Download STRAIGHTFORWARD DERIVATION OF MIE`S GRAVITATIONAL

STRAIGHTFORWARD DERIVATION OF MIE’S GRAVITATIONAL
AND INERTIAL MASSES, VIA JUST NEWTON’S LAW OF GRAVITATION
AND ENERGY CONSERVATION LAW: A NEW APPROACH TO THE END
RESULTS OF THE GENERAL THEORY OF RELATIVITY
Tolga Yarman*
Okan University, Kadikoy, Istanbul, Turkey
&
Savronik, Organize Sanayii Bölgesi, Eskisehir, Turkey
February 2006
ABSTRACT
Via “Newton’s law of gravitation” between two “static masses”, and the “energy conservation laf”, in the
broader sense of the concept of “energy” embodying the “relativistic mass-energy equivalence”, on the one side,
and “quantum mechanics”, on the other, we were able to derive the end results of the general theory of relativity.
Yet we ended up, seemingly, with the violation of the principle of equivalence, the basis of the general theory of
relativity. Thus, through a straightforward approach, we found mgravitational  minertial /  2 , where  is the usual Lorentz
dilation factor. In the aim of establishing a theory fully compatible with the special theory of relativity, Mie in
1912 had reached the same conclusion, though through an “inverse problem set up”. Not having postulated an
“alternative principle of equivalence”, he failed to carry his theory any further, which presumably led Einstein to
ignore Mie’s theory. We have happened to achieve what Mie, apparently could not, and have well ended up with
the “end results” of the general theory of relativity, via just Newton’s laf of gravitation and energy conservation
law, thus without being in the need of the principle of equivalence.
Keywords: Mass Deficiency, Gravitation, Quantization, Special Theory of Relativity, General Theory of
Relativity
INTRODUCTION
I felt enthusiastic when I saw the article by de Haas† by the end of 2004, in AFBL [1]. He
reported with merit that under a Lorentz transformation, Mie’s 1912 “gravitational mass”
behaves identical to de Broglie’s 1923 clock-wise frequency; the same goes for Mie’s
“inertial mass” and de Broglie’s wave-like frequency [2].
This statement did in fact somewhat summarize my results,‡ which I derived through a
completely different set up than those of Mie’s [3,4,5], as well as those of de Haas’s, and
published in the previous number of AFBL [6], backed up by my earlier work [7,8,9,10,11].
It was a pity that previously I did not get across Mie’s work.§
*
[email protected]
http://www.ensmp.fr/aflb/AFLB-294/aflb294m171.htm
‡
http://www.ensmp.fr/aflb/AFLB-293/aflb293m137.htm
§
http://www.physics.nl/ (pdf files of Mie’s work, reassembled by de Haas)
†
1
The fact remained that I had happened to have derived Mie’s findings, not as a result of an
“inverse problem”, but as a result of straightforward common physical considerations, and
mainly energy conservation.
In what follows, first I will briefly summarize Mie’s work [1]. Then I will summarize my
work.
Not having postulated an “alternative principle of equivalence”, Mie is reported to have failed
to carry his theory any further, which presumably led Einstein to ignore Mie’s theory.
We have happened to achieve what Mie, apparently could not, and have well ended up with
the “end results” of the general theory of relativity, via just Newton’s laf of gravitation and
energy conservation law, thus without being in the need of the principle of equivalence.
We will conclude with a discussion about this issue.
MIE’ WORK
Following cumbersome and unsuccessful trials, Mie proposed to establish a universal Lorentz
invariant Hamiltonian energy density H , function of the electromagnetic potential energy, as
well as the gravitational potential energy. Let E 0 the total energy of the object at hand of rest
mass m 0 ; its rest volume is V0 .
Then

V0
HdV0  E 0  m 0 c 02 ,
(1)
along with the usual relationship E 0  m 0c 02 [12,13]; c 0 is the speed of light in empty space.
If the object is brought to a uniform, translational motion, its volume V0 will become V, so
that
1
(2)
dV  dV0 ,

because of the relativistic contraction in the direction of motion,  being the usual Lorentz
dilation factor.
Then E 0 will be transformed as

V
HdV 
1
1
HdV0  E 0 ,

 V

(3)
since H 0 is set beforehand, to be Lorentz invariant.
2
On the other hand, through the motion, the rest mass m 0 , will be transformed to become
m inertial , the inertial mass, so that
m inertialc 02  m 0 c 02  E 0 .
(4)
Here Mie, regarding the object at hand, had in fact considered the usual relativistic inertial
energy density E inertial , to write in the first place,

V
E inertialdV  E 0 .
(5)
Mie interpreted H as pointing to the gravitational mass m gravitational , and E inertial as pointing to
the inertial mass m inertial [5].
Thus,
m gravitational c 02   HdV 
V
1
E0 .

