(gravity theory, general and special relativity theory, quantum theory), one can construct ... G p. 2 = ; M kg p = 218 1
TOWARD QUANTUM COSMOLOGY Babic Ladislav V.Nazora 2, 40000 Cakovec Croatia
ABSTRACT In this work I propose a simple model of universe which unifies micro and macrocosmic scale of nature, on the basis of quantum - relativistic principles, into unique evolutionary process; from the seed - quassielementary particle PLANCKON as the most early stage of cosmic evolution, up to fruit - universe with current characteristics which surrounds us. Model is compatible with standard model of universe. 1. INTRODUCTION The idea of unity of micro and macrospace is common today between both cosmologists and quantum physicists. At the time of the “origin” (however we understand it), which happens at the subatomic dimensions, theory of relativity (c) and gravity (G) meets quantum effects ( ). There is idea that at those and at even smaller, so called Planckonian dimensions, known physical laws cease to be valid. From the other side, today’s cosmological theories commence to contemplate development of the universe out of singularity - state of infinite density and infinitesimal dimensions. Considering it is needed to eliminate infinities in theory either by technical (theoretical) or observational means (let us say c=const.). I commenced my contemplation out of the state determined by Planckonian parameters ( P , Tp, Mp, p, Ep, c). Besides the idea about the unity of nature I have been led by the idea about its essential simplicity, and by that its intelligibility. 2. PLANCKON It is known that from the fundamental physical constants (G, c, h), which characterise not only important particular natural effects than whole parts of physics (gravity theory, general and special relativity theory, quantum theory), one can construct a certain physical magnitudes which numerical values represent (as we look upon) boundary values or convergence points of before mentioned theories from the wider point of view encountering points of macro and microspace. Some of these are: 2p = Tp =
G c3
p c
;
;
p = 161 . 10 35 m Tp = 5.38 10 44 s
Planck length
(1a)
Planck time
(1b)
1
M p2 =
c ; M p = 2.18 10 8 kg G
Planck mass
(1c)
Planck density
(1d)
kg ) m3
(1e)
and deduced magnitudes like:
p =
Mp 3p
; p = 5.2 10 96
(we´ll define it as p =
pE = p c 2
p =
1 Tp
Mp 4 3 3 p
; p = 1.86 10 43 Hz
M pc2 k
;
p = 12 . 10 96
Planck energy density
E p = Mc 2 = p
p =
kg m3
(1f)
Planck circular frequency
. 10 9 J ; E p = 196
. 10 32 K ; p = 14
Planck energy
Planck temperature
(1g)
(1h)
(1i)
We´ll define a hypothetical elementary particle, named PLANCKON, which is characterised by listed (and not listed) physical values. For further contemplation, some characteristic properties of planckon are important. a) Radius of planckon can not be minor than . Let us presume that position and impulse uncertainties of planckon are given respectively by: 1/ 2 G x = = 3 , c c p = M p v = v G If we apply approximate uncertainty relations
1/ 2
.
xp , we can write: xp =
hence,
v, c
v c.
2
Accordingly, planckon can not be “placed” (can’t be smaller) at dimensions minor than , for otherwise velocities of motion could be larger than c. b) Gravity energy of planckon equalises its proper energy (we can prove it easily by relations (1))
E gp Ep
=
GM p2 p
= 1 .
M pc2
c) Planckon is a mini black hole. From the previous property follows Schwarzschield radius RSCH =
2GM p c2
p =
GM p c2
, which is smaller than
for the same mass.
d) Total energy of planckon, as the sum of gravitational and proper energy is equal to zero (property follows from b)) U = E gp + E p = 0 .
(2)
On that basis we'll consider that all other forms of energy (except gravitational) are “closed” within dimensions of planckon constituting its (part) inner energy. 3. EXPANDING OF THE UNIVERSE (standard model) Yet since the Hubble time (Doppler effect) observations of galaxies based by general relativity theory are showing a simple truth - galaxies are getting away from each other with relative speeds which are proportional to their mutual distance: v = Hr
(3)
Parameter H is usually called Hubble constant (though its value is decreasing during the expansion of universe) and measured values are H = (40 80)
km . s Mpc
Reciprocal value of this parameter (for H=80 km/s.Mpc) 1 3.9 1017 s = 12.2 10 9 year , H is called Hubble time and represents characteristic time of universe expansion. Age of the universe is something smaller: TH =
TS = TH
1/2 energy dominates = 2/3 matter dominates
3
Hence we can write:
HTS = .
