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The Copernicus Constant K

and Size Limit Z of Atomic Nucleus

“ Looking for a simplest theory that is consistent with experimental data “

N. Copernicus . ( year 1543)

Applying the Copernicus Constant K to our expanding Universe we can get

many interesting results , the Mach’s principle is embedded in our theory.

Two problems of cosmological constant can be resolved naturally. The

cosmological constant is determined by kinematics of de Sitter spacetime.

Λ = 3*c^2/R^2 = 3*c^2*K^2/lp^2 = 3*Ho*Ho

It is wonderful that the size limit Z of atomic nuleus in our

observable Universe follows the condition

Z<= 1/α = 10^124 *Λ*lp^2 /(3*c^2)

(Warsaw University, NCBJ, Le Sy Hoi )

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The Copernicus constant K is dimensionless constant, which related

Hubble constant with G constant , temperature , entropy and energy

of observable universe with fine structure constant , CMB and large

numbers of Paul Dirac .

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We have got it from the simple Einstein de Sitter model (at

critical mass) .

K = m planck * Ho*G/c^3

Ho = 67, 0022457 km/s(Mpc)

Ho = 2,171416509/10^18 s

(Ho is Hubble constant for expanding universe)

The new results of European Space Agency Planck Mission

(March 21,2013) tell us that

Hubble Constant = 67,8 + - 0,77 km/s(Mpc)

1. Introduction to the cosmological constant

Einstein’s original field equations are

Rµν – ½ R*gµν = 8π*G*Tµν/c^4 (1)

On very large scales the universe is spatially homogeneous and isotropic,

which implies that its metric takes the Robertson Walker form

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Ds^2 = - dt^2 +a(t)^2 * Ro^2 *[dr^2/(1-k*r^2) + r^2*dΩ^2] (2)

Where dΩ^2 = dθ^2+ sinθ^2*dφ^2 is the metric on a two-sphere. The

curvature parameter k takes on values +1,0, -1 for positively curved, flat

and negatively curved spatial sections respectively. The scale factor

characterizes the relative size of the spatial sections as a function of time

a(t) = R(t)/Ro, where the subscrip 0 refer to a quantity at present time. The

redshift z undergone by radiation from a comoving object as it travels to us

today is related to a(t) as following

a= 1/(1+ z) (3)

The energy momentum sources may be modeled as a perfec fluid, specified

by an energy density ρ and isotropic pressure p in its rest frame

Tuv/c^4 = (ρ+ p/c^2) Uµ*Uv +p*gµν/c^2 (4)

Where Uµ is the fluid four- velocity. To obtain a Robertson Walker solution

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of Einstein equation, the rest frame of the fluid must be that of a comoving

observer in the metric. In that case Einstein equations reduce to the two

Friedmann equations

H^2 =( dR(t)/dt)/R(t))^2 = 8π*G*ρ/3 – k*c^2/R(t)^2 (5)

Where we have introduced the Hubble parameter H= (dR(t)/dt)/R(t) and

[D(dR(t)/dt)/dt]/R(t) = - 4π*G*(ρ+ 3*p/c^2)/3 (6)

Einstein was interested in finding static solution H=0, but (6) implies that

d(dR(t)/dt)/dt will never vanish in such a spacetime if the pressure p is also

nonnegative. Einstein therefore proposed a modification of his equations to

Rµν- R*gµν/2 + Λ*gµν = 8π*G*Tµν/c^4 (7)

Where lambda Λ is a new free parameter, the cosmological constant.

With this modification, the Friedmann equations become

H^2 = 8π*G*ρ/3 – k*c^2/R(t)^2 + Λ/3 (8)

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(D(dR(t)/dt)/dt)/R(t) = - 4π*G*(ρ+ 3*p/c^2)/3 + Λ/3 (9)

These equations admit a static solution with positive spatial curvature

(k=1) and all parameters rho ρ , p and lambda Λ nonnegative.

The discovery by Hubble that the universe is expanding had eliminated static

model. The 1998 discovery found out that the expanding is accelerated . It

has been criticized that any small deviation from a perfect balance between

the terms in (9) will rapidly grow into a runaway departure from the static

solution.

How to explain it ?

2. New Friedmann equations

We chose that Λ = 3* ( c/R(t))^2 = 8π*G*ρ, by that way we have got the

new Friedmann equations for positive spatial curvature k=1

H^2 = 8π*G*ρ/3 (10) or

H^2 = Λ/3 (10’)

(D(dR(t)/dt)/dt)/R(t) = 4π*G*(ρ – 3*p/c^2)/3 (11)

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By our choosing value for lambda Λ , equation (10) is equivalent to

equation (10’). The equation (10) says that Universe is expanding, with

speed depends on energy matter density. The curvature of spacetime is

hidden by cosmological constant lambda Λ.