(6)
Therefore,
minertial  m 0 ,
(7)
and
m gravitational 
m0
.

(8)
Hence,
m inertial
.
2
(relationship predicted by Mie, in 1912)
m gravitational 
(9)
Mie concluded that the motion undergone by matter, affects the “gravitational mass”, just like
the “inertial mass”. Even the internal energy of the elementary constituents making up an
object would cause the inertial mass of these constituents to increase, but by the same token
their gravitational masses to decrease [5]. Thus heat, or any extra energy that would be
somehow acquired by the object, would affect its gravitational mass all over.
Mie came to the conclusion that the Newtonian assumption (already carefully questioned by
Newton himself) about the equality between inertial mass and gravitational mass, becomes
invalid in a moving frame. So the equivalence between these two masses is altered under a
Lorentz tranformation, and could not work as a basis, in any trial to establish a theory of
gravitation consistent with the special theory of relativity. Mie did not postulate an alternative
principle of equivalence, and failed to carry further away his approach [1], which apparently
encouraged Einstein to ignore Mie’s theory, whilst developing the general theory of relativity
[14,15].
3
PRESENT APPROACH: THE END RESULTS OF THE GENERAL THEORY OF
RELATIVITY, VIA JUST NEWTON’S LAW OF GRAVITATION, ENERGY
CONSERVATION AND QUANTUM MECHANICS
In a previous work [6], we presented a whole new approach to the derivation of the Newton’s
Equation of Motion, which well led to the end results of the general theory of relativity, were
the velocity of the object at hand, not negligible as compared to the velocity of light.
Thus, we started with the following postulate, in fact nothing else, but the law of conservation
of energy, though in the broader relativistic sense of the concept of “energy” [12,13].
Postulate: The rest mass of an object bound to a celestial body amounts to less than its rest
mass measured in empty space, the difference being, as much as its binding energy
vis-à-vis the gravitational field of concern.
A mass deficiency conversely, via quantum mechanics (whose basis, i.e. the wave equation,
together with, briefly speaking, the de Broglie relationship, is already fully consistent with the
special theory of relativity, and basically energy conservation law), yields the stretching of the
size of the object at hand, as well as the weakening of its internal energy, via the quantum
mechanical theorems we have proven [6,8,9,10]; they are stated right below.
Theorem 1: In a “real wave-like description” (thus, not embodying artificial potential
energies), if the masses mi0, i = 1,..., I of different constituents involved by the
object, are allover multiplied by the arbitrary number  , then concurrently,
a) the total energy E0 associated with the given clock’s internal motion of the
object, is increased as much, or the same, the period T0 of the motion associated
with this energy, is decreased as much, and b) the characteristic length or the size
R 0 to be associated with the given clock’s motion of the object, contracts as
much; in mathematical words [7],**
(mi0, i = 1,..., I)  (  mi0, i = 1,..., I)
T
R
   E 0   E 0 , [ T0  0 ],  R 0  0   .

