THE
GENERAL EQUATION OF MOTION VIA THE SPECIAL
THEORY OF RELATIVITY AND QUANTUM
MECHANICS
Tolga
Yarman
Iºýk University
Address:
Maslak, Istanbul, Turkey
August
2002
ABSTRACT
Herein we present a
whole new approach to the derivation of the Newton’s Equation of Motion. This, with
the implementation of a metric imposed by quantum mechanics, leads to the
findings brought up within the frame
of the general theory of relativity (such
as the precession of the perihelion of the planets, and the deflection of light nearby a star). To
the contrary of what had been generally achieved so far, our basis consists in
supposing that the gravitational field,
through the binding process,
alters the “rest mass” of an object
conveyed in it. In fact, the special theory of relativity already imposes such a
change. Next to this fundamental theory, we use the classical Newtonian
gravitational attraction, reigning
between two static masses. We have
previously shown however that the 1/r2 dependency of the
gravitational force is also imposed by the special theory of relativity.
Our metric is (just like the one used by the general
theory of relativity) altered by the gravitational field (in fact, by any field the “measurement
unit” in hand interacts with); yet in the present approach, this occurs via
quantum mechanics. More specifically,
the rest mass of an object in a
gravitational field is decreased as much as its binding energy in the field. A mass deficiency conversely, via quantum mechanics yields the stretching of the size of the object in
hand, as well as the weakening of its
internal energy. Henceforth one does
not need the “principle of
equivalence” assumed by the general theory of relativity, in order to
predict the occurrences dealt with this theory.
Thus we start with
the following interesting postulate, in fact nothing else but the conservation of energy, though in the
broader relativistic sense of the concept of “energy”.
Postulate: The rest mass of an object bound to a
celestial body amounts less than its
rest mass measured in empty space,
and this as much as its binding
energy vis-à-vis the gravitational
field of concern.
This yields (with the familiar notation), the
interesting equation of motion driven by
the celestial body of concern, i.e.
here 
is the mass of the celestial body creating the gravitational field of
concern; G is the universal gravitational constant;
points to the location picked up on the
trajectory of the motion; 
is the tangential velocity of the object
at 
;
is the speed of light in empty
space.
The differentiation
of this relationship leads to
This differential
equation is the classical Newton’s
Equation of Motion, were 
,
negligible as compared to 
(the speed of light in empty
space).
The streching of
lengths in a gravitational field, is
equivalent to the slowing down of light, throughout, as referred to a distant
observer. Based on this, the above differential equation can be transformed in
regards to the distant observer. The mathematical manipulation in question,
together with the related solution, will be undertaken in our next article.
1.
INTRODUCTION
Herein we present a whole new
approach to the derivation of the Newton’s Equation of Motion, as well as
the findings brought up within the
frame of the general theory of relativity
(such as the “precession of the perihelion of the planets”, and the “deflection of light nearby a star”).
To the contrary of what had been
generally achieved so far, the basis
adopted herein consists in supposing that the gravitational field, through the binding process, alters the “rest mass” of an object conveyed in it.
In fact, the special theory of
relativity astonishingly, far and wide overlooked, imposes such a change.
Next to this fundamental theory, we use the classical Newtonian gravitational attraction
reigning between two static masses.
We have previously
shown however that the
1/r2 dependency of the gravitational force is also imposed by the
special theory of relativity.[1]
Furthermore the metric coming into play in this work is
(just like the one used by the general
theory of relativity) altered by the gravitational field (in fact, by any field the “measurement
unit” in hand interacts with); yet in the present approach, this occurs via
quantum mechanics. In effect, the solution of even a non-relativistic
quantum mechanical description, given that “potential energies existing in nature”
are considered, bears a casing, in perfect harmony with the special theory of
relativity. This is to say, regarding
the internal dynamics of a wave-like
object, “space” (i.e. the size of the
object), “time” (period of the internal dynamics of concern), and “mass” (the mass, to be associated with the
wave-like object, working as the “pendulum mass” of its internal dynamics),
are structured in such a way that their interrelation remains Lorentz
invariant (i.e. invariant, were the
object brought into a uniform translational motion).
Thus, as we shall see, based on the
special theory of relativity, the rest
mass of an object in a gravitational field should decrease as much as its binding energy in the field.
A mass deficiency conversely, via quantum mechanics, yields a stretching of its size, as well as the
weakening of its internal energy. This is how the metric coming into play
is altered by the field.
Therefore the basis of the approach undertaken herein,
shrinks down to only the special theory of relativity.
Henceforth one does not need the “principle of equivalence” assumed by
the general theory of relativity, in
order to predict the occurrences dealt with this theory.[2]
We predict them through the general
equation of motion established herein (thus, essentially based on the special
theory of relativity, only).
