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On the Origin and Nature
of
G'ravitational Fields
Julian M. Avery
March 1959
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Gravitational Fields
Table of Contents
troduction
nopsis
:iex to Part I - Follows Synopsis
General Relativity Theory and Einsteins Spherical Model
of the Universe
:iex to Part II - Follows Page 21
The Source of Gravitational FieldSand the
Interpretation of the Gravitational Constant In terms of
the Properties of Nucleons
lex to Part III - Follows Page 47
Practical Applications
ferences - Follows Page 66
Julian M. Avery
March 1959
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INTRODUCTION
The author has long believed that nucleons (protons and neutrons)
must be the ultimate source of the gravitational fields associated with
ponder able matter, and that an understanding of the nature and prope rtie!'
of gravitational fields should therefore be sought not only on the cosmo-logical
scale, but also by studying the physical properi.ie.; of nucleons. It
has also seemed apparent that attempts to develop a unified field theory
have been hampered rather than aided by the complexity of the mathematical
procedures which have been used. An example in point is Heisenberg's
recent equation which few scientists seem to understand and which no one
appears to be able to put to practical use.
In this paper the mathematics of tensor analysis and of quantum
mechanics have been stuc:> 1usly avoided in favor of the simplest possible
treatment from the mathematical point of view. The methodti used is
analogous to that u·-.ed by Bohr in his early presentat10n of atomic theory.
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have been very difficult if not impossible to develop the present-day
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mathematics of quantum mechanics. Even today when physicists wish
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to picture the: 'Eitom in concrete terms, they turn to Bohr's "mechanical .
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concepts set forth.
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SYNOPSIS
0 Part I of this paper is concerned with General Relativity Theory
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as applied particularly to Einstein's spherical model of the Universe. It
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is shown that if the numerical value of the gravitational constant is assumed
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0 to be identical with that of an acceleration which characterizes the general
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0 gi·avitational field of the Universe, acceptable ·.·;;.l~es for the mass, radius
0 •
0 and average density of matter in the Universe can be calculated wtthout
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0 reference to astronomical data or their interpretation.
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0 The suggestion is offered that the red- shift of light from distant
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0 galaxies is due not to a recessional velocity, but to a progressive loss of
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() energy by a photon as it traverses space due to an intrinsic property of
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() the general gravitational feHd.
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0 The concept of the curvature of space is then replaced by a rotational
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photon as it traverses space, and of the path of gravitational "lines of
force". The Universe as observed from any point of origin is then pictured
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of being concentrated in an ever-expanding spherical "skin" or shell. .An'1
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since this model of the spherical Universe is not peculiar to any specific
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0 point of origin, the concept is offered of a super- Universe which comprises
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0 an infinite number of individual Universes each phcnon1cnologically limited
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0 by the same radius of curvature from any point of origin, endlessly over-
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0 lapping one another throughout all space . In conclusion the suggcst1on 1s
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() offered that the general gravitational field which permeates all space 1s
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u therefore possess properties in commcn with eleLtromagnct1c fields.
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u may be tr cat e d ,1 s equ iv al en t.
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The inertial properties of the nucleon are then interpreted in terms
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0 mechanics theory, including the-magnetic moment of the nucleon.
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0 '!'he gravitational properties of the nucleon are then explored anrl_
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0 compared with its inertial properties, in terms of the fictive pole strength
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0 and radius, and of other fundamental constants. This leads to general
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the properties of the general gravitational field. This leads to a discussion
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and int,, rpretation of the "big number" which has c1.ttracted the attention of
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0 Synopsis - 4
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0 gravitational constant is then presented.
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0 Basis is provided for suggesting that the numerical value of the
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0 gravitational constant may be slightly in error and it is shown that such a
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0 an interpretation of the gravitational constant in terms of atomic constants
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together with the product of the dielectric constant and the constant of
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magnetic per me ability, which lertds farther support to the supposition that
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0 gravitational fields and electromagnetic fields have something in common.
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13 concepts and expressions set forth in Part I and Part 11.
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Synop,:;is - 5
Properties of the gravitational and inertial fields of the nucleon are
shown to be related to the properties of the general gravitational ·field. The
inertial mass of the nucleon (and that of ponderable matter) is in~er~reted.
The specific acceleration of the gravitational field of the nucleon ic; determined,
and the relation between the inertial and gravitational properties of matter
are discussed. The kinetic energy of the nucleon and of ponderable matte:..·
due to translatory motion are shown to be related to the properties of the
general gravitational field. In this connection, the product of the dielectric
c onstant and the constant of magnetic permeability are shown to play a
suggestive role.
The "preferred" radius Rh is pre!:ented as a kind of boundary
between nuclear and gravitational fields and phenomena. Characteristics
of the gravitational field of the nucleon, and the relation between its 1n-ertial
and gravitational fields are presented. It is suggested that basis
ha s been laic:I for the development of a unified field theory.
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The displacement of the path of a ray of light grazing the surface of
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the Sun is .o;hown to be due (the relativistic part of it) to the rotational
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PART I
General Relativity Theory
and
Einstein's Spherical Model of the Unive:se
A. Interpretation of the Gravitational Constant
Calculation of the Rad.ius, Mass an~ Density of the Universe
B . The Red Shift of Light from Distant Galaxies
Interpretation in Terms of Progressive Loss of Energy
Concept of the Expanding Universe Challenged
C. The Physical Nature of the Universe
The Curvature of Space in General Relativity Theory
Rotational Characteristics of the General Gravitational
Field
The Universe from any point of Origin Phenomenological
Limited
The Concept of the Super Universe
The Grav1tat1onal Field as the "Carrier" of Light
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0 Interpretation of the Gravitational Constant
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0 Gravitational theory on the cosmic scale is still dominated by
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0 Einstein's General Relativity Theory, but much confusion nevertheless exists
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energy cor:respond1ng to its total mass and we see that the length
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u Probably the most distinctive characteristic of the grav1tat1onal field
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u associated with a ponderable body is the accelerat10n which it imposes upon
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The simple Newtonian expression for the gravitational force between
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its center of gravity, in dynamic equilibrium.
Again, if M = 1, L =1:
I•1serting the expression (1) for G:
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and from (1):
(8) G
Consideration of (7) and (8) in light of (5) and (6) suggests the possibili
that nume ncally:
(9)
By calculation:
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the probable margin of error with latest estimates of the radius of the Universe
from astronomical data.
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1ight years. As Sandage has pointed out, many uncertanties
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remain, but nevertheless the result obtained in (10) is a remarkably close check
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But in Einstein's model the 4eodesic RG is the length of the path which a ray
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of light must follow, and it is this distance which must be compared with
Sandage' s estimate.
It 1s now necessary to determine whether the assumption
I".{A..,... :: IJ~(J "/ ' 1. ( numerically) is also acceptable, and we calculate:
( 12)
C I(~ LI f'; = 1 /
which is m keeping with current ideas about .the total mass of the Universe.
The volume of Einstein's model is given by:
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( 13)
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and the density ~ is therefore: (f,. u-: .r ·. ' ,) ;
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This again is closely in accord with recently published estimates of
the density of total matter in the Universe.
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Using the simple Newtonian expressions we then have for various
gravitational phenomena:
( 16) ~ 1·1, M,a G-l1t"
M.i C -t -:. )r - )t ~
l' /vf 14 t- '-
/vf 1 a"" )f
M, If '& - II - .:!.1:- /"fl"- '-'t,
Err /'-1. /<1 t 6- t /.-'11 > ~
= = /'1.11. C. ,.
'- /vf,.., '-
( 17)
where M
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and MS. are interchangeable.
The acceleration which characterizes the gravitational field of mass M at
distance L is:
( 18) = !J._y;- - L ._
These relations bring out clearly the fact that the inertial and grav-itational
properties of matter, as exhibited in our limited terrestrial ex-perience,
are dependent upon the properties of the general gravitational field
()f the Universe. Thus the inertial and gravitational properties of matter
<;t e rn from the mutual interaction of all matter in the Universe, as Mach s11ggcstc,
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It is now clear that the role of the gravitational constant G in General
Relativity Theory 1s:
(I) Numerically, it represents the magnitude of a unit of acceleration
which chc.. .·actenzes the general gravitational field of the Universe, and \1'1! -.,.;h
is also the acceleration which characterizes the gravitational held of unit mass
at unit distance.
