The Best Proof of the Existence of Correlation between Gravitational and Inertial masses ……… page 9 of “ Mathematical Foundations of………
(44 )
M i = n i2 m i 0 (min ) which shows the quantization of inertial mass;
ni
is the
inertial quantum number.
n in the quantized expression of M g by
We will change
n g in order to define the gravitational quantum number. Thus we have
(44a )
M g = n g2 mi 0(min ) Finally, by substituting
mg
given by Eq. (43) into the
relativistic expression of M g , we readily obtain
mg
Mg =
1−V 2 c2
(
= M i − 2⎡ 1 − V 2 c 2 ⎢⎣
)
− 12
− 1⎤ M i ⎥⎦
By replacing M i into the differential equation above by the expression given by Eq. (46), and expanding in power series, neglecting infinitesimals, we arrive, at:
(45)
By expanding in power series and neglecting infinitesimals, we arrive at: 2
Since
V
Mi
c2
1 − V 2 c 2 > 0 , the equation above can be rewritten
as follows:
⎛ V2 M g = ⎜1 − 2 ⎜ c ⎝
⎞ ⎟M i ⎟ ⎠
(46 )
Thus, the well-known expression for the simple pendulum
(Mi Mg)(l g) , can be rewritten in the following
period, T =2π
=E ⎟ − ⎜ ⎟ +r ⎜ r ⎝ dt ⎠ ⎝ dt ⎠ dϕ r2 =h dt where M i is the inertial mass of the Sun. The term E = − GM i a , as we known, is called the energy constant; a is the semiaxis major of the Kepler-ellipse described by the planet around the Sun.
=
M g = 1−
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: 2 2 2GM i ⎛ dr ⎞ 2 ⎛ dϕ ⎞
2 2 2GM g 2GM g ⎛ dr ⎞ 2 ⎛ dϕ ⎞ =E + ⎜ ⎟ +r ⎜ ⎟ − r r ⎝ dt ⎠ ⎝ dt ⎠
Since
V = ωr = r (dϕ dt ) , we get
T = 2π
⎞ ⎟ ⎟ ⎠
which is the Einsteinian equation of the planetary motion. Multiplying this equation by dt dϕ 2 and
(
remembering that
M g and
M i . The reason is due to the fact that, in the case of penduli, the ratio
V
2
2c
2
is less than
10
−17
, 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 7 one part in 10 . Posteriorly, the Eötvos experiment has been
9
repeated with accuracy of one part in 10 . In 1963, the experiment was repeated with an even greater accuracy, one 11 part in 10 . The result was the same previously obtained. 2 In all these experiments, the ratio V 2c 2 is less than 10
10
−11
−17
d ϕ )2 = r 4 h 2
, which is much smaller than the accuracy of
, obtained in the previous more precise experiment.
)
, we obtain
⎛ r 4 ⎞ 2GM g r 2GM g r ⎛ dr ⎞ ⎜⎜ ⎟⎟ + r 2 = E⎜ 2 ⎟ + + 2 ⎜h ⎟ h c2 ⎝ dϕ ⎠ ⎝ ⎠ Making
3
r =1 u ,
u
4
and multiplying both members of the
, we get
2
Now, it is possible to learn why Newton’s experiments using simple penduli do not found any difference between
(dt
2
equation by
for V << c
⎞ ⎟ ⎟ ⎠
2 2 2GM g 2GM g r ⎛ dϕ ⎞ 2 ⎛ dr ⎞ 2 ⎛ dϕ ⎞ r E + − = + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ r c 2 ⎝ dt ⎠ ⎝ dt ⎠ ⎝ dt ⎠
form:
l ⎛⎜ V2 1+ 2 g ⎜⎝ 2c
⎛V 2 ⎜ ⎜ c2 ⎝
2GM g u 2GM g u ⎛ du ⎞ E ⎜⎜ ⎟⎟ + u 2 = 2 + + h h2 c2 ⎝ dϕ ⎠
3
which leads to the following expression
d 2u dϕ 2
+u =
GM g ⎛ 3 u 2h2 ⎞ ⎜1 + ⎟ h2 ⎜⎝ c2 ⎟⎠
In the absence of term 3h
2 2
u
c2
, the integration of the
equation should be immediate, leading to 2π period. In order to obtain the value of the perturbation we can use any of the well-known methods, which lead to an angle ϕ , for two successive perihelions, given by
2π +
6G 2 M g2 c2h2
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
M g and M i . ………..