(4)
As velocities larger than the speed of light are forbidden, out of (3) we get: TH =
RS , c
(5)
where RS is magnitude called radius of universe. Hence it is clear why TH is called characteristic time. This would have been age of the universe if it had expanded uniformly at the speed of light at all times. Future of the universe (for ever expanding - open universe, boundary case - flat universe, big crunch - closed universe) depends on the ratio of its mean density and so called critical density K . This one is in direct connection with Hubble parameter
8 GK . 3 Expression is relativisticaly valid if we interpret K as total equivalent energy density. Results of current measurements are piled up around: H=
= 3 10 28
(6)
kg m3
for mean, and around:
K = 12 . 10 26
kg , m3
for critical density of the universe. There are three alternatives that emerge from questions if total energy of the universe is U0. These are: a) b) c)
< K > K = K
universe will expand for ever (U > 0) contraction will replace expansion (U < 0) boundary case (U = 0)
Current observations are pointing at model a) while more and more theoreticians are tending to case c). Theory of the “big bang” (further - standard model) presents following orders for basic characteristics of the universe: age TS 20 10 9 year radius RS 20 10 9 l. y. mass M S 1053 kg We shall not consider details of standard (and inflatory) model.
4
4. OBSERVATIONS Let us designate with p , Tp , M p , p radius, characteristic time, mass and density of planckon and with RS , TH , M S , K corresponding dimensions that characterise universe. Let us make following relations:
p K
RS 10 61 , p
(7a)
TS 10 61 , Tp
(7b)
MS 10 61 , Mp
(7c)
10122 = (10 61 ) . 2
(7d)
This is enough to connect characteristics of hypothetical elementary particle PLANCKON with characteristics of real universe, forcing us to build following simple model: 5. PLANCKONIAN MODEL OF THE UNIVERSE On the basis of preliminary contemplation, presumed unity of nature which realises in the unity of micro and macrocosmos, and necessity that every unified theory (model) satisfy as relativistic as quantum principles, we make following presumptions: 5.1 QUANTUM CONDITIONS a) Along the edge of the universe it is possibly to arrange only integer number of Compton wave lengths that corresponds to mass of planckon:
2 RS = N
p=
;
p
h . M pc
Hence,
R S = N p
N = 1,2,.... N natural number
From now on, reduced wavelength p =
(8)
p
we shall call radius of planckon or 2 Planck length. Temporary and approximately we shall use
N = 10 61 5
as present value of quantum number N. Furthermore, all characteristic dimensions we use to designate the universe will be denoted with index N - with eventual exceptions at continuity (N>>1) when they change almost continuously. Therefore instead of RS , TH , M S , K we write R N , TN , M N , N etc. Magnitudes belonging to the basic stale of the universe (N=1) will be alternatively - according to circumstances - denoted as p , Tp , M p , p respectively R1 , T1 , M 1 , 1 and corresponding. Accordingly, relations (8) will look as: R N = N p .
(8a)
b) By dividing relation (8a) with speed of light we get:
p RN =N . c c Since R N / c is actually Hubble time and p / c Planck time, we write: TN = NTp .
(9)
Actually, it is more accurate to define:
TN = from where
RN RN0 c
TN = (N N 0 )Tp .
(9a)
Elementary quantum time Tp will be called chronon. Contemporary
Hubble
time
(in
accordance
with
planckonian km . TN = 17 10 9 years responds to Hubble constant H N = 57.3 s Mpc
theory)
c) Let us postulate M N = NM p .
(10)
d) By multiplying (10) with c 2 we get that the relation of proper energies of universe and planckon is: E N = NE p .
(11)
e) Out of (10) follows: MN RN3 N . = 3 Mp p p
6
When we apply quantum condition for radius of universe we get:
p
N =
N2
.
(12)
Let us mention that by bringing quantum conditions in relation (6) of standard model, we get similar expression for critical density of universe:
p
N =
. 2N 2 Expression (12) follows out of law of energy conservation (see further) wherefrom we can see that this is critical density indeed. Numerically,
N = K = 12 . 10 26
kg . m3
f) Connection between gravitational energies of universe and planckon
E gN E gp
GM N2 RN N2 R N5 = = 2 5 =N, GM p2 p p p E gN = NE gp .
(13)
g) Let us determine how does intensity of gravitational field of universe transform along with presumption about independence of gravitational constant on time (see GM N further). Out of N = 2 follows: RN
G=
N R N2 MN
=
N N 2p Mp
.
