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Volume 3 PROGRESS IN PHYSICS October, 2005

Relations Between Physical Constants

Roberto Oros di Bartini

This article discusses the main analytic relationship between physical constants, and

applications thereo to cosmology. The mathematical bases herein are group theoretical

methods and topological methods. From this it is argued that the Universe was born

rom an Inversion Explosion o the primordial particle (pre-particle) whose outer radius

was that o the classical electron, and inner radius was that o the gravitational radius

o the electron. All the mass was concentrated in the space between the radii, and was

inverted outside the particle through the pre-particles surace (the inversion classical

radius). This inversion process continues today, determining evolutionary changes in

the undamental physical constants.

Roberto di Bartini, 1920s(in Italian Air Force uniform)

As is well known, group theor-

etical methods, and also topolog-

ical methods, can be eectivelyemployed in order to interpret

physical problems. We know o

studies setting up the discrete in-

terior o space-time, and also rel-

ationships between atomic quant-

ities and cosmological quantities.

However, no analytic relati-

onship between undamental phy-

sical quantities has been ound.

They are determined only by ex-

perimental means, because there

is no theory that could give a the-oretical determination o them.

In this brie article we give the results o our own study,

which, employing group theoretical methods and topological

methods, gives an analytic relationship between physical

constants.

Let us consider a predicative unbounded and hence

unique specimen A. Establishing an identity between thisspecimen A and itsel

A A , A1

A= 1 ,

Brie contents o this paper was presented by Pro. Bruno Pontecorvo

to the Proceedings o the Academy o Sciences o the USSR (Doklady Acad.Sci. USSR), where it was published in 1965 [19]. Roberto di Bartini (1897

1974), the author, was an Italian mathematician and aircrat engineer who,

rom 1923, worked in the USSR where he headed an aircrat project bureau.

Because di Bartini attached great importance to this article, he signed it

with his ull name, including his titular prefx and baronial name Oros

rom Orosti, the patrimony near Fiume (now Rijeka, located in Croatian

territory near the border), although he regularly signed papers as Roberto

Bartini. The limited space in the Proceedings did not permit publication o

the whole article. For this reason Pontecorvo acquainted di Bartini with Pro.

Kyril Stanyukovich, who published this article in his bulletin, in Russian.

Pontecorvo and Stanyukovich regarded di Bartinis paper highly. Decades

later Stanyukovich suggested that it would be a good idea to publish di

Bartinis article in English, because o the great importance o his idea o

applying topological methods to cosmology and the results he obtained.

(Translated by D. Rabounski and S. J. Crothers.) Editors remark.

is the mapping which transers images o A in accordancewith the pre-image o A.

The specimen A, by defnition, can be associated onlywith itsel. For this reason its inner mapping can, accord-

ing to Stoilows theorem, be represented as the superposition

o a topological mapping and subsequently by an analytic

mapping.

The population o images o A is a point-containingsystem, whose elements are equivalent points; an n-dimen-sional afne spread, containing (n + 1)-elements o the sys-tem, transorms into itsel in linear manner

xi =n+1

k=1aikxk .

With all aik real numbers, the unitary transormationk

aikalk =k

akiakl , i, k = 1, 2, 3 . . . , n + 1 ,

is orthogonal, because det aik =1. Hence, this transorm-ation is rotational or, in other words, an inversion twist.

A projective space, containing a population o all images

o the object A, can be metrizable. The metric spread Rn

(coinciding completely with the projective spread) is closed,

according to Hamels theorem.

A coincidence group o points, drawing elements o the

set o images o the object A, is a fnite symmetric system,which can be considered as a topological spread mapped

into the spherical space Rn. The surace o an (n + 1)-dimensional sphere, being equivalent to the volume o an

n-dimensional torus, is completely and everywhere denselyflled by the n-dimensional excellent, closed and fnite point-containing system o images o the object A.

The dimension o the spread Rn, which consists only othe set o elements o the system, can be any integer n insidethe interval (1N) to (N 1) where N is the number oentities in the ensemble.

We are going to consider sequences o stochastic transit-

ions between dierent dimension spreads as stochastic vector

34 R. Oros di Bartini. Relations Between Physical Constants

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October, 2005 PROGRESS IN PHYSICS Volume 3

quantities, i. e. as felds. Then, given a distribution unction

or requencies o the stochastic transitions dependent on n,we can fnd the most probable number o the dimension o

the ensemble in the ollowing way.

Let the dierential unction o distribution o requencies in the spectra o the transitions be given by

() = n exp[2] .