(2) Dimensionally:
( l 9) G
·.v)nch expresses G in terms of the acceleration '-'llhich characterizes the gravi-tc.
1 10:'.,d field of mass !vt at distance L, as in Newton's theory.
This interpretatic.n c·f G pern1its, as we have seen, calculation of the
:"ctdi ,:s, mass and density of the Universe from the numerical value of G without
',y1r.g tpon astronorr.1cal data and their interpretation. Although these results
'·· r.: ',·1:>c:d nn Ein:;tcin':, spherical model of the Universe, it seems likely that
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the principles involved can be applied to the geometry of any other model, such
as the hyperbolic space. But for reasons which will appear later, Einstein's
model {v,rith suggested modifications) rnRy well be the correct one,
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The Red Shift of Light from Distant Galaxies
The basis of the concept of the Expanding Universe is· the observed
fact tl1at the light from distant galaxies shows a progressive "red shift'' as
the distance increases, the increase 1n wave length and the corresponding
c:ecrease in frequency being directly proportional to the distance, Interpre!;.-
tion of this phenomenon in terms of tre Doppler effect yields the relations:
(20) f I'
whert:! 6 f is the change in frequency of a selected spectral band, f is the
normal frequency, and V is the presumed velocity of recession of the galaxy,
C being the speed of propagation, i.e. the ·speed of light. Then, since the
f req,1e nc y of the photon dee rease s with the distance (i. e. with the presumed
incrr· ,:i ::.e of the velocity of recession of the source of emission), the fr ,!quency
f a t d i stance L is:
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The limit of this"recessional velocity''is of course the velocity of
light, and it is assumed that this limit is approached as the distance approaches
the outer limits of the Universe. The resultant formula for the ,·adius of the
Universe is:
{22)
-V
where L is the distance of the galaxy, V is its presumed recessional velocity
and R is the radius of the Universe.
Estimates of the radius of the Universe based on this method have been
progressively increased from an initial value of about 2 x 1027 cm to the
latest (1958} estimate by Sandage (based on the work of Hubble and Baade}, of
about 1. 23 x 10 28 cm or 1. 3 x 10 10 light years. This is very close to the value
28 10
l. 35 x 10 cm or 1. 42 x 10 light years which we calculated from the
acceleration
t.
C =-
~(r
Let us suppose that this red shift is due not to a recessional velocity,
bu t to :i progr c ssi,:t:! loss of energy by a photon as it traverses space. Since
th e triin s lat1onal velocity of the photon h a s the constant value c and its energy
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is given by:
{23)
where/, is Plancks constant and/ is its frequency, it is evident that the
energy of the photon at distance L is
Since the path of the photon is a geodesic, the distance we have designated
as R is ev1dentl y RG ;:: JJ:" whence:
.2, re. ""'
(2 5)
As L approaches RG the energy of the photon approaches zero, and at the
distar.ce RG it will have lost all of its energy.
If we relate the corresponding decrease in frequency to a constant
acceleration with resultant increase in velocity, and identify this acceleration
as :
(26)
c~
= -/?G
we have
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0
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so that the assumed velocity V:.. approaches C as a limit, as in the theory of
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the expanding Universe.
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But if we relate it to a loss of energy (deceleration) which is reflected
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0 of the photon approach zero.
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0 It is suggested that this is the true explanation of the red shift, rather
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0 than the concept of the expanding universe. If this is true, then the observable
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()
0 light years, which is the maximum distance a photon can travel before it has
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prcp;:,.gat1on of light and the properties of the general gravitational field, a
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-15-
The Physical Nature of the Universe
In attempting to describe the physical nature of the Universe, and
especially in order to rationalize the theory of the Expanding Universe, cos-mologists
generally picture it as a kind of spherical shell which originated
from the explosion of a primordial "kernel" into which was compressed the
total matter of the Universe. Galaxies, as Gamow puts it, are like raisins
embedded in the spherical surface of an expanding plum pudding, or as
Eddington put it, they are like dots on the surface of an expanding bubble. Thus
the distances between galaxies are constantly increasing as the expansion
progresses.
The radius of curvature of this model of the Universe is the distance we
r.ave icl. ~ntif1 e d as Ru, and it is therefore comparable to the Euclidean r-adius
R of a t h ree dimensional spherical surface, The path of a ray of light from one
gaL.1xy to another 1s pictured as lying in this spherical surface, which contains
'. !-.c gcode ,; ics or "straight" lines of this geometry. Thns the distance .:.-- ;i '.. / .
rq)r1· .-v ,t s "on e quarter of the r!istance a round" this s pherical surfac e , a nd
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J
8
8 -16-
0
0 since the sphere is expanding, both Ru and RG are constantly increas'.ng. And
0
0 if the mass of the Universe is constant, it follows from the expression for RG
0
0 that the numerical value of G 1s constantly decreasing.
0
0 This can all be fairly readily comprehended, but the picture 1s not yet
0
0 complete. In Einstein's four-dimensional space the three "Euclidean" axes of
0
0
0
this hypothetical spherical surface rotate about one another, and this is not
0
0
easy to comprehend. In fact so astute a cosmolog.st as Eddington, after des-
0
0 cribing this concept of the Universe, wrote:
0
0 "Having said so much in disparagement of the picture of our three-
0
0 d1mens1onal space contorted by curvature in f.".ctitious directions, I must now
0
0 mention one application in which it is helpful. We are assured by analysis
0
0 that in one important respect the picture s not misleading. The curvature,
()
u or bending round of space, may be sufficient to give a "closed space" - space
0
(_)
I in which 1t is impossible to go on indefinitely getting farther and farther u
Iy'
y
<)
a",1Y f!"om the starting point. Closed space differs from an open infinite space
1:: th,· s.trnc way that the :,urf;icc of a sphere differs from ;, plane infinite space." '< '-/
(' \
)
8
0
0
0
0
C
C
C
C
C
C
C
C
C
C
() ~[ . )
"--1
C
C
C
C
C
C
C
c(
C(
Lf
(_,(
I
Li
·~
-17-
We have already challenged the concept of the Expanding Universe on
the general ground that it is not the only poss '..ble explanation of the "red-shift",
and have suggested as an alternative the concept of progressive loss of energy
by a photon as it traverses space.
We now challenge the expanding universe concept on the ground that
!he picture of the entire matter of the Universe being confined within a spherical
5hell which rotates simultaneously about three Euclidean axes or coordinates
=> not only intellectually incomprehensible, but unnecessary and not in accord
.~'ith observed facts. For example, if the Earth is located in such a surface,
,nd we point a telescope along this surface, we observe light coming from
!alaxies and traveling along geodesic paths within the surface. But how thick
s this surface, and what do we observe if we point our telescope radially
·..:tward or inward? Experience teaches that we see the same thing that we
,bserve if we point the telescope tangentially.
Let 1,1s suppose that the Universe is essentially Euclidean, w ith galaxies
:,, tributed uniformly througho11t the vol11me of the sphere of radius of curva-r
e Ru. And 111stcad of the rotation of the three coordinate axes about one
)
\
.J J I
1._ , --Q
J
.,
:)
21
"v
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
()
(J
0
0
(_j
(J
u
lo
v
I u
\...)
-18-
another, we substitute the concept that the path of a ray of light is rotated
with respect to all three axes as it traverses space, and that this rotation is
due to a rotational property which is a character.'.stic property of the general
gravitational field.
It was shown earlier that the general gravitational Celd is character-
1zed by an acceleration If we link this acceleration
to an angular velocity. then:
(28) a.(.\. :
where ~'1... is an angular velocity which characterizes the general grav:tational
fteld.
The time required for a ray of light to traverse the distance Re
1S t = ~ C.
and the corre spond1ng angle of rotation is
(29) (/J,._ t ; I
l:,:it in this 4-d1mens1onal geometry the radian is
(radian)
7T
;i... - 9 0 () instead of
·17. 4° as in Euclidean geometry. Hence the angle of rotation is 90°, and the
9
Q
(~
y
I --Cl
-19-
direction of propagation of the photon has therefore be en rotated through an
0 angle of 180 , as can be shown by a simple geometric construction.