The equation (10’) says Universe is expanding with speed depends on

cosmological constant lambda Λ. The curved spacetime is made by hidden

matter. The existence of matter is shown by the curvature of spacetime. We

can say matter is another side of curved spacetime and vice versa. By

equation (10’) we understand the difficults to look for dark matter and dark

energy today.

The expanding of our universe is accelerated if ρ > 3*p/c^2 . We DO NOT

need negative pressure. We see that the pressure p ~ ρ/R(t)^2 , so that after

t = t critical we always get (11) positive, the expanding is accelerated.

The Mach’s principle that matter determines inertia and geometry

of spacetime is embedded in our theory by the relation 8π*G*ρ =

3*(c/R(t))^2. The matter energy density and cosmological constant

lambda decides geometry of our spacetime.

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So we need a new relativity for spacetime with positive lambda. The natural

candidate is de Sitter spacetime. In de Sitter spacetime the cosmological

constant lambda Λ is related to a special length parameter l through

Λ = 3*(c/l )^2 (12)

3.Fundamentals of de Sitter relativity

The de Sitter space (left) and anti de Sitter (right)

The de Sitter space is a vacuum solution of Einstein equation with

cosmological constant lambda Λ given by

Λ = (n-1)*(n-2)/2*l^2 with n = number dimension

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With n=4 (hyper surface in 5 dimension Minkowski space ) we get

Λ = 3/l^2

A de Sitter spacetime, which will be denoted dS(4,1) is a homogeneos

Space defined as the quotient between de Sitter and Lorentz group

dS(4,1) = SO(4,1)/L (13)

It can be viewed as a hyper surface in the pseudo Euclidean space E(4,1)

Immersed in a five dimensional pseudo Euclidian space E(4,1) with Cartian

coordinates (XA) =(X0, X1,X2,X3,X4), it is defined by

(X0)^2 – (X1)^2- (X2)^2 – (X3)^2 – (X4)^2 = - l^2 (14’)

ηab*Xa*Xb - (X4)^2 = - l^2 (14)

Where ηab = diag(1,-1,-1,-1)

With l the de Sitter length parameter, related to the spacetime curvature . It

should emphasize that there are infinite spaces and groups , one pair for

each value of l (or Λ).

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In stereographic coordinates {xa}, the generators of the de Sitter Lie

algebra are written as

Delta xa = εcd* Lcd*xa – ε b*Πb*xa (15’)

Lab = ηac*xc*Pb – ηbc*xc*Pa (15)

And

La4 = l*Pa – Ka/(4*l) (16)

Where Pa = da=d/dxa (17)

and Ka =(2*ηac*xc*xb- σ^2*δab)*Pb (17’)

( σ^2 = ηab*xa*xb=(xo)^2 -(x1)^2 -(x2)^2 -(x3)^2 )

( εcd and ε b are parameters of de Sitter transformations)

Are respectively the generators of translations and proper conformal

transformations. Generators Lab refer to the Lorentz subgroup, whereas the

remaining La4 define the transibility on de Sitter spacetime. It is convenient

to define the generators

Πa = La4/l = Pa – Ka/(4*l^2 ) (18)

Which are usually called de Sitter translation generators.

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From the algebraic point of view, the change from Poincare to de Sitter is

achieved by replacing Pa by Πa. The relative importance of translations

and conformal transformations is determined by the value of l , that is by

the value of cosmological constant. We would like to emphasize that in de

Sitter special relativity only translations are violated, the Lorentz subgroup

remaining as a physical symmetry.

4. Horizons and Fundamental Constants

In static coordinates (t, r, θ, φ) the de Sitter metric is written as

Ds^2 = (1-r^2/l^2)*c^2*dt^2 – dr^2/(1-r^2/l^2) – r^2*(dθ^2 + sin^2θ*d φ^2)

(19)

In this form, it reveals an important property of the de Sitter spacetime, the

existence of a horizon at r = l. It is well known from the Schwarzschild

solution, there is a relation between gravitational horizons and

thermodynamical properties. Spacetime with horizons present a natural

analytic continuation from Lorentz to Euclidean signature, obtained by

making t to it.

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For example, one can associate to the de Sitter horizon the entropy

Sds = kb*Ah/(4*lp^2*pi) (20)

Where Sds is the entropy of de Sitter spacetime

kb = Boltzman constant

Ah = 4π*l^2 is the area of the horizon

Lp = (G*hn/c^3)^1/2 is the Planck length

Hn = reduced Planck constant

C = light speed

5. Ordinary matter and Cosmological Term

How does a physical system give rise to a local lambda Λ ?

