Vasily L. Yanchilin
Gravitation and Quantum Mechanics
PACS 03.65.-w; 95.35.-k
Vasily L. Yanchilin. Gravitation and Quantum Mechanics. Editirial-Publishing Center
of Novosibirsk State University. Novosibirsk, Russia, 2000
Abstract
In this work a new model of space-time is proposed where the velocity of light and
Planck’s constant depend on the gravitational potential. According to this model, the
existence of inertial reference systems is directly connected to all the masses in the Universe.
An idea of Chaos beyond the Universe is introduced: a state of space-time with degenerate
(uncertain) laws of motion. The proposed model considers all the branches of mechanics from
a unified point of view.
The author has an idea and plan of experiments. A book containing a more detailed account
of the subject will be soon published. Interested persons can communicate with the author via
Email: yanchilin@mail.ru
 Vasily. L. Yanchilin, 2000
1. Introduction
The motion of material bodies in the world takes place in the gravitational field created by
the huge mass of the Universe. In this work a hypothesis is put forward that the motion of
physical objects (described either by classical mechanics, relativistic theory or quantum
mechanics) is the result of gravitational interaction of these objects with all the masses of the
Universe. Let us consider the fundamental principles of mechanics.
1. The foundation of Newtonian mechanics is the law of inertia stating the following: the
velocity of a free-moving body remains constant. This law has been formulated on an
experimental basis and does not have any theoretical explanation. In this concern, Richard
Feynman wrote: "But the motion to keep the planet going in a straight line has no known
reason. The reason why things coast forever has never been found out. The
law of inertia has no known reason." [1].
2. The foundation of the special theory of relativity is the principle of constancy of the
velocity of light in vacuum. This means that the velocity of light is the same for all observers
irrespective of their motion. This, at first sight, paradoxical statement can be explained by the
fact that both time and distance obey a special law of transformation from one inertial
reference system to another (the Lorentz transformations). Einstein considered the constancy
of the velocity of light as an intriguing challenge: «It seems unbelievable to me, that the cause
of any process (for example, the propagation of light in vacuum) could be considered to be
independent of all other processes in the world» [2].
3. The general theory of relativity is based upon the equivalence principle stating that a
gravitational field and a non-inertial reference system are indistinguishable at a local scale.
The foundation of this principle is the experimental fact that the inert and gravitational masses
are equal. The question why these masses are equal remains unanswered.
1
4. The foundation of quantum mechanics is the uncertainty principle, which reflects the fact
that a quantum object (for example, an electron) does not have a definite trajectory of motion.
Plank’s constant determines the uncertainty of the motion of an electron (and any other
quantum object). It is still unknown why the laws of micro-world have the probability
character.
All the principles mentioned above are independent of each other. There is neither unity
nor connectivity between them even though they reflect properties of one and the same world.
Does any connection exist between the law of inertia and the uncertainty principle or between
constancy of the velocity of light and equality of the inert and gravitational masses?
Several questions about the Mach principle exist in modern physics. The Mach principle
states that inertial reference systems have causal relationship with the system of fixed stars. In
other words, the inertia of a body manifests itself only when it is accelerated relative to all
massive objects of the Universe. While developing the general theory of relativity Albert
Einstein expressed his hope that the Mach principle would be embodied into his theory. At
that time he wrote: "My masses carry out a motion that is at least in part causally determined
by the system of fixed stars. The laws which govern the motion of masses in my environment
are also determined by the fixed stars" [3]. "In a consistent theory of relativity there can not
exist any inertia with regard to "space" but only inertia of the masses with respect to each
other. So if I remove a mass sufficiently far from all other masses of the world, the inertia of
the said mass is bound to approach zero. We try to formulate this condition
mathematically"[4].
However, when the theory had been built it appeared that the theory
was not consistent with the Mach principle [5]. Therefore the Mach principle is still an
unsolved problem. The following are the arguments from Berkeley Course of Physics: "The
existence of an inertial reference system suggests a difficult and unanswered question: What
effect does all of the other matter in the universe have upon an experiment done in a terrestrial
laboratory? ... The (opposite to Newton's) point of view, that only acceleration relative to the
fixed stars has any significance, is a conjecture commonly called Mach's principle. Although
there is neither experimental confirmation nor objection to this point of view, some physicists
including Einstein have found this principle to be attractive a priory. Others have not found it
attractive. This is a matter for speculative cosmology. If one believes that the average motion
of the rest of the universe affects the behavior of any single particle, a number of related
questions present themselves without offering any clues to the answers. Are there other
relations between the properties of a single particle and the state of the rest of the universe?