Thus in the light of (27) a ray of light in traversing the geodesic RG
has not only lost all of its energy, but it has also been rotated (with respect
to all three axes), so that its direction of propagation is tangential to the
surface of the sphere Ru. It follows that the surface of the sphere Ru
represents a limiting distance of observation from the point of origin 0.
We have thus another definition of the "size" of the Universe.
We note in passing that this concept provides a complete answer to
Olber' s paradox.
The gravitational energy of the Universe, as has been seen, is given
by: - I.:;;~ :
from which one deduces that gravitational "lines of force" also follow a
geodesic path, which is presumably identical with that followed by a ray of
ligh t. If this is true then gravitational lines of force from any point of origin
)
) J -1
J
J
:)
8 -20-
0
0
0 become tangential to the surface of the sphere Ru at.the geodesic distance R0 .
0
0 Thus the problem of the indefinitely large gravitational force is also done
Q .
0 away with.
0
0 Since the surface of the sphere Ru is the boundary both of observation
0
0 and for the action of the gravitational field ce·n.te1·..e,.{i at 0, it follows that the
0
0 Universe from any point of origin O is phenomenologically limited to a sphere
b
0 of radius of curvature Ru.
b
0 But the point of origin chosen was not unique by definition (as it must
0
0 r •')
~)
t)
'1
be in the expanding Universe theory), so the question arises: what follows if
we choose another point of reference, say for example a point on the surface
of the hypothetical sphere? Presumably exactly the same conditions hold: we
Q μ observe another Universe of exactly the same nature, which in part overlaps
~ 0
the one centered on 0, but largely comprising a different volume of space. We
u thus picture a super Univen,e, unlimited as to size or boundaries, comprised
v
0 of ..in infinite number of individual Universes phenomenologically limited to the
I u
0 r ;,ci 1us of curvature Ru and overlapping one another endlessly throughout all
I~
I
( '
)
""")'
J
J
--:)
)
~
J
8
0
0
0
0,
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
()
0
0
u
u
0
0
\)
1u
v
-21-
space. This at least has the merit of being an intellectually satisfying concept
of the Universe, and it may well be true,
In closing this section, it seems appropriate to offer another proposal.
The close relation between the propagation of light through space and the
properties of the general gravitational Feld suggest the possibility that the
general gravitational field is rn fact the "carrier" of light i.e. the physical
means by which it is propagated through space. If this is true then since
light is an electromagnetic phenomenon it follows that the gravitational field
also is related to electromagnetic phenomena -- a possibility which has long
been the subject of hopeful speculation.
)
)
J
:)
~
r")
C)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 8 \)
Q
Q
Q
~j
Q
y
ly
y
y
/\
\q
"
PART II
The Source of Gravitational Fields
and the
Interpretation of G in Terms of the Properties of Nucleons
A. The Source of Gravitational Fields
Nucleons Postulated as the Source
Strength of Field Proportional to Number of Nucleons
Gravitational Equivalence of Pr,otons and Neutrons
The ."Common Denominator" Electromagnetic Property
B. Inertial Properties of the Nucleon
Inertial Field Energy of the Nucleon
Gravitational Field Energy of the Nucleon
The Pole Strength O;ic and Radius RJ< as related to
the Properties of Nucleons
Magnetic Moment in Terms of Q~ and R,* .
C. Gravitational Pr ope rtie s of the Nucleon
D .
Strength of the Gravitational Field Produced by Nucleon
Corresponding Angular Velocity of the Field
The Radius R~ and the Pole Strength Q "7
Intrepretation of G in Terms of the ·Physical Properties of
Nucleons
Review of Previous Derivation of G
Interpretation of G in Terms of R 1( , Q;t , M 'I
Interpretation of Gin Terms of R.., and Q ")
__)
:)
:)
:)
0
8 I ()
,)
6
()
c>
C)
()
(~
Q
\~ 'I
E.
The "Big Number" ~ 1040
Interpretations of the "Big Number"
Selection of Preferred Expression for G
The Numerical Value of G
-8
The Numerical Value Q::: 6. 58 x 10
Interpretation of the "Big Number" as c4
Expression for G in Terms of c4
- 'I, f .,. -9' Substitution of &vt;.. •,. -
-8
The Possibility that G :. (6. 58 0. 01) x 10
Progress Toward a Unified Field Theory?
__)
")
~'
:)
:)
:)
=:)
8
J
0
0 g
)
)
)
)
.../
Q
y
( ,
-22-
The Source of Gravitational Fields
The strength of the gravitational field associated with a ponderable
body is directly proportional to its mass i.e. to the amount of matter which
it contains. This is true regardless of the physical state of the matter
(solid, liquid,. or gaseous), its temperature or density, and the extent of
;ubdivision of the particles of which the matter is composed, even to the
Joint of complete ionization into atomic nuclei and electrons, as in the case
,f stars. An overwhelmingly large proportion of the mass of an atom is
epresented by the mass of the atomic nucleus, and an overwhelmingly large
roportion of the mass of an atomic nucleus is :.n turn repre!-ented by the
1ass of the individual nucleons (protons and neutrons) of which it is compo!-ed.
is therefore postulated that nucleons must be the ultimate source of the
~avitational fields associated with ponderable bodie!-, and that the gravita-
::mal field of a ponderable body is the summation in space of the individual
·avitational fields of the nucleons comprised within the matter of which it i'.i
mposed.
C
C
C
C
C
C
C
C
C
C
C
/'"'
\___
C
C
0
0
0
)
.._)
0
0
,l ou
u
( )
-23-
In support of this view, we may express the acceleration which
characterizes the gravitational field of mass M at distance L in terms of
the nucleon:
{30}
where a {M, L} is the acceleration induced by mass M at distance L, Mn
is the mass of the nucleon, and N is the number of nucleops comprised
rn
within the matter of which mass M is composed.
Obviously the expression
{ 31}
represents the acceleration which characterizes the field of a single
nucleon at distance L. The strength of the field of mass M as measured by
the acceleration which it induces, is therefore equal to the strength of the
field of an individual nucleon multiplied by the number of nucleons comprised
within the matter of which mass M is composed. In other words, the
gravitational field associated with mass M is the summation in space of the
:ndividual gravitational fields of the nucleons compri5ed within its matter,
)
)
')
~
A
)
C
C
C
~
C
C
C
(;
0
0
0
0
0
\_.,)
J
u
0
u
u
u
iu
( J
-24-
as was postulated.
This conclusion may be questioned on the ground that expression (30)
is not unique to mass Mn since any other unit of mass Mx may be substituted
with the same general result. Nevertheless it is unique with respect to mass
I
' Mn because this is the only unit of mass which actually gives (by the ratio.61)
M,,,, l the number of parHcles which are the source of the gravitahonal Held. The
I expression J',1 (where M<- is the mass of the electron) for example, does I />f~ not give the number of electrons comprised wHhin mass M.
ratio~ (where M~ is the mass of the hydrogen atom) give the number of
IV/~
Nor does the
hydrogen atoms comprised within mass M unless the mass is composed entirely
of hydrogen. And in that case the result is also the number of protons (i.e.
'.lucleons} comprised within the mass.
The strength of the field of mass M at distance L is also characterized
by an angular velocity w(/.11 L) and an "orbital" velocity CA..; ( /vf, L) )( L given
by:
w 1 L t
-::.
L-
(32} C,l)( /'1, L) :: I Mj:~
wCMJ t) iC/_.: r~&-•
C
r"
\...,
C
C
C
C
C
C
C
Ci
C
C
C
0
0
0
0
___)
0
0
u
u
u
u
( J
-25-
where VO is the orbital velocity of a body revolving about mass M at distance
L in dynamic equilibrium.
For the field of a single nucleon we then have:
(33)
tV-111 (~) =-I :~~
a;li ~) ;cL --I¥= JI; (N~r.lt•':)
Thus the angular velocity and the orbital velocity which characterize
the gravitational field of mass M at distance L are both proportional to the
square root of the number of nucleons comprised within mass M.
The reader may be bothered at this point by the fact !hat we have not
distinguished between protons and neutrons, despite the fact that the proton
carries unit (positive) electronic charge whereas the neutron is electrically
neutral. But gravitational fields are certainly not electrostatic in character,
the proton and neutron have very nearly identical masses, and we may there-fore
assume that as sources of gravitational fields they are equivalent.