I n 1, the mathematical expectation or the requencyo a transition rom a state n is equal to

m() =

0

n exp[2] d

2

0

exp[2]d

=n + 1

2

2

n+1

2

.

The statistical weight o the time duration or a givenstate is a quantity inversely proportional to the probability o

this state to be changed. For this reason the most probable

dimension o the ensemble is that number n under which theunction m() has its minimum.

The inverse unction o m(), is

n =1

m()= S(n+1) = TVn ,

where the unction n is isomorphic to the unction o thesuraces value S(n+1) o a unit radius hypersphere locatedin an (n+1)-dimensional space (this value is equal to the

volume o an n-dimensional hypertorus). This isomorphismis adequate or the ergodic concept, according to which the

spatial and time spreads are equivalent aspects o a maniold.

So, this isomorphism shows that realization o the object Aas a confguration (a orm o its real existence) proceeds rom

the objective probability o the existence o this orm.

The positive branch o the unction n is unimodal;or negative values o (n + 1) this unction becomes sign-alternating (see the fgure).

The ormation takes its maximum length when n =6,hence the most probable and most unprobable extremal dis-

tributions o primary images o the object A are presented inthe 6-dimensional closed confguration: the existence o the

total specimen A we are considering is 6-dimensional.Closure o this confguration is expressed by the fnitude

o the volume o the states, and also the symmetry o distrib-

ution inside the volume.

Any even-dimensional space can be considered as the

product o two odd-dimensional spreads, which, having the

same odd-dimension and the opposite directions, are emb-

edded within each other. Any spherical ormation o n di-mensions is directed in spaces o (n + 1) and higher dim-ensions. Any odd-dimensional projective space, i immersed

in its own dimensions, becomes directed, while any even-

dimensional projective space is one-sided. Thus the orm

o the real existence o the object A we are considering is a(3 + 3)-dimensional complex ormation, which is the producto the 3-dimensional spatial-like and 3-dimensional time-like

spreads (each o them has its own direction in the (3 + 3)-

dimensional complex ormation).

One o the main concepts in dimension theory and combi-

natorial topology is nerve. Using this term, we come to the

statement that any compact metric space o n dimensionscan be mapped homeomorphicly into a subset located in a

Euclidean space o (2n + 1) dimensions. And conversely, any

compact metric space o (2n + 1) dimensions can be mappedhomeomorphicly way into a subset o n dimensions. Thereis a unique correspondence between the mapping 7 3and the mapping 3 7, which consists o the geometricalrealization o the abstract complex A.

The geometry o the aorementioned maniolds is determ-

ined by their own metrics, which, being set up inside them,

determines the quadratic interval

s2 = 2n

nik

gik xixk, i, k = 1, 2, . . . , n ,

which depends not only on the unction gik o coordinates iand k, but also on the unction o the number o independentparameters n.

The total length o a maniold is fnite and constant,

hence the sum o the lengths o all ormations, realized in the

maniold, is a quantity invariant with respect to orthogonal

transormations. Invariance o the total length o the orm-

ation is expressed by the quadratic orm

Nir2i = Nkr

2k ,

where N is the number o entities, r is the radial equivalento the ormation. From here we see, the ratio o the radii is

R. Oros di Bartini. Relations Between Physical Constants 35

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R

r2= 1 ,

where R is the largest radius; is the smallest radius, realisedin the area o the transormation; r is the radius o spherical

inversion o the ormation (this is the calibre o the area). Thetransormation areas are included in each other, the inversion

twist inside them is cascadedR r

2= Re ,

R = r,

r

2= e .

Negative-dimensional confgurations are inversion im-

ages, corresponding to anti-states o the system. They have

mirror symmetry i n = l(2m 1) and direct symmetry in =2(2m), where m = 1, 2, 3. Odd-dimensional confgurat-ions have no anti-states. The volume o the anti-states is

V(n) = 41Vn

.

Equations o physics take a simple orm i we use the LTkinematic system o units, whose units are two aspects l andt o the radius through which areas o the space Rn undergoinversion: l is the element o the spatial-like spread o thesubspace L, and t is the element o time-like spread o thesubspace T. Introducing homogeneous coordinates permitsreduction o projective geometry theorems to algebraic equi-

valents, and geometrical relations to kinematic relations.

The kinematic equivalent o the ormation corresponds

the ollowing model.

An elementary (3 + 3)-dimensional image o the objectA can be considered as a wave or a rotating oscillator,which, in turn, becomes the sink and source, produced by

the singularity o the transormation. There in the oscillator

polarization o the background components occurs the

transormation L T orT L, depending on the directiono the oscillator, which makes branching L and T spreads.The transmutation L T corresponds the shit o the feldvector at /2 in its parallel transer along closed arcs o radiiR and r in the afne coherence space Rn.

