particular, studied. In this case, the terms responsible for dynamic and irreversible processes in Eqs. (1) or (2) are equal to zero so that Eq. (2) is reduced to the following shape, see Eq. (2.2) in [3]
r = j(Unj - aik^Ui J) ds, i, k = 1 2. (4)
In this particular case, the field does not depend on time so that the integration contour C can be arbitrarily modi-
fied into any other contour in this elastic body, with the value of the integral being invariant. In Ref. [5], this property was directly used by this author. It is common for the integrals of analytical functions of a complex variable, which is physically due to the energy conservation law, as well.
In paper [8] published in 1968, Rice called the integral in Eq. (4) as the /-integral and so, later many called it Rice's integral.
Moreover, Eq. (4), as if by EUREKA, appeared in the very beginning of Rice's paper [8] as some /-integral without a citation of works [2, 3]. Rice just proved its path-independence using the divergence theorem and utilized it only to study the strain concentration at a groove bottom, to Neuber's test. Its application to the cohesion model of brittle fracture looks very doubtful because, e.g. in glass, the size of the cohesion zone is about the interatomic distance, see [4]. But, this mistake was corrected in the nick of time.
In Ref. [3], for the particular case of linearly elastic homogenous isotropic materials, Irwin's equation of the crack-driving force was derived from Eq. (4), which made the foundation of the mechanics of brittle fracture.
In the case of incompressible power-law hardening materials, it was proven that the corresponding local problem was self-similar so that the stresses near the open-mode crack-tip were found to be in the following form [3]
oik =
1
42n
K/ fik (9), X = -
1
1 + 8'
D = 2aS8, r = 2n(8)aK1+8, f22(0) = 1.
(5)
Here D and S are the second invariant of the strain and stress tensors, respectively, a and 8 are some elastic constants, n(8) is a dimensionless function, and K: is the stress intensity factor.
According to Iliushin's theorem, this solution is valid also for the power-law hardening elastic-plastic materials in the case of simple loading [3]. For incompressible linearly elastic materials, when 8 = 1, aE = 1.5, X = -0.5, and n(1) = 1, equation for r in Eqs. (5) coincides with the corresponding Irwin's equation (E is Young's modulus).
For some unknown reason, this solution was called the HRR solution by the names of the authors of papers [9-11] who numerically calculated functions fik(9) in Eqs.(5).
In Ref. [3], the initiation of the crack tip in a perfectly plastic material was considered by means of Eq. (1), and it was found that the energy flow into the crack-tip is equal to zero at the very beginning of the loading and crack growth [5, 12]. In other words, the crack starts on growing at the very beginning of loading. Besides, in [5, 12] the empiric Paris' law of the fatigue crack growth, valid for the low and moderate values of stress intensity factor, was derived from the r-concept.
Later, in [13-15] the coupled nanomechanics of the dislocation generation and dislocation-affected crack
growth was advanced. It allowed this author to calculate the diagrams of the crack growth versus stress intensity factor for the loading of cubic crystal lattices and to find out, particularly, the critical value of stress intensity factor at the onset of pop-in. The minimum stress intensity factor necessary to initiate a crack growth proved to be many times less than the fracture toughness.
In [3, 5, 12, 14, 16], the crack growth in linear vis-coelastic materials was also considered. It was shown that, in this model, the crack growth is determined by the elastic component of the material response, and that an account for the process zone at the crack tip is necessary in order to describe a more complicated behavior (see also [17] for some comments on [3]).
Dynamic crack propagation in elastic materials was studied in papers [5, 18] where the corresponding equation for the energy flow into the crack tip was found. The invariant integral of Eq. (1) was, in fact, used in [18] because original Ehelby's integrals are static.
Lots of new important applications of the invariant integral in Eq. (1) as well as many new invariant integrals of physics and many new laws resulting from these invariant integrals have been found out since then. For an account of some results based on these integrals, one can see works [6, 7, 14, 15, 19, 20-26]. For example, the following new laws of physics were derived:
- the generalized Archimedes' law of buoyancy which takes into account the surface tension of liquids [21];
- the generalized Coulomb's law for electric charges moving at a relativistic speed [19, 20];
- the generalized Newton's law of gravitation which takes account of the Dark Energy and describes the expansion of the universe in the cosmological scale, see [6, 7, 19, 22, 25, 26];
- the law of the fracking of porous rocks by gas/oil flows [6, 7];
- the new law of the crack growth in brittle materials based on the consideration of the process of crack growth as the process of the transfer of elastic energy into heat, without to use the surface energy of solids [6, 7];
- the new law of rolling that refuted and replaced the old Coulomb's law of rolling that proved to be wrong [23, 24];
- the laws for crack-driving forces as applied to joints of any combination of membranes, plates and solids of different materials as well as to fatigue, corrosion and hydrogen embrittlement [12, 14, 15, 19, 21, 27].