The proton and the neutron also differ with respect to another property -
:r.agnetic moment - which must be taken into consideration. The magnetic
~oment of the proton is 2. 785 nuclear magnetons and that of the neutron is
6
Q
j -r
Q
(
j
-26-
-1. 95 nuclear magnetons. The magnetic moments are therefore of opposite
orientation and have somewhat different numerical values. The nuclear
magneton in terms of which these magnetic moments are measured is:
(34)
where h is Plancks unit of angular momentum and e is the electronic
charge in e. s. u.
We then have for the two particles:
(35) == /,17)< ,t'tJ'/.A,,,c
( He.t..frei 1-f) : -tJ.~j I< h ,e
- ~,rμ,.c
and we postulate that so far as their gravitational properties may be concerned,
the proton and the neutron possess a "common denominator" magnetic moment
( 36)
which may be related to a "kernel" or "core" or to a "shell" which is common
to both particles,
This matter is important, because physicists and cosmologists have
.ong thought that there must be some presently obscure relat~on between
J
:)
:)
~
=>
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
jQ
I
~)
()
I c>
()
c')
c)
18 lq
I \
-27-
gravitational and electromagnetic phenomena. It follows that if nucleons are
· indeed the ultimate source of gravitational fields their magnetic (or electro-magnetic)
properties must be closely related to their gravitational properties.
We shall discuss this in detail later, but at the moment all that need be said
or can be said is that insofar as the magnetic (or electromagnetic) properties
of the proton and neutron are related to their gravitational properties, they
must be equivalent and thus must have a "common denominator" of some
kind, such as is set forth in (36).
-28-
Inertial Properties of the Nucleon
Since the strength of the gravitational field produced by a nucleon is
oportional to its inertial mass Mn, it is to be expected that the energy of
gravitational field is proportional to the energy associated with its rest
LSS, which we shall identify as its inertial (field) energy as distinguished
m its gravitational (fidci) energy.
Thus if the inertial energy of the nucleon is given by
(37) E.
gravitational energy should be:
(38)
re Q ;)( is a pole strength and R~ is a radius to be identified, and W, jo
the respective angular velocity and angular momentu~ which are related
1e energy of the particle for each case.
Thus the inertial and the gravitational field energies of the nucleon
related by the expression
( 39)
1
:; H11 C I'
)
J
)
J
)
:)
8
8
0
()
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
()
0
C)
0
,:)
0
-~
u
__ )
u
CJ
u
u
i
-29-
So little is known about the physical nature of either the proton or
I
. the neutron that the values of Q ~ and R 7( are uncertain, and it is even
questionable whether it is proper to apply this method of classical physics
to the inertial properties of the nucleon. Nevertheless if we do so, certain
· conclusions can be drawn, which are of value in the present study.
Thus if Q )( = e 1 1t follows that
(40)
' This ''classical" radius of the nucleon is over 100 times smaller than the
-111
length ( ,"'\.. /o c,,.,) which is thought by many physicists to be the shortest
. length which has a valid physical meaning.
A more likely interpretation (at least for present purposes) is that:
( 41)
·,·here ;) .. ~ ..f:J_ is the Compton wave length of the nucleon. We then have: '1 μ,, c:.
~)('
~,, 'Z. 4-L he.. ~r... f! =- ; (42) "' = ..I.t2.-,!.i. ,t. ,r ~
'L ··here cX; ~!le.: !1t .- -I is the fine structure constant of Ac.. G. 117
'pectroscopy and ~ is the orbital velocity of the hydrogen electron.
)
8 ~)
\)
8 <)
c;_)
()
<)
8 ()
¢
y
8
Q
(j
v
u
~
l l ! i
t
,"!··t
~ 1
t,~
I
~
-30-
In terms o{ 1:1:i:e:-quantum mechanics the inertial energy o{ the nucleon
is then:
(43}
where a,; is the angular velocity and is the angular
momentum associated with the inertial energy of the nucleon.
The magnetic moment o{ the nucleon is then:
(44) ..,(,,/. ~)( " R.:at -- e;,c. _&. ,1.-,;-
As- - _A_ ..L
/.. 'l{ /f,111 (! 1 rr )c M11 ~
which is in accord with the expression derived in (36) as representing the
"common denominator" magnetic moment o{ the proton and neutron.
This interpretation of the inertial energy and therefore of the inertial
properties of the nucleon appears to be acceptable. We have no'w to determine
how it works out in relation to its gravitational properties.
)
)
)
)
)
:)
:)
:::)
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
J
..)
0/
u
0
u
u
-31-
Gravitational Properties of the Nucleon
Since the gravitational fields associated with ponderable bodies are
of Coulomb type, and we have undertaken to relate such fields to the physical
properties of the nucleon, we may assume that there is a radius R ?t
related to the gravitational properties of the nucleon such that:
a,, (1.) M~ &!-_ ;..,,, " 6- ~ ~
)< a-_
(45) - t., ._ ~;e ... I-..,_,
w '1. L., z. w f If'~~ -L- = ~a.'- : ~ J.,,
where d.11 (L) is the strength of the gravitational field of a single nudeon
at distance L as measured by the acceleration which it induces; R;t is a
radius presumably on the order of nuclear radii which is to be identified
'. ind evaluated; &<., is the angular velocity associated with the acceleration
a_h (L).
The corresponding specific properties of the gravitational field of
the nucleon determined by substituting L:;:. R,r, are:
~" l"'1 ~ ~ G.() ~ I? '1,. I( (46) -- = "' a: :: ?(.)J i "L )t' if"'" z. re"
W11 -.. y ~;;),~
)
)
)
)
:)
:)
:)
:)
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v
\J
v
0
0
u
\J
-32.-
We see by inspection that neither (45} or (46) yields a unique value
for Ri( , and that '1'1 and~ are dependent upon the numerical value of R,r .
From the discussion of the previous section {41) it seems possible
~ - ...L that R;< may be interpreted as ,t.'1i _ ,lr;-M,.C. But for reasons which
will appear later there appears to be a more likely candidate for this dis-tinction
which we shall now identify as:
(4 7)
This radius is the complete counterpart of the orbital radius Rff of
the hydrogen electron:
A t.
{48)
where Me,. is the mass of the electron and V /1 is its orbital velocity. Thus
• R>1 is in fact the radius of gyration of the proton about the center of gravity
. of the hydrogen atom, and therefore it has at least one physical significance
in atomic physics.
If, after the manner of (37), we express the inertial energy of the
nucleon in terms of this radius and a pole strength:
-33-
(49)
The numerical vah.1~ of Ob is almost identical with that of the
gravitational constant, though of course its dimensions are entirely different.
The thought tuggests itself that this may be something more than a mere
numerical coincidence. and we shall see later that it probably has an important
physical significance. In fact it seems possible that the two values are
actually equal, and that the numerical value of the gravitational constant is
-1 -e
6. 58 x 10 instead of 6. 66 x 10
The groundwork has now been laid for an interpretation of the
Q
gravitational constant in terms of the physical properties of nucleons. 8
y
l
.,
-34-
Interpretation of G in Terms of the
Physical Properties of Nucleons
In Part I of this paper the gravitational constant G was interpreted
(8) in terms of General Relativity Theory and, more specifically, in terms
of the properties of Einstein's spherical model of the Universe:
(50)
where RG =f RLA.- is the geodesic corresponding to the radius of curvature
- ..s-t. R ~ of the Universe, /.if""- is the total mass of the Universe, and ~ - ~(r
is an acceleration which characterizes the general gravitational field of the
Universe. It was postulated that tt/..tA- is numerically equal to G, from which
it follows that /1/A. is numerically equal to the square root of RG. It was
shown that these postulates result in acceptable values for RGJ M ~ and
Ac .,c,.. , the average density of matter contained in the Universe.
Since our objective is to express G in terms of the physical properties
of the nucleon as discussed in previous sections, we might expect by
analogy an expression of the form:
( 51)
L
-35-
where R;ic is either,..1
11
, ;l'1 or R'n as previously defined, andfPis a
~1,
dimensionless function which must be evaluated and identified.
From the equation:
{5l)
it is evident thatf is numerically the bverse of the ratio
Setting RX.: R rJ = 2. 89 Y. 10 - lZ cm we find by calculation:
I
"!