A
change, through the binding process, in the rest mass of an object interacting with a
gravitational field, though, seems somewhat clear. Indeed, the special theory of
relativity predicts such an occurrence as, for example, the proton and the electron, when bound to each other in
the hydrogen atom, weigh less than the sum of the proton and the electron, carried away from each other; the mass deficiency in question is (by taking the speed of light, unity),
exactly equal to the binding energy of the proton and the electron in the
hydrogen atom, i.e. 13.6 ev, based on
the fundamental relationship[3]
(Energy released, or acquired) =
(Magnitude of the algebraic increase in the mass)
x (Speed of light in empty space)2
.
(1)
So,
contrary to the widespread opinion, the electron or the proton cannot be the same, when bound to each other; they are
different. Their internal dynamics altogether, weaken as much as 13.6 ev, when they are
bound to each other to shape up the hydrogen atom.
Many
scientists though, still firmly think that there is the “proper mass” (rest mass) and the “relativistic mass” (defined within the frame of the special
theory of relativity), and that the proper mass is, whatsoever, an
invariant which is a characteristic of matter, and that is
all.
Generally
speaking, this is unacceptable. The proper mass of a given particle on the whole
at rest may, depending on the circumstances, embody a more or less energetic internal
motion; this will, one way or the
other, affect the proper mass.
Suppose
indeed that Captain Electron (we mean,
the electron itself) is cruising in a full electric isolation, with a uniform
translational velocity. So does Captain Proton (i.e. the proton itself). They approach
to each other. Then (based on the special
theory of relativity) we would be certain that, Captain Electron in its own
frame of reference, all the way through, preserves its identity, defined at
infinity. (So will also do Captain
Proton.) If now, we remove the previous electric isolation, Captain
Electron and Captain Proton, because of the electric attraction force, they
mutually create, shall start getting accelerated toward each other. The “extra kinetic energy” they would acquire, as well as
the energy they would radiate through this process, shall be supplied by the
system made of the two. (For easy
wording, we will neglect the energy emitted by radiation, without though any
loss of generality.) The total energy of Captain Electron and Captain Proton
[i.e. (the sum of their relativistic masses) x (the speed of
light)2], through the motion, shall remain constant, and equal to the equivalent of the sum of
their initial relativistic masses. (Otherwise, the energy conservation law
would be broken.) Let us suppose for simplicity that in the latter case (where we have no electric isolation),
they start, far away from each other, at
rest; then their initial relativistic masses are, essentially identical to
respectively their rest masses. If now the accelerating Captain Electron, say in Captain
Proton’s frame of reference, hurts an obstacle and looses all the kinetic
energy, it would have acquired through the attraction process; thence, it must
concurrently dump a portion of its
rest mass, and this, as much as the amount of the kinetic energy it would have
piled up, on the way.[*]
Thus, we
cannot say that the proton and the electron are the same, after we have
retrieved from the system made of the two, a given amount of energy, no
matter how much. The greater is the energy extracted, the harder will be the
harm caused in their internal dynamics, consequently in their proper masses
defined at infinity.
This is
exactly what happens when, say the hydrogen atom is formed, except that the
electron, as referenced to the proton is not anymore at rest, but possesses a
given amount of kinetic energy; still an energy of 13.6 ev is needed, to carry
the electron away from the proton, back to infinity.
It is thus
clear that as referenced to the
proton, or (since the proton is much
too big as compared to the electron), practically the same, as referenced to the laboratory system,
the hydrogen electron’s proper (rest) mass, is altered as much.
Likewise,
the daily production of thermal energy, is due to the transformation of a minimal part of the mass entering in
reaction, into energy. Thus the reaction products weigh less than the reactants,
and this, as much as the energy produced throughout.
The fuel,
i.e. coal, petroleum, uranium, plutonium, anything, in a power plant of, say
3000 MWthermal, continuously
working for a period of one year, thus producing an energy amounting to 3000
MWthermal x year, at the end of this period, weighs less, as much as
the equivalent of the energy output in question, i.e. [based on the equivalence
between mass and energy], about 1 kg. This is of course insignificant as
compared to millions of tons of coal or petroleum that would be fired into the
plant of concern, but well detectable as compared to about a ton of plutonium-239, or uranium-235
needed to be depleted in a nuclear power
plant of 3000 MWthermal through a period of one year.
In a
similar way, a compressed spring
should be heavier than the “same spring”
when stretched out; or the gas in a
room at a high temperature should weigh more than the “same gas” at a lower temperature, etc.
All these,
already happen to be well established facts. Thus, any proper mass weighs less, after releasing
energy, or conversely it shall weigh more, after piling up an extra amount of internal
energy.
Therefore,
herein we anticipate that when an object is bound to a celestial body, its rest mass (measured in empty space) is
decreased as much as the binding
energy, it would have developed in the gravitational field of the celestial
body of concern.[4]
Einstein
in his general theory of relativity, considers the conservation of
the “rest masses”, instead of the
conservation of the “total
energy”.2
Yýlmaz somewhat fulfilled this gap.