The eective abundance o the pole is

e =1

2

1

4 s Eds.A charge is an elementary oscillator, making a feld

around itsel and inside itsel. There in the feld a vectors

length depends only on the distance ri or 1/ri rom thecentre o the peculiarity. The inner feld is the inversion map

o the outer feld; the mutual correspondence between the

outer spatial-like and the inner time-like spreads leads to

torsion o the feld.

The product o the space o the spherical surace and

the strength in the surace is independent o ri; this valuedepends only on properties o the charge q

4q = SV = 4r2d2l

dt2.

Because the charge maniests in the spread Rn only asthe strength o its feld, and both parts o the equations are

equivalent, we can use the right side o the equation insteado the let one.

The feld vector takes its ultimate value

c =l

t=

SV

4ri= 1

in the surace o the inversion sphere with the radius r. Theultimate value o the feld strength lt2 takes a place in thesame surace; = t1 is the undamental requency o theoscillator. The eective (hal) product o the sphere surace

space and the oscillation acceleration equals the value o the

pulsating charge, hence

4q =1

24r2i

l

t= 2ric

2.

In LT kinematic system o units the dimension o acharge (both gravitational and electric) is

dim m = dim e = L3T2.

In the kinematic system LT, exponents in structuralormulae o dimensions o all physical quantities, including

electromagnetic quantities, are integers.

Denoting the undamental ratio l/t as C, in the kinematicsystem LT we obtain the generalized structural ormula or

physical quantities

Dn = cTn,

where Dn is the dimensional volume o a given physicalquantity, n is the sum o exponents in the ormula odimensions (see above), T is the radical o dimensions, nand are integers.

Thus we calculate dimensions o physical quantities in

the kinematic LT system o units (see Table 1).Physical constants are expressed by some relations in

the geometry o the ensemble, reduced to kinematic struc-

tures. The kinematic structures are aspects o the probability

and confguration realization o the abstract complex A. The

most stable orm o a kinematic state corresponds to the mostprobable orm o the stochastic existence o the ormation.

The value o any physical constant can be obtained in the

ollowing way.

The maximum value o the probability o the state we

are considering is the same as the volume o a 6-dimensional

torus,

V6 =163

15r3 = 33.0733588 r6.

The extreme numerical values the maximum o the

positive branch and the minimum o the negative branches

o the unction n are collected in Table 2.


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Table 1

Quantity Dn, taken under equal to:

Parameter n 5 4 3 2 1 0 1 2

C5Tn5 C4Tn4 C3Tn3 C2Tn2 C1Tn1 C0Tn0 C1Tn+1 C2Tn+2

Surace power L3T5

Pressure L2T4

Current density 2 L1T3

Mass density, angular

accelerationL0T2

Volume charge density L1T1

Electromagnetic feld

strengthL2T3

Magnetic displacement,

acceleration1 L1T2

Frequency L0T1

Power L5T5

Force L4T4

Current, loss mass L3T3

Potential dierence 0 L2T2

Velocity L1T1

Dimensionless constants L0T0

Conductivity L1T1

Magnetic permittivity L2T2

Force momentum, energy L5T4

Motion quantity, impulse L4T3

Mass, quantity o mag-

netism or electricity+1 L3T2

Two-dimensional

abundanceL2T1

Length, capacity, sel-

inductionL1T0

Period, duration L0T1

Angular momentum,

actionL5T3

Magnetic momentum L4T2

Loss volume +2 L3T1

Surace L2T0

L1T1

L0T2

Moment o inertia L5T2

L4T1

Volume o space +3 L3T0

Volume o time L0T3


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Table 2

n + 1 +7.256946404 4.99128410

Sn+1 +33.161194485 0.1209542108

The ratio between the ultimate values o the unction

Sn+1 is

E =

+S(n+1)max

S(n+1)min

= 274.163208 r12.On the other hand, a fnite length o a spherical layer

o Rn, homogeneously and everywhere densely flled bydoublets o the elementary ormations A, is equivalent to avortical torus, concentric with the spherical layer. The mirror

image o the layer is another concentric homogeneous double

layer, which, in turn, is equivalent to a vortical torus coaxial

with the frst one. Such ormations were studied by Lewisand Larmore or the (3 + 1)-dimensional case.