(53)
whence one solution for (52) is:
(54)
G
the other two cases yield the expressions:
{55)
/1 '1 '- I "1. /\ ';L, I
)< - I' - G - ......:.---- ~ r ,. ri-/i/f '1 ,,hi RV ?t"'-
C. . ./7 VI I -'- .-,,')'-' C > )( - = - )( r ')< y /.//,, ll~
Q
0
9 8
6 I
_)
_)
,)
)
u
u
u
v
v
i
f
-36-
As a matter of possible interest, it was the fact that these two cases
reduce to (54) which drew attention to the possible significance of the un-orthodox
radius R>, , which the reader was promised in a previous section
would prove to be of interest.
The beautiful simplicity of {54) is somewhat marred by the presence
' ;,? t
of the factor-:;. , but the fact that J : 5 instead of unity cannot be denied,
J ~ ,
and this fact dictates the presence of the factor?. It can of course be
dispensed with by taking R )(-:= R
11
x ff, but this destroys the identity of R"?
as the nuclear counterpart of RH , as discussed previously. This small
I
imperfection in {54) is however not really important, because the factor ?
cancels out in any expressions treating with the gravitational properties of
ponderable bodies.
For example, the acceleration which characterizes the gravitational
field of unit mass at unit distance is:
(56)
which is correct, as was shown in (5) of Part I.
y
v
)
)
)
)
=>
~
l
-37-
This derivation of an expression for G in terms of R ~-, and M ;-, may .
have the air of being a tour-de-f"orce, because no basis was laid for intro-
~ duction of the factor } . But this can be supported by a somewhat different
/,(«
approach:
I(.,, ::. !<~_{
)t cf 6- - μfl M, ...
(57) £~ ' " /.;f •1 - A _/_,{.-J I /!£\ij ((' >< ~M -; . ,,,. :: ,<~ /{~ .. /1/'f •L .'If " /vf,4,,
Substituting RX:. R~:
/?!L ~ I lvf4 ,A ~L
't.
I (58) Cl -- I<.:- N~; i. /·;fa r I
I(. C -t.. t.1 G -- ' ~ - )(
l "7 - ,. - at:- .,.
Mr. J
This not only supports the introduction of the rati.o -R., i. n (5 1 ); it
R~
I
also reveals that the factor r is not an arbitrary constant, since it represents
the dimensionless product of two ratios:
/vf ,.,
( 59) -/vf '4
)' ,(/ G:... i. ':" . - )
' . \ Ii
I
Several physicists and cosmologists have commented on the "big
number" of the order of 10
40
which appears when electrostatic and gravita-tional
forces are compared, or when phenomena on the cosmic and the atomic
~cale are compared. Eddington for instance suggested that it might represent
J
:)
:)
:>
~
0
0
0
0
0
0
0
0
0
0
0
C
u
0
0
0
0
,o lou
0
0
0
0
);
ui
C>j
Lj
L,,
ul I
-39 -
which interprets the "big number" as the ratio of a fictive electrostatic
(or electromagnetic) force to the gravitational force between a pair of protons
-- or possibly between any pair of nucleons. This is close to one of
Eddington's speculations, but we can come even closer by recalling that
·, ~ ;; .,S.. and writing:
7 11 ~
By calculation:
·; whence
(64)
This expresses an order of magnitude smaller "big number" in
I.. terms of the ratio of the electrostatic to the gravitational force between
an electron and a proton, which is exactly one interpretation suggested by
Eddington.
Furthermore,
(65) G- :
from (54):
~ l C.
., - )t
/V'f,,,
I
)c -
.J
-40-
where /\,{i. is the total number of nucleons in the Universe. This is of course
a specific solution of another of Eddington' s interpretations of the ''big number".
The total mass of the Universe is then:
(66)
which checks the mass calculated in ( 12) for Einstein's spherical model.
Thus all tl,rce of Eddington' s conjectures about the significance of the
"big number" are confirmed, though the numerical value of the number differs
somewhat for the various interpretations.
( d From all this it appears that the expression:
(67)
is an acceptable interpretation of the gravitational constant in terms of the
physical properties of nucleons, and it will the re fore be adopted in what
follows. u
)
Cj
()
\r.)'
C.)
(
-< "'-(
Q
'y
1 u
1¢
('-(
l;
-41-
The Numerical Value of G
and
A New Interpi=etation of G
In a previcus sectiot} (49) attention was called to the odd circum-stance
that the numerical value of the pole strength
(68)
-8
6.58x10
is very nearly identical with that of the gravitational coa.stant. It was
i suggested that this may represent something more than a mere numerical
coincidence, and that it may in fact indicate that the numerical value of G
-8 -8
is 6, 58 x 10 instead of the accepted value of 6. 66 x 10
explore this matter further.
Consideration of the expression:
( 69) I
X - s-
We shall now
-8
':: 6. 66 X 10
suggests that there may be some meaningful relation between the number
40
l. 34 x 10 and the fourth power of the velocity of light:
(70) -- 40
8t1. ,d xlO
C)
c.)
\)
0
~)
Q
()
0 J u
J
I~
u
u
u I
By calculation:
40 40
2. 34 x 10 .:;:. G0.7Sxl0 x
whence
{71) G
-42-
1
6 == 80. 7'J. X 1040
X 3. 96 •,:<_
34.
l where /Y, 't/:: / is a dimensional factor required to preserve the
dimensional integricy of the equo..tion. Recalling that
{72) ::
' where~-::,_ =~s the pole strength corresponding to the radius b, we have:
~ I~ ~~
{7 3) -8 6.66 X 10
which is a perfect square except for the number 3. 96. Assuming that this
number should really be 4. 00 we have:
6?2 't. ~ Iv. 4 / I (74) G - 11- ; -8 - M01 "' C" y~c. :,, :j ~ X 10
- ~ >-,(_v_. !_,., I ..L -8 . 6. 58 X 10
- M,,• (, "f it y~ ..
Then since~ = 6. 58 x 10-
8
it is evident that the denominator of this ex-pression
should be numerically equal to 6. 5: x 10- 8 and we find by calculation:
(7 5) 6 •:·:· r, X 10-8
(
_(
)
()
()
l)
~
c~,
(
V u
.,_J
I
-43-
This is not an exact check, but the uncertainties in the values of the
constants involved are such that we may, safely write the tentative result:
(76) G --c;a..,< "' /~ '1 I i<-I 6, 58 X 10 -8 (:! o. 01)
- M,., z.. e, 4 'I~ -
Similarly it is evident that
I .&- ><
I -8 (to. 01) (77) - IC°
,?~ - 6, 59 X 10 μ,,'"' r
which provides a new method for calculating RG and yields a value slightly
larger than was previously calculated,
It seems very unlikely that these relations could be the result of mere
numerical coincidence, even though there is no apparent reason why the
pole strength ~"' and the quantity ( /11"1
1>- C:. Y ;t C,, ~ ) should both be
numerically equal to the gravitational constant.
In the classical physics of electricity and magnetism {as well as in
optics) the numerical value of the product of the dielectric constant t£ and
the constant of magnetic permeability,t{ {for vacuum) is
(78) [;A :: / :- f().;, ;./• yo I ; )
where the dimensional factor on the right indicates that the dimensions are
those of the inverse square of a velocity i.e. the velocity of light.
)
\
tJ
6 8 ()
8 ()
(J
~
(J
J
u
u
(j
u
J
I \
-44-
The dimensions of /: and~ separately have always been a subject
of controversy, and the productt'A(as Planck and others have pointed out)
is given the dimensions of the inverse square of a velocity in order to
regularize the dimensions of the expression in which it appears. In the
present case we are dealing with a pure number and therefore assume that
it is justifiable to assume thatc!'.M may here be used as a pure number. As
such it represents, for example, the square of the ratio of unit velocity Vt
to the velocity of light c. In terms of relativity theory it represents the ratio
of the incremental rrass corresponding to the kinetic energy of a body moving
at unit velocity to its inertial mass. Using!A<,in this sense we write:
(70)
-~1.. "l
G - /v111 x~ 6 -8 (-: e;., ,) 1,. - • 58 X 10
'I"-
62.:,, -"t. 6. 58 X 10-8 (.1~.11)
- ,~ -- whence: /Jf., 't ~""" 1-
(80)
1
"t ::-
f )
)
)
)
\....)