3. Applying invariant integral of cosmology
Invariant integrals are equivalent to some conservation laws, from which the integrals are derived. That's why the invariant integrals published in [6, 7, 12, 14, 19, 21, 25, 26] can be applied for the solution of any problem of mathematical physics of the corresponding field without using
the differential equations of the field because the field differential equations are derived from same conservation laws and are a consequence of these invariant integrals. The formulation of the problems of mathematical physics in terms of invariant integrals is more general than the common statement of boundary value problems in terms of differential equations because it allows one to solve singular problems unresolved in the framework of the differential equations alone. Besides, the invariant integrals can be very effective toward the direct formulation of some models and theories of mathematical physics.
In Refs. [25, 26], such a direct neoclassic approach to cosmology was undertaken to study the history, size and shape of the universe. This approach was necessitated by the flatness of the large-scale universe proven as a result of the Supernova Cosmology Project and by the recent numerous probes of the WMAP and PLANCK satellite missions made for many years by several independent teams of astrophysicists so that the general relativity appeared to be unrelated to the large-scale universe.
In Refs. [19, 22, 25, 26], the law of energy conservation for the coupled cosmic-gravitational field under study was written in the form of the following invariant integral
1 - 2Q ) + A^
r=f
8nG
dS,
(6)
i, k = 1, 2, 3.
Here rk is the external energy spent to move the cosmic-gravitational matter inside closed surface S on unit length along axis xk, x1, x2, and x3 are the Cartesian coordinates in the Euclidian space, x1, x2, x3) is the potential of the coupled cosmic-gravitational field (per unit mass), S is an arbitrary closed surface of integration, nk is the kth component of the unit normal vector to S, G is the gravitational constant, and A is the cosmological constant. If there are no field singularities inside S, then rk = 0.
3.1. Basic astrophysical data
The scale of about 100 MPc was accepted as the elementary scale in which the matter of the universe can be assumed to be homogeneous and isotropic. This large-scale universe was under study in papers [25, 26]. It was shown that the flat universe in the large scale was the Aristotelian/ Ptolemaic universe that can be considered static during the Homo Sapience lifetime, with the Minkowski metric of the flat space-time being reduced to the Euclidian metric in this large scale [25, 26]. The physical matter of the universe was characterized by some distribution of mass-energy in space-time, with mass-energy and space-time being the basic dual properties of the matter.
The matter of the universe consists of the ordinary matter, the Dark Matter and the Dark Energy. According to the most accurate data of the Planck satellite mission the ordinary matter makes up 4.9%, the Dark Matter 26.8% and the
Dark Energy 68.3% of the physical content of the universe. The ordinary matter and the Dark Matter are gravitational: they tend to collapse into some clots of different size. The clots of the ordinary matter are stars, galaxies, quasars, pulsars and other observable objects.
The Dark Matter is almost unobservable; it is mainly concentrated in the central Black Holes of galaxies and in dead neutron stars where almost all emissions like photons are locked by gravity forces. The only exposure effect of the Dark Energy is the extension of the gravitational matter; no other effects of the Dark Energy have been observed. The Dark Energy is uniformly distributed in the universe, with the density being equal to cosmological constant A.
3.2. The NEOC cosmology of the expanding universe
In Refs. [6, 7, 19, 21, 22, 25, 26], the matter of the universe and its interaction laws were defined in the Euclidian space by the field potential and by the invariant integral of Eq. (6). The universe assumed to be a homogeneous and isotropic sphere in the Euclidian space was found to develop according to the following evolution equation
d2R 4n , ^ 4n n3
—r- = — GR(A - p), where M = — pR3. dt2 ^ 3
(7)
Here R is the radius of the universe, t is the time, p is the density of the gravitational matter of the universe (the ordinary matter plus the Dark Matter); and M is the total gravitational mass of the universe assumed to be constant. The density A of the Dark Energy was also assumed to be a constant of the expanding but unrevolving universe. As a matter of fact, Eq. (7) represents the equilibrium equation of the forces acting upon an imaginary probe mass, a gauge, on the edge of the universe.