Conditions o stationary vortical motion are realized i

V rotV = grad , 2vds = d ,

where is the potential o the circulation, is the mainkinematic invariant o the feld. A vortical motion is stable

only i the current lines coincide with the trajectory o the

vortex core. For a (3 + 1)-dimensional vortical torus we have

Vx =

2D

ln

4D

r

1

4

,

where r is the radius o the circulation, D is the torusdiameter.

The velocity at the centre o the ormation is

V =uD

2r.

The condition Vx = V, in the case we are considering,is true i n = 7

ln4D

r= (2 + 0.25014803)

2n + 1

2n=

= 2 + 0.25014803 +n

2n + 1= 7 ,

D

r= E =

1

4e7 = 274.15836 .

In the feld o a vortical torus, with Bohr radius o the

charge, r = 0.999 9028, the quantity takes the numericalvalue = 0.999 9514 . So E= 14e

6.9996968 =274.074996.In the LT kinematic system o units, and introducing therelation B = V6E/ = 2885.3453, we express values o allconstants by prime relations between E and B

K = E B,where is equal to a quantized turn, and are integers.

Table 3 gives numerical values o physical constants, ob-

tained analytically and experimentally. The appendix gives

experimental determinations in units o the CGS system (cm,

gramme, sec), because they are conventional quantities, not

physical constants.The act that the theoretically and experimentally obtain-

ed values o physical constants coincide permits us to suppo-

se that all metric properties o the considered total and unique

specimen A can be identifed as properties o our observedWorld, so the World is identical to the unique particle A.In another paper it will be shown that a (3 + 3)-dimensionalstructure o space-time can be proven in an experimental

way, and also that this 6-dimensional model is ree o logical

difculties derived rom the (3 + 1)-dimensional concept othe space-time background.

In the system o units we are using here the gravitational

constant is

= 14 l0

t0 .

I we convert its dimensions back to the CGS system, so

that G =

l3

mt2

, appropriate numerical values o the physic-

al quantities will be determined in another orm (Column 5 in

Table 3). Reduced physical quantities are given in Column 8.

Column 9 gives evolutionary changes o the physical quanti-

ties with time according to the theory, developed by Stanyu-

kovich [17].

The gravitational constant, according to his theory,

increases proportionally to the space radius (and also the

world-time) and the number o elementary entities, accordingto Dirac [18], increases proportional to the square o the

space radius (and the square o world-time as well). There-

ore we obtain N = T2mB24, hence BT

112m .

Because Tm = t00 1040, where t0 10

17 sec is the

space age o our Universe and 0 =c = 10

23 sec1 is the

requency o elementary interactions, we obtain B 10103 =

= 1013 1000.

In this case we obtain m e2 T2m B24, which

is in good agreement with the evolution concept developed

by Stanyukovich.

Appendix

Here is a determination o the quantity 1 cm in the CGS

system o units. The analytic value o Rydberg constant is

Roberto di Bartini died beore he prepared the second paper. He died

sitting at his desk, looking at papers with drawings o vortical tori and drat

ormulae. According to Proessor Stanyukovich, Bartini was not in the habit

o keeping many drats, so unortunately, we do not know anything about

the experimental statement that he planned to provide as the proo to his

concept o the (3+3)-dimensional space-time background. D. R.Stanyukovichs theory is given in Part II o his book [17]. Here T

0m

is the world-time moment when a particle (electron, nucleon, etc.) was born,

Tm

is the world-time moment when we observe the particle. D. R.


