|
Reanalysis of the Hubble Law in General Relativity solves the cosmic acceleration phenomenon.
I shall outline the demetrification of General Relativity as an example for demistification
of theoretical physics.
Now this simply means that the complexities of say the mathematics of tensors, string theory
or the determinants of particle physics become encompassed in the same mathematics upon whom they are themselves built and
constructed upon.
So the multidimensional approach of the tensor is reduced to the common vector and the modern physics
they represent mathematically becomes reborn in the classical , say Newtonian approach of their own historical development.
What
I am saying here, is that it is the nature of space and time themselves, which allows their demetrification.
The Einstein
Universe evolved from the Newton Universe and much of Newton's physics is simply rewritten in the developed form of the classical
approach.
I am saying that the ever growing and evolving complexity in the attempt to model reality in terms of space
and time must eventually reach a point of simplified space and time, namely the origins of space and time themselves as physical
parameters describing reality.
It is this point of decomplexification, which I have termed demetrication.
It
develops somewhat like this.
The Einstein Tensor and associated metrics describe the curvature of space in a multidimensional
system of partial differential equations, which relate the parameters of the variables or coordinates of space and time through
the presence of inertia, described by density and mass=densityxvolume.
The metrics then lead to equations of motion
in terms of not space and time as commonly understood, but as fractional change or percentages of them.
This is why
the redshifts and the dimensionless quantities like the (velocity v/lightspeed c) ratio formulations are so important in the
relativities.
But these fractional-change-ratios themselves are analysed in terms of Newtonian dynamics, who in a sense
swallow up the percentage changes in the more familiar evolution of displacement vecors, velocities and accelerations.
But
the complexity in say General Relativity to model physical reality, is the interpretation of the percentages in a physically
meaningful way and methodology.
Now the great realisation relative to me, is that the percentages are dimensionless
and thus a similar approach to dimensionless space and time allows the demetrification of both space and time in the percentages
of Quantum Relativity.
Quantum Relativity was constructed just on the premise of space and time being dimensionless
and thus naturally must encompass into whatever complexities the metrics might evolve into.
It is like the future physics
having linked to its own past.
Because my scientific experience and intuition must by necessity be rooted in that past,
I, like all of you find myself in a continuing process of self-obscuration, trying to decode the past meeting its own future.
But
this future is subject to imagination and scientific dreaming or intuition and that is what created the passion to construct
Quantum Relativity.
So I shall illustrate how the dynamical equations derived from the metrics and solved as the percentages
(called expansion parameter, deceleration parameter and redshift based on the Hubble Constant and proportional rates of change)
by fundamentally classical methods of calculus, can be encompassed by the demetricated approach.
This demetricated
approach then does not change the solutions of the dynamical equations as such, but swallows up the metrics of say a cosmological
constant related to pressure in percentage changes in one form of variables, to variables which specify the percentage changes
in other terms.
So you have an asymptotic (static) solution for the dynamics of the Einstein-de Sitter universe which
geometrically defines the curvature evolution as function of Newtonian time t mapped as the geometric solution of the same
Einstein-de Sitter universe as a function of demetricated time t=n/Ho in Quantum Relativity, with n=Hot
being dimensionless.
Because the metric t-evolution engages possible additional parameters, such as the cosmological
constant or quintessence and the pressure also as functions of time t, which complicate the dynamical equations to be solved;
there are a number of possible cosmologies of say varying curvatures, resulting in the dynamical solutions.
So I shall
indicate those conventional dynamical solutions say in the Einstein-de Sitter cosmology and the Friedmann-Le Maitre cosmology
as the most appropriate and this will be necessarily somewhat laborious as it must use the standard cosmological approach.
But
the present dilemmas in cosmology will also become apparent.
Is there a cosmological constant or a quintessence as
the vacuum 'dark energy'?
Or is it or the pressure of matter term 0 and is the Omega 1 and twice the deceleration parameter?
Those
questions are answered by Quantum Relativity in the mapping of the geometries of the metrics onto the nonmetrics of the percentage
changes.
This assimilates the relationships between the Omega and the Lambda and the Hubble Constant in an unified
approach and naturally crystallises the evolution of the universe in a given parametric definition of the scalefactors described
by General Relativity as a function of metric time.
In other words the big question in Quantum Relativity is not about
beginnings and endings, but where are we NOW.
The birth and projected death of the universe are given as initial boundary
conditions in the superbrane parametres from which the subsequent cosmology develops.