χ
This, together with our postulate, yields at once the next two theorems.
Theorem 2: A wave-like clock in a gravitational field, retards via quantum mechanics, due to
the mass deficiency it develops in there, and this, as much as the binding energy
it displays in the gravitational field; at the same time and for the same reason, the
space size in which it is installed, stretches as much.
**
Note that as the “overall mass” of the object increases by the arbitrary factor  , its internal dynamics speeds
up as much; or the same its de Broglie wave-like frequency is increased as much [17]. One can show that, only
if such characteristic is drawn, and this already at rest, can the internal dynamics all together slow down as
much, where the object brought to a uniform, translational motion,  then being the usual Lorentz dilation
factor.
4
Theorem 3: A wave-like clock interacting with any field, electric, nuclear, gravitational, or
else (without loosing its “identity”), retards as much as its binding energy,
developed in this field; at the same time and for the same reason, the space size
in which it is installed, stretches as much.
This can further be grasped rather easily, as follows. The mass deficiency the wave-like object
displays in the gravitational field (or in fact, any field with which it interacts), weakens its
internal dynamics as much, which makes it slow down. Thence, we arrive at the principal
results, we just stated.
Here, we made use of the classical Newtonian gravitational attraction law, yet with the
restriction that, it can only be considered for static masses.
Luckily we are able to derive the 1/r2 dependency of the “classical gravitational force”
between “two static masses”, here again, based on just the Special Theory of Relativity [6].
This can be achieved easily by noting that the quantity [force] x [mass] x [distance] 3 is
Lorentz invariant.†† (In fact dimensionally speaking, it amounts to the square of the Planck
Constant, which in return is Lorentz invariant.) On the other hand, it is known that the electric
charges are Lorentz invariant. (If not, say in excited atoms, energetic electrons would exhibit
electric charge intensities different than the electric charge intensity of the electrons at the
ground level, which is not the case.) Now suppose we have a “dipole” of a given mass at rest,
bearing a given length r at rest. Coulomb’s Force reigns between the electric charges. Suppose
we assume that Coulomb’s Force is, as usual, expressed as proportional to the electric charges
coming into consideration, also to 1/rn, where though we do not know, a priori the exponent n.
Suppose then we bring the dipole to a uniform translational motion, along the direction
delineated by the line connecting the electric charges making it. Since then, [mass] x [length]
remains invariant, it becomes evident that the Lorentz invariance of [force] x [mass] x
[distance]3 shall hold, only if Coulomb Force, dimensionally behaves as charge 2 / r n , n being
exclusively 2, given that charges are Lorentz invariant.
Note that the same holds, if “charges”, in question, are gravitational charges; in this case
however, the product of charges should be considered together with the universal gravitational
constant.
Thus, the framework in consideration is fundamentally based on the “Special Theory of
Relativity”.
Our metric (just like the one used by the general theory of relativity) is altered by the
gravitational field (in fact, by any field the “measurement unit” in hand interacts with); though
in the present approach, this occurs via quantum mechanics.
Henceforth one does not require the “principle of equivalence” assumed by the general theory
of relativity, as a precept in order to predict the end results of this theory.
Let then m 0 be the mass of the object in consideration, at infinity. When this object at rest, is
bound to a celestial body of mass M , assumed for simplicity infinitely large as compared to
††
The dimension of “force”, is as usual, [mass] x [length] x [period of time] -2.
5
m 0 ; this will be diminished as much as the binding energy coming into play, through the
binding process, to become m(r), so that [6]
m( r )  m 0 e   ( r ) ,
(10)
where (r ) is
( r ) 
GM
;
rc 02
(11)
G is the universal gravitational constant; r is the distance of m(r) to the center of M , as
assessed by the distant observer.
We would like to recall that G is not Lorentz invariant, though classified as a “universal
constant”. One can immediately see this, as follows: Dimensionally speaking, GMm (r ) is
equivalent to (electric charge)2; but the electric charge intensity is Lorentz invariant; thus so
must be the quantity GMm (r ) ; “mass” is not a Lorentz invariant quantity; hence neither G
can be, though the product GMm (r ) is.
Now suppose that the object of concern is in a given motion around M ; the motion in
question, can be conceived as made of two steps, just the way we did: i) bring the object from
infinity to the given location, at rest, ii) place the object in orbit, at the given location.
The first step yields a decrease in the mass of m 0 as delineated by Eq.(10). The second step
yields the Lorentz dilation of the rest mass m(r) at r, so that the overall mass m  (r ) of the
object in orbit becomes
m  (r ) 
m( r )
v2
1  20
c0