He derived the “exact solution” of
the “accelerated elevator”, and to
his great surprise, found out that Einstein’s fields equations were not
satisfied; this was the
beginning of Yýlmaz’s efforts towards a more consistent theory, though along the
same direction as that drawn by Einstein.[5],
[6],[7]
At any rate, Einstein’s general
theory of relativity leads to the fact that, his original relativistic “mass-energy relationship” [i.e.
Eq.(1)], does not hold between gravitational coordinate values of energy
and mass, in any perceptible way.[8]
We do not have such an annoyance, since we derived our results essentially based
on Einstein’s “mass-energy relationship”
obtained within the frame of the special theory of relativity, and not the
principle of equivalence assumed by the general theory of
relativity.
Thence one can propose the following
postulate, in fact nothing else, but the
energy conservation law, where though, as introduced by the special theory
of relativity, [energy] and [(mass) x
(speed of light)2] [cf. Eq.(1)], are no different from each other.
Postulate: The rest mass of an object bound to a
celestial body, amounts less than its rest mass measured in empty space, and this as
much as its binding energy vis-à-vis
the gravitational field of concern.
It is important to note that, on the
contrary to what the general theory of relativity eventually formulates, as we
shall see, here, it is question of a
decrease of mass in a gravitational field, and this is interestingly, just
as much as the mass increase (due to
acceleration) formulated by the former theory.1
Note further that, so far there had
been no measurement of mass in a gravitational field; thus a measurement of mass
at two different altitudes on Earth, can furnish a verification of our guess.
According to recent developments[9],[10]
this indeed seems possible, if one proposes to achieve the measurement of mass through the
measurement of the Rydberg Constant[†],
already measured with a precision of 10-14, whereas a precision of
10-13 appears to be good enough for a difference of altitudes of
103m. Note further that, based on our approach, the classical red shift due to gravitation,
is nothing else, but an overall mass decrease (due to the gravitational binding), of
the emission source.
Below, we first sketch how the
gravitational binding energy reduces the rest mass of an object bound to the
celestial body in consideration
(Section 2). Then, we recall the quantum mechanical theorems we have established previously (Section
3). An elaboration on the gravitational binding energy follows (Section 4). The
change of the rest mass of an object in a gravitational field, together with the
Lorentz mass dilation, due to the local motion, yields the general equation of
motion (Sections 5 and 6). A conclusion follows (Section
7).
Next, taking into account how unit
lengths, quantum mechanically stretch
in a gravitational field, one is able to obtain the precession of the perihelion of Mercury
(or anything as such), as well as the
deflection of light grazing a
celestial body; this shall constitute the content of our next article.
2. THE GRAVITATIONAL BINDING
ENERGY
At this stage, we have to evaluate
the gravitational binding energy. For
this purpose we have to use the expression for the gravitational force.
Herein we consider only the gravitational force between two static masses.
Since
we aim ultimately at deriving a result obtained within the context of the
general theory of relativity, without
having to rely on it, we better should not plainly borrow, the expression
for the gravitational force (between two
static masses), with its classical
empirical form, from Newton,[11]
since this (were the case, that of a weak
gravitational field), is formally, well manufactured by the general theory
of relativity.
Therefore (and luckily) we derive the 1/r2
dependency of the gravitational force between two static masses, here
again, from the special theory of relativity.1
EB =
;
(2)
here 
is the mass of the host body binding the object of mass
,
as measured in empty space, R0 the distance of the mass 
to the center of the host body, and G the universal gravitational constant.
In reality
(based on the discussion presented above,
in Section 1), changes continuously throughout. One can, as we shall soon see,
easily elaborate on this.
When the object of mass 
is bound to the gravitational field,
decreases to become m.
Thus
m
, m
=
;
(3)
is determined out of Eq.(1), more
specifically
(
- m) 
= EB .
(4)
Given EB, 
(smaller than unity)
becomes:
=
1-
=
1-
.
(5)
3.
THEOREMS PREVIOUSLY ESTABLISHED
The
approach presented herein becomes very interesting, if one recalls the following
theorem established elsewhere.[12],[13],[14],[15],[16],[17]
Theorem 1:
In a “real wave-like description” (thus, not embodying artificial potential energies), composed of I electrons and J nuclei, if the (identical) electron masses
mi0, i = 1,..., I and different nuclei masses mj0, j =
1,…, J, involved by the object, are overall
multiplied by the arbitrary
number 
, then concurrently, a) the total energy E0 associated
with the given clock’s 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 
to be associated with the given clock’s
motion of the object, contracts as much; in mathematical words this is
{
[(mi0, i = 1,..., I) 
(
mi0, i = 1,..., I) ],
[
(mj0, j = 1,…, J) 
(
mj0, j = 1,…, J) ]
} Þ { [
], [
],
[
] }
.
(6)
Then,
following the above derivation, we come at once, to the next
theorem.
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.
This can
further be grasped rather easily as follows. The mass deficiency the wave-like object
displays in the gravitational field weakens its internal dynamics as much.
Thence, we arrive at the two principal results, we just stated.
Note that, according to the approach
presented herein, the classical
gravitational redshift and a related
mass decrease, occur to be concomitant quantum mechanical effects.