From the evolution equation (7), it follows that [25, 26]
TT2 ( 1 dR f 8n_( 1. ^ „ c2
H2 =1--I = — GI p +—A I-K—r, where (8)
I R dt I 3 I 2 1
R2
Kc2 = R2
(
4n
GA +
2MG
\
- H2
(9)
Here H is the Hubble parameter, and R0 and H0 are the value of the radius of the universe and the Hubble parameter at the current epoch.
Equation (8) almost coincides with the basic equation of the modern FLRW and ACDM cosmological models derived from the general relativity which is based on the wrong assumption of the curved/non-flat universe ( they differ only in the physical meaning ofK). The solution of the differential equation (8) provides the radius of the universe in terms of time which describes the evolution of the universe [25, 26].
From Eqs. (7) to (9), it follows that the expansion of the universe was first decelerating when p > A, and then accelerating when p < A. In particular, the Big Bang of the universe was found to be described by the following simple asymptote [25, 26]
R = (2MG)1/31 -1
2/3
P = -
6nGl2
when t ^ 0. (10)
At the Big Bang the cosmological constant had no effect.
The current age TU of the universe was found in terms of T, the Poincare number a and the Hubble number P, where
13 3M 3H2
T = .
4nGA
a = P0 =
ß = JH^-1. (11)
4nGA
A 4nAR0i
Based on the observational data of the Supernova Cosmology Project, and the WMAP and PLANCK satellite missions, these parameters can be taken to be equal to [25, 26]
A = 0.6 -10-26 kg/m3, H0 = 68.5 k^s per MPc,
T = 2.4-1010 years, a = 0.46, P = 1.8.
At the present epoch the universe is accelerating because a < 1. The current age of the universe was calculated to be equal to 12.3 billion years which is amazingly close to the age found in the FLRW and ACDM models based on the wrong assumption of the general relativity, see [25, 26].
Also, according to the NEOC cosmology the mass of the stable Black Holes was found to be greater than 6.7 times the Sun mass while the mass of the live neutron stars and dead dwarfs to be less than this critical value.
However, any theory using the mystic Dark Energy cannot be considered satisfactory until the physical meaning of the Dark Energy is clarified.
3.3. The modified NEOC cosmology of the expanding and revolving universe
The evolution equation (7) represents the equilibrium equation of a probe mass on the edge of the universe subject to the inertia force, the gravity 4n/ 3pGR, and the repulsion force of the Dark Energy 4n/3AGR. In papers [25, 26], it is suggested that the universe is revolving so that the repulsion force of the Dark Energy is, as a matter of fact, the centrifugal force of rotation at the angular velocity ro such that
2 4n ro2 =— GA.
(12)
—18 —1
It provides that = 10 s which means that at this angular velocity the universe would have turned only about 30° for 12.3 billion years of its history! Evidently, such a slow rotation as well as the center of rotation which is the center of the universe cannot be determined for the short lifetime of the Homo Sapience existence. Nevertheless, this hypothesis allows us to get rid of the mystic Dark Energy and, therefore, deserves to be studied.
Any rotation of a large set of many gravitating masses makes the set take the shape of a spheroid which symmetry axis coincides with the axis of rotation (like galaxies and planetary systems shaped as some flattened spheroids). The
rotation of the universe came up as a result of a "birth defect" at the Big Bang so that at some initial stage the angular momentum of the probe mass on the edge of the universe was controlled by the moment of some eccentricity force proportional to the gravity of the universe. The moment arm of this force about the axis of rotation was assumed to be proportional to the radius of the universe. As a result, for this initial stage one more evolution equation of the revolving universe came out [25, 26]
R d(aR2) = R2 Pe. dl T2
(13)
Here a(t) is the angular velocity of the universe, R* is the radius of the universe at a specific epoch t = T* of its history, and Pe the dimensionless eccentricity parameter, a new constant characterizing the birth defect and the eccentricity of the universe caused by this defect.
According to Eqs. (12) and (13), the angular velocity and the cosmological constant varied with time. It is reasonable to assume that due to the rotation the universe should, at least at the initial stage of a very dense state, have the shape of an oblate or prolate spheroid, with the axis of rotation being the main axis of the spheroid. For the expanding and revolving universe, the evolution equation (7) must be modified in order to take into account the rotation and the spheroidal shape of the universe.
3.4. The initial stage of the universe history
Suppose that at the initial stage of its history, just after the Big Bang, the homogeneous universe revolving around axis x3 is inside the following spheroid
Xi + X2
X2 R 2
= 1.