I shall not describe this here,
but this is the Quantum Relativity, not searching for unification, but being unified, allowing its symmetry breaking and deunification
as underpinning mathematical and theoretical premise.
So why am I doing this?
Daily I ask myself, why am I doing
this? Have I not got something better to do?
The answer is no.
Despite the misgivings of some and the appreciation
of others, I am doing this for all of us.
I simply feel this is a work that must be done and perhaps I must do it alone;
despite my continuing call and ask for help of assistance, noone appears so far to have grasped the nature of this sufficiently
to make enough sense of it to meaninful critisise it in contribution and coauthorship.
And yes, I know that some of
you have grasped the philosophy underpinning Quantum Relativity wonderfully and are resonating with this work in fraternity.
So
the cosmic family is growing, there is joy and exhuberance in the spirit of the metaphysics which connects us all.
But
here I am talking of a scientific paradigm change on the greatest possible scale and that is not achievable by the philosopy
which is the basis for all true religion, which is gnosis by poetry and beautiful literature and words of depth and meaningful
essence.
So the hard work is to present the mathematics of the demetrication to the world and these forums allow me
and us to do just that.
Can you see what we are doing here?
We are a spiritual family of, well, 'spirits' finding
ourselves in embodiment, and being ultimately rather uncomfortable in this situation, we are here struggling, arguing and
attempting to recognise ourselves and to find recognition.
This search for recognition must be understood for what
it truly is.
It is not some egocentricity running wild, but as my friend Cisco has said, it is necessary to be or appear
selfish and self-centred to recognise oneself first.
For how can one recognise the family of the spirit, if one doesn't
know oneself?
And as my friend Pythagoras has said: 'Man know thyself, then shall you know the Universe and God".
So
it doesn't matter at all if someone reading all this and understanding the 'beautiful physics' can somehow break through the
elitism of the old entrenched science orthodoxy to begin its own metamorphosis from the human science of the earthbound caterpillar
to the starhuman science of the cosmic butterfly.
What matters is that we, and all of you through me and me through
all of you recognise this work as our contribution towards this metamorphosis.
Without any of you this work would not
be what it is, and the family of spirits knows this very well.
The ones attuned to that are the Philosophers of Quantum
Relativity.
With these words I'll end the introduction and begin the revision process.
Tony B.
......Revision under engagement.....the following is under construction.....
So I shall present
a decisive little paper which shall shed light on the cosmological conundrums of the 'missing mass' and the accelerating universe,
using the field equations of General Relativity directly.
I shall derive the following:
1. The reformulation
of Hubble's Law regarding the Hubble-Constant as the ratio of recessional velocity over displacement, underpinning modern
cosmology and often subject to controversy.
2. The precise cosmological redshift, where the cosmic acceleration begins
to take over from the previous deceleration and a fact which is accentuated in the scientific literature.
3. The demetricated
expression for the evolution of the universe in parametric terms.
I shall try to derive those expressions from first
principles and add commentary for the mathematically inexperienced reader however.
There are a couple of references
at the end, which should provide enough background data to the topics discussed.
The references do not show photograps
in this rtf-file here, but can be found on the web as indicated.
(1) http://www.pnas.org/cgi/content/full/96/8/4224
(2)
http://www.physicstoday.org/pt/vol-54/iss-6/p17.html
(3) http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
(4)
http://astronomyonline.org/default.asp?Cate=Home
(5) http://www.talkorigins.org/faqs/nri.html
(6) http://users.rcn.com/wcri/wcri/Big%20Bang%20Text.htm
(7)
http://users.rcn.com/wcri/wcri/index.htm
1. The Reformulation of the Hubble Law
Let
us state the Einstein Field Equation underpinning all of cosmology.
The Einstein-Riemann Tensor Guv=Ruv-guvR/2 =
-8pG*Tuv/c^2 relates the Riemann-Metric guv with scalar tensor R to the Ricci-Tensor Ruv for a stress-energy density tensor Tuv. The Weyl Curvature
in Ruv preserves volume as a tidal shear effect, whilst the Ricci
Curvature acts on the density and changes the density and so the volumes.
The Weyl Curvature Nullification hypothesis
of Roger Penrose (Oxford University, UK) shows, that the Weyl Curvature must become 0 at the threshold between General Relativity's
metrics and the 'singularity' of quantum mechanics for the selfconsistency of the physical universe to hold in its inertial
parameters.