m0e  ( r )
v2
1  20
c0
;
(12)
v 0 is the tangential velocity of the object at r.
The total energy of the object in orbit, i.e. m  (r )c02 must remain constant [16,17,18], so that
for the motion of the object in a given orbit, we finally have
m  (r )c 02  m 0 c 02
e 
v2
1  20
c0
 Constant ;
(13-a)
(relationship previously written by the author)
v 0 (the relative velocity of m 0 in regards to M ), and c 0 , in our approach, remain the same
for both the local observer and the distant observer, similarly to what is framed by the special
theory of relativity.
6
Amazingly the general theory of relativity predicts (as furnished by Reference 18)
m  (r )c 02  m 0 c 02
1  2
1
v
c
2
0
2
0
 Constant ,
(13-b)
(relationship presented byLandau and Lifshitz)
which coincides up to the second order with our prediction, i.e. Eq.(13-a); yet there does not
seem any easy way to interpret the numerator, above in Eq.(i), whereas not only that it is
possible to ascertain what the numerator of Eq.(13-a) is all about, but also the set up of this
latter equation is evident.
It is worth to note that one arrives to Eq.(13-b) of the general theory of relativity, thorugh
cumbersome settings and integration operations, whereas we arrived to (13-a), following our
approach, in just three lines, and already as an intgral form.
Now, the differentiation of Eq.(13-a), leads to

dv
GM  v 02 
1  2   v 0 0 .
2 

dr
r  c0 
(14)
This equation can be put in the form
2
GM  v 02  r 0 dr ( t )