·__J
J
.)
)
u
0
I tJ
J
I \
-45-
1.
The gravitational pole strength /Jf11 G .Z.. of the nucleon is thus ex-the
first time a link is uncovered between the gravitational properties of
matter and the electromagnetic properties of matter and of space, which
strongly su.ggests a close connection between gravitational and electro-
· magnetic phenomena.
Interesting side-lights of these relations are the following interpre-tations:
_-t &... 'i oC, - 71., )t ~ (81) /:A )I. ----- ..
rt~ s- ,L71 ~" J
and I ,
Mir(. i.. M,, (.- i E ,,/..,(_ ~ r;z ~ ;t ()(._ ..
~"1
~ 6!,,'\
Z-The
first expression represents<£~ {or rather its reciprocal) as the
big number according to Gamow's suggestion. The second expression
inteprets et,(. as the ratio of the gravitational pole strength of the nucleon to
an electromagnetic pole strength, modified by the factor ;;;x:...
r
q ~
d
d
Cf q
0
d
~ C,. C
C c~
C}
("
',-..----.
(.____
C?
0
d
0
d
()
0
u
"-,)
j'
()
u
u
\.)
I I
I
I '
' I
1
wc.:c,.e,.;.z;;z)R.;.Qt;_. . . .o ,;.- .. --. •
-46-
A singi!icant .conclusion is that i£ expression (79} is acceptable, the
numerical value 0£ the gravitational constant can be calculated from atomic
constants and the velocity 0£ light ( _!_}
{t.AJ-without
actually measuring it
in the laboratory.
The appearance 0£ ~ in so many o{ our expressions is perhaps dis-concerting
because we can at present o££er no explanation. It is comforting
however to recall Eddington' s prediction that the constant d.. would find
an honored place in the deveiopment 0£ an acceptable unified field theory,
If it is true that G:: 6. f:: x 10- 8( .±.o. 01~ it follows that the correspond -
ing values for Ra, Rt1,,,, M~ and/='~ are slightly dif£erent than those cal-
-8
culated in Part I from the value G :: 6. 66 x 10 . But this is 0£ little practical
importance £or our present purposes because the change is only approximately
1%.
One might suspect that this hypothetical smaller value for G supports
the theory o! the expanding universe, But according to this theory the rate
of change in the value 0£ G is approximately 50% in 10 10 years, so that this
J
:)
)
)
:)
\
'-
'.")
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
u
0
)
\,_)
u
u
v
( \
-47-
could not possibly provide an explanation. We must leave this problem for
physicists in the hope that either the numerical value of G or that of one or
more atomic constants, will be found sufficiently in error to explain the
matter.
We are now ready to proceed with a study of the physical properties
of the gravitational field of the nucleon, and thus of gravitational fields in
general.
)
)
0 ).
()
0
6
0
0
u
\__)
0
u
lo lu
0
{ '
(A)
PART III
Practical Applications
Gravitation and the Inertial Properties of Matter
Interpretation of the gravitational field energy of the nucleon
Interpretation of the inertial field energy of the nucleon
Interpretation of the inertial mass of t.he nucleon
The specific acceleration of the gravitational field of the nucleon
Relation between inertial and gravitational properties of matter
Interpretation of the kinetic energy of the nuclc•.m and of
ponderable matter, as related to the general gravitational
field.
-'- Introduction of (;';« into expression for kinetic energy.
(B) The Gravitational Field of the Nucleon
(C)
Final expression for G in terms of a preferred radius R~ .
Suggestions that this radius represents a boundary between
gravitational and nuclear phenomena
Characteristics of the gravitational field of the nucleon
Relations between the inertial and gravitational fields of
the nucleon.
Relation between the inertial and gravitational fields of the
nucleon, and the general gravitational field of the Universe.
Basis laid for the development of a unified field theory?
The Magnetic Moment of the Earth
Elsasser's hypothesis
The E'lac: .ett-Wilson hypothesis
(',
(
C
(
(
\.,.
(:
(:
.....,
,,,
......
"' \.,. r
~
t
1
)
j
j'
)
)
)
u
\)
.J
lJ
u
J
\
PART III {Continued)
Discussion of Blackett's expression for the magnetic moment
of the Earth in terms of concepts presented in this paper.
Suggestion that the magnetic properties of the Earth may be
due to its axial rotation (or the rotation of its gravitational
field) with respect to the general gravitational field.
Possibility that the BlacketU.Wilson hypothesis is fundamentally
sound despite lack of supporting astronomical data.
(D) The "Displacr.ment" of a Ray of Light Grazing the Sun (not finished)
(E) Rough Concept for the Derivation of Gravitational Field Equations
(not finished)
}
>
)
)
)
~
~
:)
C)
n
o ·
0
0
0
0
0
0
0
0
0
0
0
0
0
0
()
u
0
(J
0
()
(_J
u
u
0
~
0
V
\....)
u
u
u
u
( \
-49-
The inertial field energy of the nucleon (M "">1 c 2) is equal to the
gravitational field energy of the nucleon reacting with the entire mass of the
· Universe at distance Ra.
~
Then by transposing C :
The inertial mass of the nucleon is identical with the mass correspond-
, ing to the gravitational fic:d energy between the nucleon and the entire mass of
the Universe at distance RG.
Further, from (82)
(84)
The specific acceleration which characterizes the gravitational field
of the nucleon is equal to the acceleration which characterizes the general
gravitational field of the Universe.
And as was shown in (56) the strength of the field of mass M at distance L
as measured by the acceleration which it induces is:
(8 5)
:J
\
.J
:)
:)
~
~
r\
'-.J -50-
0
0 All of this is strong support for the concept that the inertial and
0
0
0
gravitational properties of matter . are closely related, and that the general
0
0
gravitational field plays a major role in the inertial properties of ponder-
0
0 able matter. One surmises that the kinetic energy corresponding to the
0
0 motion of a ponderable body may be due to interaction between the gravita-
0
0 tional field or the electromagnetic field of the body and the general gravita-
0
0 tional field.
0
0 This the kinetic energy of a nucleon in motion at velocity V is
0
0 given by:
0
0 (86)
0
0
0
which is of course completely comparable to the kinetic energy of an
0 electron:
0
0
0
(8 7)
-.J
__)
The since we have from (83)
\ .. )
v i (88 A) !;~ (,vt.4,1~ .. 01) I /vf,._ M'7 G // 'l
-: - )C - :;., (( (r "' '\ u I
u Ii
I
and from (82) in the light of (86):
u i
I
' u I ( '
(88 B) {: ( N v.c. /~~ ,-) I /.,,(~ ( t-
;c !B= : i( j' /( JL_t- t, ,:i, 11~ I<~ G, '1.,.
•• •
•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •
-50-
All of this is strong support £or the concept that the inertial and
gravitational properties of matter . are closely related, and that the general
gravitational field plays a major role in the inertial properties of ponder-able
matter. One surmises that the kinetic energy corresponding to the
motion of a ponderable body may be due to interaction between the gravita-tional
Held or the electromagnetic field of the body and the general gravita-
, · tional field •
This the kinetic energy of a nucleon in motion at velocity V is
given by:
which is of course completely comparable to the kinetic energy of an
electron:
The since (9,,. t. - /vf '1 (. z.
t?,,., - we have from (83}
(88 A) /;~ (Nt.1. c. /u1-1) I = -~
and from (82) in the light of (86):
(88 B)
J
' :)
:)
:)
J
C)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
\.J
0
u
)
v
u
u
u
I '
I --~
-51-
These expressions are interesting for several reasons:
The kinetic energy of the nucleon is expressed by {86) in terms of the
motion of a pole strength or "charge" comparable to the classical expression
for the kinetic energy of an electron in motion. This may perhaps be obvious,
but the possibility of expressing the kinetic energy of ponderable matter in
this manner seems to have been overlooked. It is clear of course that the
kinetic energy of any body of mass M at velocity V is simply the kinetic
energy of a nucleon multiplied by the number of nucleons comprised within
the body. The expressions of (88} clearly relate the kinetic energy of a
ponderable body to the motion of the gravitational field associated with the
body with respect to the general gravitational field. This is especially clear
from:
(88 C} EK (nucleon)
L
where Ml'! G ;a.is the gravitational pole strength of the nucleon.