Will the charge on the electron, or its mass, or the interaction energy between nucleons
change if the number of particles in the Universe or their density were somehow altered? So
far, the answer to this deep question of the relation between the distant Universe and the
properties of single particles, remains unanswered." [6].
In this work we propose a theoretical model of time-space which accounts, from the
one hand, the influence of all masses in the Universe on the laws of motion according to the
Mach principle and, from the other hand, sheds a new light on all the four mentioned
principles of mechanics.
2. The essence of the proposed cosmological hypothesis
Basic statements:
1. Let us consider a region of space that is far away from all the masses of the Universe and
where gravitational fields are negligible. In this region test bodies will travel randomly and
without a definite trajectory. The total energy of test bodies will approach zero. Let us call
this state of space-time as Chaos.
2
2. The gravitational field created by all the masses of the Universe decreases the uncertainty
of the motion of bodies. Therefore, the effect of uncertainty can be observed only in the micro
world.
From these statements, the model of time-space is postulated, which differs from the
existing in modern physics in the point that the velocity of light с and Planck constant h
depend on the value of the gravitational potential Ф, created by all the masses of the
Universe:
с2 + Ф = 0
h2Ф = const
(1)
(2)
The potential Ф is normalized so that it approaches zero at a remote distance from all the
masses of the Universe. Hence, Ф < 0.
3. Explanations
The following commentaries are not a logical basis of the hypothesis proposed above. The
purpose of these explanations is to reveal the physical sense of our model of space-time.
1. A free body moves at a constant velocity relative to an inertial reference system. The
Mach principle states that inertial reference systems exist in space due to existence of the
large masses of the Universe. From this point of view, there are no inertial reference systems
at a large distance from all the large masses of the Universe. The existence of an inertial
reference system is the core of the first law of motion. This law is the foundation for
formulation of other laws of motion. Therefore, we assume that all laws of motion are
determined by the large masses of the Universe, and at a large enough distance from them all
the laws of motion are uncertain.
Since we do not know ahead how test bodies would move in the area without certain laws of
motion it is reasonable to assume that the test bodies will not have a definite trajectory of
motion at a sufficiently far distance from the large masses. And vice versa, as they approach
the large masses of the Universe the uncertainty in their motion will decrease. For instance, in
the vicinity of the Earth this kind of uncertainty is observed at the level of micro world.
2. Let us consider a moving electron. Plank’s constant determines the uncertainty of its
motion. Using the above-mentioned assumptions, we conclude that remoteness from the large
masses of the Universe means a higher uncertainty for the motion of an electron (i.e. a higher
value of Planck’s constant). Existence of the large masses of the Universe restricts this
uncertainty. We can also assume that the electromagnetic field created by the electron charge
counteracts this restriction (the smaller a vicinity where the electron is localized the higher
energy of the electromagnetic field created by the electron). Therefore, for a general case,
Planck’s constant h is a function of the gravitational potential Ф and the value of the electron
charge е. Using this fact and dimensionality reasons we obtain the following equation (CGS
system of units):
h2 = ke 4/Ф,
(3)
where k is a dimensionless coefficient. From Eq.(3) we obtain Eq.(2).
3. We assumed that all the laws of motion are determined by the distribution of all the
masses of the Universe. This means that properties of space-time are also determined by that
distribution because we are able to tell something about these properties only while observing
the motion of bodies. Properties of space-time are determined by such values as the velocity
3
of light and Planck’s constant. Therefore, we assume that the velocity of light с also depends
on the value of the gravitational potential Ф. Using dimensionality we obtain the equation:
с2 = -аФ
(4)
where а is a dimensionless coefficient. Later we will show that а = 1.
4. To describe the motion of matter in terms of space-time coordinates physical objects are
required, which are subject to laws of classical mechanics, within certain accuracy. A logicaly
closed mechanics consisting of entirely quantum objects is impossible [7]. Therefore, at a
large distance from all the masses of the Universe, due to an increase of the uncertainty in the
motion of bodies, space and time degenerate, i.e. cease to exist as a physical reality.
5. Consider the formulation of the law of inertia: a free-moving body moves in a straight
line and has a constant velocity. There is a kind of contradiction in this statement. It appears
that a free-moving body is unfree in the sense that it is unable to deviate from the straight
line – neither right nor left. It cannot increase or decrease its velocity. Its path in space and
time is strictly determined. The motion of this body should be rather called the unfree motion.
A question arises which still has no answer: what is the cause determining the motion of such
a body?
