……… page 9 of “ Mathematical Foundations of………
( ) ( )4402 miniii mnM =
which shows the quantization of inertial mass; in is the
inertial quantum number.
We will change n in the quantized expression of gM by
gn in order to define the gravitational quantum number. Thus
we have
( ) ( )amnM igg 4402 min=
Finally, by substituting gm given by Eq. (43) into the
relativistic expression of gM , we readily obtain
( ) ( )45112
1
2
1
22
22
ii
g
g
McVM
cV
m
M
⎥⎦
⎤⎢⎣
⎡ −−−=
=
−
=
−
By expanding in power series and neglecting infinitesimals, we
arrive at:
ig M
c
V
M
2
2
1−=
Since 01 22 >− cV , the equation above can be rewritten
as follows:
( )461
2
2
ig M
c
V
M ⎟⎟⎠
⎞
⎜⎜⎝
⎛ −=
Thus, the well-known expression for the simple pendulum
period, ( )( )glMMT giπ2= , can be rewritten in the following
form:
cVfor
c
V
g
l
T <<⎟⎟⎠
⎞
⎜⎜⎝
⎛ +=
2
2
2
12π
Now, it is possible to learn why Newton’s experiments using
simple penduli do not found any difference between gM and
iM . The reason is due to the fact that, in the case of penduli,
the ratio 22 2cV is less than 1710− , which is much
smaller than the accuracy of the mentioned experiments.
The Newton’s experiments have been improved upon
(one part in 60,000) by Friedrich Wilhelm Bessel (1784–1846).
In 1890, Eötvos confirmed Newton’s results with accuracy of
one part in
710 . Posteriorly, the Eötvos experiment has been
repeated with accuracy of one part in
910 . In 1963, the
experiment was repeated with an even greater accuracy, one
part in
1110 . The result was the same previously obtained.
In all these experiments, the ratio 22 2cV is less
than
1710− , which is much smaller than the accuracy of
1110− , obtained in the previous more precise experiment.
Then, we arrive at the conclusion that all these experiments
say nothing in regard to the relativistic behavior of masses in
relative motion.
Let us now consider a planet in the Sun’s gravitational
field to which, in the absence of external forces, we apply
Lagrange’s equations. We arrive at the well-known equation:
h
dt
d
r
r
GM
dt
d
r
dt
dr i
=
=−⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛
ϕ
ϕ
2
2
2
2 2
E
where iM is the inertial mass of the Sun. The term
aGM i−=E , as we known, is called the energy constant;
a is the semiaxis major of the Kepler-ellipse described by the
planet around the Sun.
By replacing iM into the differential equation above by
the expression given by Eq. (46), and expanding in power
series, neglecting infinitesimals, we arrive, at:
⎟⎟⎠
⎞
⎜⎜⎝
⎛+=−⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛
2
22
2
2 22
c
V
r
GM
r
GM
dt
d
r
dt
dr gg E
ϕ
Since ( )dtdrrV ϕω == , we get
2
2
2
2
2 22 ⎟⎠
⎞⎜⎝
⎛+=−⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛
dt
d
c
rGM
r
GM
dt
d
r
dt
dr gg ϕϕ E
which is the Einsteinian equation of the planetary motion.
Multiplying this equation by ( )2ϕddt and
remembering that ( ) 242 hrddt =ϕ , we obtain
22
3
2
4
2
2 22
c
rGM
h
rGM
h
r
r
d
dr gg ++⎟⎟⎠
⎞
⎜⎜⎝
⎛=+⎟⎟⎠
⎞
⎜⎜⎝
⎛
Eϕ
Making ur 1= , and multiplying both members of the
equation by
4u , we get
2
3
22
2
2 22
c
uGM
h
uGM
h
u
d
du gg ++=+⎟⎟⎠
⎞
⎜⎜⎝
⎛ E
ϕ
which leads to the following expression
⎟⎟⎠
⎞
⎜⎜⎝
⎛ +=+ 2
22
22
2 3
1
c
hu
h
GM
u
d
ud g
ϕ
In the absence of term 2223 cuh , the integration of the
equation should be immediate, leading to π2 period. In order
to obtain the value of the perturbation we can use any of the
well-known methods, which lead to an angleϕ , for two
successive perihelions, given by
22
226
2
hc
MG g+π
Calculating per century, in the case of Mercury, we arrive at an
angle 43” for the perihelion advance.
This result is the best theoretical proof of the accuracy of
Eq. (45).
Let us now consider another consequence of the
existence of correlation between gM and iM . ………..
The Best Proof of the Existence of Correlation
between Gravitational and Inertial masses