Another interesting relation results from the introduction of
into these expressions (79) {80}:
I
)( -f'
J
:)
J
J
:)
J
')
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
___)
)
I '
-52-
whence:
{89)
Further explorations of these relations and concepts is bE:yond the
scope of' this paper, but it is hoped that their presentation here may
, stimulate theoretical study as to their physical meaning,
J
:)
=>
=>
:)
J
:)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(J
0
f \
.L.
-53-
Gravitational Field of the Nucleon
We have postulated that the gravitational field of a ponderable
body is the summafion in space of the gravitational fields of the nucleons
comprised within the matter of which the body is composed. If this is
truP, ".i.n understanding of the physical nature of the gravitational fiei.c..~ of
the nucleon should teach us a great deal about the nature of gravitational
fields in general.
The inertial properties of the nucleon have been discussed in
terms of several "models" of the nucleon, and the relation between the
inertial and gravitational properties of the nucleon have been roughly
outlined. It remains to develop thi~ theme more intensively in the hope
of determining what the gravitational field of the nucleon "looks like",
and what is the nature of the transition from its electromagnetic inertial
field to its gravitational field.
The expression finally selected for the gravitational constant was
(62) and (65):
(?... f
-;,, - ;e(r S'
J
)
:)
:)
)
J
8
C)
-54-
0
0
0 where:
0
0
0
0
In (47) (48) it was pointed out that the radius R,, is the nuclear
0
0 counterpart of the radius R~ of the hydrogen atom, and that it is in fact
0
0 the radius a< gyration of the proton about the center of gravity of the
0
0 hydrogen atom. There is however no obvious connection between this
0
0 fact and either the gravitational or the inertial properties of the nucleon.
0
0 But there is another reason for the selection of R'>) which appears to give
0
0 it at least same semblance of validity.
0
0 Recent research into the nature of the proton and neutron indicates
0
0
0
that each has a somewhat nebulous distribution of electric charge or
u
0
electric current, and that in both cases these fall off rapidly at distances
0
---'
greater than about 10- 13 cm and are practically nil at distances substantially
j
0
-12
less than 10 cm. Studies of the enormously strong nuclear forces show
J
v that they also fall off rapidly beyond about 10-13 cm and are practically
u [
u f u I
I '
-12
non-existent at 10 cm. Further, the radii of atomic nuclei are found
_)
)
:)
:)
)
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
~
\..)
u
v
u
u
u
u
( \
-55-
-lZ to have an upper limit of something less than 10 cm.
We therefore postulate that the radius R '>? may represent a -kind
of physical boundary between nuclear phenomena per se and gravitational
phenomena. It seems unlikely that gravitational fields as such exist
within the atomic nucleus, but we know that beyond the nucleus, they ex-tend
throughout all space and obey the laws of the Coulomb type field.
The concept therefore is that the radius R'>1 represents a zone or boundary
of transition of the electromagnetic inertial field of the nucleon into its
gravitational field. This is the real reason for the selection of R'>7 in ,~ preference to -;.-.,;,. . or I) ,, for the interpretation of G.
Thus with respect to its gravitational properties we picture the
nucleon as a particle of radius R ")') , without forgetting the fact that in
reality, with respect to its inertial properties, it probably should be
pictured as having a radius of ~and possibly a core with radius1.,which
~Tr
represents most of its inertial mass.
.)
l
:)
)
)
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
__)
( \
-56-
If the inertial (electromagnetic) field of the nucleon is pictured in
relation to the radius R ~ we have for this field (as was shown earlier)
'.he following properties:
(90)
where t,t,
1 and are the angular velocity
and angdar momentum associated with this field. The corresponding
:ipecific acceleration is:
(91)
The gravitational field energy of the nucleon (54) (55) is presumably:
(92) EG (nucleon) -
If this energy is set equal to a,,,! P.._: one is tempted to assume that
and , but it is not clear what should be
I
ione about the factor - . Ignoring this factor for the moment we have for r
the corresponding specific acceleration in terms of Wz.. /f"'i:
J
)
)
:)
:)
8
8
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
()
0
lo
,o
u
(.. __,,,
u
u
u
u
lJ
v
G
-57-
(93)
The correct expression for the acceleration as was shown in
32) is:
(94)
md the correct expression for the specific angular velocity is, by
;ubstituting ~~ ~ R~ in (46)
(95) C,41,, = r .
:'he correct expression for the specific angular momentum is therefore:
/?., :: a;,, M11 .A\ E.
(96)
-= μ,, I< ,?. t ~ s. ; !'I-: ;
"I /?;, /24- f
-: /4'1 C ~'1 I tA f,
vhence:
vhich is correct.
From this it is clear that the tentative assumption made in (92)
9 3) with respect to ~L and Pi. are not tenable, and that so far as the
)
)
)
)
)
:)
:)
.~
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
u
u
u
u
u
<~)
()
u
u
u
( \
-58-
properties of the gravitational field of the nucleon are concerned, they
cannot be related directly to the axial spin of a particle of radius R>,
In terms of the properties of the inertial field we then have for
the gravitational field:
(7) / ~· - .!::. I I?.,, : J - .s. r;;-:-1 ~~ - II~ It."->'' - Rerf "i';•;
(jlf : /-4~ C I( '1 /C / ~ ,_;,
a_ : _£_'-~ {( M /t .J_ ':' _£; IC j_
., /(~ /t~ f' It~ .r
Thus the "transition" !actor is / %:. ,c ? for both U
111 ~ , and
the square of this factor for~- This seems clearly to support the
concept that the gravitational field of the nucleon is created by the inter-action
of its electromagnetic inertial field with the general gravitational
field of the Universe.
But if the general gravitational field is the summation in space of
the gravitational fields of the individual nucleons, and the gravitational
field of the individual nucleon results from interaction between its electro-magnetic
field and the general gravitational field we have the familiar hen -
and - egg problem. The solution would seem to rest in the fact that the
J
_)
:)
J
:)
,~~·
8
C)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 1g
lo
'o
0
0
0
I~ I
V
)
(_)
u
(J
u
u
u
u
I \
l
-59-
total gravitational energy of the Universe is equal to the energy corres-ponding
to its inertial r,:iass, as was deduced in (2). This could be interp-reted
to mean that the general gravitational field is the summation in space
/Vi.( ( I-.,."-5
of the electromagnetic fields of the individual &1-e-et-rons acting over the
distance RG.
I£ this is true, a direct relation has been uncovered between electro-tna~
netic and gravitational phenomena. This is a subject beyond the scope
of the present paper, and we leave it to competent scientists as a problem
which, if solved by suitable mathematical treatment, should lead to progress
toward the development of an acceptable unified field theory. Interpretation
of the gravitational constant, and of the gravitational field of the nucleon and
of the Universe in terms of physical constants such as .b.. , ~ ,
,tq- e and
the like, should certainly be helpful in that connection.
)
'.)
)
.)._ ,
:)
8
8
C)
0
0
0
()
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v
v
0
u
u
u
-60-
The Magnetic Moment of the Earth
Since the time of Gilbert, who first showed that the Earth as a
.ole acts as a gigantic magnet, many attempts have been made and
potheses presented to explain this phenomenon. In recent times
sasser -has presented the hypothesis that the magnetic field of the Earth
created by electric currents generated by the slow flow of matter in
, fluid core. Blackett presented an hypothesis relating it to the axial
tation of the Earth and the gravitational properties of the matter of which
~ Earth is composed, which he fould had previously been proposed by
ilson based on a suggestion of Schuster. We shall concern ourselves
re with Blackett' s hypothesis.
This hypothesis is based on a fact discovered independently by
ackett, who compared the ratio of the magnetic moment of the Earth to
; angular
M(~r1AJ
momentum fo the similar ratio for the Bohr magneton
/J(~ .. ,,~J
and found that the numerical relation between the two corresponded
ry closely to the ratio of the "gravitational mass" of an electron to its
J
:)
:)
:)
)
')
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
u
u
v
v
(J
u
u
()
u
-61-
I
μ~ G_1.
e. :iarge in e. s. u. , thus:
(98) ,I.,{ ( ~a. ,. fj) = /?
tl{t;o.,.1-li)
here 114 ,
is a numerical constant of magnitude about~"·
Expressi.ng the axial angular momentum of the Earth in terms of
s equatorial angular velocityt'<,i and radius R, and the value_.At (Earth)
25 7. 9 x 10 , Blackett found:
(99) _,,,I.A. {Earth)
here ?'/ is a factor correcting for non-uniform rotation (i.e. slippage)
f the fluid core of the Earth, and I{ is a factor correcting for non-uniform
ensity. Setting ks 0. 88 as estimated by astrophysicists the value of ~ ,{>
s calculated by Blackett is 0. 263 and the corresponding value of 11 is
. 299.