For G not to depend on N, it has to be:
N =
p N
(14)
It turns out that out all of these quantum conditions we can comprehend planckon as the initial (basic) state (N=1) in discreet and quantified development of universe. 5.2 ENERGY CONSERVATION LAW To be in keeping with former conclusions as with relation (2), for total energy of planckon we write:
7
U N = E gN + E N = E gp + E p = U p = 0
(15)
As by this as by former selection of mean density of universe N = K (which directly emerges from preceding relation) one can see that our universe belongs to boundary case c). If we observe the universe in two states (characterised by quantum numbers N 1 and N 2 ) we may write: U N 2 U N1 = ( E gN 2 E gN1 ) + ( E N 2 E N1 ) = 0 . Hence, GM p2
( N 2 N1 ) M pc = ( N 2 N1 ) . p change of proper 2
energy of universe
(15a)
change of gravitational energy of universe
Universe is “indebted” at its gravitational energy - its proper energy increases on the account of decreasing the last one. Total energy is always zero, so there is no contradiction with energy conservation law. 5.3 ELECTRICAL CHARGE OF UNIVERSE (PLANCKON) From the conclusion that planckonian stage is just the initial stage of universe development and the fact that total charge of universe is zero, follows Qp = 0
(16)
Electrical charge of planckon is zero. 5.4 SPEED OF LIGHT AND GRAVITATIONAL CONSTANT a) Speed of light c=
N p p RN . = = TN NTp Tp
b) Gravitational constant G=
E gN R N
=
( E gp N )( N p )
=
E gp p
. ( NM p ) M M p2 Since c and G do not depend on N we will conclude that these constants are not changing an expansion of universe. Indeed, at derivation of relation (13) we implicitly presumed independence of gravitational constant on time. But, if (11) is valid, then out of energy conservation law (15) directly follows conclusion (13). 2 N
2
8
5.5 RELATIVISTIC EXPANSION Through quantum conditions, our model is directly built-in in quantum mechanics. But what is with relativity? As universe is at the state of expansion, we shall set issue in this way. Since we are participating in the expansion too, we shall observe occurrences from the “edge” of the universe, which is moving at the speed v. a) Change of the universe radius Let us presume that we are watching from the centre as planckon is expanding. If we measure radius p , for an observer from periphery it has to be larger (because lengths are contracted for the central observer):
p
RS =
.
v2 1 2 c
(17)
As the shock front of universe is moving at the speed close to the speed of light, the expression
1
=
v2 1 2 c
,
is surely large. Let us postulate:
=
1 v2 1 2 c
= N.
(18)
In that way relation (17) is formally taking over form of quantum condition (8). As from (18) we get
1 v N2 = c 2 1 2 , N
(19)
for large N 10 61 speed of expansion is practically light speed ( v c ). As we see, speed of expansion is changing discreetly and furthermore (with eventual exceptions at continuity) we denote it as v N .
b) An observer from periphery is perceiving universe with volume
p Vp 4 3 4 VS = RS = = = N 3V p , 3 3 v N2 3/ 2 v N2 3/ 2 1 2 1 2 c c 3
9
so, V N = N 3V p .
(20)
c) Characteristic (Hubble) expansion time Hubble time for an observer from the “edge” of universe is TH =
RS 1 = c c
TH =
then,
p v N2 1 2 c
Tp v N2 1 2 c
Tp
=
v N2 1 2 c
,
,
(21)
TN = NTp
So, age of the universe depends on how much its speed of expansion is smaller than speed of light. Velocities of expansion for N=1,2,3,4,10,100 are shown in TABLE 1. From the initial stage ( TH = 0 , "the beginning” of universe) universe is expanding at speed which is taking over discreet values of light speed. In advanced stages of evolution (N>>1) speed changes almost continually. Yet for nine - fold planckonian time ( TN = 9Tp ), speed of expansion amounts 99,5% of light speed. Because of matter inertia this is physically more convincing than that what quantum conditions implicate without relativistic treatment - namely, that speed of expansion is equal with speed of light since the beginning (although there is speed of expansion v comprised in expression for proper energy - relation (15)). We can compare universe in this initial stage with “overclocked” microprocessor at high frequency
p = 186 . 10 34 GHz , which is during increasing of its complexity, gradually decreasing as the result of its development: p N = N d) Mass of universe Peripheral observer can see that all parts expanding planckon are moving away from him and can establish increased mass
10
MS =
Mp
,
v N2 1 2 c
(22)
respectively: M N = NM p .