(14)
Here k is the eccentricity parameter of the spheroid expressed as follows: for the oblate spheroid
k = V 1 -e2, 0 < e <1, k<1, (15)
for the prolate spheroid
k= , 1 , 0<k<1, k>1. (16)
V1-K2
The mass of this spheroid is equal to
M = — kpR3. (17)
The components Xj, X2, and X3 of the gravitation force upon a unit point mass on the surface of this spheroid are equal to [28]: for the oblate spheroid
X = -2npGx¿V 1 -e2--I -VT
2 arcsin e
-e +-
i = 1,2,
(
X3 = -4np Gx3 -3
e
for the prolate spheroid
-VT
e arctg
(18)
(19)
X, =-2TCpGx-1 f K--Vl-K2 ln1^
i 31 2 1 -K
i — 1, 2,
X3 = -2npGx3
K
1 -K3
-2K+ln
1+ K 1-K
(20)
(21)
for the sphere, all Eqs. (18) to (21) provide 4n
X, = — pGXi, i = 1,2,3.
From Eqs. (18) to (21), we can also derive the following useful asymptotes: for the thin disk of radius R, when e^ 1:
X1 = X2 = 0, X3 = -3
GM ~Rr
, x1 = x2, x3 ^ 0, (23)
X1 =-
3n GM
4 R2 '
X 2 =X3 =0,
X1 — R, X2 — X3 — 0,
(24)
for the equivalent cylinder of radius R and length L >> R, when k ^ 1:
X1 — X2 — 0, X3 — —GGM-ln{1 - k),
9 L2
X1 — X2 — 0, X3 — L —
4R
(25)
3V—
X1 = -2npGR, X2 = X3 =0,
3 /-7 (26)
x1 = R = — L\ 1 -k , x2 = x3 = 0.
According to Eqs. (18) and (20) the gravitation force
upon a unit probe mass at the edge of the universe when
2 2 2 x3 = 0, x + x2 = R is equal to
GM F = -r|-
R
where for the oblate spheroid
n = —(-W1 -e2 + arcsin e), 2e
for the prolate spheroid
3
n — 2
L 1 + k
K- — (1 -K2)ln-
2 1 -K
(27)
(28)
(29)
Coefficient n describes the effect of eccentricity of the universe. For the oblate spheroid, it monotonously grows from n = 1 when e = 0 (sphere) to n = 3n/ 4 when e = 1 (thin flattened spheroid). For the prolate spheroid, it monotonously decreases from n = 1 when k = 0 (sphere) to n = 3R/L for the spheroid length L >> R, when k ^ 1. For a mass on the edge of the oblate universe, this effect monotonously grows when eccentricity e increases so that the gravity of the thin disk is 3n/ 4 = 2.35 times greater than the gravity of the sphere of the same mass and radius. The gravitation force upon the unit probe mass at x1 = R, x2 = x3 = 0 on the surface of a very long prolate spheroid of mass M is equal to 3GM/(RL) where L is the length of the spheroid.
Let us place a probe mass at point xj = x2 = 0, x3 = XR on the axis of rotation at the edge of the universe. Because the centrifugal force is zero at this point, due to Eqs. (14), (19), (21) and (22) the equilibrium equation of the probe mass takes the following shape
£ <XR) = -Z ^
dt2 R2
(22) where for the oblate spheroid
z—
e arctg
V—
for the prolate spheroid
31-K2
z — ~ 3
-2 K + ln
1+K
(30)
(31)
(32)
2 k3 ^ 1 -k
For the oblate spheroid, coefficient Z monotonously grows from Z = 1 when e = 0 (sphere) to Z = 3 when e ^ 1 (thin spheroidal disk).
For the prolate spheroid, coefficient Z monotonously decreases from Z = 1 when k = 0 (sphere) to 3X-2 ln X ^ 0 when X ^ ^ (k ^ 1) for a very long spheroid.
For the probe mass at x3 — 0,
X? + X2 — R2, due to
Eq. (27) the equilibrium equation has the following shape
(33)
d2 R
2„ MG — a1 R-n—r-dt2 R2
Here n is defined by Eqs. (28) and (29).
The equation system, Eqs. (13) and (28) to (33), describes the evolution of the expanding and revolving universe in time, with R(t), a(t) and e(t) or K(t) characterizing the change of the angular velocity, size and shape of the universe in terms of time at the initial stage of its development.