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Table3

Parameter

Notation

Str

uctural

fo

rmula

K=

EB

Analyticallyobtainednumericalvalues

Observednumericalvalues

inCGS-system

Structural

formula

Dependence

ontime

LT-systemofunits

CGS-system

inCGS

Sommerfeldconstant

1/

1

/2E

2

1

0E0B0

1.3

70375

10

2

l0t0

1.3

70375

10

2

1.3

70374

10

2cm

0gm

0sec0

1 2

E

const

Gravitationalconstant

1/

4F

2

2

1E0B0

7.9

86889

10

2

l0t0

6.6

70024

10

8

6.6

70

10

8cm

3gm

1sec

1

TmT

0m

Fundamentalvelocity

c

l/t

20

0E0B0

1.0

00000

10

0

l1t1

2.9

97930

10

10

2

.997930

10

10cm

1gm

0sec

1

C

const

Massbasicratio

n/m

2

B/

21

1E0B1

1.8

36867

10

3

l0t0

1.8

36867

10

3

1.8

3630

10

3cm

0gm

0sec

0

n m

n mTmT

0m

1

12

Chargebasicratio

e/m

B6

20

0E0B6

5.7

70146

10

20

l0t0

5.2

73048

10

17

5

.273058

10

17cm

2 3

gm

2sec

1 2

em

em

TmT

0m

1 2

Gravitationalradius

ofelectron

r/2B12

2

1

1E0B

12

4.7

802045

10

43

l1t0

1.3

46990

10

55

1.3

48

10

55cm

1gm

0sec

0

S

const

Electricradius

ofelectron

e

r/

2B6

2

1

1E0B

6

2.7

53248

10

21

l1t0

7.7

72329

10

35

Se

Se

T

0mTm

1 2

Classicalradius

ofinversion

r

R

20

0E0B0

1.0

00000

10

0

l1t0

2.8

17850

10

13

2

.817850

10

13cm

1gm

0sec

0

r

const

Spaceradius

R

2

B12r

21

1E0B12

2.0

91961

10

42

l1t0

5.8

94831

10

29

10

29

>10

28cm

1gm

0sec

0

R

RTmT

0m

Electronmass

m

2

c2

20

0E0B

12

3.0

03491

10

42

l3t2

9.1

08300

10

28

9.1

083

10

28cm

0gm

1sec

0

m

m

T0mTm

Nucleonmass

n

2rc2/B11

21

1E0B

11

5.5

17016

10

39

l3t2

1.6

73074

10

24

1.6

7239

10

24cm

0gm

1sec

0

n

nT

0mTm

11

12

Electroncharge

e

2ec2

20

0E0B

6

1.7

33058

10

21

l3t2

4.8

02850

10

10

4.

80286

10

10cm

3 2

gm

1 2

sec

1

e

eT

0mTm

1 2

Spacemass

M

2Rc2

22

2E0B12

1.3

14417

10

43

l3t2

3.9

86064

10

57

10

57

>10

56cm

0gm

1sec

0

M

M

T0mTm

Spaceperiod

T

2

B12t

21

1E0B12

2.0

91961

10

42

l0t1

1.9

66300

10

19

10

19

>10

17cm

0gm

0sec

1

T

TT

0mTm

Spacedensity

k

M/

2

2R3

2

2

3E0B

24

7.2

73495

10

86

l0t2

9.8

58261

10

34

10

31cm

3gm

1sec

0

k

k

T

0mTm

2

Spaceaction

H

M

c2R

24

4E0B24

1.7

27694

10

86

l5t3

4.4

26057

10

98

cm

2gm

1sec

1

H

const

Numberofactual

entities

N

R/

22

2E0B24

4.3

76299

10

84

l0t0

4.3

76299

10

84

>10

82cm

0gm

0sec0

N

N

T2m

T2 0m

Numberofprimary

interactions

A

NT

23

3E0B36

9.1

55046

10

126

l0t0

9.1

55046

10

126

cm

0gm

0sec

0

NT

NMTmT

0m

3

Planckconstant

m

cEr

20

1E1B

12

2.5

86100

10

39

l5t3

6.6

25152

10

27

6

.62517

10

27cm

2gm

1sec

1

T

0mTm

Bohrmagneton

b

Er2

c2/4B6

2

2

0E1B

6

1.1

87469

10

19

l4t2

9.2

73128

10

21

9

.2734

10

21cm

5 2

gm

1 2

sec

1

T

0mTm

1 2

Comptonfrequency

c

c/

2Er

2

1

1E

1B0

5.8

06987

10

4

l0t1

6.1

78094

10

19

6.1

781

10

19cm

0gm

0sec

1

c

const

F=

E/(E1)=

1.0

03662


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[R] = (1/4E3)l1 = 3.0922328108l1, the experime-

ntally obtained value o the constant is (R)=109737.3110.012cm1. Hence 1 cm is determined in the CGS system

as (R)/[R] = 3.54880411012l.

Here is a determination o the quantity 1 sec in the CGSsystem o units. The analytic value o the undamental ve-

locity is [ c ] = l/t = 1, the experimentally obtained valueo the velocity o light in vacuum is (c) = 2.997930 0.00000801010cmsec1. Hence 1 sec is determined in

the CGS system as (c)/l [c ] = 1.06390661023t.Here is a determination o the quantity 1 gramme in the

CGS system o units. The analytic value o the ratio e/mc is[e/mc ] = B6 = 5.77014601020l1t. This quantity, mea-sured in experiments, is (e/mc)=1.7588970.000032107

(cmgm1)12 . Hence 1 gramme is determined in the CGS

system as(e/mc)2

l[ e/mc ]

2= 3.2975325101015l3t2, so CGS

one gramme is 1 gm (CGS)= 8.351217107cm3sec2 (CS).

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