In our model of space-time this body is not free because it moves in the strong gravitational
field created by all the masses of the Universe. The huge mass of the Universe restricts the
uncertainty in motion of this body, thus determining its path in space and time. A body can
move free only beyond the gravitational field of the Universe. But in this region space and
time degenerate into Chaos.
4. The limits of applicability of the general theory of relativity
The Newtonian theory of gravitation assumes that interaction between bodies takes place
instantly. However, the special theory of relativity made it clear that any interaction is carried
over with the final velocity. Therefore, a need had appeared to change the Newtonian theory
of gravitation. Einstein had solved this problem in his general theory of relativity. It is known
from experiments that the gravitational mass of a body equals to its inert mass. Einstein
proposed the following interpretation of this fact. For the uniform gravitational field,
everything occurs in the same way as for a space without gravity but with the inertial
reference system being substituted with a reference system with acceleration relative to the
latter. In a general case, the effect of gravitation is reduced adding a curvature to space-time
near massive bodies [8]. The equations of the general theory of relativity include neither the
mass of the Universe nor its density distribution [8]. In the framework of the general theory of
relativity, all the masses of the Universe have no influence on phenomena in space near the
Earth or anywhere else. In other words, space-time exists independently of all the masses of
the Universe because material bodies can influence only on its geometry [5].
From the standpoint of the hypothesis proposed in this paper, the gravitational field of the
Universe determines the most fundamental parameters of space-time: the velocity of light and
Planck’s constant. Therefore, the equations of the general theory of relativity are applicable
only in the case when the gravitational potential changes slightly, |Ф|«|Ф|, and, therefore,
changes of the velocity of light are negligible. For example, for the classic relativistic
gravitational tests in Solar system where |Ф|/|Ф|  10-6 [9] corrections inserting by Eq.(1)
and (2) in the general theory of relativity do not surpass one millionth. For a more general
case, one should consider a variation of the velocity of light and Planck’s constant caused by
a change in the gravitational potential.
From this point of view, a new interpretation of the mechanism of gravitational interaction
can be given. Let us consider behavior of a quantum object in a gravitational field. Let it be
4
localized in the vicinity of point A. There exists a probability that after a period it will be in
the vicinity of point B. There also exists a probability of the reverse transition. In space with
the constant gravitational potential these two probabilities are equal to each other. Now let us
consider a case when the absolute value of gravitational potential in point B is higher than in
point A (ФB  ФA but |ФB|  |ФA|). This means that the value of Planck’s constant (i.e. the
uncertainty in motion of this quantum object) is higher for point B than for point A.
Therefore, the probability of transition from A to B is higher than the probability of the
reverse transition. In other words, gravitational interaction possesses a quantum-mechanical
nature.
5. Equality between the inert mass and the gravitational mass
1. From Eq.(4), one can see that at a far distance from all the masses of the Universe the
velocity of light approaches zero. In other words, the energy of any body outside a
gravitational field is equal to zero. That is, the energy of any body has gravitational origin.
Therefore the well-known equation of energy conservation for a body moving in a potential
field Е + U = const, becomes an identity:
Е + U  0,
(5)
where Е is the total energy of the body and U is the potential energy of the body in the
gravitational field of the Universe. In the zone of Chaos U  0, Е  0. Let the body of the
rest mass m0 travels at the velocity v in space with the constant gravitational potential Ф.
Then, Е = m0с2, where  = 1/(1-v2/с2). From Eq.(5) and (4) we obtain:
U = -E = - m0с2 = аm0Ф.
(6)
It follows from Eq.(6) that the gravitational mass of a body (which is a coefficient in front of
the potential Ф) is: mgr = am0 = amin , where min = m0 is the inert mass of the body.
Therefore from the assumption that the velocity of light is a function of the gravitational
potential Eq.(4), we obtain the fact of the proportionality between the inert and gravitational
masses. For the applied system of units, а = 1. Eq.(1) gives us a new view on the Einstein
formula Е0 = m0с2: the energy of a body at rest is equal to the work performed over the body
by the gravitational field created by all the masses of the Universe.
For a body moving in the gravitational field the work performed over this body is
dA = -mgrdФ
Therefore the change in the energy of the body is equal to:
dE = d(m0с2) = dA = -mgrdФ
Accounting that mgr = min = m0, we have the following:
m0dс2 + с2d(m0) = -m0dФ
2
Applying Eq.(1) we have с d(m0) = 0. Hence, we can write the following:
m0 = const
(7)
For a body moving in the gravitational field its complete (relativistic) mass is constant. Since
Е = m0с2 then:
Е/с2 = const
(8)
6. Correlation to the experiment
1. In modern physics Planck’s constant and the velocity of light are fundamental constants.
The most accurate measurements give us the value of the velocity of light (CODATA, 1973):
с = 29979245800  120 cm/sec. In the proposed model of space-time the velocity of light (as
well as Planck’s constant) is a variable value. We can tell about consistency of this model if a
5
variation of the velocity of light as follows from Eq.(1) is not higher than 120 cm/sec in
terrestrial conditions.
