Cosmic Quantum Mechanics

Afri an Journal Of Mathemati al Physi s Vol

2

No 1 (2005)1-10

A Cosmi Quantum Me hani s Arbab I. Arbab Department of Physi s, Tea hers' College, Riyadh 11491, P.O.Box 4341, Kingdom of Saudi Arabia.

and

Comboni College for Computer S ien e, P.O. Box 114, Khartoum, Sudan

Abstra t We presented a model for uni ation of ele tri ity and gravity. We have found a onsistent des ription of all physi al quantities pertaining to the system. We have provided limiting values for all physi al values. These values are neither zero nor in nity. Our universe is des ribed at all times by the four dimensional onstants ; h ; k; G only. The remnant of va uum remains at all epo hs with dierent values. The present osmologi al hierar hy and puzzles are justi ed as due to the onsequen es of osmi quantization developed in this work. The missing energy in the universe an be resolved if one onsiders the

ontribution of the gravito-ele tromagneti ounterparts besides the observed mass in the universe. Keywords:

osmology: quantum-uni ation, quantum me hani s, gravity.

world su h problems should not be present. Consequently, a quantum treatment should remove the singularity problem by allowing all physi al quantities to have a limiting values; neither zero nor in nity. The ele tromagneti ontribution arising from gravitational system is very genuine and should be taken into onsideration. Su h a ontribution ould oset the dieren e between the presently observed and anti ipated energy density of the universe. This amounts to say that the dark energy problem is no longer a problem. The quantum nature of the whole universe is evident in the a

eleration of the osmi uid that permeates the spa e time at dierent s ales. In parti ular, at present time there should be a uniform a

eleration of this osmi uid of the order of 10 10m s 2 permeating the whole universe. Su h a value is observed in Casimir experiment and by the Pioneer satellite. In this work, we provide the limiting values of the universe at dierent stages. We have written all physi al quantities representing the universe in terms of the four fundamental onstants, viz.,

; h  ; k; G. The universe at dierent levels is gov-

I. INTRODUCTION

Many attempts have failed to unify gravity with quantum me hani s. We propose here a new approa h for uni ation of gravity, ele tri ity and magnetism. It is based on the idea that the gravito-ele tri ee ts an not be ignored at large s ale. This is a hieved by de ning an appropriate Plan k onstant that takes are of the large s ale ee t. In this sense, one onsiders the universe to have a quantum nature present at all levels. The quantum nature is manifested by gravitationally bound system only, whi h we all osmi system. General relativity predi ts a singularity at the Big Bang and within a Bla k Hole. A Bla k Hole is understood when quantum analysis is developed for it. We have found that all osmi systems require a quantum treatment as well. So in a real quantum

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1

Arbab I. Arbab

Afri an Journal Of Mathemati al Physi s Vol 2 No 1 (2005)1-10 same as it is now (Arbab, 2001b). Assuming all fundamental parti les, with rest mass, harbor one or more of these quanta (in virtual form) at their ores, where va uum tension is maximal, the result would be a short range warp 'bubble' enveloping all parti les with mass. Periodi reversal of this quanta's eld would give rise to avity os illator behavior, subje ting its host parti le to an alternating polarity warp metri , whose intensity would mat h the ele tromagneti eld. If the phase of this periodi ity syn hronizes with the y li al a

eleration/de eleration for es of an ele tron in ellipti al orbit about its nu leus, and the warp- eld is always aligned along the ele tron/nu leus axis, then the ele tron would follow a sinusoidal, time-like geodesi through spa e-time, negating syn hrotron radiation. The os illations of this natural, mi ro-warp eld are therefore proposed to be the essen e of de Broglie matter waves, whi h are the basis of stable, non-radiating atomi orbits, and the starting point for wave me hani s. The volumetri variations, within the warp 'bubble', are proposed to alternate between Minkowski spa e and the extra dimensions, giving rise to bipolar relativisti syn shifts (relativity of simultaneity), due to the resulting bi-dire tional linear translations between parti les. Consequently, all fundamental parti les will appear to rapidly os illate between the past and future at de Broglie frequen ies, but average to the lo al present. For bound gravitational obje t, the spa e time inside the obje t and out side is dierent. Thus, parti les moving inside these obje ts experien e an a