In the current model, the initial stage means that 0< t < T* and 0 < R < R* where t = 0, R = 0 corresponds to the Big Bang, and t = T*, R = R* corresponds to a specific epoch that characterizes the transition of the growth of the universe from the initial stage to the final stage. The values of T* and R* should be found using the conditions of the smooth transition.
The final stage is characterized by one evolution equation, Eq. (33), where a = œ = const is determined by Eq. (12) and coefficient n has to be found from the smooth transition conditions that can be written as follows:
when t = T*: a = œ, R = R*, [e] = 0, [R] = 0. (34)
The last two equations require the continuity of the expansion rate and shape of the universe at the transition epoch.
Let us study the initial stage controlled by the system of the following three equations:
d2R 2 D MG d2 . _ MG TT = a R-n—r, TT(XR) =
dt2 R2 dt2 R2
r ±(ar2) — r- pe.
dr t2 e
Here coefficient X is a function of e or k given by Eqs. (15) or (16).
The solution of this equation system is given by the following theorem:
Theorem 1. The exact solution of the equation system (35) satisfying the initial boundary value conditions R ^ 0, a ^ ^ when t ^ 0, is given by the following functions
R = i(p)(MG)1/312/3, a = S(p)t, X = X(p), (36) t < T»,
where functions i(p), 8(p), and X(p) are to be found, and
p=
MGT*2 PR 3
(37)
Proof. According to Eqs. (35) and to the initial boundary value conditions, the problem-solving functions R, a, and X can depend only on t (s), MG (m3/s2) and PeR*3/T*2
3 t 2
(m /s ). From here, based on the simple analysis of dimensions, it follows that the solution of the equation system (35) satisfying the initial boundary value conditions can be written in the shape of Eqs. (36) and (37). The theorem is proven.
According to Eqs. (36), the universe occasionally shaped as an ellipsoid at the Big Bang retains its eccentricity as well at a certain initial stage of its development.
Substituting R, a, and X in Eqs. (35) by the corresponding functions in Eqs. (36) provides the following three algebraic equations
^p) = Rp)0"(p) -n^ -(p) = -9 X
p) = i( p)S2 ( p) -nil'2 ( p) = - V2 ( p), (38)
pS(p)i (p) = 3.
(39)
The first two smooth condition equations (34) are met, if
R* = ^(p)(MG)1/3T*2/3, 0(p) = roT*. (40)
Then, other two smooth condition equations (34) are satisfied automatically.
Five algebraic equations, Eqs. (38) to (40), serve to determine five unknown functions p, 0, R* and T*.
Let us transform three equations (38) and (39) to the following ones
0(p)=3X- ^3(p)=9X, p2=77^. (41)
3pZ 2X Z(nX-Z)
Using Eqs. (40) and the first equation (41) let us find R* and T* in terms ofp and X
R*3 = MX, T* = . (42)
Substituting R* and T* in Eq. (37) by Eqs. (42) provides
p = —. (43)
P 9PeZ ^ '
Now, using Eq. (43) and the last equation (41) we find the characteristic equation which determines the eccentricity of the universe in terms of the original "birth defect"
coefficient Pe
40.5Pe2Z = nX-Z. (44)
Let us analyze Eqs. (15)-(16), (28)-(29) and (31)-(32) in terms of e or k. Evidently, for the oblate spheroid we have X < 1 and Z > n so that the characteristic equation (44) cannot be met. For the prolate spheroid, we have X > 1 and n > Z so that Eq. (44) can be satisfied, and we come to the following theorem
Theorem 2. The universe has the shape of the prolate spheroid which eccentricity k is uniquely determined in terms of the "birth defect" coefficient Pe by the following characteristic equation
Pe2 = -2e 81
X(K) n« -1 ( ) Z (K)
0 <K< 1.
(45)
Here functions X(k), n(K), and Z(k) are given by Eqs. (16), (29) and (32).
Function C(k) given by the following equation
C(K) = X(K) ^-1 Z(k)
(46)
monotonously grows when k increases, from C = 0 when k = 0 (sphere) to infinity when k ^ 1 (cylinder) so that the "birth defect" coefficient equals zero for the spherical, "defectless" universe. And so, one and only one value of eccentricity k corresponds to any value of parameter Pe. The theorem 2 is proven so that at a certain initial stage of its development the universe in this model represented the prolate spheroid having constant eccentricity K(Pe) defined by Eqs. (45) and (46).
4. The origin of everything: the superphoton hypothesis
The Big Bang theory and the NEOC cosmology cannot answer the question of the origin of the universe because these theories are phenomenological. Most advantageous and alluring is the idea that the current universe was born from a certain primordial elementary particle. Let us analyze this idea.
