Einstein's
Theory of Relativity
versus
Classical Mechanics
by Paul Marmet
Chapter Six
Geometrical Illustration of the Advance
of the Perihelion of Mercury.
6.1 - Conditions Controlling
the
Geometrical Shape of an Orbit.
The advance
of the perihelion of Mercury given in equation 5.46 was calculated
using
perturbations of individual parameters. This advance can also be
illustrated
using geometrical considerations. Newton stated the universal law of
gravitation
which predicts an exact quadratic gravitational field around a mass.
Newton
has shown that in the gravitational field around a central body, all
masses
move in elliptical orbits independently of the mass of the orbiting
body.
According to classical mechanics, the necessary condition to get an
exact
elliptical orbit is for the mass to move in a gravitational field whose
intensity decreases exactly as the inverse of the square of the
distance
R from the central mass:
 |
6.1 |
There are
several
measurements showing that this quadratic decrease of the gravitational
field
is followed quite accurately in nature. At a distance RM(o.s.)
from
M(S)(o.s.), the field is given by:
 |
6.2 |
where
G(o.s.)
is the number of outer space units of the gravitational
constant
and M(S)(o.s.) is the number of outer space units of the solar
mass.
Equation 6.2 implies that the Sun generates an exact quadratic
gravitational
field (in outer space units) in which Mercury is submerged.
Although the
inverse quadratic law is generally accepted, a very slight deviation of
that law was first suggested by Aseph Hall in 1894 [1].
Since we have seen that the mass of a body changes when it is moved
into
a gravitational potential, we can show that such a slight change of
mass
leads to an effect equivalent to the slight change of the quadratic
function
suggested by Hall.
Classical
mechanics shows that a massive body travels in an elliptical orbit when
the force F rather than the field between the central mass and the
orbiting
mass decreases as the square of the distance. Let us consider Newton's
equation (written in a correct way, contrary to equation 5.1):
 |
6.3 |
Since the
mass
of Mercury changes with its distance from the Sun, it is incorrect to
believe
that the force between the Sun and Mercury still follows an inverse
quadratic
function of that distance. Even if the gravitational field around a
central
mass decreases exactly as the square of the distance, the total force
between
Mercury and the Sun does not decrease at the same rate as the field.
The
trajectory of a planet whose mass decreases when it gets deeper in a
gravitational
field corresponds exactly to the problem of a non quadratic force
around
a central mass. Using classical mechanics we can calculate the new
geometrical
shape of the orbit when the force (not the field) between the Sun and
Mercury
is non quadratic.
However,
when
we consider the proper parameters of the observer moving to different
distances
from the Sun, the gravitational field (defined as the
force
divided by the proper mass) calculated from equation 5.9 is not
quadratic
for the observer traveling between different locations from the Sun.
Consequently,
using the parameters existing where Mercury interacts with the
gravitational
field leads to an apparent non quadratic field (since the proper
mass of Mercury is constant for a Mercury observer).
Using either
the non quadratic force as seen by an outer space observer that takes
into
account the change of mass of Mercury or the apparent non quadratic
force
given by equation 5.9 (with constant proper mass) leads to a similar
advance
of the perihelion of Mercury. However, these calculations are
incomplete
because other fundamental phenomena, like the change of mass as a
function
of the velocity of Mercury on its orbit, are not taken into account.
Changes
of length and clock rate due to Mercury's velocity and gravitational
potential
should also be taken into account.
Since we
have
already calculated the total precession in equation 5.46, we will limit
our demonstration here to the change of one parameter using only the
change
of mass of Mercury as a function of its distance from the Sun. We will
use only the perturbation of this parameter and show that it is one of
the contributions to the geometrical precession of the ellipse which
can
be illustrated in a classical experiment that can be done in a
laboratory
using a simple apparatus.
6.2 - The Change of Mass
of
Mercury.