e) Density of matter in universe Some observer can establish density of universe:
Mp
S =
v N2 1 2 c Vp
MS = VS
v N2 1 2 c
, 3
respectively, v N2 S = p 1 2 , c
namely
N =
p N2
(23)
.
f) Proper energy of universe Observer from the shock front of expanding universe establishes its proper energy ES = M S c2 , ES =
M pc2 v N2 1 2 c
=
Ep v N2 1 2 c
,
(24)
so, E N = NE p .
g) Creation (limit) temperature To proper energy of planckon expressed as on equivalent thermal energy M p c 2 = k p ,
11
corresponds temperature
p = 14 . 10 32 K . At temperatures close to (larger than) this “limit temperature” creation of planckon is possible out of thermal radiation, respectively, at its annihilation (if we may call so “decay” of planckon) radiation temperature would be just the same. Likewise, peripheral observer can connect proper energy with thermal energy: M S c 2 = k S ,
wherefrom, by application of formula for relativistic mass transformation we get:
S = respectively,
p v N2 1 2 c
,
N = N p .
(25)
(25a)
This amounts N = 14 . 10 93 K . This result should be interpreted in this way; if all mass of present universe would turn to energy, resulting radiation would have temperature of 14 . 10 93 K . Indeed, from M S c 2 = k S ,
emerges exactly this temperature (for M S = 2.18 1053 kg ). h) Power Change of proper energy of universe (which is happening on the account of gravitational energy) per unit of time, we call power: PN =
E N (N 2 N 1 )E p . = TN (N 2 N 1 )Tp
Follows, P=
Ep Tp
= 3.65 1052 W .
(26)
Power does not depend on the age of universe (therefore, we left out index N). Every second 3.65 1052 J of decreased gravitational energy turn into increased proper energy of universe. In other words, every second additional mass is created from gravitational energy (let us put aside kinetic energy for a moment) M N =
P 35 kg . 2 = 4.1 10 s c
(27)
12
Today, this amounts
M N = 19 . 10 18 . MN
19 . 10 18 mass of our universe. In a space of a cubic metre, M N kg = 14 . 10 44 3 , VN m s of “new matter” (for R N = 17 10 9 l. y. ) originates per one second, which we can present like this: - every second in a box with an edge of 418 km, one proton is being created . 10 27 kg ) ( m p = 167 - in a cubic metre one proton is being created each 2.3 10 9 years - about 24 protons are being created in one second per one packet of planckonian mass ( M p ) (it is obvious that proton represents mass that is equivalent to it). While transformation power of proper energy does not depend on time, it is different with power per mass unit of universe: PN* =
Ep Ep 1 E N . = = M N TN M p NTp M p TN
(28)
It is inverse proportional to Hubble time.
. 10 60 "Before" (N=1) it was PN* = 17 . Today ( N = 10 61 ) it is PN* = 017
W . kg
W . kg
Difference is showing us that when universe was “young” evolutionary processes were unproportionally more intensive than at its mature age. Considering power and proper energy of universe, our model is showing how quantum generating of matter is equivalent with increase of mass, because of its relativistic motion. We see how connection between planckonian characteristic and corresponding characteristics of universe is being established by simple relativistic transformation. We believe that, up to now, considerations clearly show that our model is satisfying basic principles of quantum and relativistic theory. Illustration of some, up to now, stated issues and comparison of standard and planckonian model are presented in TABLE 2. Parameters of planckonian universe are clearly listed in TABLE 5.
13
6.FURTHER CONTEMPLATION a) Spin of universe On the basis relations (1a)-(1c) we can express Planck constant in the following way: 2p M p 1 R N2 M N . = = 2 Tp TN N Follows: L MN = N 2 L p = N 2 . (29) In the basic state (N=1) speed of expansion is zero, and respectively, it is the same with impulse of universe (planckon). This fact is not changing later for - because of symmetrical, radial expansion - impulses of diametrically opposite parts cancel each other. Therefore, we presume that expression (29) presents proper angular amount of movement - we shall call it mass spin of universe. Namely for the spin of universe we have to take into consideration spin connected to gravitational field. From the structure of expression it is obvious that “bare” planckon (without considering gravitational field) belongs to bosons. If we use classical relation: R N2 M N 2 L MN = I N N = M N R N N = , TN follows: 1 N = . TN As spin is structural characteristic of elementary particle, so is mass spin of universe characteristic magnitude connected to its expansion, and not to rotation around some axis. We see that mass spin of universe is not preserved. “Before” (N=1)
LM1 = Lp = .
Today ( N = 10 61 )
LMN = (10 61 ) 2 .