From the position of the generally-recognized standard model of the modern physics, everything is built from elementary particles, the "zoo" of which consists of 61 "species" including quarks, neutrinos, photons, electrons, glu-ons, muons, and others, plus their antiparticles, the Higgs boson and the hypothetical graviton. Every elementary particle is characterized by its mass, charge, spin and by some other properties of the second order. In principle, every particle can be split into some other elementary particles in a fission process being controlled by the conservation laws of energy, momentum and angular momentum.
Of most interest are stable elementary particles which lifetime is infinite. These are photon, gluon, neutrino, electron and some quarks. Mass and charge are very elusive properties, being easily converted into energy. Neutrino has been recently proved to have mass 0.32 ± 0.08 eV/c2
which is the least among all particles with some mass. Therefore, photons and gluons that the only ones that have zero masses can be the first in the row to pretend to the role of the primordial elementary particle. However, gluons "glue" massive quarks and hence can be omitted from this consideration.
It is reasonable to suggest that everything came from a primordial photon we will call the superphoton. The superphoton had zero mass, zero charge and the spin equal to +1 or -1, if the current universe rotates. If the current universe does not rotate, a pair of the superphotons of different spin could be the original universe.
Based on the energy conservation law the following equation is valid
E2 = m2c4 + p 2c2. (47)
Here E, m, andp are the total energy, mass and magnitude of the momentum of any elementary particle, and c is the speed of light in vacuum.
For the zero mass superphoton, we have
p = —, E = hv, X = -. (48)
c v
Here X and v are the wave length and frequency of the
superphoton, and h is Planck' constant equal to 6.6 X
x10-34 J-s.
Let us estimate the total energy of the universe at the Big Bang including only the ordinary matter and the Dark Matter. We assume that the total energy is equal to the kinetic energy plus the rest energy what is strictly right only for velocities much less than the speed of light. The rest energy of the universe is equal to Mc2 where M is the gravitational mass of the universe (the ordinary and Dark matter). The kinetic energy at the Big Bang in the NEOC cosmology depends only on M, MG, and c, see Eqs. (10). Hence, it is equal to ZMc2 where Z is a number which can be estimated using the assumption that at the Big Bang the velocity v(r) varied linearly with radius and achieved the speed of light at the edge of the universe. (As a reminder, in the NEOC model the velocity of matter inside the universe was out of consideration). Then, we get
R R
J 0.5pv2 (r) d V = 2nc2 J (r/R)2 r 2dr = 0.3Mc2. (49)
0 0
Thus, the total energy of the universe at the Big Bang was equal to 1.3Mc2 in this approximation assuming that heat energy of the universe is much less than its mass energy. This energy should be the same at any time in the history of the universe, if there have been no losses of energy as we assume.
According to the current knowledge, the mass of the ordinary matter of the universe is equal to about 1053 kg so that based on the last data of the Planck mission the total mass of the gravitational matter of the universe including the Dark Matter is 31.7/4.9 times greater, that is equal to M = 2 -1054 kg. From here and Eqs. (49), it follows that
the primordial superphoton had the following energy, frequency and wave length
E = hv = 1.3Mc2 = 2.3-1071 J,
v = L3ME2 = 3.5. 101041j g, (50)
h
X=- = 1.4-10-96m. v
The following possible chain of physical transformations of the superphoton leading to the current universe can go through myriad scenarios which all can well comply with modern physical knowledge.
If gravitons exist, the supergraviton of the same energy, frequency and wave length can be this primordial universe.
5. Conclusion
As applied to two very different fields of physics, this paper illustrates the power and scope of the method of invariant integrals that can serve as an alternative to other methods of mathematical physics based on differential equations and conservation laws. One field under consideration is the earlier fracture mechanics originated in 1966 to 1967, and the other field is the modified neoclassic cosmology that is capable to address some basic problems of the origin, shape and development of the universe. To address the mystery of the Big Bang a new hypothesis of a primordial superphoton is advanced.
Acknowledgments
This author thankfully recalls the invaluable communications of the early 1960s with late B. Kostrov, these communications pushed the author forward onto his invariant integral of fracture mechanics. A thank you goes as well to G. Sih who persuaded the author to undertake this venture.
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Поступила в редакцию 05.02.2016 г., после переработки 08.12.2016 г.
Сведения об авторе
Cherepanov Genady P., Prof., The New York Academy of Sciences, USA, genacherepanov@hotmail.com