This value for 'i? indicates an enormous amount of "slippage", which
eems incredible after hundreds of millions of years of axial rotation of
:1e Earth. A more reasonable value for '7 can be obt a ·.ned by introducing
)
) ) J.
)
:)
:)
:) -62-
8
0 f
the factor -.,,. in a purely arbitrary manner. Also, a more recent figure
0
0 for the magnetic moment is.,,t{ = 8. 04 x 10
2 5, a little more than 1% higher
0
0 than Blackett's figure. Using Blackett' s figure ,1,: 0. 88 we then find
0
0 (100) ~ (Earth) -- 0
0 where 11 - 0. 94, which seems a much more reasonable value than 0. 30.
0
0
0
And we observe in passing that if 11 :1, corresponding to ~ slippage, the
0
0
corresponding value of k is 0. 83 which is probably within the margin of
0
0 error of the value estimated by astrophysicists. In any case we may for
0
0 present purposes write for the angular momentum of the Earth:
0
0 ( l O l) ;a (Earth) = ?" n h ~ M If z.;: t<J, M ~ £
0
0 where~ is the "effective" angular velocity and R 1 is the "effective" radius
0
0 allowing for the fact that we are dealing with a solid rotating sphere of
0
0 non-uniform density and probably non-uniform angular velocity. Then
.J
0 using expression (67) for G in terms of the pole strength On:
\..)
()
u
()
v
u
u
J
) ) .J
)
:)
~
'.)
'.)
0
0
0
0
0 {l 02)
0
0
0
0
0
0
0
0 From this we deduce that each nucleon within the mass of the
0
0 Earth develops, due· to the axial rotation of the Earth, a magnetic pole
0
0 strength
0
0 ( l O 3}
0
0
0
~ and that the equivalent average length of the dipole is 12r. so
0
0
that the magnetic moment of the Earth is:
0
0
0
0
or, as was shown earlier, if ')1 = 0.299:
u
u
luf _ )' 4'I )t
/?,
p_ (Earth} - -- μ,, ~
iu
v
u
1 u
I~ .-----
)
) ) r_J
:)
:)
:)
8 -64-
8
0
0 In terms of quantum mechanics, remembering that c0
11
0
0 we have for the effective pole strength of each nucleon
0
0
0
0
0
0
( I 05) Then, from (62) 1 I ~., ,. J . ;w,~ f ... 't
we have ~tr 'f' ~~
0
0
0
I
( I 06) ,/111 -- M~ ~ 1 )t ~ t:,
0
0
which expresses the magnetic pole strength of the nucleon due to the
0
0
axial rotation of the Earth in terms of its gravitational pole strength.
0
0 The thought suggests itself that the magnetic properties of the
0
0 Earth as a whole are due to its axial rotation with respect to the general
0
0 gravitational field.
0
0
0
If the factor / ~.,. "? is expressed in terms of e{ M. , we have
0 from (80)
~
I )
10
0
( l 07)
0 which shows again that the magnetic pole strengthA,.~ is related to the
u
' u lu
properties of the general gravitational field.
I / I
)
:) :) L~
:)
:)
~
0 -65-
0
0 Thus in keeping with the hypothesis of Blackett and Wilson, the
0
0
0
magnetic moment of the Earth can be expressed in terms of magnetic
0
0
effects developed by the matter of which it is composed, due to its axial
0
0 rotation. We have expressed this in terms of the magnetic pole strength
0
0 developed by each nucleon and related this in turn to the axial rotation
0
0 of the Earth (or of its gravitational field) with respect to the general
0
0 gravitational field.
0
0 Blackett believed that this same expression for the magnetic
0
0 moment of a rotating cosmic body applied to all such bodies, including
0
0 the planets, Sun and stars in general. On the basis of astronomical
0
0 data this belief has been challenged. We venture to suggest that Blackett' s
0
0
0
hypothesis is probably basically correct, and that the interpretation of
0 astronomical data is at fault because of several factors. One is that the
0
_)
v values of 71 and,k for the stars (including the Sun) and most of the planets
u
0 are very uncertain indeed, and that the measurements forA, C<,J and~
~
u for the stars are subject to a substantial margin of error. It also seems
u
u
I '
)
) d__J )
)
:)
:)
8 -66-
8
0 probable that massive turbulence in stars may so affect their magnetic
0
0 fields as to destroy the significance of Blackett's expression so far as
0
0
0
numerical value of the observed magnetic moment is concerned. But
0
0
this does not necessarily destroy the basic soundness of his hypothesis,
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
u
()
u
u
u
u
u
) -·-- ----·---.
)
)
)
)
:)
:)
~
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v
u
u
u
v
I~
I ' ,
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
l 3.
References
Albert Einsten - "The Meaning of Relativity"
Third Edition - Princeton University Press, 1950
Sir Arthur Eddington - "The Expanding Universe"
Cambridge University Press - 1946
Symposiu~ on "The Universe" -
Scientific American - September, 1956
Papers by Robertson, Fowler, Baade, Cost,
Minkowski, Gamow, Hoyle, Sandage,
Meyman and Scott, Ryle, Dingle
"Theories of the Universe" - The Free Press - 1957
Papers on ''Modern Theories of the 'Jni verse" by
Einstein, Hubble, DeSitter, Eddington, Lemaitre,
Milne, Robertson, Gamow, Bondi, Sciama, Hoyle
Margaret and Geoffrey Burbidge - "Formation o~ Elements in Stars"
Science - Agust 22, 1958
George Gamow - "Dissertation on Fundamental Constants" -
Physics Today - January, 1949
W. Heisenberg - "Nuclear Physics" -
Philosophical Library - 1953
Otto Oldenburg - "Introduction to Atomic Physics" -
McGraw-Hill - 1954
John C. Slater - "Modern Physics" -
McGraw-Hill - 1955 ..
A. W. Rucker - "On the Suppressed Dimensions of Physical Quantitie
Philosophical Magazine - May 22, 1889
Paul R. Heyl - "Gravitation - Still a Mystery" -
Scientific Monthly - May, 1954
S. B. Treiman - "The Weak Interactions" -
Scientific American - ?
R. E . Pereils - "The Atomic Nucleus!' -
Scientific American - ?
- ---·-- _______ ___...
)
)
:)
:)
:)
8
8
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
u
0
0
u
u
10
1-._)
0
u
G
u
{
•1 J~
· ..
-2-
14. Gell-Mann and Rosenbaum - "Elementary Particles" -
Scientific American - ?
15. Robert Hofstadter - "Interactions of Electrons with Nuclei" -
Re search Reviews {U.S. Navy Dept.) April, 1958
16. Lawrence Wilets - "Shape of the Nucleons" -
Science - February 13, 1959
17. John J. Grebe - Periodic Table for Fundamental Particles" -
Annals New York Academy of Sciences - Sept. 15, 195
18. A. T. Gresky - "New Physical Constants from Dimensional Analysis" -
Journal of Franklin Institute - Feb. 1958
19. A. T. Gresky - "Explanation of the Electric and Mag ..... '.;ic
Constants and Units" -
Journal of Franklin Institute - {date ? )
20. Max Planck - "Theory of Electricity and Magnetism"Second
Edition - MacMillan Co. - 1949
21. P. M. S. Blackett- "The Magnetic Field of Massive Rotating Bodies" -
Nature - Vol. 159, 1947
22. Walter M. Elsasser - " The Earth as a Dynamo" -
Scientific American - ·1
23. P.A. M. Dirac - "Quantum Mechanics and the Aether" -
Scientific Monthly - March, 1954
24. R. H. Dicke "New Research and Old Gravitation" -
Science - March 6, 1959
25. L. I. Schiff - "Neutrino Theory of Gravitation" -
Science - May, 1958