Let us look (slightly awkward) at spin connected with gravitational field of universe. LgN t
= FN R N =
GM N2 RN . R N2
Therefore:
LgN
NGM p2 GM N2 = t = N . RN p
For field spin we get:
LgN
N 2 GM p2 . = p Tp
As GM p2 = c and
p Tp
= c , we get:
14
LgN = N 2 .
(29a)
Accordingly, spin of gravitational field of universe is quantitatively equal and directionally opposite to mass spin of universe. Spin of universe is therefore a sum of there two spins: L N = L MN + LgN = N 2 N 2 = 0 ,
LN = 0
(29b)
Therefore, proper angular momentum of universe is a magnitude which value is maintained - at each moment it is equal to zero. And therefore, “dressed” planckon belongs to bosons (with spin equal to zero). However, how to explain that mass spin increases proportionally to N 2 ? Suppose that planckon with spin (“naked” planckon) on N-th level is “shattered” in N 2 parts, each with spin . As mass (and proper energy) of universe is not increased N 2 -fold but N-fold, should it be (?) concluded that out of N 2 particles, N of them do have, while N 2 N do not have still mass, even more - that these are virtual particles. For example, at the state with quantum number N=10, there are 90 virtual +10 mass =100 particles . Since, for small quantum numbers, universe is at the state where every particle interacts with each other, which requires that for each mass particle (planckonian mass conglomerate) we have N-1 interaction transmitters - total N ( N 1) = N 2 N particles. Let us mention that all interaction transmitters are bosons, which are at high temperatures (> 3 1015 K ) particles without still mass (TABLE 3). These presumed virtual particles as such cannot be directly identified in an experiment, however they contribute to the mass of planckon (universe). Following question is obvious: how does fermions originate - particles with half-fold spin (protons, neutrons, electrons, quarks,…)? By “eruption” of planckon (at 10 32 K ), considering that in its basic eV state it presents depot of enormous energy density of 6.75 10131 3 (twice as much m if we take into consideration gravity), these bearers of interaction and probably (?) X-bosons are being released first. By cooling of universe, originally unique interaction is decomposing. Above the temperature of creation - characteristic temperature for a particular particle (neutron 10.903 1012 K , proton 10.888 1012 K , electron 5.930 10 9 K ,…) - above which it can be created from thermal radiation - it is coming to copious creation of fermions and bosons - particles and antiparticles (in pairs, because of electrical charge conservation law) - which annihilate later on, until a certain surplus of not annihilated matter remains - constructive element of our universe. In phase of nucleon-synthesis there should have originated present relation of nuclear particles (barions) and photons, which is significant for further evolution of universe:
S=
number of photons 10 9 . number of barions
It is considered that present number of barions in the universe is of magnitude:
15
number of barions =
Mp MN =N 1080 . mp mp
Example: After how much time does the mass spin of universe change for a 1%? From (29) follows:
dL MN 2dN = = 0,01 . L MN N Therefore it is
dN = 0,005 . This change of N comprehends that universe is growing N
old,
dTN dN = = 0,005 , TN N respectively dTN = 8.5 10 7 years (for TN = 17 10 9 years). Note: When finding dL MN and dTN we were using continuity approximation. Namely, we can always change differences with differentials if dN1. b) Let us show that relation
1 v N2 = c 2 1 2 , N considering our quantum conditions, has a form of Hubble expansion law. Taking p into account that c = , we can transform this relation into: Tp
(
)
v N2 = R N2 2p H N2 ,
(30)
which has: 1) for R N >> p
v N = H N RN c , form of universe expansion law. 2) At early stages of evolution, when R N is comparable with p , shock front of universe is expanding with velocity which does not have a form of Hubble’s law. c) Mean speed of expansion
1 v N2 = c 2 1 2 . N
16
In continuity approximation (N>>1) we have,
N 2 1 dN . N
N
c N 1 1
vN = Because of
N 2 1 N
follows:
N
c vN = dN c , N 1 1 vN c .