Let us
consider
the change of force on Mercury due to its change of mass as a function
of its distance from the Sun. Equation 4.25 shows how the absolute mass
of a kilogram decreases when getting closer to the Sun. Consequently,
the
total mass of Mercury decreases by the same ratio. From equations 4.39,
4.40 and 4.41, the mass of Mercury (in outer space units) follows the
relationship:
 |
6.4 |
Equations
6.4
and 4.25 give:
  |
6.5 |
or:
  |
6.6 |
Using
equation
6.6 in 6.3 gives a force equal to:
  |
6.7 |
which is
equal
to:
  |
6.8 |
Let us
define:
 |
6.9 |
Equation 6.8
becomes:
  |
6.10 |
Equation
6.10
shows that the gravitational force is the difference between a
quadratic
and a cubic function. It is known that in a quadratic field, an
elliptical
orbit with a small eccentricity (first order expansion) follows the
equation
r = a(1 + ecosq) (a is the semi-major axis
and
e is the eccentricity). This equation implies two components: a
tangential
component of constant radius a and a radial component of amplitude aecosq.
Since Kepler's third law predicts the same period (first order) for
orbits
having the same average radius with or without eccentricity, both the
tangential
and the radial components lead to the same period in a quadratic field.
However, in
the case of a non quadratic field (cubic term in equation 6.10), the
period
of oscillation of the radial component becomes longer than the period
of
the circular tangential component. Of course, a circular component does
not 'feel' the field gradient. Because the cubic radial component of
oscillation
has a longer period, there is a continual shift of phase between the
periods
of the tangential and of the radial components.
Consequently,
the cubic term in equation 6.10 which does not follow Kepler's
quadratic
gradient of force, is responsible for the precession of the ellipse
because
the radial component, having a longer period, becomes out of phase with
the circular component. It is the difference of period between the
tangential
and the radial components of motion that produces the precession of the
ellipse. We also notice that it is the radial component of oscillation
which is most affected by the change of parameters resulting from
mass-energy
conservation.
Let us
examine
the bracket on the right hand side of equation 6.10. Using a series
expansion,
we can show that it is mathematically equivalent to a simple
exponential
form given by:
 |
6.11 |
in which the
exact
value of e is:
 |
6.12 |
A very good
approximation
to the first order (with n = 1) gives:
 |
6.13 |
Combining
equations
6.10 and 6.11 gives:
  |
6.14 |
where e
is always positive. Equation 6.14 shows that, because of the decrease
of
mass due to mass-energy conservation, the force F
between
Mercury and the Sun no longer decreases exactly as the square of the
distance.
The change of mass of Mercury as a function of its distance from the
Sun
is responsible for the change of power of RM
from 2 to 2+e. Therefore even if the
gravitational
field
affecting
Mercury decreases exactly as the inverse of the square of the distance
as written in equation 6.2 (as in a perfect Newtonian field), the
gravitational
force
is not Newtonian as shown in equation 6.14. Let us reconsider now the
trajectory
of bodies submitted to a force decreasing with a function which is
slightly
different from 1/R2.
6.3 - Orbital Shapes and
Gravitational
Force Gradients.
We have
calculated
in equation 6.10 the force on Mercury as a function of the distance RM.
The
corresponding
gravitational
potential
VM(o.s.) is
obtained
by the integral of equation 6.10. This gives:
  |
6.15 |
The orbit
followed
by a mass submitted to the potential described by equation 6.15 has
already
been calculated [2, 3]. Using temporarily Goldstein's notation [2],
the solution of equation 6.15 is a precessing ellipse with a velocity
of
precession equal to:
 |
6.16 |
where W(sec)
is in radians per second of time. Transforming Goldstein's notation
into
ours, we have m = M(M)o.s.(o.s.) and h = (G(o.s.)M(S)(o.s.)M(M)o.s.(o.s.)k1)/2.
t
is the period of translation of Mercury around the Sun. The angular
momentum
l
in equation 6.16 is:
  |
6.17 |
where dq/dt
is the angular velocity. Therefore, from equation 6.9 and the
definitions
above, we have:
 |
6.18 |
From
equations
6.16, 6.17 and 6.18, we have:
 |
6.19 |
Let us
transform
the precession W(sec) given in radians per
second
for radians per circumference W(circ). We
obtain:
 |
6.20 |
By
definition,
the period t equals:
 |
6.21 |
Equation
6.21
in 6.20 gives:
 |
6.22 |
Newton's law
shows
that the force of gravity FG is equal
to
the centrifugal force FC in a
circular
orbit (the eccentricity has not yet been taken into account). We have
the
fundamental equations:
  |
6.23 |
Equation
6.23
gives:
 |
6.24 |
Equations
6.24
and 6.22 give:
 |
6.25 |
Equation
6.25
gives the velocity of precession of an ellipse for the case of a
perfect
quadratic field in which the orbiting mass changes with its position in
the gravitational potential, due to mass-energy conservation.