A 0 Weyl curvature means that the Lorentz Contraction of a tangential displacement vector travelling around
a 'wormhole singularity' or Wolford-Centre as Black Hole event horizon must dewarp itself at that wormhole perimeter in accompanying
invariance of the scalar orthogonal radius vector not subject to the Lorentz contraction of Special Relativity in say a rotating
system.
We shall describe this Weyl-Limit as a superbrane parameter negating the mathematical singularity of General
Relativity in a minimum superstring condition: lps=2prps.
The Einstein-Riemann
Tensor then represents 16 partial nonlinear differential equations, which as a system lead to the most basic and general solution
for a gravitational field in the standard-isotropic-metric: dt^2=B(r)dt^2-A(r)dr^2-r^2dq^2-r^2sin^2qdf^2, where A(n) and B(n) are the functions (of r) to be determined for the static solutions of the metric.
The metric
then leads to the differential equations, which describe the evolution of the universe in dynamical equations with respect
to the scalefactor R=aRo as the 'size' or radius
of the universe.
In the Friedmann-Lemaitre cosmological model the universe is homogenous and isotropic, that is it
doesn't change in its uniform composition in any directional sense. Here the cosmological constant Lambda (L) is taken as positive for various curvatures k and a nonzero pressure p.
In the Einstein-de Sitter cosmological model this universe has also constant
curvature k and a zero cosmological constant (Lambda L) and pressure p.
The equations of motion for the
two models can be derived as follows:
Density r(r)=M(r)/V(r);
M(r)=4pr(r).r^3/3; dM=4pr(r)r^2.dr
r(doubledot)=-GM(r)/r^2 + Lr/3 for a(doubledot)ro=-4pr(r)G.aro/3 +Laro/3 and a(doubledot)/a=-4pGr/3 + L/3
Since acceleration r(doubledot)=d(r(dot)^2/2)/dr, integrating r(doubledot) gives:
r(dot)^2/2=(a(dot).ro)^2/2=GM(r)/r +L(aro)^2/6
+ constant with constant of integration given in r(dot)=c for r=ro. and so constant=-c^2 for c^2=c^2+c^2+constant .
{This will give L=3Ho^2 in a de Sitter cosmology of
a massless universe with Ho=c/Rmax in demetrication}.
So r(dot)^2=(a(dot).ro)^2=2GM(r)/r + L(aro)^2/3 - c^2.
This
becomes: {a(dot)/a}^2=2GM(r)/r(aro)2 - c^2/(aro)^2 + L/3=8pGr/3-c^2/(aro)^2+L/3............. (FLM1)
Now consider the
universe's expansion to be adiabatic, that is thermodynamically closed. Energy E and the pressure (P) variation with respect
to Volume V sum to 0 change in the 'heat content Q' (or enthalpy H=U+PV for internal heat content U).
dQ=dE+PdV=0;
dE/dt+PdV/dt=0 for E=M(r)c^2=4pSrR^3c^2/3 and total density Sr=rmatter +pressure
d{SrR^3}/dt=-(3P.R^2/c^2).dR/dt
=R^3.r(dot)+3R^2.r.R(dot) for
r(dot)+3r.(a(dot)/a)=-(3P/c^2).(a(dot)/a) and
the dynamical equation:
r(dot)+3(a(dot)/a){r+P/c^2}=0....... (FLM2)
The combined Friemann-Lemaitre equation of motion
for matter density r then
is:
{a(dot)/a}^2 = 8pG/3{r+3P/c^2} - c^2/(aRo)^2 + L/3..........................................................(1)
The
Equation of motion in the Einstein-de Sitter cosmology then sets P=L=0 and a constant curvature k=1/Ro^2=0
in the Friedmann-Lemaitre model for:
(a(dot)/a)^2=8pGr/3 = H^2 with R=aRo=aRmax
Solving for a(dot)^2=2GM/aRmax^3 via Sqrt(a).da=Sqrt(2GM/Rmax^3)dt leads to
a.Rmax=Cuberoot{9GM/2}.t^[2/3] for a limiting boundary condition ao=0 for to=0; (which we shall see is actually n=nps=Ho.tps for a=1/(1+Rmax/lps).
Using H=a(dot)/a, 9GM/2=(aRmax)^3/t^2 for H=Sqrt(4/9t^2)=(2/3t) and the Hubble Time becomes
1/H=3t/2 as the age of the universe for time t.
We shall
show that this Hubble Time actually represents the completion of a Hubble-Oscillation and so the LIGHTPATH R(n)=ct where Rmax necessarily represents a semiwavelength as the distance
between the even nodes 0,2,4,6.. and the odd nodes 1,3,5,7,....