Thus in fact, contrary to what the general theory of relativity ultimately
considers, we expect a mass decrease
in a gravitational field (and not a mass
increase).
It is of course impressive to notice
that the foregoing reasoning is not restricted to gravitation only. It should hold in any kind of
interaction where the wave-like clock, develops a binding, thus undertakes a
mass deficiency (without of course,
loosing its identity), as described above; in such a case, EB
becomes the binding energy of the
wave-like clock to either field (electric, magnetic, nuclear, gravitational,
whatever) of concern, our finding holds.1 So, quite on the
contrary to the prevailing opinion, the gravitational field is not any different
than other fields, in affecting the clocks. Thus, one can establish the
following simple theorem,
generalizing the previous one.12
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.
Let us now elaborate on the binding energy.
4. ELABORATION ON THE GRAVITATIONAL
BINDING ENERGY
In calculating the binding energy
EB, at the level of Eq.(2), we had tacitly assumed that the wave-like
clock of original mass m0, looses only an insignificant part of it, through the
binding process. Otherwise, Eq.(7) should be written as follows:
1
,
(7)
,
(8)
=
,
(9)
at a distance R0 from the
center of the host celestial mass M0, via the usual
definition
.
(10)
The outcome EB of Eq.(9)
is zero when 
is at infinity; EB becomes
more and more important as 
increases. Yet there appears to be no singularity at all (unless m0 when transplanted nearby
M0, is somehow
degenerated). This seems to be remarkable, since (based on Theorems 1 and 2)
it yields no singularity in time,
thus no “black
holes”.
Note that Eq.(9), along Eqs. (3) and
(5) leads to
.
(11)
We would like to say few words about
how we come to a mass decrease in a
gravitational field, instead of a concluding mass increase considered by
the general theory of
relativity.
Einstein’s depart point is, based on the equivalence principle, a mass increase displayed by the object,
carried away by the “accelerating
elevator”. This depart point,
though a striking idea, seems inappropriate for (chiefly, next to the reason we will develop
herein), a major reason; it is that, there is a clear asymmetry between the accelerating elevator and the gravitational field, with respect to a
distant observer. “Getting on the
accelerating elevator” (when we are nearby at rest, in empty space) and “getting on a celestial body” (from empty space), indeed are not at
all the same process (for the distant
observer), clearly at least for one thing, i.e. he has to get accelerated to be able to catch up with
the accelerating elevator, whereas he has to get decelerated in order to be able to land
on the celestial body.
The first process (within the context of the theory of
relativity) yields a mass increase,
whereas the second one, through the line we followed, gives a mass decrease (with respect to the distant
observer).
A mass decrease, through Theorem 2, yields
a unit time increase, but also a length loosening (not a length
contraction).
Thence according to the approach
developed herein, Einstein’s
transposition, of mass increase
and a concurrent length contraction
taking place in an accelerating elevator, to a gravitational field, seems to be
incorrect. We shall elaborate further, on this, below.
Nonetheless Eqs. (2), (9) and (11),
happen to be in close agreement[‡]
with the gravitational potential
furnished by the general theory of relativity.[18]
How come?
As we shall detail below, briefly for
one thing it seems that, assuming the
equality of the inertial mass and the
gravitational mass, and overlooking
the mass equivalence of the gravitational energy, constitute effects of
about the same magnitude and amazingly canceling each other; this should be how
we could reproduce practically the same result as that of Einstein, in regards
to the gravitational potential, the
precession of the perihelion of
Mercury, etc. Recall anyway that even alike predictions made by the general
theory of relativity and the theory presented herein, are not exactly the
same.
5. THE GENERAL EQUATION OF GRAVITATIONAL
MOTION IN
SCALAR FORM
Now, we are ready to derive the general equation of gravitational
motion.
The idea behind it, is strikingly
simple, and is rooted to the postulate, stated above. When an object enters into
interaction with a celestial body, its “total energy” (as conceived within the frame of the
special theory of relativity), throughout, remains the same. The extra
kinetic energy it shall acquire or it shall lose on the way, shall be accounted
by an equivalent change in its rest mass.
Henceforth, when an object falling in a gravitational field, is stopped and
the kinetic energy, it would have acquired is taken away, its rest mass (as measured in empty space) should be
decreased as much as the binding energy it would have developed in the field.
Here, to make things easier, we
tacitly assumed that, one of the interacting objects is very massive, and the
other is very small, so that we have to worry about only the small one. The one
which is massive undergoes practically no
change. The approach presented herein can be easily extended to the general
case.
In order to ease our dissertation we
shall work on a concrete basis, more specifically we will consider the planet
Mercury, in motion around the sun (without though, any loss of generality).
We can conceive Mercury’s motion (around the sun), as made of two
steps:
i) Bring it
from infinity to a given location, situated on its “elliptical” orbit around the sun; the
energy this process requires, is the magnitude of the classical potential
energy.
ii) Deliver to
it, the kinetic energy it would
display on this location. (Note that on
the orbit, the classical total
energy, i.e. potential energy + kinetic energy, is a constant of the motion.)