;
 2 1  2  
r  c 0  r0
dt 2
(15)
here r is the vector bearing the magnitude r , and directed outward, and v 0 ( t ) is the velocity
of m 0 in vector form, at time t .
Eq.(15) is the classical Newton’s Equation of Motion, were v 0 negligible as compared to c 0 .
Multiplying Eq.(15) by the constant overall mass m  (r ) at both sides we can state that
( Static Gravitational Force ) 1 
v 02
 (Overall Mass ) x ( Acceleration ) .
c02
(16)
Thence, Newton’s equation of gravitational motion, i.e. [Gravitational Force = Mass of the
Planet x Acceleration], is broken, since an extra term, i.e. e  1  v 02 / c 02 comes to multiply
the gravitational force, in its classical form.
Formally, this equation can be restored, if instead, we chose to alter the “classical
gravitational force”; but then the gravitational mass and the inertial mass, as classically
defined, shall not be identical.
Thus we came to establish the following theorem [6].
7
Theorem 4: The gravitational mass m gravitational , and the inertial mass minertial , as classically
defined, are not the same; the theory summarized herein, to formally save
Newton’s equation of gravitational motion, predicts
m gravitational  m 0 exp( ) 1 
v 02 ,
c 02
(17)
given that
minertial 
m 0 exp( ) ;
v2
1  20
c0
(18)
though undetectable for most cases we observe, m gravitational and m inertial differ.
Thence we arrive at
m inertial
,
2
(relationship predicted by the author)
m gravitational 
(19)
which is the same as that predicted by Mie as a result of his inverse problem set up, back in
1912.
Note that in Mie’s approach, the rest mass m 0 of the object measured at infinity is (contrary
to what we found), left untouched when gravitationally bound.
Let us recall that, taking into account the quantum mechanical stretching of lengths in the
gravitational field, Eq.(13-a) can be transformed into an equation written in terms of the
proper lengths [19], i.e.
exp(  0 e  0 )
v2
1  20
c0
 Constant ,
(20)
where  0 , i.e.
0 
GM
,
r0 c 02
(21)
is expressed in terms of the proper distance r0 ; note that the constant, appearing in the RHS of
Eq.(20) is different than the constant appearing in the RHS of Eq.(13-a).
The use of Eq.(20) [instead of Eq.(13-a)], will lead to the replacement of the expantional
function exp( ) , at the Right Hand Sides of Eqs. (17) and (18), by the expantional function
exp(  0e   ) .
0
8
We would like to recall that, though consisting in a totally different set up, than that of the
general theory of relativity, Eq.(20) amazingly yields results identical to those of this theory,
within the frame of a second order approximation.
DISCUSSION
Herein we were able to derive, the relationship established by Mie around the beginning of
the 20th century, though, through a totally different manner than his.
Not having postulated an “alternative principle of equivalence”, Mie failed to carry his theory
any further, which presumably led Einstein to ignore Mie’s theory. We have happened to
achieve what Mie, apparently could not, and have well ended up with the “end results” of the
general theory of relativity, and this via just Newton’s laf of gravitation and energy
conservation law, thus without being in the need of “the principle of equivalence”, i.e. the
basis of the general theroy of relativity.
Our approach seems to reinstate the broken link between the classical end results of the
general theory of relativity and Newton Mechanics. Though, it indicates that Newton’s law of
gravitational attraction is only valid for “static masses” (just like the Coulomb’s law, based on
our approach, is only valid for “static charges”). In other terms, the “gravitational mass”
entering Newton’s law of gravitational attraction, is not the “inertial mass”, entering the
Newton’s description of “force”, more precisely, [force] = [inertial mass] x [acceleration], i.e.
the Newton’s Second Law of Motion.
These two masses turn out to be different; it is that
[gravitational mass] = [inertial mass] / [Lorentz dilation factor]2,
as delineated by Eq.(19).
This result is evidently, against the phrasing of the “principle of equivalence” of Einstein
(which states that the “two masses” of concern, should be equal).
At any rate, Eq.(19) suggests that a moving particle in a gravitational field would weigh less
than the same particle at rest, in the same location in that field. V. Andreev effectively
reported at the PIRT Conference held in July 2005, in Moscow that, a pendant load irradiated
at the General Physics Institute of the Russian Academy of Sciences, by “high energy
electrons”, comes to weigh less than its untouched twin counterpart [20]. The author, right
after Andreev’s presentation, suggested that, the effect must be due to “energizing” the
“unpaired electrons” of the atoms of the load in consideration (which happened to be
duraliminium) [21]. These electrons [based on Eq.(19)], due to the high energy, they acquire
through the irradiation process, become practically “weightless”. A quick calculation indeed
proves this point of view, which shall be elaborated on, in a subsequent article.
The conclusion is that, heated electrons weigh less than normally bound electrons, and this
appears to point to a clear violation of the common interpretation of the principle of
equivalence.
9
Anyhow, through our approach we do not at all need to make use of the “principle of
equivalence”.