For define R(n)=Rmax(n/(n+1)) with n=Hot and a=(n/(n+1)) for a
lightpath of 2Rmax =ct*=2c/Ho, then this time t* given in c-invariance in say 11D/5D hyperspace will be reduced in n=2 for R(2)=Rmax(2/3)=2c/3Ho for the matter dominated cosmology in 10D/4D.
This then maps the 'nodally corrected' Hubble Law as
Ho=2c/3R(2) onto the old H=2/3t.
The
Friedmann-Lemaitre cosmology, incorporating the pressure and lambda terms solves in terms of the curvature k=1/Ro^2. We use R=aRo and R(dot)=a(dot)Ro and a(dot)/a=R(dot)/R=H and the definition of Omega (W)=r/rcritical=8pGr/3Ho^2 andwrite (1) as a COSMOLOGICAL
EQUATION:
R(dot)^2=8pGrR^2/3+LR^2/3-kc^2 .
So curvature kc^2=R^2{Ho^2W+L/3-Ho^2}=(aRo)^2{Ho^2(W-1)+L/3}..............(Curvature*)
Then
for (R/c)^2=(aRo/c)^2=(R(n)/c)^2=(Rmax(n/(n+1))/c)^2=(a/Ho)^2, k=0 iff (a/Ho)^2{Ho^2(W-1)+L/3}=0
which is the case for W=1 and L=0 OR for L(W-1)=-3Ho^2 (as frequency squared cycle units).
In the demetricated
scenario W=0.028 with a varying L as quintessence in the mass parametric and open cosmology, however encompassed by W=1 in the electromagnetic oscillatory closure.
Then as Ho=1.877728045x10^-18 1/s* , 'L'=constant=3Ho^2/0.972=1.0882292x10^-35 and of the order of the Planck Scale.
This indeed is the
experimental observation of the 'cosmological constancy'.
The curvature is 1 for 'L'=3Ho^2{2-W+2/n+1/n^2} and a function of cyclenumber n however.
Then for the initial condition of the inflaton and the
instanton, n=nps=lps/Rmax=6.259..x10^-49 and 'L'
is upper bounded in 3c^2/lps^2=2.7x10^61 frequency units.
This becomes simply 3fps^2 as the
source frequency in the demetrication for the constant 'L',
expressed in the quintessence of the variable L
in the de Broglie inflaton of the 0-node discussed later.
For n=1 and the first odd node at the semi-wavelength for the Hubble Oscillation 'L'=5.26x10^-35 frequency units for the 0 approximation.
At the completion of the Hubble Oscillation n=2 and 'L'=3.41x10^-35
frequency units and decreasing towards 0 after infinite time t=n/Ho.
This then solves the cosmological constant dilemma in the superpositioning of spacetimes.
Next
we define 1+z=a/ao specifying the deceleration
parameter q.
q=-[a(doubledot)/a]/[a(dot)/a]^2=-(a(doubledot)a)/(a(dot))^2n=-(a(doubledot)/a)/H^2 in say Taylor-Expansion a(t) about t=to, that is some
initial time to where a=ao and H=Ho, which in the demetrication is a
double value Rmin=lps=c/fps,Ro=Rmax=c/Ho for k=0,1
for no=nps,infinity limit and ao=no/(no+1) or
lps/Rmax and 1 respectively.
a(t)=a(to)+a(dot)(to)[t-to]+(a(doubledot)(to)/2)[t-to]^2+...=ao{1+Ho[t-to]-(qo/2)Ho^2[t-to]^2+...}.
So 1+z=1+Ho[t-to]+(-qo/2)Ho^2[t-to]^2+...=a/ao=fo/f=Ho/H.
Then density ro=(a/ao)^3.W.rcritical=(a/ao)^3.W.3H^2/8pG...........(boundary
density).
This becomes (1+z)^2={(1+v/c)/(1-v/c)} in demetrication
with v=c/(1+n)^2=R(dot).