eleration whi h is dierent from those moving outside. For instan e, the tension of spa e-time inside the nu lear region is enormously redu ed,

ompared to the outside tension. This may elu idate the asymptoti free nature of quarks residing inside baryons. The surfa e tension of the nu leus is the same as that of the whole universe. In 1984 DerSarkissian suggested that a osmi version of ordinary quantum me hani s may be responsible for the observed physi al properties of galaxies. Agob et al (1998) used a fra tal spa e time has shown that the Solar System is a quantized system. They have found a osmologi al Plan k's onstant for the galaxies of the order of  g  1067Js. With this huge value the expe t a rah dio emission to dominate galaxies. A similar form of osmi quantum me hani s was suggested independently by Co ke (1984). Re ently (Arbab, 2004; 2001a) we have shown that the hierar hi al problems of the matter buildup of the universe is resolved with the idea of large s ale quantization. In this work, we provide the lower and upper limits of our physi al quantities. They are neither zero nor in nity. Consequently, in nities an not o -

erned by this set of equations. Thus, the universe appears at these stages the way it understood be ause it is the only way it ould. It turns out that some of the physi al quantities are relativisti , quantum, gravitational and ele tromagneti . This is evident from the way it depends on the orresponding onstants. We remark that the speed of light does not depend of the size of the system under onsideration. However, the Plan k's onstant depends on the size of the system sin e its unit is M L2T 1. Therefore, it is large for larger systems and small for smaller systems. Hen e, we expe t that its value for a ma ros opi to be very large (large M and L). The in lusion of G into the model is to represent gravity (mass), or equivalently to represent spa e-time. In a re ent work, we have found that for every bound system (nu leus, atom, star, galaxy, and the whole universe) there is a hara teristi Plan k

onstant (Arbab, 2001a). Hen e, a parti ular system intera ts quantum me hani ally with its orresponding Plan k's onstant. With this pres ription all gravitational phenomena are interpreted in terms of quantum ones. With this remedy in mind, the gravitational systems are su

essfully des ribed. Hen e, any bound gravitational system exhibits the quantum nature (phenomena) if it is fully understood. We have seen that an obje t (mass) whether harged or not exhibits the ele tromagneti phenomena. That is be ause an ele tromagneti eld is asso iated with every gravitational bound system. Thus, an (un) harged mass in a gravitational eld intera ts as if it were a

harged mass pla ed in ele tromagneti eld. Its

orresponding ele tromagneti elds andvoltage  7  12  5  12 1 are E = h Gk2 ; B = h Gk2 and V = G4 k 2 : At the present time, some osmologists believe that a new gravitational phenomena is thought to show up in extra dimension. An extra dimension of  10 17 m is thought to allow an exponential in rease in the strength of gravity to where it would mat h the strength of the ele tro-weak and strong for es at the remarkably modest energy of about 1 TeV. This is nearly 16 orders of magnitude below the Plan k s ale, whi h is 1019 GeV, the dogmati ally assumed uni ation energy of all nature's for es. If the extra dimension on ept is valid, then gravity should parti ipate equally with the strong and ele tro-weak for es in the synthesis of exoti new quanta in the supersymmetry mass range. Arguments are advan ed to support the thesis that at least one of these quanta should be endowed with a pseudo-gravity eld, whose strength is equal to the ele tromagneti eld in its low energy form. We, however, have shown in an earlier work that the gravitational onstant at Plan k's time is the 2

Arbab I. Arbab

Afri an Journal Of Mathemati al Physi s Vol 2 No 1 (2005)1-10

ur in our physi al world, i.e., no ultraviolet no infrared atastrophes in our theories. Thus, we noti e that no fundamental onstant an be set to zero ( 6= 0; h 6= 0; et ) as this would violate the osmi quantum hypothesis. Sin e the a tion of the universe is very large in omparison with Plan k onstant, one an use the WKB approximation to write down a eld theoreti model for su h an approa h. This is feasible sin e our osmi system anti ipated to involve large numbers quantization.

