28:02 Pauli, RT, paragraph 22, Geometry of the Real World, "So far we have assumed that the form ds^2 is a definite form. In the real space-time world, this does not take place, since ds^2 in normal form has three positive and one negative term." On the Protouniverse: 0.in RT the main invariant is the 4-interval (a mathematical description of the constant c), however, it could offer another invariant value based on another physical constant. 1.Comparing with Einstein's equations of 1915, we find a=-c^3/16πG. Strictly speaking, in order to determine the constant a, it was necessary to make a transition to the Poisson equation. Thus, a rigorous derivation of Einstein's equations can be given. The transition to the non-relativistic limit allows us to determine a constant factor for the integral of the gravitational field according to: R[(0)^0]=(4πG/c^2)p; Δφ=-pc^3/4a=4πGр. And a=(1/16π)m(pl)w(pl). 2.From Kepler's third law follows: M/t=v^3/G, where M/t=I(G)=[gram•sec^-1] is the gravitational current. By the way, in SR: I(G)=inv; this follows from the Lorentz transformations: m=m(0)/√(1-v^2/c^2) and ∆t=∆t(0)/√(1-v^2/c^2). Hence, obviously, we have I(G)=m/t=m(0)/t(0)=inv. 3.Therefore, the Poisson equation can be written as: ∆g(00)=8πGT(00)/c^4, where g(00) is the time component of the metric tensor (for a weakly curved metric the time component of the energy-momentum tensor: T(00)~=pc^2). This equation is true only in the non-relativistic case, but it is applicable to the case of a homogeneous and isotropic Universe, when Einstein's equations have only solutions with a time-varying space-time metric. Then the energy density of the gravitational field: g^2/8πG=T(00)=pc^2 [~=(ħ/8πc^3)w(relic)^4= =1600 quanta/cm^3, which is in order of magnitude consistent with the observational-measured data (~500 quanta/cm^3)], where the critical density value determining the nature of the model is: p=(3/8π)H^2/G. Hence it follows: g~πcH. 4.Expansion is a special kind of motion, and it seems that the Universe is a non-inertial frame of reference that performs variably accelerated motion along a phase trajectory, and thereby creates a phase space. And according to the strong equivalence principle: g=|a*|=πcH [=r(pl)w(relic)^2], and w(relic)^2=πw(pl)H. Thus H=1,72*10^-20sec^-1. {By the way, at t(universe)= πт(pl), w(“relic”) was =w(pl); at 1/”H”= t(universe)=380000 years, w(“relic”)/2π was =3.5*10^14 Hz} 5.Intra-metagalactic gravitational potential: |ф0|=πGmpl/λ(relic)=[Gm(pl)/2c]w(relic), where the constant Gm(pl)/2c is a quantum of the inertial flow Ф(i) = (½)S(pl)w(pl) = h/4πm(pl) (magnetic flux is quantized: = h/2e, Josephson’s const; and the mechanical and magnetic moments are proportional).Thus, the phenomenon can be interpreted as gravity/inertial induction. 5(a).The basic formula QG of the quantum expression of the Newtonian gravitational potential is: ф(G)=-Ф(i)w, where w is the frequency of the quanta of the gravitational (~ vibrational) field. (b).“Giving the interval ds the size of time, we will denote it by dт: in this case, the constant k will have the dimension length divided by mass and in CGS units will be equal to 1,87*10^-27", Friedmann, (On the curvature of space, 1922). [The ds, which is assumed to have the dimension of time, we denote by dт; then the constant k has the dimension Length Mass and in CGS-units is equal to 1, 87.10^ ± 27. See Laue, Die Relativitatstheorie, Bd. II, S. 185. Braunschweig 1921.] (c.)Apparently, the following expression takes place: μ(0)ε(0)Gi=1, which means that Gi=с^2 where i is inertial constant, i=1,346*10^28[g/cm]; or k°=1/i=7,429*10^-29[cm/g]: (d.)k(Friedmann)/k°=8π; where k°=r(pl)/m(pl)=r(G)/2m(0); i=m(pl)/r(pl)=(1/c)m(pl)w(pl), (e.)w=[r(G)/r]w(pl). That is ф(G)=-[Gm(pl)/2c]w=-(½)[w/w(pl)]c^2=-(½)(√Għ/c)w=-Ф(i)w.