(31)
Mean speed of universe expansion (later on) is neglectfully lower than speed of light. d) Total momentary acceleration of expansion, a N = a gN + a SN ,
is a resultant of an opposite directed vectors - deceleration which is caused by gravitation and acceleration which is caused by expansion of universe from the basic state. In continuity approximation it amounts: dv c (32) aN = N 3 . dTN N Tp Today it is of magnitude a N 10 131
m s2
therefore, absolutely neglectful. For N it tends to zero. Gravitational deceleration (see 5.1 g)) of expansion is given with a gN =
GM p N
2 p
=
c . NTp
Present value ( N = 10 61 ) amounts a gN = 5.6 10 10
v N = 0 ) was
(32a)
m while the initial value (N=1, s2
m . s2 We can get the total acceleration of expansion by deriving relation (3): a g1 = a gp = 5.6 1051
aS =
dv dr dH . =H +r dt dt dt
With help of (4) follows:
17
v2 1 aS = 1 . r
(32b)
As this is nothing else than (32) if we put r = R N and v c , we get 1 c c 1 = 3 . R N N Tp
Therefore,
N2 . N 2 1
N =
(32c)
In our model N >1, while today N 1 . For acceleration of expansion without the influence of gravitation follows: a SN = a N a gN ,
c c , + 3 N Tp NTp
a SN =
c , NTp hardly some more than value of gravitational deceleration. Mean values of accelerations are: a SN
(32d)
For acceleration without the influence of gravitation N
a SN
c c 1 = dN = ln N , N 1 1 NTp NTp
a SN
Today a SN 7.9 10 8
c ln N . Tp N
(32e)
m . s2
For gravitational deceleration
a gN = Today a gN = 7.9 10 8
c ln N . Tp N
(32f)
m . s2
Mean value of total acceleration
aN = 0 .
(32g)
18
Let us look more closely at first two “tacts” of expansion (N=1 and N=2). For N=1 ( v N = 0 ) acceleration of expansion is zero. For N=2: 1 1 3 c 1 2 c 1 2 c v N 2 1 4 , aN = = = 2Tp Tp TN Tp a N (1 2) = 4.8 1051
m . s2
This is clearly presented in TABLE 4. At the “beginning” (N=1) there is no expansion. Gravitation keeps planckon gathered with its enormous field. Total acceleration of expansion is zero. When expansion starts (because of extremely increased fluctuation of vacuum inside of planckon) initial acceleration of expansion exceeds gravitational deceleration by absolute value. Total acceleration expansion is larger than zero ( a N >0) which is, clearly, necessary for the expansion to start. Furthermore, gravitational acceleration is decreasing (by absolute value) faster than expansion acceleration so that resultant acceleration is always positive. e) Dimensions and future of the universe By knowing the expansion acceleration we can determine radius of the universe. Say, from gravitational deceleration c c2 m a gN = = = 5.6 10 10 2 , NTp NR N s one gets value for current “radius” of the universe: R N = 17 10 9 g. s. ,
(clearly, this follows directly from (8)). As total acceleration of expansion is hardly larger than a zero, it is following that universe will expand endlessly, what we have already concluded earlier from energy contemplation. Let us notice that all numerical values we get from the approximate value of quantum number N which is accepted as agreed upon. f) When did a mass of universe amount 1%0 of present mass? Let today’s mass of universe be M N = NM p while some earlier mass is M N0 = N 0 M p M N0 MN
=
N0 1 = . N 1000
19
Hence follows N 0 =
N = 1058 . Universe was then 1000 TN o N 1 = 0 = , TN N 1000
accordingly, TN , 1000 thousand times younger than today. For today’s age of the universe of approximately TN = 17 10 9 years , we get the age of TN 0 = 17 10 6 years . TN 0 =
7. UNCERTAINITY RELATIONS Let R1 = p and p1 = 0 (because v1 = 0 ) be uncertainties of radius and impulse of universe in its initial state (N=1). Consequently,
R1 p1 = 0 ,
(33)
uncertainty relations for radius and impulse (as conjugate variables) are not valid for universe as a whole in its initial state. For proper energy, meanwhile we have E1 = p , while uncertainty of characteristic time is T1 = Tp , so that E1 T1 = p Tp = ,
(33a)
for these two variables uncertainty relations are valid in initial (N=1) and in incited states. As total energy of universe is U N = 0 , it is
U N TN = 0 .
(33b)
For total energy of universe and its characteristic time of expansion, uncertainty relations are not valid. We can observe energy and temporal evolution of universe by use of relativistic laws: Uncertainties of proper energy and Hubble time in later phase of evolution (by use of continuity approximation) are:
( ),
E N = E N2 E N
respectively,
2
( ).