time, regarding the universe as a sphere, one nds the same value. This for e is attributed as due to quantum va uum u tuations of the ele tromagneti eld. Hen e, one realizes that su h a quantum nature does still exist, and has now be ome sizable. One an de ne a gravitational harge as r

(2.5) = Gk M ; for a system whose gravitational mass is M . Moreover, we have shown re ently that the Plan k onstant for large s ale system is de ned by (Arbab, 2001a) q

II. THE MODEL

2

P ; (2.6)  = GM h

where MP is the osmi Plan k's mass. This equation represents a bi-pass from ele tri system to gravitational system. So if some phenomena is known in one system the orresponding quantity will be expe ted to take pla e for the other system. Spa e-time is onne ted by strings whose tension is de ned by

In order to unify gravity with ele tri ity and magnetism one requires that only fundamental

onstant des ribing these domains should appear. These systems are des ribed by the following onstant: G ; k ; ; h :

(2.1)

With this pres ription, one an de ne the quantum ee t of gravitational and ele tromagneti systems. In order for gravity to unify with ele tri ity, they should have on e had same strength. This would mean that one had at some time the equation Gm2

= kq2 ;

(2.2)

where k = 41 0 is the ele tri al onstant, whi h means that gravitational and ele tri for es between elementary parti les with mass m and

harge q were equal. We argue that this for e however remains un hanged ( onserved). Its value at Plan k's time and today is the same; and all other for es are derived from it. Its value at Plan k's time is FP

2

P = Gm  1043 N; r2 P

r T

is also appli able. If the ee tive Plan k's area is really in reasing ass the universe expands, that suggests the universe will be ome more and more gravitationally quantized larger s ales. The Plan k's area is given by the produ t of the lassi al gravity radius and the quantum radius as   GM h  A= : = Gh : (2.9)

(2.3)

2

and its value today is F0

2

0 = GM  1043 N; R2 0

4

= 8 G ; (2.7) This value happens to be very huge. It implies that the spa e-time is in redibly sti (T  1043N) and no stress-energy density an make it bend no matter how big it is. The string has a duality prin iple that for a physi al quantity r the relation  1 ; h r0 = (2.8) T

M

3

Now we see that this area at Plan k's time (Apl ) and at the present time (A0 ) are respe tively AP  10 69 m2 ; (2.10) and A0  1052 m2 ; (2.11) respe ting the above mentioned duality. Thus, the two theories would be appli able. We see that the same formula governs the mi ros opi as well as ma ros opi worlds. Therefore, the two worlds are

omplementary to one another. Now de ne the following physi al quantities des ribing our system as follows:

(2.4)

where R0  1026 m, rP  10 35 m, mP  10 8 kg and M0  1053 kg. It seems the gravitational for e tiding the universe is onserved. In 1948 Casimir has found an attra tive for e between two 8h A, where A is the plates of the form F = 480 L4 area of the plate and L is the separation between the two plates. If one al ulates this for e for the present time with the osmi Plan k's onstant (h  1087J:s, see the next se tion) and Plan k's 3

Arbab I. Arbab

Afri an Journal Of Mathemati al Physi s Vol 2 No 1 (2005)1-10

A. Ele tromagneti quantities

Q

These quantities provide limiting values for a

essible physi al quantities. A

ording to our hypothesis, one an write this in terms of our fundamental onstant as B



=

Gh k

1 2 2



I

=

6 Gk

 12

m

E

=

 21

7 k hG2

(2.15)

V

=

4 k G

 21

I

(2.16)

:

 21

5 k

= h G2 : The magneti ux density is given by B

=



k h

 12

(2.17)

11

 12

= h G3 : The surfa e energy is given by

U

= A =



Gh

3

 12

S

=

7 hG3

 21

=

:

2

(2.25)





8 : G2 h

(2.26)

Gh3

5

 12

(2.27)

:

 2

G

(2.28)

:

 3

G

(2.29)

:

The ele tri ondu tivity is de ned by 

=



5 hk2 G

 12

:

(2.30)

The a

eleration of the quantum uid lling the spa e-time is giving by a=

(2.20)