Then
(1+z)^2=(n^2+2n+2)/(n^2+2n)=1+2/{(n+1)^2-1} and
a/ao=1+z=Sqrt{1 + 2/[(n+1)^2-1]}=Sqrt{1+2/[(c/v)-1]}..............................Expansion-Redshift-Parameter
For the boundary conditions , given by ao then, (1) is written with k=(a.Ho/c)^2{W-1+L/3Ho^2}:
(ao(dot))^2=8pGroao^2/3-kc^2+L.ao^2/3=8pGroao^2/3-(aHo)^2.(W-1)+(ao^2-a^2)L/3}
So (a(dot)/a)^2=Ho^2{W.a/ao-(W-1)}+((ao/a)^2-1)L/3.................Friedmann-Lemaitre Equation of motion
For W=1=2q and L=0 (a(dot)/a)^2=Ho^2(1+z)
For W=0.028 and k=1, L=3Ho^2( Rmax/a+0.972) for the curvature limit k=1/Rmax^2=1.
2a(doubledot)/a=-{8pG/3c^2}S[ri*c^2+3pi]......................(1)
(Raychaudhuri Equation)
Here G is the Gravitational Constant, c the lightspeed r(rho) is
the mass-density and p is the pressure, both summed over the universe (Si).
I shall attempt to simplify the Raychaudhuri formula
in demetrication.
It uses a radius-scalefactor aR(to)=R(t),
a velocity a(dot)Ro=v(t)=dR(t)/dt and an acceleration a(doubledot)Ro=a(t)=dv(t)/dt=d^2(R(t))/dt^2.
It then defines the Hubble-Constant as the ratio of 'Ho'=v(t)/R(t)=a(dot)/a as the Hubble-Law.
We shall show, that this ratio, which is termed epoch-dependent, is
indeed not a constant, but varies as a function of inverse square relative to a true NODAL HUBBLE CONSTANT, which we here
denote as Ho to differentiate it from the changing Hubble Constant
as H(t)=v(t)/R(t).
Now this scalefactor R(t)=a.Ro means that some 'size' of
the universe at time to was characterised by the scale Ro, which has a, called the expansion parameter, naturally getting bigger in time.
But what about this original
size of the universe Ro?
Contemporary cosmology says that
Ro was the size of a 'grapefruit' and has now grown to the size of
the Hubble radius of magnitude between 10 and 20 Billion lightyears.
Now the expansion parameter a does indeed increase
as we shall show, but the Ro is actually a maximum size, which R(t)
is increasing towards asymptotically in an oscillating series expressible in the sequence 0/1, 1/2, 2/3, 3/4, 4/5,...,n/(n+1)
thus eternally approaching but never reaching unity 1.
So the contemporary model is mathematically correct, but then
begins to assume the Hubble Relation to DECREASE in the H=a(dot)/a definition as a constancy, albeit epoch dependent.
If
a increases, as it must, then a(dot) must increase in the same proportion for constant H(t).
Equation (1) then says
that a(doubledot)/a=-4pG/3{r+3p/c^2}/H^2=deceleration
parameter q........(1*),
because differentiating a as dimensionless fraction for the expansion of scalefactor R(t)
with respect to time t is a(dot)H and differentiating again for acceleration is a(doubledot)H^2
for the correct units for acceleration in displacement per time squared.
q=-(a(doubledot)/a)/(a(dot)/a)^2=-a(doubledot)a/a(dot)^2=L(nps)/(Rmax.fps^2)=GoMo/(c^2.Rmax^2) and qo=Mo/(2Mcritical)=ro.Vo/(2rcritical.Vmax)={4pGoro/3Ho^2}(Ro/Rmax)^3..........(14)
Then
the standard cosmology reappears in distributing the density ro(Ro/Rmax)^3=r+3P/c^2, that is
Pressure P=(c^2/3){ro[Ro/Rmax]^3-r},
P=0 for ro=r given by the mass seedling Mo and the Omega of 0.028=2qo which assumes Ro=Rmax in the superposed curvatures of the boundary conditions
k=0,1.
So we see that the Lamda Quintessence
is incorporated into standard cosmology as the evolution of the universe bounded in its curvatures.
This means, that
the evolving cosmology uses a negative hyperbolic curvature for a open universe, yet bounded in its asymptote of k=0 for the expansion parameter a=n/(n+1) approaching 1 in infinite time
t=n/Ho.
This superposes the fixed W=2qo for
the required Euclidean flatness onto the quintessential L from
the inflaton nps=lps/Rmax.
Curvature k=R^2{Ho^2W+L/3-Ho^2}=(aRo/c)^2{Ho^2(W-1)+L/3}
then becomes naturally -1 for W='L'=0 and Ho=c/Rmax=c/Ro, implying that a=1 that is the limit
for k=0 and k=1.
But this cosmological constant 'L' is NOT the same as the quintessence and can be set equal to 0 just as originally proposed by Albert Einstein.