TN = TN2 TN
2
Since, N
2
( )
1 1 E = NE p dN N 2 E p2 , N 1 1 3 2 N
respectively, N
EN =
( )
1 1 NE p dN NE p , N 1 1 2
it follows,
20
3 NE p . 6 Similar, for uncertainty of characteristic time follows: E N =
TN =
3 NTp . 6
Hence follows:
N2 . 12 This product of uncertainties has a least possible value in initial state (N=1) , E N TN = 12 E N TN =
(33c)
which is not accurate (because continuity approximation is not valid for this domain) in truth, but therefore it is presented more accurate by relation (33a). As the “initial” impulse of universe (for v1 = 0 ) is equal to zero, and as it is because of symmetrical and radial expansion - the same furthermore, in later phases of evolution it is valid: R N p N = 0 . (33d) Thus follows that in our model universe is relativistic, semiquantum “object” whose evolution we can observe by using classical (relativistic) and quantum methods. 8. AGE OF THE UNIVERSE As speed of expansion is not speed of light, a question is set whether it is justified to identify characteristic time of expansion ( TN ) with age of the universe ( TS ). Let us look: 2
1 dR N = v N2 = c 2 1 2 . dt N Hence it is (in continuity approximation; N>>1): N
TS = Tp 1
which gives
dN 1 1 2 N
,
TS ( N 1)Tp TN
(34) so, identification is justified; completely in harmony with N 1 and relation (4). 9. SYNTHESIS By leading us with ideas about unity and simplicity of nature and mutual convergence of relativity theory (c), general relativity theory (gravitational theory, G) and quantum theory ( ) in the moments of birth of macrocosmos out of microcosmos, we pointed one simple (half?) quantum-relativistic model of universe. 21
Physical magnitudes which in the essence of this model, and out of which all physical magnitudes should be derived (clearly, in a theory much more complex than ours) are Planck magnitudes - primarily P , Tp, Mp. As our idea comprehends, it should not be searched for elementary particle named PLANCKON in the universe, considering that it does not exist as such, individually. Planckon actually represents the earliest phase in the evolution of universe; hence protouniverse. It is miniature black hole, which is kept from falling to pieces by extremely powerful gravitational forces. Therefore all forms of energy (except gravitational) are captured inside its volume and are (probably?) equal to energy of vacuum fluctuations - which forms proper energy of planckon. Total charge of early universe (planckon) is zero while its spin is zero too. In accordance with quantum statistic, planckon belongs to bosons. Unknown reason (of sudden extremely increased vacuum fluctuation?) is causing a destruction of its stable structure and acts as a detonator for the creation of universe it initialises limitless expansion of vacuum fluctuations - Hubble expansion of universe, releasing in this way until then confined energies, in a mini black hole. Relativistic expansion, with speed that is close to speed of light, (for 4.4 10 43 s speed of expansion already values 99.5% of light speed) is causing (on the account of indebtedness at gravitational energy) increase of mass and proper energy of universe. At the moment of planckon disintegration temperature is of magnitude 10 32 K and furthermore decreases in harmony with laws of radiation and interaction of radiation and matter. All physical magnitudes are changing discontinuously, in harmony with quantum conditions (alike Bohr - de Broglies´) but soon this discontinuous development (for small values of quantum number N) is crossing over in almost continuous change (for vast values of N; N>>1). Present age, radius and mass of the universe (so as other physical magnitudes) depend primarily upon how much does a present speed of expansion differ from the speed of light. In case that gravitation does not prevent expansion, these magnitudes tend to be infinite while density of kg universe - from initial planckonian density ( p = 12 . 10 96 3 ) tends toward zero. In m our model, total momentary acceleration of expansion is insignificantly larger than zero - universe is, although decelerated by activity of gravitational forces, expanding with average speed that is almost equal to speed of light, so that expansion will not turn to contraction in the future. Let us emphasise once more; values of all physical characteristics of difference - between speed of expansion and speed of light. Upon author’s opinion, this short work has two restrictions. First restriction is backwards - it says nothing about preplanckonian phase of the universe (neither does author know how to come up to that). In this way all speculations on that, if this is even possible, up to those that it is (are known physical laws valid then?), are still actual. Toward second restriction, optimistic relation is possible. All variety of postplanckonian interactions, including an explanation of whole spectre of elementary particles, decomposition of unique interactions on four known nowadays (weak, strong, electromagnetic and gravitational), synthesis of hydrogen and deuterium, creation and evolution of galaxies, …, is not considered in this work. But present level of knowledge and creative human potentials are unquestionably a match to it. We referred less attention to strict values of latest observational data than to some principle questions. Reader can surely retrieve that independently.
22
LITERATURE 1) S.Hawking, "A brief History of time", Bantam Press, London 1988. 2) C.W.Misner, K.S.Thorne, J.A.Wheeler, "Gravitation", W.H.Freeman&Co., San Francisko 1973. 3) I.Supek, "Teorijska fizika i struktura materije", kolska knjiga, Zagreb , 1992. 4) S.Weinberg, "Prve tri minute svemira", MISL, Zagreb, 1997.