7 Gh

 12

:

(2.31)

The a

eleration of a harged parti le (of harge q and mass m) in an ele tri eld (E ) is given by

The surfa e mass density is de ned as 

=



Q=

(2.19)

:

7 : G2 h

The mass ow rate is de ned by

The surfa e tension ( spa e-time stiness onstant) is de ned as 



=

(2.18)

:



The gravitational eld is de ned as

The magneti eld is de ned as 

(2.24)



The moment of inertia of a gravitating mass about its enter is given by

The potential dieren e is given by 

:

B2 k

The amount of energy emitted per unit time per unit area (energy ux) is given by =

:

(2.22)

(2.23)



=

=

P

The ele tri eld E is de ned as 

:

:

The pressure is given by

(2.14)

:

 12



5

= h G2 This an be written as m

This is de ned formally by (2.13) B = IA ; where I is the urrent owing around the loop whose area is A. Using eq.(9) the above equation yields, 

=

10 h2 G3 k

The magneti (ele tri ) eld ontribution to mass density is given by

(2.12)

;



a=

(2.21)

q E; m

(2.32)

where E is de ned above. Using eq.(5) this equation yields

The ele tri harge density is de ned as 4

Arbab I. Arbab

Afri an Journal Of Mathemati al Physi s Vol 2 No 1 (2005)1-10

a=

  12 G k

Moreover one nds that the mass and the diusion

onstant are \ anoni al onjugate" to ea h other, i.e., DM = h : (2.40) It has been emphasized by Kozlowska and Kozlowski (2003) that as time goes one the universe be omes more and more quantum on large s ale by allowing h ! 1. They on luded that the prevailing thermal pro ess for thermal phenomena in the universe (that taking pla e on large s ale) is the diusion.

(2.33)

E;

valid for all gravitationally bound system. This a

eleration oin ides with the de nition 

a=

GM R2



(2.34)

;

for a gravitational system with mass M and radius R. We remark here the spa e-time (va uum) a

elerate due to its very nature. This a

eleration is required to allow the matter to be pla ed in it. That is be ause there is a limiting mass that an be pla ed 2at a given region. This is given by the quantity G mentioned above. The energy embedded in this spa e time de ays to give the matter we observe today. However, the de ay (transfer) rate is3 limited to the value governed by the quantity G . The spa e-time a

elerate to give more spa e for the reated matter to be pla ed in. Thus, spa e-time (va uum/quantum) should have a definite geometri stru ture. Thus, spa e-time represents a state of a minimum energy. Hen e, energy

an not be destroyed ompletely. The remnant of it will orrespond to spa e-time. The minimum energy (ground state) may not be noti eable. But its ee t an be observed by the way in whi h the primeval matter is reated in the universe. So this minimum energy state would provide us with a universal referen e for the motion of matter. One an de ne a apa itan e of a gravitoele tromagneti system as follows 

 12

hG C= k 2 3

III. PLANCKIAN DOMAIN

We al ulate the above quantities at Plan k's times. Now we see that BP

IP

= EC =



4 Gk

 12

:

EP

D=

Gh

 12

:

D=

2 k



:



=

BP

 21

44

J=T ;

 1025 Amp :

(3.1)

(3.2)

7 k hG2

 12

 1061 V=m ;

(3.3)

=



5 k hG2

 12

 1053 T:

(3.4)

The Plan kian magneti ux density is given by

(2.36)

P =



k h

 12

 10 17 Wb:

(3.5)

The magneti (ele tri ) eld ontribution to mass density is given by

(2.37)

Using eq.(31) one nds that Da = 3 : (2.38) This means that a highly a

elerating obje t is less diusing, and vi e versa. One an also write the equation relating the ele tri al ondu tivity () to the diusion onstant (D) as 

=

6 Gk

 10

whi h is a typi al Plan kian eld. The Plan kian magneti eld density is

One an de ne a diusion onstant (area/se ) as 



 12

Gh2 k

The Plan kian ele tri eld intensity is given by

The harge per unit length (q ) is given by EC , or q

=

and

(2.35)