The
demetricated expansion parameters then become:
a=n/(n+1); a(dot)=Ho/(n+1)^2 and a(doubledot)=-2Ho^2/(n+1)^3.
Now solving qo=Wo/2=Mo/2Mcritical=0.01405..
DEFINES no=qo at redshift z=7.477 and gives in the definition of (14) q(n)=2n; (generally for n=a/(1-a), (aq(n)=2 for
a=X for the acceleration redshift discussed later).
So q(no)=2qo=2no=Wo for a radiation dominated universe with
r=3P/c^2 and as the boundary condition for the superposed curvatures.
The deceleration parameter of the standard
cosmology so defines this limiting threshold between a radiation dominated universe with qo=Wo and a
matter dominated universe for which qo=Wo/2.
q(n)=2n and q(a)=2a/(1-a)
the latter function increasing q by 2 for any half-cycle Hubble Oscillation.
q(0)=0; q(1/2)=2; q(2/3)=4; q(3/4)=6;
q(a=n/(n+1))=2/(1/n)=2n.
Now this shows that for any n >1/4 the universe exceeds the threshold q(n=0.25)=1/2 and
the standard cosmological parameter begins to diverge from its stationary value qo.
n=0.25 defines a universe of age 1/4Ho of so 4.225 billion years and at a recessional velocity of (16/25)c for which a 64% (v/c) ratio implies
a cosmological redshift of z=Sqrt(1.64/0.36)-1=1.13437...
This indicates the 'acceleration' scenario manifested at
n=0.618..so 6.23 billion years from the qo-threshold
and so 8.67 billion years in the past.
The standard cosmological equations then are solved in writing R(doubledot)=a(doubledot)Rmax
or R(doubledot)=-GM(R)/R^2 +Q(n), where Q(n)=Q(R/(Rmax-R)) via R(n)=Rmax(n/n+1).
-2cHo/(n+1)^3=a(doubledot)Rmax=-(4pr(n)G(n)/3)R(n)+Q(n),
considering the distribution of densities and limiting value for the deceleration parameter q as discussed with Q(n) forming
the quintessence as Q(n)=Q(a/(1-a)).
This reincorporates q but not as function of the Hubble-Law.
Rather it
uses its limit in the curvatures and in W and incorporates
L in the form of a(doubledot).
-2cHo/(n+1)^3=
-(4pG(n)/3)(3M(n)/4pR(n)^3)(R(n))+Q(n)=-G(n)M(n)/R(n)^2+Q(n).
It is not required to solve for the expansion
parameter as a function of n, but one can solve for the Quintessence Q(n).
Q(n)=G(n)M(n)/R(n)^2-2cHo/(n+1)^3 = Omega-Factor + Deceleration-Milgrom-Factor=Lambda-Factor
It can be shown, that M(n)=Mo.Sqrt{Y^n} and that G(n)=Go.X^n with X=1/Y as pentagonal symmetry parameters of the cosmogenesis.
It is then evident, that the
demetrication of the General Relativity parameters in dimensionless cycletime n not ony defines the expansion parameter a,
but also redefines the Hubble Law in its boundary conditions.
Now in Quantum Relativity, Ro=Rmax and a=n/(n+1) defining the series above
as a sequence of the expansion parameter over time, which becomes demetricated in parameter n=Hot.
Then R(n)=Rmax(n/(n+1)) and a(dot)=Ho.Rmax/(n+1)^2 with a(doubledot)=-2Ho^2.Rmax/(n+1)^3.
But HoRmax=c and the metricated
acceleration parameter in General Relativity becomes reformulated as demetricated a(doubledot)=-2cHo/(n+1)^3 in
Quantum Relativity.
The equation for dynamical motion for the universe then becomes in General Relativity:
a(doubledot)/a
= -4pG/3{r+3p/c^2} and -2cHo/n(n+1)^2
in Quantum Relativity for a zero cosmological constant.
Integrating
a(doubledot)/a via acceleration A=vdv/dx=d(v^2/2)/dx with respect
to time and for r=aro and r(dot)=a(dot)ro and r(doubledot)=a(doubledot)ro and M=rV=4prr^3/3 gives:
Sr(doubledot)/(r)dr=S(GM(r)/r^3)dr and d(r(dot)^2)/dt=2r(dot)r(doubledot)=-2GM(r).d(1/r)/dt=2GM(r)r(dot)/r^2.
As r(dot)^2=a(dot)^2ro^2=2GM(r)/r -c^2, with constant of integration ro(dot)^
|