Let us then make the following casual
definitions.
or 
: distance of the sun to the
planet, at time 
(measured in terms of the local
metric)
: the planet’s rest mass at
infinity
or 
: the planet’s rest mass at a distance
,
or at the corresponding time 
,
as referred to the sun
(
)
or 
: the planet’s total relativistic mass (which is its mass at infinity decreased as
much as its binding energy, and increased based on the special theory of
relativity, due to its “translational” motion on the orbit) at 
,
and at the corresponding time 
or 
or 
: magnitude of the
tangential velocity of the planet on the orbit, at
,
and at the corresponding time 
,
as referred to the local observer
: the velocity of light in empty space (free of any gravitational
field)
or 
: dimensionless quantity defined along
Eq.(10), for the distance 
of the planet from the
sun
Equation of Motion of Mercury as
Assessed by the Local Observer
On any given natural orbit, the relativistic total energy of the object of concern,
i.e 
,
thus 
must remain constant. If the orbit is not
circular, throughout the object’s journey on the orbit, however this may be,
both 
and 
shall vary; but
,
thus 
must stay constant.
Thus starting with the energy
conservation postulate, and the above definitions, one can now write the
following equations based on, first,
Eqs. (10) and (11), yielding the decrease
of the rest mass of the planet brought from infinity, and then, the familiar relativistic mass increase with tangential
velocity on the orbit:
,
(12)
=
D ,
(13)
where D is a constant we are to determine.
Note that
remains as a constant in the case of a
circular orbit; so is 
;
thus, in this case, D is anyhow a
constant. This special case, does not advance us. Yet what is interesting, as we
propose to study, is that D is
whatsoever, a constant.
Let us explain this, a bit further.
According to our approach, 
(the total relativistic energy of the
planet) ought to be constant all along Mercury’s journey around the sun. As
the planet speeds up nearby the sun,
it is that, an infinitesimal part of its mass somehow “sublimes” to get transformed into kinetic energy, yielding the extra kinetic energy (the planet acquires as it speeds up);
as the planet slows down away from
the sun, through its orbital motion, it is that, a portion of its kinetic energy
somehow “condenses” onto its rest
mass, on the orbit.
This alternating process through the motion,
based on the special theory of
relativity, anyway, makes that the planet’s total relativistic mass (i.e. the classical rest mass at infinity,
decreased as much as the gravitational binding energy + the mass equivalent of
the kinetic energy) remains the same. This should be considered harmonious
with the fact that the planet’s classical
total energy on the orbit, is constant. We will soon elaborate on this
point.
What is this constant? It would first
be interesting to examine the case of free fall, where D (as we shall see) is interestingly
unity.
Free Fall
Consider an object originally at rest, practically at infinity,[§]
and experiencing a free fall in a
gravitational field. Let 
its rest mass, at infinity. Its binding energy EB, were it
stopped at a given altitude,
according to Eq.(9) is
,
(14)
where
represents the value of this quantity at the altitude in
consideration.
The rest mass of the object at this
altitude, according to Eq.(11), is 
.
On the
other hand, the object through its
free fall, would (up to the altitude of
concern) acquire the velocity 
,
yielding the overall mass 
,
while some of its mass content, as just mentioned, is transformed into kinetic energy. The differrence of the
corresponding energies is nothing, but the binding energy EB [given
by Eq.(14)]:
(15)
This, right away yields unity, for the constant D , appearing in Eq.(13), i.e.
(16)
Note that the classical total
energy, i.e. potential energy +
kinetic energy, through the free
fall is conserved, which is quite harmonious, with
Eq.(15).
If the falling object started at infinity with an initial velocity 
(and not at rest), than Eq.(16) would
become
(17)
Note that no matter what the direction of the initial velocity 
at infinity, or the direction of 
at the given location, we associate with
the object in hand, the above relationship is still valid.
Differential Equation of Motion as
Assessed by the Local Observer
The constancy of D can further be easily checked and fixed
for the case of Mercury, based on the actual data associated with the planet, at
a given location of it, on the orbit. [**]
For further simplicity we can recall
that the orbit of the planet is nearly circular.
Thus, based on Eq.(13), we can
write
=
D 
;
(18)
Here recall that 
is the tangential velocity of the planet,
at the location 
on the orbit (as referred to the local
observer).
Note further
that
,
(19)
where (for a reason that we shall clarify right
away) we associate with 
,
the quantity 
,
i.e. the average distance of the
planet to the sun (which happens to be
the semi-grand axis of the elliptical orbit);
is the average of 
,
and 
the mean square velocity.
It is already striking that the
second equality displayed by Eq.(18), under the assumptions in question (i.e.
small 
,
small 
),
is nothing but, the Newton’s equation of gravitational motion (in its integral form), relating the
tangential velocity 
of the planet, to its distance to the
sun.