:



mP

=



B2 k



 1097kg=m3 :

(3.6)

The surfa e tension of the universe at Plan k's time is given by

P

=



11 hG3

 12

 1078N=m:

(3.7)

The surfa e mass density at Plan k's time is de ned as

(2.39) 5

Arbab I. Arbab =

SP

Afri an Journal Of Mathemati al Physi s Vol 2 No 1 (2005)1-10



 12

7 hG3

 1061kg=m2 :

whi h is a typi al nu lear eld. The nu lear magneti eld density is

(3.8)



The pressure exerted at Plan k's time is given by =

PP



7 G2 h



 10112 N=m2 :

(3.9)

The ele tri Plan kian harge density is given by 

N =

 12

10

 1086 C=m3 ; (3.10)  2 G3 k h whi h is a enormously huge quantity. The a

eleration of the quantum uid lling the spa e-time at Plan k's time is given by QP

=

aP

=



7 Gh

 12

 1051m=s2 :

P =

8 G2 h



 10 W=m : 120

2

N

(3.12)

UN

=

IN

=

and



6 GN k

 12

 10

24

J=T ;

 105 Amp :

EN

=

7 k hG2N

 12

 1020 V=m ;

=

=



N

(4.7)



B2 k



 1015 kg=m3 :

(4.8)



 12

11 hG3N

 1018N=m:

(4.9)

GN h

3

 12

 10 12J  10 MeV: (4.10)

=



2 GN



 10 13kg=m :

(4.11)

This de nes the maximal mass that an be pla ed inside the nu lear gravitational eld. Thus the maximal mass whi h an be pla ed over a distan e of 10 15m is 10 28kg. Therefore, the mass of the nu leus we ome to know toady is the only possible mass that the nu leus an hold. That is be ause the spa e-time tension inside the nu lear region is ex eedingly weak (i.e. T  103N), in omparison with the tension outside ( whi h is  1043N). The diusion onstant for nu lear domain is

(4.3)

(4.4)

DN

The nu lear ele tri eld intensity is given by 

 10 17Wb:

This oin ides with the typi al value for the binding energy per nu leons. We would like to remark here the s ale (QCD ) for quantum hromodynami s (QCD) is found o be in this range (QCD = 66  10M eV ). The gravitational eld inside the nu lear region is

This gives a range of about 1 Fermi, that is a typi al distan e for nu leons. Now we see that BN

 12

The surfa e mass energy of the nu leus is given by

We see that the Plan k's area inside the nu lear domain is given by   GN h   10 30 m2 : AN = (4.2)

3  21

k h

We see that the nu lear density is independent of the number of nu leons present. The surfa e tension of a nu lear medium is given by

(3.11)

From an earlier work (Arbab, 2001b) we have shown that inside the nu lear region, the Newton's

onstant (GN ) is given by GN  1040 G : (4.1)

GN h2 k

=

mN

IV. NUCLEAR DOMAIN





The magneti (ele tri ) eld ontribution to mass density is given by

The amount of energy emitted per unit time per unit area (energy ux) during Plan k's time is given by 

 12

5

 1012 T: (4.6) = h Gk2 N The nu lear magneti ux density is given by BN

=



GN h

 12

 10 7 m2 =s :

(4.12)

Therefore, during the nu lear time the diused area of the nu lear onstituents is 10 30 m2 . This

(4.5) 6

Arbab I. Arbab

Afri an Journal Of Mathemati al Physi s Vol 2 No 1 (2005)1-10 The diusion onstant the nu lear medium is given by

is a typi al area of nu lear size. The surfa e mass density inside the Nu lear region is de ned as SN

=



7 hG3N

 12

 102kg=m2 :

The pressure exerted medium(quantum) is given by 

=

PN

7 G2N h



by

 1032N=m2 :

(4.13)

DN

nu lear

QN

10  G3N k h

=

2

 1026 C=m3;

(4.14)

N

4

= 8G  103N :

(4.15)

aN

=

 12

7

GN h

 1031m=s2 :

=



4 GN k

 12

 10 3C=m:

N =

8 G2N h



 10 W=m : 40

2

7

m2 =s :