23
TABLE 1. Velocity of universe expansion N
TH T p
1 2 3 4 10 100
0 1 2 3 9 99
v2 c2 0 34 9 10 15 16 99 100 9999 10000
vc 0 0.8660 0.9487 0.9682 0.9950 0.99995
TABLE 2. Characteristic time of universe expansion WEINBERG´S DATA 3
STANDARD MODEL
r/(kg/m ) 3.8. 1012 3.1. 1010 3.8. 108 99
TH /s 0.022 0.2 2 1h 15m
STANDARD MODEL +QUANTUM CONDITIONS N TN = N .TP . 41
2.8 10 3.1. 1042 2.8 .1043 5.5 .1046
PLANCKONIAN MODEL
/s
0.015 0.17 1.5 49.3m
N 5.6 . 1041 6.2 . 1042 5.6 .1043 1.1 .1047
TN = N .TP /s 0.03 0.33 3 1h 39m
Data for universe density and TH are taken from [4]. TABLE 3. Transmitters of interaction FORCE
INTERACTION TRANSMITOR
electromagnetic gravitational weak force strong force
SPIN / h
photon graviton W+, W -, Z0
1 2 1
gluon
1
STILL MASS (in masses of proton)
0 0 0 above 3.1015 K 80 -90 below3.1015 K 0
ELEKTRIC CHARGE
0 0 +, -, 0 0
TABLE 4. Acceleration of universe expansion N 1 2
agN / m/s2 - 5.6 .1051 - 2.8 .1051
aSN / m/s2 +5.6 .1051 +7.6 .1051
aN = agN + aSN / m/s2 0 +4.8 .1051
24
TABLE 5. Parameters of planckonian model of universe NAME QUANTUM NUMBER UNIVERSE RADIUS HUBBLE CONSTANT HUBBLE TIME AGE OF THE UNIVERSE MASS OF THE UNIVERSE CRITICAL DENSITY EXPANSION VELOCITY GRAVI. DECELERAT. EXPANSION ACCELERAT. TOTAL ACCELERAT. OF EXPANSION PROPER ENERGY
MARK
UNIT
EARLY UNIVERSE
N
MODERN UNIVERSE
2
1061
RN
m, l.y.
3.22. 10-35 m
17. 109 l.y.
HN
km/sMpc
5.74 .1062
57,3
TN
s, year
5.38. 10-44 s
17. 109 year
TS
s, year
5.38. 10-44 s
17. 109 year
MN
kg
4.36. 10-8
2.18 .1053
rN
kg/m3
3. 1095
1.2 .10-26
VN
c
0.866. c
c
agN
m/s2
- 2.8. 1051
- 5.6. 10-10
aSN
m/s2
7.6 .1051
5.6 .10-10
aN
m/s2
4.8 .1051
10-131
EN
J
3.92. 109
1.96. 1070
GRAV. ENERGY OF UNIVERSE TOTAL ENERG. OF UNIVERSE POWER
EgN
J
- 3.92. 109
- 1.96.1070
UN
J
0
0
PN
W
3.65. 1052
3.65. 1052
POWER PER MASS UNIT MASS SPIN OF UNIVERSE SPIN OF GRAV. FIELD OF UNIV. TOTAL SPIN OF UNIVERSE
PN*
W/kg
1.7 .1060
0.17
LMN
4
N2
LgN
-4
- N2
LN
0
0
Parameter values of modern universe are mostly approximate because of insufficient knowledge about current value of quantum number N.
25
CONTENTS
ABSTRACT
1
1. INTRODUCTION
1
2.PLANCKON
1
3.EXPANDING OF THE UNIVERSE
3
4.OBSERVATIONS
5
5.PLANCKONIAN MODEL OF THE UNIVERSE
5
- 5.1 QUANTUM CONDITIONS
5
- 5.2 ENERGY CONSERVATION LAW
7
- 5.3 ELECTRICAL CHARGE OF UNIVERSE - PLANCKON
8
- 5.4 SPEED OF LIGHT AND GRAVITATIONAL CONSTANT
8
- 5.5 RELATIVISTIC EXPANSION
9
6.FURTHER CONTEMPLATION
14
7.UNCERTAINTY RELATIONS
20
8.AGE OF THE UNIVERSE
21
9.SYNTHESIS
21
Literature
23
Tables
24
26