The Earth moves around the Sun in an elliptic orbit, so the distance between them varies in
a year. This has to change the gravitational potential created by the Sun on the Earth surface.
Therefore the value of the velocity of light changes in a year. Let us estimate using Eq.(1) the
maximal possible variation of the velocity of light с caused by a variation of the
gravitational potential  Ф: (с + с)2 + (Ф + Ф) = 0  с2+ 2сс +(с)2 + Ф + Ф = 0 
2сс + (с)2 + Ф = 0, Since с « с then we can neglect the term (с)2 and have:
с = - Ф/2с
(9)
By analogy we obtain a variation of Plank’s constant:
h = hФ/2с2
(10)
The maximum variation of the Earth-Sun distance during annular motion of the Earth is
L= 5 1011 cm. From that we can estimate a variation of the gravitational potential (under
condition that L«L): Ф = GМsL/L2, where G=6,67 10-8 g-1cm3s-2 is the gravitational
constant, Мs = 2 1033 g is the mass of the Sun, and L = 1.5 1013 cm is the distance between
the Earth and the Sun. As a result, we have the following:
|  с| = GМsL/2сL2
(11)
After simple calculations we have | с| = 5 cm/sec. Thus, the proposed cosmological
hypothesis does not contradict the experimental date about constancy of the velocity of light.
2. For an experimental verification of the proposed hypothesis the velocity of light and
Plank’s constant need to be measured with a high accuracy in a year. For this a variation of
the gravitational potential, which is the result of the motion of the Earth around the Sun and
the Universe expansion have to be taken into account. If Eq.(9) and Eq.(10) are correct then it
will be possible to measure the rate of the Universe expansion directly.
From Eq.(9) we can calculate the velocity of light in different areas of the Solar system.
Obviously, the largest change we expect near the Sun: с = 3 104cm/sec. At the orbit of
Mercury the velocity of light will differ by с = 300cm/sec. This is a small variation, but still
higher than the measurement accuracy of modern days, so this can be verified experimentally.
With the increase of height over the Earth surface by 2 meters the relative variation of
Planck’s constant (10) and the velocity of light (9) is about h/h = -c/c = 10 -16
3. From Eq.(1) we can calculate the total mass of the Universe. The gravitational potential
created by all the masses of the Universe in near-Earth space is estimated by formula
Ф= -GM/R
where М is the total mass of the Universe, and R is the size of the Universe. As a result, we
have the following equation for the mass of the Universe:
M = с2 R/G
(12)
28
10
56
Taking R  10 cm and с = 3 10 cm/sec we obtain М  10 g. The obtained result can
be compared with data from “Tables of physical quantities” [10]: М = 1054 - 3 1056 g.
7. Quasars and black holes
1. In astrophysics there exists a problem of origination of so-called strong red shift for
quasars. It is the most surprising feature of these objects. All the other problems connected
with quasars are, in fact, linked to this feature [11]. It is widely accepted that in the expanding
Universe the velocity of light and Planck’s constant are invariable. Therefore the red shift to
be observed in emission spectra of quasars can be explained only by the Doppler effect of
recession of galaxies.
6
We can conclude from Eq.(1) and Eq.(2) that for the expanding Universe the velocity of
light decreases and Planck’s constant increases. This is also the reason for the red shift. It can
be seen from Eq.(8) that the energy of a photon moving in the expanding Universe will
decrease due to reduction in the value of the velocity of light. This means that the
cosmological red shift is only partly caused by the Doppler effect. Therefore the rate of
galaxies’ recession is actually smaller and the age of the Universe is larger than it was
assumed in modern cosmology.
2. In the general theory of relativity the velocity of light is the absolute constant. This
allows existence of such heavy objects that light cannot leave their gravitational field. These
objects are called black holes [9]. Note that classical mechanics has its own concept of a black
hole: it is a massive body with the escape velocity, which is higher than the velocity of light.
In the proposed model of space-time black holes do not exist because the velocity of light
increases in the vicinity of massive bodies Eq.(1). Therefore huge masses, which are sources
of powerful electromagnetic radiation, can exist. For example, they can be expected to exist in
active nucleus of galaxies. Spectra of their radiation are expected to have a strong shift to the
infrared region.