It is not that we question the principle of equivalence; it is that we are in no way bound to
utilize it. Whether it is valid or not, we do not really bother with such a question.
It happens that we still attract severe reactions, when we say we do not need the principle of
equivalence. Some people think we must need it (and, as we have seen, we really do not need
it).
The fact that we do not use it, does not, on the other hand, constitute any contradiction with
the theory of relativity, more precisely with the special theory of relativity, which constitutes
our main framework.
Quite on the contrary, our approach, remedies the inconsistencies, otherwise coming into
play.
In any event, in order to dissolve the upset we, unwillingly, create regarding the principle of
equivalence, this principle ought be analyzed and clarified.
As we have just stated; according to Newton, the “gravitational mass”, is the mass to be
plugged in the expression of “Newton’s law of gravitational attraction”, whereas the “inertial
mass” is the mass to be plugged in the expression of “Newton’s Second Law of Motion”.
Newton himself, before anyone else, doubted about the equality of these two masses (and we
find indeed that they are not equal).
As very many authors openly state, the principle of equivalence appears to state that these two
masses are equal.
But the general theory of relativity rejects the concept of “force”, which in Newton’s
description, made of “mass” and “acceleration”, involves the “inertial mass”; the general
theory of relativity also rejects the concept of “Newton’s law of gravitational attraction”,
which involves the “gravitational mass”.
Thus the original ground considered by Newton, to appraise the two masses in question, is
totally wiped out.
On the other hand, “inertia”, by definition, is a “characteristic”, an object develops as a
“resistance” to any variation, in its motion. Thus the concept of “inertia” is associated with the
change, the object undergoes, through a given “motion”.
Regarding the principle of equivalence, the “motion” of concern is, “that of the accelerated
elevator”, as assessed by an outside observer, fixed in regards to distant stars.
Thus Einstein considers the “rest mass” of an object, lying on the floor of the elevator,
accelerating (as assessed by the outside fixed observer) “upward” (along the direction drawn
from the floor to the ceiling of it).
10
This is what constitutes the basis of the principle of equivalence; accordingly, the “effect of
acceleration” on the object of concern, is considered to be equivalent to the “effect of
gravitation” (yielding the acceleration, in question). This indeed seems to be a striking
analogy. But here, by analogy, so far we only describe a “rest mass in a gravitational field”.
We do not describe a “mass in motion, in a gravitational field” (such as a planet around the
sun, as originally visualized by Newton, in regards to the equivalence of inertial mass and
gravitational mass).
Through Einstein’s analogy, in order to describe a “mass in motion in a gravitational field”,
we have to go back to the inside of the accelerated elevator, and see what happens to the
object originally at rest on the floor of this, say, if it is thrown, in a given direction.
Only the comparison of the “rest mass of the object originally lying on the floor of the
elevator”, and the “relativistic mass of this object brought to a given motion inside the
elevator”, in regards to a given observer (to be precised), can (via Einstein’s analogy) bring an
answer to the question introduced by Newton, regarding the “gravitational mass” and the
“inertial mass”; it is obvious that these are not either equal within the frame of the general
theory of relativity. This is mathematically displayed by Eq.(13-b), yeld by the general theory
of relativity.
Thus anyone fond of the principle of equivalence should not really be frustrated right away,
when we say that the “gravitational mass” and the “inertial mass” are not really equal. It is
that they are not equal in the way Newton first questioned. Unfortunately, as mentioned, there
are quite a number of books, which are inevitably misleading when they introduce the
principle of equivalence, through Newton’s original question, and they aim to provide an
answer to it, in a domain where Newton’s original tools are demolished by Einstein.
*
Albeit, our approach restores the incompatibilities arising between the special theory of
relativity, and the general theory of relativity, such as the breaking of the fundamental
relativistic result, [energy] = [mass] x [speed of light]2 [22]. Along the same line, it does not
allow the energy conservation law, nor the momentum conservation law to break down
(contrary to what happens within the frame of the general theory of relativity).
Furthermore, because it leaves unnecessary the usage of the “principle of equivalence”, i.e.
the basis of the general theory of relativity, it provides us with a whole different horizon.
In particular, gravitationally bound clocks shall, according to our approach, retard as implied
by the decrease of their internal energy as much as the gravitational binding energy coming
into play; thus not only this, but, “clocks anyway bound to any field”, they interact with,
should also retard. For instance, an electrically charged clock, bound to an electric field, must