* 6. And а*=-2πcа/M(universe), what is F=M(universe)а*=-2πса=-с^4/8G=-(⅛)F(pl). I(G)=M(universe)H=m(pl)w(pl)/8π=c^3/8πG=-2a (~ the "dark" const~inv), where M(universe)=E/c^2 is the full mass of the Universe, and the total energy E is spent on creating a phase-quantized space-time: m(pl)w(pl)=8πM(Universe)H { w(relic)^2=πw(pl)H. 7.That is: Δφ=-pc^3/4a= рс^3/2M(universe)H^2. And Δφ=4π[с^3/Gm(pl)w(pl)]H^2= 4πH^2; which is evidence of a phenomenon: spontaneous Lorentz transformations. 8.Thus; Δφ(0)/Δφ=w(pl)^2/H^2~10^126, where Δφ(0)=4πw(pl)^2; the best prediction. ---------------- *) - So, ф(G)=-[w/w(pl)]c^2/2, where ф(G) - is Newtonian gravitational potential, r(n')=nλ/π=(n+n')2r(pl)l , the corresponding orbital radius, w - the frequency of the quanta of the gravitational field (space-time); - obviously, the quanta of the field are themselves quantized: λ=(1+n'/n)λ(pl) = 2πc/w, where n'/n=M/2∆m: system gravity unpacking ratio, n'- the orbit number (n'=0,1,2,3…). 1.Obviously, on the horizon {r=r[r(G)]}, n'=0] the "door" is closed, however, the quanta [λ=λ(pl)] can go out singly and form the first and all subsequent half-orbits (n'=1,2, 3 ...) during the time t(0)=r/c=2nт, where т=1/w, т=((1+n'/n)т(pl), spending part of their energy on it each time. And it is this mechanism that provides the step-by-step formation of a variable gravitational field: variably accelerated expansion of spacetime as a phase space: |a|=g=πc^2/L, where L[=πr^2/r(G)] is the length of the phase trajectory (of course, the quanta coming through the "window" are also rhythmically restored: from the source- protouniverse). 2.Moreover, the parameter r(0)=r(G)-r(pl)=(2n-1)r(pl), defining the interval of the formation of the system, at n=0, when r=r(G)=0 (for example, the state of the "universe" before the Big Bang) turns out to be a quite definite quantity: r(0)=-r(pl). In the area [(-rpl) - 0 - (+rpl)] there is an implementation of external forces, "distance": (-rpl)+(+rpl)=0 (≠2rpl). 3.The phase velocity of evolution v'/π= r(pl)w/π; m(0)=(c/2G)rv', where v'=v^2/c. The angular momentum: L(p)=|pr|=n^2ћ [const for all orbits of the system; at n=1: L(p)=ћ] and moment of power: M(F)=dL(p)/dt(0)=nћw/2=-E(G)=E*, where t(0)=r/c, E*- energy of self-action. 4.The gravitational field is characterized by a spontaneous flow: J*=(v'/π )(1/4π) g^2/G, where v'/π- phase velocity of field evolution. 5.Entropy (here: a measure of diversity/variety, not ugliness/disorder) of the system: S=πε(pl)r(t)=(n+n')k, where k is the Boltzmann constant. Obviously, on the horizon entropy=min and with fundamental irreversibility, information is preserved (+ evolves, accumulates). 6.Accordingly, m=m(pl)/(1+n'/n), where m=ħw/c^2, is the quantum of the full mass: M=n'm [
28:02 Pauli, RT, paragraph 22, Geometry of the Real World, "So far we have assumed that the form ds^2 is a definite form. In the real space-time world, this does not take place, since ds^2 in normal form has three positive and one negative term."