8. Conclusion
When Einstein had created the general theory of relativity he pointed out the defects of
classical mechanics: space and time can exist without other material bodies. In other words,
space and time are absolute. However, Einstein’s general theory of relativity inherited the
same defect: space-time can exist without material bodies [5]. In the general theory of
relativity, as in Newtonian mechanics, “acceleration relative to the space” has the absolute
character [12].
Our model of space-time eliminates this defect. The basic idea of the proposed
cosmological hypothesis is that space-time is created by all the bodies of the Universe. At a
large distance from all the masses of the Universe the laws of motion degenerate and spacetime transforms into Chaos. For Chaos the energy of test bodies approaches zero Eq.(1) and
the uncertainty in their motion increases to infinity Eq.(2). This model satisfies the Mach
principle and makes possible its experimental verification. In addition, it provides a unified
view to the main principles of mechanics. For terrestrial conditions the difference between our
model and the conventional one stays within accuracy of modern experiments Eq.(11).
Acknowledgments
The author expresses his gratitude to Oleg Antzutkin and Alexander Bulgakov for
discussions and the support.
Sergei Kiyanov and Andrei Sherstyuk translated the work into English. Vanessa Sheffes
translated the German quotations from Einstein’s works.
7
REFERENCES
1. R. Feynman, The Character of Physical Laws, M.I.T. Press, Cambridge, Massachusetts,
1965, p. 14.
2. A.
Einstein,
Prinzipielles
zur
verallgemeinerten
Relativitätstheorie
und
Gravitationstheorie, Physikalische Zeitschrift, 15 (1914) 176-180. "Es erscheint mir
unglaublich, daß der Ablauf irgendeines Vorganges (z. B. der der Lichtausbreitung im
Vakuum) als unabhängig von allem übrigen Geschehen in der Welt aufgefaßt werden
könne", s. 176.
3. A. Einstein, Zum Relativitäts-problem, Scientia (Bologna), 15 (1914) 337-348. "Meine
Massen führen eine Bewegung aus, welche wenigstens zum Teil vom Fixsternsystem
kausal bestimmt wird. Die Gesetze, nach denen sich Massen in meiner Umgebung
bewegen, werden mitbestimmt durch die Fixsterne", s. 345.
4. A. Einstein, Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, Königlich
Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte, (1917) 142-152. "In
einer konsequenten Relativitätstheorie kann es keine Trägheit gegenüber dem "Raume"
geben, sondern nur eine Trägheit der Massen gegeneinander. Wenn ich daher eine Masse
von allen anderen Massen der Welt räumlich genügend entferne, so muß ihre Trägheit zu
Null herabsinken. Wir suchen diese Bedingung mathematisch zu formulieren", s. 145.
5. A. Einstein, Prinzipielles zur allgemeinen Relativitätstheorie, Annalen der Physik,
55(1918) 241-244. "Nach den Gleichungen (1) wäre also im Widerspruch mit dem
Machschen Postulat ein G-Feld denkbar ohne jede erzeugende Materie", s. 243. ("In
contradiction to Mach's postulate, it therefore follows from the equation (1) that a G-field
can be assumed to exist without any generating matter". p. 243).
6. Ch. Kittel, W. D. Knight, and M. A. Ruderman, Berkeley Physics Course, v.1,
Mechanics, Education Development Center, Inc., New-York – San-Francisco,1965, 3.3,
p. 63.
7. Ландау Л.Д., Лифшиц Е.М. Теоретическая физика т.3, М.: Наука, 1989, §1.
(L.D. Landau and E.M. Lifshits, Course of Theoretical Physics, Quantum mechanics, v.3,
Pergamon Press, Oxford, 1976, §1).
8. Ландау Л.Д., Лифшиц Е.М. Теоретическая физика т.2, М.: Физматгиз, 1962, §82;§95.
(L.D.Landau and E.M. Lifshits, Course of Theoretical Physics, The classical theory of
Field, v.2, Pergamon Press, Oxford, 1975, §82;§95).
9. Misner C.W., Thorne K.S., Wheeler J.A., Gravitation, Freeman, San Francisco, 1973,
§16.2; §38.1; §33.1.
10. Таблицы физических величин, М.: Атомиздат, 1976, с. 987.
11. Mitton S., Exploring the galaxies,Faber and Faber Limited 3 Queen Square, London,
1976, §10.3.
12. Фок В.А. Теория пространства, времени и тяготения, М.: Гостехиздат, 1955, §96.
(Fock V., The Theory of Space, Time and Gravitation, Pergamon Press, London, 1959,
§96).
8