Let us consider a de Sitter spacetime with l=2*lp= 2*(hn*G/c^3)^1/2. The

corresponding cosmological term is

Λ p = 3*c^2/4*lp^2 (21)

We define the Planck energy density

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ε p = 3* mp*c^2/(4π*2^3*lp^3)

= 3*mp*c^2/(32π*lp^3) (22)

With mp = (c*hn/G)^1/2 the Planck mass , equation (21) gets the form

Λ p =( 8π*G/c^2)* ε p (23)

This equation gives the local value of the cosmological term as a function of

the energy density of the physical system. Please remember epsilon p here

is NOT dark energy but matter energy density.

For small energy densities lambda will be very small, spacetime will

approach Minkowski spacetime, and de Sitter special relativity will

approach Einstein special relativity, whose kinematics is governed by the

Poincare group.

Our universe has the value of lambda as

Λ = 3*c^2/R^2 = 3*c^2*K^2/lp^2 (24)

Where K = Copernicus constant = 11,70623761/10^62

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K = mp*Ho*G/c^3

Hubble horizon = R= lp/K ~~ 1,38*10^26 m

Lp = planck length

C = light speed

It is wonderful that Geometry of our spacetime decides the value of Fine

Structure Constant α as following

1/α = 10^124 *Λ*lp^2 /(3*c^2) (24’)

6. Some physical consequences of de Sitter Relativity.

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Replacing Poincare by de Sitter as the group governing the spacetime

kinematics leads to a relation between the energy density of any physical

system and the local value of lambda Λ. When applied to the whole

universe, that relation is able to predict the value of cosmological constant.

By that way, cosmological constant is no more an independent

parameter. It is determined by the spacetime kinematics.

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Now we can understand the mechanism for Inflation

We can see from (10) or (10’) the matter density or lambda Λ are very large

at Big Bang, that’s why cosmos was expansion with

R(t) = R(to)*exp((t-to)*(Λ/3)^1/2) or

R(t) = R(to)*exp((t-to)*( 8π*G*ρ/3)^1/2)

We can see that two problems of cosmological constant (see Steven

Weinberg 2000) are resolved naturally in our theory.

We would like to emphasize that de Sitter relativisty is invariant under a

simultaneous rescaling of mass, energy and momentum, and it is valid at all

energy scales.

7.Geometry of curved spacetime and size limit Z of atomic

nucleus

In physic, the fine structure constant α (also known as Sommerfeld’s

constant) is a fundamental physical constant characterizing the strength of

the electromagnetic interaction between elementary charged particles with

the electromagnetic field by the formula

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4π*ε0*hn*c*α = e*e

Being a dimensionless quantity, it has the same numerical value in all

systems of units.

α = e*e/(4π* ε0 *hn*c)

α = 7,2973525663/ 10^3

1/α = 137,03599914

where:

e is the elementary charge

hn = h/2π is the reduced Planck constant;

c is the light speed;

ε0 is the electric constant or permittivity of free space;

Fine structure constant gives the maximum positive charge of the central nucleus that will allow a stable electron-orbit around it. For electron around the nucleus with atomic number Z,

m* v2/r = 1/4πε0 (Z*e2/r2).

The Heisenberg uncertainty principle, momentum/position uncertainty relationship of such an electron is just m*v*r = hn. The relativistic limiting value for v is c, and so the limiting value for Z is reciprocal of fine structure constant ~ 137.

Z < =1/α = 10^124 *Λ*lp^2 /(3*c^2)

We can guess that a very long time ago (some billion years ago, when lambda Λ had got bigger value) there were bigger atomic nuclei in our Universe ?

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ACKNOWLEDGEMENTS

The author Lê sỹ Hội would like to express his gratitude to Prof. (

Warsaw University ), Prof. ( NCBJ Polska ) and Prof. Trần Hữu Phát

for the helps with the work . The work is supported by Warsaw University,

NCBJ and Poland Government . The author also would like to thank the

International Centre for Theoretical Physics(ICTP) many friends and

family for helps . God bless us .

P.S.

Cosmological constant 2015 from Wikipedia

Observations announced in 1998 of distance–redshift relation for Type Ia

supernovae[8][9] indicated that the expansion of the universe is accelerating. When

combined with measurements of the cosmic microwave background radiation these

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implied a value of ,[10] a result which has been supported and refined by more recent

measurements. There are other possible causes of an accelerating universe, such as

quintessence, but the cosmological constant is in most respects the simplest solution.

Thus, the current standard model of cosmology, the Lambda-CDM model, includes the

cosmological constant, which is measured to be on the order of 10−52 m−2, in metric

units. Multiplied by other constants that appear in the equations, it is often expressed as

10−52 m−2, 10−35 s−2, 10−47 GeV4, 10−29 g/cm3.[11] In terms of Planck units, and as

a natural dimensionless value, the cosmological constant, λ, is on the order of

10−122.[12]

Our calculation gives cosmological constant = 3*Ho*Ho = 1,414513424/10^35 ( /s*s)