come to slow down, just like clocks bound to gravitational field, slow down. More
specifically, a muon decay’s rate, when bound to an atomic nucleus, retards as much as the
binding energy, coming into play [23].
Moreover, we come to be able to establish a natural link between our approach and quantum
mechanics, otherwise hindered by the general theory of relativity.
11
In fact, the author was recently able to derive the de Broglie relationship, based on the main
idea presented herein, i.e. the mass deficiency delineated by the bound particle, regarding
either an electric field or a gravitational field, though inevitably inducing an interaction, at
tachyonic speeds [24, 25].
These happen to be the heartening harvests of our approach.
In short, our approach appears to restore the broken or non-existing links, or annoying
incompatibilities, between different disciplines of atomic physics, quantum mechanics,
relativity, and celestial mechanics, whereas, it is the desire of all of us to establish just a
“whole package of conception” for the “unique nature” existing out there, whether it is
question of “micro aspects”, or “macro aspects”, of it [21, 23, 26, 27].
ACKNOWLEDGEMENT
The author is indebted with gratitude to Dr. V. Rozanov, Director of the Laser Plasma Theory
Division, Lebedev Institute, Russia. The author would further like to extend his heartfelt
gratitude to Dr. C. Marchal, Dr. N. Veziroglu, Dr. O. Sinanoglu, Dr. X. Oudet, Dr. S. Kocak,
and Dr. E. Hasanov. Without their unequally sage understanding and ceaseless support, this
controversial work could not have come to daylight. The author would also like to extend his
gratitude to Professor E. Betak who made a considerable effort to secure a fair review of the
present article.
REFERENCES
[1]
E. P. J. de Haas, The Combination of de Broglie’s Harmony of the Phases and Mie’s
Theory of Gravity in a Principle of Equivalence for Quantum Gravity, AFLB, Volume
29, Numéro 4, 2004.
[2] L. de Broglie, Recherches Sur La Théorie Des Quanta, Annales de Physique, 10e Série,
Tome III, 1925.
[3]
G. Mie, Ann. Phys. 37, 511-534, 1912.
[4]
G. Mie, Ann. Phys. 39, 1 - 40, 1912.
[5]
G. Mie, Ann. Phys. 37, 1 - 66, 1913.
[6]
T. Yarman, The General Equation of Motion Via the Special Theory of Relativity and
Quantum Mechanics, AFBL, Volume 29, Numéro 3, 2004.
[7]
T. Yarman, Invariances Based on Mass And Charge Variation, Manufactured by Wave
Mechanics, Making up The Rules of Universal Matter Architecture, Chimica Acta
Turcica, Vol 27, 1999.
12
[8] T. Yarman, F. A. Yarman, The de Broglie Relationship is in Fact a Direct Relativistic
Requirement - A Universal Interdependence of Mass, Time, Charge and Space,
DOGA – Turkish Journal of Physics, Scientific and Technical Research Council of
Turkey, Volume 16 (Supplement), 1992, 596-612.
[9] T. Yarman, A Novel Systematic of Diatomic Molecules Via the Very Special Theory of
Relativity, Chimica Acta Turcica, Vol 26, No 3, 1998.
[10] T. Yarman, An Essential Approach to the Architecture of Diatomic Molecules. Part I.
Basic Theory, Optics and Spectroscopy, Volume 97 (5), 2004 (683).
[11] T. Yarman, An Essential Approach to the Architecture of Diatomic Molecules. Part II:
How Size, Vibrational Period of Time And Mass Are Interrelated?, Optics and
Spectroscopy, Volume 97 (5), 2004 (691).
[12] A. Einstein, Ann. Phys., 17, 891, 1905.
[13] A. Einstein, Ann. Phys., 20, 627-633, 1906.
[14] A. Einstein, Ann. Phys., 49, 1916.
[15] A. Einstein, Phys. Z., 19, 115, 1918.
[16] L. D. Landau & E. M. Lifshitz, The Classical Theory of Fields, Chater 10, Paragraph
88, Butterworth & Heinemann.
[17] H. Yilmaz, Einstein, the Exponential Metric, and a Proposed Gravitational MichelsonMorley Experiment, Hadronic Journal, 2: 997, 1979.
[18] H. Yilmaz, Towards a Field Theory of Gravity, Nuovo Cimento, 107B: 941, 1992.
[19] T. Yarman, The End Results of General Theory of Relativity, via Just Energy
Conservation And Quantum Mechanics, Communication to Professors V. Rozanov,
Director of Laser Plasma Theory Division, Lebedev Institute, Moscow, Russia, and to
Professor A. A. Logunov, Director of Particle Physics Laboratory, Protvino, Russia,
2005.
[20] D. Yu. Tsipenyuk, V. A. Andreev, Physical Interpretations of the Theory of Relativity
Conference, Bauman Moscow State Technical University, 4 – 7 July 2005.
[21] T. Yarman, V. B. Rozanov, Physical Interpretations of the Theory of Relativity
Conference, Bauman Moscow State Technical University, 4 – 7 July 2005.
[22] R. A. Mould, Basic Relativity, Springer-Verlag New York Inc., 1994.
[23]
T. Yarman, The End Results of the General Theory of Relativity, via Just Energy
Conservation and Quantum Mechanics, Physical Interpretation of The Relativity
Theory, Moscow, 3 – 7 July 2005.
13
[24] T. Yarman, Tachyonic Interaction or the Same de Broglie Relationship, Part I:
Electrically Bound Particles, Article Under Review in Foundation of Physics.
[25] T. Yarman, Tachyonic Interaction or the Same de Broglie Relationship, Part II:
Gravitationally Bound Particles, Article Under Review in Foundation of Physics.
[26]
T. Yarman, V. Rozanov, N. Zaim, F. Özaydın, The Architecture of Triatomic and
Polyatomic Molecules, Physical Interpretation of The Relativity Theory, Moscow, 4 – 7
July 2005.
[27] T. Yarman, A Novel Approach to the Bound Muon Decay Rate retardation: Metric
Change Nearby the Nucleus, Physical Interpretation of The Relativity Theory, Moscow,
3 – 7 July 2005.
14