On the Protouniverse:
0.in RT the main invariant is the 4-interval (a mathematical description of the constant c), however, it could offer another invariant value based on another physical constant.
1.Comparing with Einstein's equations of 1915, we find a=-c^3/16πG. Strictly speaking, in order to determine the constant a, it was necessary to make a transition to the Poisson equation. Thus, a rigorous derivation of Einstein's equations can be given.
The transition to the non-relativistic limit allows us to determine a constant factor for the integral of the gravitational field according to: R[(0)^0]=(4πG/c^2)p; Δφ=-pc^3/4a=4πGр.
And a=(1/16π)m(pl)w(pl).
2.From Kepler's third law follows: M/t=v^3/G, where M/t=I(G)=[gram•sec^-1] is the gravitational current. By the way, in SR: I(G)=inv; this follows from the Lorentz transformations: m=m(0)/√(1-v^2/c^2) and ∆t=∆t(0)/√(1-v^2/c^2). Hence, obviously, we have I(G)=m/t=m(0)/t(0)=inv.
3.Therefore, the Poisson equation can be written as: ∆g(00)=8πGT(00)/c^4, where g(00) is the time component of the metric tensor (for a weakly curved metric the time component of the energy-momentum tensor: T(00)~=pc^2).
This equation is true only in the non-relativistic case, but it is applicable to the case of a homogeneous and isotropic Universe, when Einstein's equations have only solutions with a time-varying space-time metric. Then the energy density of the gravitational field: g^2/8πG=T(00)=pc^2 [~=(ħ/8πc^3)w(relic)^4=
=1600 quanta/cm^3, which is in order of magnitude consistent with the observational-measured data (~500 quanta/cm^3)],
where the critical density value determining the nature of the model is: p=(3/8π)H^2/G. Hence it follows: g~πcH.
4.Expansion is a special kind of motion, and it seems that the Universe is a non-inertial frame of reference that performs variably accelerated motion along a phase trajectory, and thereby creates a phase space.
And according to the strong equivalence principle: g=|a*|=πcH [=r(pl)w(relic)^2], and
w(relic)^2=πw(pl)H. Thus H=1,72*10^-20sec^-1.
{By the way, at t(universe)= πт(pl), w(“relic”) was =w(pl); at 1/”H”= t(universe)=380000 years, w(“relic”)/2π was =3.5*10^14 Hz}
5.Intra-metagalactic gravitational potential:
|ф0|=πGmpl/λ(relic)=[Gm(pl)/2c]w(relic), where the constant Gm(pl)/2c is a quantum of the inertial flow Ф(i) = (½)S(pl)w(pl) = h/4πm(pl) (magnetic flux is quantized: = h/2e, Josephson’s const; and the mechanical and magnetic moments are proportional).Thus, the phenomenon can be interpreted as gravity/inertial induction.
5(a).The basic formula QG of the quantum expression of the Newtonian gravitational potential is: ф(G)=-Ф(i)w, where w is the frequency of the quanta of the gravitational (~ vibrational) field.
(b).“Giving the interval ds the size of time, we will denote it by dт: in this case, the constant k will have the dimension length divided by mass and in CGS units will be equal to 1,87*10^-27", Friedmann, (On the curvature of space, 1922).
[The ds, which is assumed to have the dimension of time, we denote by dт; then the constant k has the dimension Length Mass and in CGS-units is equal to 1, 87.10^ ± 27. See Laue, Die Relativitatstheorie, Bd. II, S. 185. Braunschweig 1921.]