(4.20)

9 G3N h

 12

 109 Ns=m2 :

(4.21)

3

This gives a range of about 108m, that is a typi al distan e for stars. Now we see that BS

=

and IS

=





 12

Gh2S k

6 Gk

 21

 1042 J=T ;

 1025 Amp :

(5.2)

(5.3)

The nu lear ele tri eld intensity is given by 

(4.17)

7

 12

(5.4) = h Gk 2  1017 V=m ; S whi h is a typi al nu lear eld. The nu lear magneti eld density is ES

(4.18)

BS

Thus, for a nu lear dimension one has a harge of an order 10 3  10 15  10 18C, whi h is the

harge of the nu leus. The amount of energy emitted per unit time per unit area (energy ux) in the nu lear region is given by 

=



S

The harge per unit length in the nu lear region is given by q

 10

For su h a system (Globular Cluster) one has a

orresponding Plan k's onstant hS  1052 Js. We see that the Plan k's area inside the star domain is given by   Gh  S  1017 m2 : (5.1) A =

This oin ides with the value al ulated for the quarks on ned in side hadrons. It is s thought that a quark-antiquark is made if one tries to separate strongly intera ting parti les, in whi h ase the string tension is broken. The a

eleration of the quantum uid lling the spa e-time inside the nu leus is given by 

 12

V. STAR DOMAIN

(4.16)

N

GN h

This value suggests that the nu lear onstituents move freely in this nu lear medium. This may elu idate the fa t that quarks are free inside hadrons.

thus having the same magnitude as the nu lear mass density. This implies that inside the nu leus both ele tri ity and gravity dominate. We

al ulate here the ele tri eld of an ele tron whose radius is  10 15m. This is given by 2 E = ker  1020 Vm 1 and its mass density is 14 kg m 3 , and its harge density is e = m r 3  10 e e = r3  1026 C m 3 . Comparing these values with the above data one sees that an ele tron as a single system resembles a nu leus. The nu lear tension is given by TN



One an de ne a oeÆ ient of vis osity for the nu lear medium as

The ele tri nu lear harge density is given by ! 12

=

=



5 k hS G2

 12

 109 T:

(5.5)

The star magneti ux density is given by S =



k hS

 12

 1027 Wb:

(5.6)

The magneti (ele tri ) eld ontribution to mass density is given by

(4.19) 7

Arbab I. Arbab mS

Afri an Journal Of Mathemati al Physi s Vol 2 No 1 (2005)1-10 

=

B2 k



 108kg=m3 :

The gala ti magneti ux density is given by

(5.7)



=

S

11 hS G3

 12

 1035N=m:

 0:1 C=m3 ; (5.9)  2S G3 k h This implies that inside the stars ele tri ity is onsiderable. The a

eleration of the quantum uid lling the spa e-time inside the stars domain is given by =

QS

=

aS



7 GhS

 12

 10 m=s : 8

S

=

9 G3 hS

 21

 10 Ns=m : 27

2

QG

(5.11)

=

aG

3

BG

=

and IG

=



6

 12

 10 J=T ; 58



 10 5kg=m3 :

(6.7)



11 hG G3

 12

 1027N=m:

(6.8)



10 2 3 hG G k

 12

 10

33

C=m3 ;

(6.9)

=



 12

GhG

 1025 m2 =s:

(6.10)

=



7 GhG

 12

 10 1m=s2 :

(6.11)

The oeÆ ient of vis osity in this region is

This gives a range of about 1017 m, that is a typi al distan e for galaxies. Now we see that  21

B2 k

This an be ompared with value obtained by Agob et al., whi h is 1:9  1026 m2 =s. The a

eleration of the quantum uid lling the spa e-time inside the galaxies is given by

For su h a system one has a Plan k's onstant  G  1068 Js. We see that Plan k's area inside the h gala ti domain is given by   Gh  G A =  1033 m2 : (6.1)

Gh2G k

(6.6)

whi h is a vanishing small quantity. The diusion

onstant (area/se ) for this system is

VI. GALACTIC DOMAIN



 1035Wb:

The ele tri gala ti harge density is given by

DG

G

=

G

The oeÆ ient of vis osity in this region is 

 12

The surfa e tension of a gala ti medium is given by

(5.10)

2



=

mG

 12

10

k hG

The magneti (ele tri ) eld ontribution to mass density is given by

(5.8)

The ele tri nu lear harge density is given by 



G =

The surfa e tension of a star medium is given by

G

(6.2)

=



9 3 G hG

 12

 1018 Ns=m2 :

(6.12)

VII. COSMIC DOMAIN

 1025 Amp :

(6.3)

(6.4)

Here the system is des ribed by the Plan k's onstant h  1087 Js. We see that the Plan k's area inside the nu lear domain is given by   Gh   1052 m2 : A = (7.1)

whi h is a typi al gala ti eld. The gala ti magneti eld density is

This gives a range of about 1026 m, that is a typi al distan e for our present universe. Now we see that

Gk

The gala ti ele tri eld intensity is given by EG

=

BG



=

7 k hG G2



 12

5 k hG G2

 1010 V=m ;

 12

 10 T: 2

(6.5)

B

8

=

3



Gh2 k

 12

 1077 J=T ;

(7.2)

Arbab I. Arbab and I

Afri an Journal Of Mathemati al Physi s Vol 2 No 1 (2005)1-10



=

6 Gk

The mass ow rate is de ned by

 12

 10 Amp :

(7.3)

25

Q

E

=

 12

7 k h G2

 1 V=m ;

(7.4)

whi h is a typi al osmi eld. The osmi magneti eld density is B



=

5 k h G2

 12

 10 T:

S

(7.5)

8

 =

k h

 12

 1044 Wb:

=



B2 k



 10 26kg=m3 :

(7.6)



11

(7.7)

 21

(7.8) = h G3  1018N=m :

The surfa e energy of the universe is given by U

=



Gh

3

 12

 1070J :

(7.9)

The amount of energy emitted per unit time per unit area in whole universe is given by  =



8 G2 h



 1W=m2 :

Q

(7.10)



=

G

 1027 kg=m :

(7.12)



7 h G3

 12

 102kg=m2 :

(7.13)

=



7 G2 h



 10 9N=m2 :

(7.14)

=



10 2 3 h G k

 12

 10

36

C=m3 ;

(7.16)

Again, this implies that the present universe an't be dominated by ele tri ity today. The a

eleration of the quantum uid lling the spa e-time at present's time is given by

We see that the va uum energy ux today is in redibly small in omparison with one at Plan k's time. It is 120 orders of magnitude smaller. The gravitational eld is de ned as  2

 1035kg=se :

Comparing this with the Plan k value one nds   PP = hh  10122 : (7.15) P Hen e, not only the osmologi al onstant today is 122 orders of magnitude, but several other osmi quantities. One therefore should not be puzzled by the smallness of the osmologi al onstant, but by the whole other osmi quantities as well. This is a manifestation of a osmi quantization of our universe at all levels. It therefore very natural to observe these hierar hies in our physi al world. I think be ause of these hierar hies our universe is unique, and without them we might not have a universe lasting for 10 - 15 billion of years! The ele tri osmi harge density is given by

The surfa e tension of a osmi medium is given by

=

P

The magneti (ele tri ) eld ontribution to mass density is given by m

G

The pressure exerted by va uum(quantum) at the present time is given by

The osmi magneti ux density is given by 

 3

This implies the universe developed its entire mass during a time of 1018 se : We therefore see that the universe appears the way it is, be ause it is a highly

onstrained system. The surfa e mass density of the whole Universe at the present time is de ned as

The osmi ele tri eld intensity is given by 

=

a

(7.11)

=



7 Gh

 12

 10

10

m=s2 :

(7.17)

We therefore expe t all obje ts to have experien ed a uniform a

eleration due to expansion of the osmi uid lling the whole universe. Thus, every obje t will experien e this a

eleration as far as it oats on spa e-time. However, su h an a

eleration is observed in Casimir experiments and re ently observed by Pioneer satellite. The diusion

onstant for the osmi domain is given by

This de nes the maximal mass that an be pla ed in gravitational eld. Thus the maximal mass whi h an be pla ed over a distan e of 1026 m is 1053kg. Therefore, the mass of the universe we observe toady is the only possible mass that the universe an hold. 9

