LET IT BEE POSTULATET THAT THE COSMOLOGICAL CONSTANT
HAVE SUCH A VALUE THAT: IF GRAVITATIONALCONSTANT=0, : HUBBLE PARAMETER IS CONSTANT!
THE HUBBLE PARAMETER IS CONSTANT WHEN THERE IS NO GRAVITATIONALCONSTANT!
IF THE SCALE-FACTOR OF THE COSMOS IS EXPONENTIAL!
THEN THE HUBBLE PARAMETER IS CONSTANT!
THE MASS OF TWO TEST-PARTICLES IS PROPORTIONAL:
m1 ~ m2
THE GENERAL GRAVITATIO FORCE IS:
F= -e.M.m1.G/r^2 + C.[SM].m1.r
M IS THE MASS OF THE COSMOS OUT TO THE TEST PARTICLE!
e is einstein, is : e=1 if GRAVITATION, e=-1 if ANTIGRAVITATION.
m1 is the mass of the test-particle
r is the distance out to the test-particle.
THE SECOND TERM IS THE COSMOLOGICAL FORCE!
IT IS PROPORTIONAL THE SUM OF ALL MASS IN COSMOS=[SM]
C IS THE CONSTANT OF PROPORTIONALITY!
F=(m1.v)' THE FORCE IS TIME DEIVATE OF THE MASS TIMES VELECITIO!
v = H.r introducing the HUBBLE PARAMETER!
GIVES:
F=(m1.H.r)' =+m1'.H.r +m1.H'.r +m1.H.r'
GIVES:
-e.M.m1.G/r^2 + C.[SM].m1.r = +m1'.H.r +m1.H'.r +m1.H.r'
USING THE POSTUALTE: IF G=0, H'=0
IF G=0:
-e.M.m1.0/r^2 + C.[SM].m1.r = +m1'.H.r +m1.0.r +m1.H.r'
+ C.[SM].m1.r = +m1'.H.r +m1.H.r'
C.[SM]=L/3 INTRODUCING THE COSMOLOGICAL CONSTANT!
+ L/3.m1.r = +m1'.H.r +m1.H.r'
THIS IS THE VALUE THE COSMOLOGICAL CONSTANT HAVE!
GIVES:
-e.M.m1.G/r^2 = +m1.H'.r
LET'S WRITE C=|C|.g ,,,,,,, g is the sign of C.
+ L/3.r = +(m1'/m1).H.r +.H.r'
+ L/3/H-H= +(m1'/m1)
THE RHS IS THE SAME OF ALL TEST-PARTICLES!
m1 ~ [SM]
m1'/m1 = [SM]'/[SM] = L'/L
GIVES:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ L/3/H-H= L'/L (1)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
M/(8PI/6.r^3)=rho
GIVES:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-e.8PI/6.G.rho = +H' (2)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
M=(8PI/6.r^3).rho
|C|.g .[SM]=L/3
[SM] ~ M ,WITH POSITIVE PROPORTIONALITY CONSTANT!
GIVES:
[SM] ~ u.M ,,,,,,,,,,,u>0
GIVES:
|C|.g .u.(8PI/6.r^3).rho =L/3
|C|.g .u.(8PI/6.r^3).rho =L/3 |C|>0 u>0 8PI/6>0
GIVES:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
.g.q.r^3.rho =L/3 ,,,,,,,,,,,q>0 (3)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
v = H .r
GIVES:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
r'=H.r (4)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
MATEMATIJCAL ANALYSIS:
THE FORMULAS:
VARIABLES:
L IS THE COSMOLOGICAL CONSTANT!
H IS THE HUBBLE PARAMETER!
rho IS THE DENSITY!
r is THE SCALE-FACTOR!
CONSTANTS:
e=EINSTEIN
G=GRAVITATIONNBAND
8,PI,6:NUMERICAL CONSTANTS
g=sign OF THE L;
q is POSITIVE!!!!;;;;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ L/3/H-H= L'/L (1)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-e.8PI/6.G.rho = +H' (2)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
.g.q.r^3.rho =L/3 ,,,,,,,,,,,q>0 (3)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
r'=H.r (4)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
**END MATEMATIJCAL ANALYSIS, END**
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