(c.)Apparently, the following expression takes place: μ(0)ε(0)Gi=1, which means that Gi=с^2 where i is inertial constant, i=1,346*10^28[g/cm]; or k°=1/i=7,429*10^-29[cm/g]:
(d.)k(Friedmann)/k°=8π; where k°=r(pl)/m(pl)=r(G)/2m(0);
i=m(pl)/r(pl)=(1/c)m(pl)w(pl), (e.)w=[r(G)/r]w(pl). That is ф(G)=-[Gm(pl)/2c]w=-(½)[w/w(pl)]c^2=-(½)(√Għ/c)w=-Ф(i)w.*
6. And а*=-2πcа/M(universe), what is F=M(universe)а*=-2πса=-с^4/8G=-(⅛)F(pl).
I(G)=M(universe)H=m(pl)w(pl)/8π=c^3/8πG=-2a (~ the "dark" const~inv), where M(universe)=E/c^2 is the full mass of the Universe, and the total energy E is spent on creating a phase-quantized space-time:
m(pl)w(pl)=8πM(Universe)H
{
w(relic)^2=πw(pl)H.
7.That is: Δφ=-pc^3/4a=
рс^3/2M(universe)H^2.
And
Δφ=4π[с^3/Gm(pl)w(pl)]H^2=
4πH^2; which is evidence of a phenomenon: spontaneous Lorentz transformations.
8.Thus;
Δφ(0)/Δφ=w(pl)^2/H^2~10^126, where Δφ(0)=4πw(pl)^2; the best prediction.
----------------
*) - So, ф(G)=-[w/w(pl)]c^2/2, where ф(G) - is Newtonian gravitational potential, r(n')=nλ/π=(n+n')2r(pl)l , the corresponding orbital radius, w - the frequency of the quanta of the gravitational field (space-time); - obviously, the quanta of the field are themselves quantized: λ=(1+n'/n)λ(pl) = 2πc/w, where n'/n=M/2∆m: system gravity unpacking ratio, n'- the orbit number (n'=0,1,2,3…).
1.Obviously, on the horizon {r=r[r(G)]}, n'=0] the "door" is closed, however, the quanta [λ=λ(pl)] can go out singly and form the first and all subsequent half-orbits (n'=1,2, 3 ...) during the time t(0)=r/c=2nт, where т=1/w, т=((1+n'/n)т(pl), spending part of their energy on it each time. And it is this mechanism that provides the step-by-step formation of a variable gravitational field: variably accelerated expansion of spacetime as a phase space: |a|=g=πc^2/L, where L[=πr^2/r(G)] is the length of the phase trajectory (of course, the quanta coming through the "window" are also rhythmically restored: from the source- protouniverse).
2.Moreover, the parameter r(0)=r(G)-r(pl)=(2n-1)r(pl), defining the interval of the formation of the system, at n=0, when r=r(G)=0 (for example, the state of the "universe" before the Big Bang) turns out to be a quite definite quantity: r(0)=-r(pl).
In the area [(-rpl) - 0 - (+rpl)] there is an implementation of external forces, "distance": (-rpl)+(+rpl)=0 (≠2rpl).
3.The phase velocity of evolution v'/π= r(pl)w/π; m(0)=(c/2G)rv', where v'=v^2/c.
The angular momentum: L(p)=|pr|=n^2ћ [const for all orbits of the system; at n=1: L(p)=ћ] and moment of power: M(F)=dL(p)/dt(0)=nћw/2=-E(G)=E*, where t(0)=r/c, E*- energy of self-action.
4.The gravitational field is characterized by a spontaneous flow: J*=(v'/π )(1/4π) g^2/G, where v'/π- phase velocity of field evolution.
5.Entropy (here: a measure of diversity/variety, not ugliness/disorder) of the system: S=πε(pl)r(t)=(n+n')k, where k is the Boltzmann constant. Obviously, on the horizon entropy=min and with fundamental irreversibility, information is preserved (+ evolves, accumulates).
6.Accordingly, m=m(pl)/(1+n'/n), where m=ħw/c^2, is the quantum of the full mass: M=n'm [