The
usual form of the equality of concern is[19],[20]
;
(20)
here 
is the classical mass of the planet and a,
the semi-grand axis of the elliptical orbit of
this; 
;
for the case of Mercury.
Throughout
the approach presented herein, we have assumed the sun infinitely big as compared to Mercury,
this being the reason for which the mass of the latter does not appear in our
relationships. Below, we shall continue to set all our relationships, that
way.
It is
further interesting to note that Eq.(20) is nothing but the classical energy conservation equation;
thus it states that, on the orbit (classically), the total energy of the
planet, is conserved.
Classically, the magnitude of the
total energy is the energy one has to spend in order to remove the planet
bearing a velocity 
,
on the orbit at a distance 
to the sun, from its actual position, to
infinity. It is composed of, on the one hand the potential energy, of magnitude 
(which is the energy one has to spend in
order to remove the planet of mass 
,
at rest, from a distance 
to the sun, to infinity), and on the
other hand the kinetic energy
.
Thus
Eq.(20) states that the magnitude of
the classical total energy,
i.e. the sum of 
and 
,
on the orbit, must be constant and equal to 
.
Having
started with Eq.(13), the “relativistic
energy conservation equation”, it should be natural, as well as fulfilling
to land at the “classical energy
conservation equation”, for small velocities and weak gravitational
fields.
Thus for
Mercury, D (considering the assumptions in
question), shall be given by
.
(21)
Note
that, because 
is small, D is very close to unity. Though the
divergence, as small as ~10-8 from unity, is still
essential.
At any rate, following Eq.(13)
(giving that the RHS of this equation, is constant), we expect that the total differential of 
,
must vanish.
Thence, by differentiating Eq.(13), we arrive at
the rigorous equation, regarding the
revolution of the planet around the sun, or anything as such:[††]
.
(22)
(written by the author, in the local
frame of reference)
This relationship is interesting in
many ways. First of all when 
(as compared to the velocity of light)
is negligible, or similarly when 
is small, it reduces right away to the classical Newton’s equation of gravitational
motion. This can be checked immediately by differentiating Eq.(20), which is
a scalar form of Newton’s equation of gravitational
motion.
Eq.(22) can further account for the
precession of the perihelion of a planet
(or anything such), as well as the deflection of light nearby a celestial
body, though it is derived through a totally different approach than that of
Einstein; the predictions in question shall be elaborated in our next
article.
6. THE GENERAL EQUATION OF GRAVITATIONAL
MOTION IN
VECTOR FORM
From a rigorous mathematical point of view, one
may argue about the following.
-
One does indeed land, from Newton’s equation of gravitational
motion written in vectorial form, to Eq.(20),19,20 thus
also to Eq.(22), in the case the cruise velocity
of the object in hand is small as compared to the velocity of
light. But can we really obtain from the scalar Eq.(22), a corresponding equation in vector form,
similar to Newton’s (vectorial) equation
of gravitational motion?
The answer is
- Yes.
After all one may right away note
that our derivation is similar to that achieved in obtaining Newton’s equation
of gravitational motion, through the classical energy conservation assumption,
i.e. the classical Hamiltonian way;19 through such an approach the
differentiation of Eq.(20) (where we only
have to know the fact that the classical total energy amounts to a “constant”,
clearly without having to know the value of it), yields a scalar differential equation, but which can be converted, as
we shall soon swiftly derive, into Newton’s equation of gravitational motion in
the usual vectorial form.
The energy conservation, in the
broader sense of the concept, covering the equivalence of energy of mass, then
well, and as shown below, similarly leads to our Eq.(22) and to the corresponding equation in vectorial
form.
Thus consider the general case, where the magnitude 
,
of the velocity vector 
,
changes continuously, all along the motion in question.
In any case, through the infinitely
small period of time 
,
we have as usual
.
(23)
Obviously
and 
are not oriented in the same direction;
is oriented along the direction of the
motion on the orbit, whereas 
is directed toward the sun.
The infinitesimal increase 
in the “magnitude” of
,
i.e.
,
(24)
is generally different from 
,
the “magnitude of the infinitesimal increase” in 
,
though 
and 
become
equal, if the motion were a one dimensional motion.
Note that
vanishes in the case of a circular orbit. Recall however that our
original equation, i.e. Eq.(13), in this case becomes trivial; thence the
differentiation of it, does not provide us with any additional information.
According to the definitions we have
made along Eqs. (23) and (24), one can show that, 19,20 the classical
Newton’s equation of gravitational motion, i.e.
,
(25)
(the classical Newton’s equation
of gravitational motion, in vector form)
yields well
,
(26)
(the classical Newton’s equation
of gravitational motion, in scalar form)
and vice-versa.
Here is a quick proof of this last,
vice-versa, statement.
Eq.(26) can be classically written
as
;
(27)
this equation expresses that the decrease in the potential energy and the
increase in the corresponding kinetic
energy are equal to each other.
Recall that here 
is the classical mass of the planet, and
the magnitude of the gravitational force
between the sun and the planet, at the given location.