Arbab I. Arbab D

=



Afri an Journal Of Mathemati al Physi s Vol 2 No 1 (2005)1-10

Gh

 12

 1034 m2 =s :

the present osmi quantities are related by the Plan kian ones by a fa tor that depends on Plan k

onstants of the two systems. This fa tor takes into a

ount the smallness and the vastness of the atomi and osmi realms when ompared to ea h other. However, sin e GP = G0 (Arbab, 2001b), one de nes this fa tor as:

(7.18)

Therefore, during the osmi time the diused area of the osmos onstituents is 1052m2 . This is a typi al area of osmi size. It has been emphasized by Kozlowska and Kozlowski (2003) that as time goes one the universe be omes more and more quantum on large s ale by allowing h ! 1. They

on luded that the prevailing thermal pro ess for thermal phenomena in the universe (that taking pla e on large s ale) is the diusion. One an de ne a oeÆ ient of vis osity for the osmi medium as 

=



9 G3 h

 21

 109 Ns=m2 :

N

  T 

:

=

 h  h



 1061

(8.1)

;

so that the mass, density, a

eleration and pressure of the universe are   1  ; (8.2) M = (N ) M ;  = 0

(7.19)

0

P

 

1 a0 = N

This means that today the osmos are moving freely and that the ideal uid approximation is valid for the present era. We however, see that the vis osity oeÆ ient for the nu lear medium and

osmos are the same. This implies that the quantity G3 h = G3N h is onserved. Thus the universe at the very large and the very small s ales is governed by the same rules and shows a similarity as regards to its surfa e mass density, surfa e tension and vis osity. This is be ause we have the interrelations: = ,  = S . One also an write, using eqs.(7), (37) and (73), the relation that D=

s

N2

aP ;

P0

=



P

1



PP ;

N2

and; the radius, age, vis osity and ele tri eld of the universe are R = (N ) RP ;

t0

P ;

E0

 

1 0 = N

= (N ) tP ; = 1 E N

(8.3) P

:

Hen e, the present osmi quantities are big(or small) when measured in Plan kian units. The main reason behind this is that the universe is a very quantum (restri ted) system. We observe that most of the present quantities are 120 orders of magnitude smaller than their Plan kian ounterparts. There must be some onspira y between the fundamental onstants that an maintain a riti al universe at all times. We therefore, argue that the physi al laws that govern the universe at its birth are still lingering behind.

(7.20)

This shows that the diusion onstant is inversely proportional to vis osity oeÆ ient. Using eqs.(38), (7), (19) and (37), eq.(116) yields  2  

a ; (7.21) = a and

= G G whi h would mean that spa e time a

elerates faster in a more vis ous medium than a less one. The vis osity of the universe at Plan k's time was very enormous (1070 Ns=m2 ) due to existen e of so many intera ting parti les whi h mimi s a vis ous

ow. Sin e the vis osity of the present universe is very small (109 Ns=m2 ), whi h is 61 orders of magnitude smaller than its Plan k value, one would expe t that this phenomena had played a great role in bringing the homogeneous and isotropi universe we ome to observe now.

IX. REFERENCES

Agob, M. et al., Aust. J. Phy. V.51,9, 1998. Arbab, A.I., Gen. Rel. Gravit. V.36, Nr.11, 2465, 2004. Arbab, A.I., Spa etime & Substan e, V.2, 55, 2001a. Arbab, A.I., Spa etime & Substan e, V.2, 51, 2001b. DerSarkissian, M., Lett. Nouvo Cimento, V.40, 390, 1984. Co ke, W.J., Astrophys. J. Lett. V.23, 239, 1983. Kozlowska, J.M., and Kozlowski, M., http://lanl.arxiv.org/abs/astro-ph/0307168.

VIII. THE HIERARCHICAL UNIVERSE

We see from our analysis that the hierar hi al stru ture of our universe is due to that fa t that 10