But evidently
,
(28)
given that the gravitational binding energy is path independent.
In fact as we shall soon recall, one
can further, directly prove Eq.(28). In this equation
is the gravitational force (in vector form); 
is the location vector defined along 
;
and 
are
the same quantities; one can thus write the definitions
,
(29)
.
(30)
The negative sign at the RHS of
Eq.(28) arises from the fact that, as 
increases, the force counteracts, making
the cosine of the dot product
negative (or the same, as 
“decreases”, the force acts in speeding
up the motion, making the cosine of the dot product,
positive).
We now rewrite Eq.(27), dividing its
both sides by 
:
,
(31)
where we made use of the usual
definition of 
,
i.e.
.
(32)
Let us multiply both sides of Eq.(31)
by 
[cf.
Eq.(23)], and rearrange it:
;
(33)
here 
is the angle between 
(directed toward the sun), and 
(tangent to the orbit). [Bear in mind
that 
is identical
to
.]
One can on the other hand, easily
show that
,
(34)
checking at once the case of the circular motion, for which 
,
and 
;
one can moreover note that this also checks well the sign of 
for an elliptic orbit.[‡‡]
(Note indeed that for an elliptic orbit, one has
<0,
when 
<0,
and 
>0,
when 
>0.)
Furthermore
is directed toward the sun, along the
same direction as 
.
This makes that Eq.(26) written in scalar form, yields well Eq.(25) written in
vectorial form (c.q.f.d.).
Based on the foregoing information,
it becomes clear that starting with Eq.(22), we can obtain the vectorial equation
.
(35)
(the general equation of gravitational
motion written
by the author, in the local frame of
reference)
Unless
is small, this relationship displays an
amazing feature; it is that the “classical gravitational mass” and the
“inertial mass” are not the same. We
shall elaborate on this in what follows.
Regarding the motion of a planet around the sun, the classical energy
conservation, via the Hamiltonian approach yields well Newton’s second law of
motion, i.e.
Gravitational
Force = 
x Acceleration,
(written out of Newtonian approach,
based on energy conservation)
or the same,
Gravitational
Field (Vector) = Acceleration (Vector).
(written out of Newtonian approach,
based on energy conservation)
The approach presented herein, via
the relativistic energy conservation, clearly, does not yield Newton’s second
law of motion; it yields something else.
In order to draw a one to one comparison between the frame
we just sketched [through Eqs. (25) – (35)], and our approach, we
would like to rewrite Eq.(22), out of Eq.(13), and reexamine
it:
.
(36)
[Eq.(22), rewritten by
differentiating Eq.(13)]
The LHS of this equation expresses
the infinitesimal change in the gravitational binding energy of the object in
motion.
The RHS conversely expresses the
infinitesimal change in the kinetic energy of the “overall mass” 
;
recall that this mass remains constant throughout [cf. Eq.(13)]. Note that the
change on the kinetic energy, is solely due to the change on the
velocity.
Thence by rereading Eq.(36), along
the derivation of Newton’s equation of gravitational motion [Eq.(26)], we can
state that
;
(37)
(the general equation of
gravitational motion written by the author)
here the gravitational force, next to
the sun’s mass (assumed at rest),
embodies the overall mass, 
of the revolving object.
Eq.(37), reduces to Eq.(22), once one
divides both of its sides by the overall
mass.
Eq.(37), based on the analysis made
on Eq.(36), seems the natural way of presenting our result. Accordingly one uses
the same mass, i.e. 
, to multiply both the gravitational field vector and the acceleration vector. But then Newton’s
equation of gravitational motion, i.e. [Force = 
x Acceleration] is broken.
Formally, this can be saved if
instead, we choose to alter the “classical gravitational force”; but
then the gravitational mass and the inertial mass, as classically defined, shall not be
same.
We conclude on this
below.
7.
CONCLUSION
The
essence of this article was, based the energy conservation, in the broader sense
of the concept, embodying the equivalence
of mass and energy, to derive a general equation of gravitational motion,
more specifically
.
(35)
(the general equation of gravitational
motion written
by the author, in the local frame of
reference)
This
becomes the Newton’s equation of motion, only if
is small as compared to the velocity of
light. In our next article, we shall see, how this equation can cover up the
basic predictions envisaged by the general theory of relativity, provided that
one takes into consideration the fact that the mass deficiency due to the binding, alters via quantum mechanics, unit lengths, unit periods of time, etc, along Theorem
1, presented above.
The way
it stands though, the principle of
equivalence about the gravitational mass and the inertial mass, in general,
seems in trouble.
This principle is anyway severely
questioned.[21],
[22],
[23]
Nonetheless we can formally save Newton’s equation of
gravitational motion, by redefining
the gravitational
mass.
Thus
consider the classical formulation of Newton’s equation of gravitational motion,
tuned along the special theory of relativity, i.e. with the familiar
notation11
Classically
expressed Gravitational Force
= [d (Momentum of the object in motion, due to
gravitation) /dt0].
(38)
To ease our
expression, let us continue to consider, say Mercury of mass
,
defined at infinity, in its
motion around the sun
of mass M0, without however any loss of
generality.
Note that here it is assumed that we
are positioned at Mercury. Things will be seen differently, when we will be
positioned at a distance far away from the sun’s gravitational field. This
latter situation shall be undertaken in our next article.
Comparing Eqs. (35) and (36), the
mass 
,
pertaining to the planet, and entering the formulation of the momentum of the planet, shall be
;
this corresponds to the classical
inertial mass; it is a constant
of our approach, therefore it
comes out of the differentiation
operation on the momentum.
Let us then call 
,
a gravitational mass pertaining to
the planet, taking part in the usual gravitational force acting between the
sun and the planet, so that
.
(39)
[Eq.(38) written via the introduction
of a gravitational mass]
This latter
equation becomes the same as Eq.(35), if we propose to
write
.
(40)
(gravitational mass that would take
part in the classical gravitational force expression, as assessed by the local
observer)
Our result, at any rate, leads us to
a straightforward conclusion, albeit totally against the prevailing wisdom; it
is worth to state it as a separate theorem.
Theorem 4: The gravitational mass 
,
and the inertial mass 
, as classically defined, are not the same;
the theory presented herein, to formally save Newton’s equation of
gravitational motion, predicts 
,
given that
;
though undetectable, for most cases we observe, 
and 
differ.
The equality of the gravitational
mass and the inertial mass, based on the approach presented herein is an
approximation which is acceptable, only if the velocity of the object in motion
is small, as compared to the velocity of light in empty space.
It is interesting to note that, all
the highly precise measurements
regarding the relative divergence of these two masses, are performed on Earth (where the observer is moving with
Earth), so that the precision
they produce, no matter how fine this may be, should be considered, as
misleading. In effect, since the gravitational mass, as stated by Eq.(40),
depends on the velocity, one should not rely on the experiments in question, any
more then he should count on the null result of the Michelson Morley
experiment[24]
(which, being performed on Earth, fails
to detect the motion of Earth around the sun, or else). In other terms, the
principle of relativity (the main
ingredient of the special theory of relativity), forbids that we can on
Earth, detect any such difference, based on the velocity of motion in question (since otherwise we should be able to tell
accordingly, how fast we are cruising in space, and we
cannot).
Not knowing that the equality of
gravitational mass and inertial mass, is only approximate, one may still insist
(just the way it is done regarding the
experiments in question) that, such an equality can well be established on
Earth. But the rotational velocity 
of Earth around itself is 1667 km/hour.
Hence one should attain a precision of 
,
i.e. better than 2.6 x 10-12, whereas the
highest precision reached so far, is bearly, this much.
On the other hand, measurements based
on a possible polarization of Earth and the Moon, through their motion around
the sun (on which we can indeed rely),
require a precision of ~10-8 (which is the related
ratio of 
to
),
whereas the precisions actually reached (~ 10-4), happen to be far
below this. [25],
[26]
Note further that, even through the
fastest observable celestial motions, such as that of binary stars, around each
other (where the objects move with speeds
around 1000 km/s), the difference between the gravitational mass and the
inertial mass, still remain undetectable.
In contrast, it is astoundingly
interesting to note that Eq.(22) can be obtained from the following equation
bearing the same form as that of the
classical Newton Equation of Motion, i.e. Eq.(26):
,
(41)
This means
that, if the local mass 
were given by
,
(42)
instead of that given by Eq.(11)
(i.e. 
),
and if the local relativistic effect due to the translational motion of the
object of concern can be ignored [since the momentum quantity, expressed as
under the differentiation operation at
the RHS of Eq.(17) clearly does not cover the effect due to the translational
motion of the object], only then we
could claim that the principle of equivalence holds, i.e. the gravitational
mass and the inertial mass are the same.
But this is not the case; that is,
through the approach presented herein, Eq.(42) is incorrect; furthermore the local relativistic effect due to the
translational motion of the object, in general, cannot be ignored.
Thus according to the approach
presented herein, the principle of
equivalence must be incorrect.
ACKNOWLEGMENT
The author
is especially grateful to Dr. Christian Marchal, Directeur Scientifique de l’ONERA, France, and to Dr. Xavier Oudet,
Editeur, Les Annales de la Fondation Louis de Broglie, France; without their
unequally sage understanding and encouragement, this controversial work could
not come to daylight. The author would further extend his deep gratitude to Dr.
V. Rosanov, Director of the Laser Plasma Theory Division, Lebedev Institute,
Russia Academy of Sciences, Dr. N. Veziroðlu, Director of the Clean Energy
Research Institute, University of Miami, Dr. O. Sinanoðlu, from Yale University,
Dr. ª. Koçak, from Anadolu University, Eskiºehir, Dr. E. Hasanov from Iºýk
University, Istanbul, who provided him with great time of discussions about the
subject.
Thanks are
also due to my Dear Student Fatih Özaydýn, who has kindly helped the typing of
the manuscript.
REFERENCES