6.4 - Identity of
Mathematical
Forms.
We find that
the advance of the perihelion of Mercury obtained with the perturbation
method used by Einstein and by us in equation 5.46, has the same
mathematical
form as equation 6.25 which clearly corresponds to the precession of an
elliptical orbit. There are two obvious differences. Since we have not
taken into account the eccentricity of the orbit, the term 1-e2
is naturally missing in equation 6.25 as explained in section 5.10.
Other
similar parameters are ignored here since we do not take into account
the
perturbations explained in section 6.1. If we take into account these
perturbations,
other similar terms will be added and the full precession will be found
as obtained in chapter five. The aim of the present demonstration is
only
to illustrate the reality of the classical precession of the ellipse in
the case of a non quadratic force.
6.5 - Illustration of
Trajectories
in Potential Wells.
When the
force
on a planet moving around the Sun decreases as the square of its
distance
from the Sun, it travels on a perfect ellipse. However, due to
mass-energy
conservation, the exact intensity of the force does not decrease as the
square of the distance. As seen in equation 6.14 the force follows the
relationship:
|
6.26 |
The
trajectory
of a particle submitted to equation 6.26 is an ellipse as illustrated
on
figures 6.1 and 6.2. In figure 6.1, a smooth conic surface is built (in
the Earth gravitational field) in such a way that the height above the
ground increases as the negative of the inverse of the square of the
distance
from the central axis. This corresponds to e
= 0 in equation 6.26. In this case, the potential energy of a ball
sliding
(without friction) on the surface increases according to the inverse
quadratic
function from the center. If we throw a ball on the surface, we can get
a circular orbit at various distances from the center. Using a
different
initial angular momentum, one can observe a stationary
elliptical
orbit as drawn on figure 6.1.
Figure 6.1
Demonstration of a mass moving in an elliptical orbit
in a quadratic potential well changing as 1/r2.
However,
if
the shape of the cone is different (see figure 6.2) so that the
potential
increases more rapidly than the inverse square of the distance
(corresponding
to equation 6.26 with e ¹
0), after throwing a ball, we see that the axis of the elliptical orbit
precesses just as observed for Mercury in its orbit around the Sun. The
cause of that classical precession on that apparatus is (in part) the
same
as the cause of the precession of 43 arcsec per century of Mercury. Of
course, this demonstration assumes that the friction and the rotation
of
the ball are negligible.
Figure 6.2
Demonstration of the precessing orbit of a mass moving
in a potential well changing as 1/r(2+e).
This
shows
that the advance of the perihelion of Mercury is not caused by space or
time distortion. It is simply a beautiful demonstration of classical
mechanics
that predicts precessing orbits giving the shape of a rosette.
6.6 - Validity of the
Classical
Model.
We have
found
above that there is a perfect mathematical agreement between the result
calculated in equation 5.46 and the result predicted using Einstein's
mathematics.
Moreover, those results are in perfect agreement with the observations
of the advance of the perihelion of Mercury.
In order to
arrive to his equation, Einstein, needed several new hypotheses called
Einstein's relativity principles. Let us compare the hypotheses used by
Einstein with the ones used in this book to find the Lorentz
transformations
and the equation for the advance of the perihelion of Mercury. This
comparison
is important if we wish to apply Occam's razor which gives a preference
to the theory that requires the minimum number of hypotheses.
Einstein's
theory requires many new hypotheses, for example:
1) the
reciprocity
principle which is not compatible with mass-energy conservation as
showed
in section 3.9;
2) the
hypothesis
that the acceleration produced by a change of velocity is
undistinguishable
from the acceleration due to gravity (see chapter ten);
3) the non
conservation of mass-energy in general relativity.
Einstein
then
arrived at the consequences that space and time can be distorted,
contracted
and dilated. In fact, Einstein's model not only requires new physical
hypotheses,
it also requires "new logic" which is not compatible with the natural
understanding
of nature. Classical logic can no longer be applied in relativity. In
this
book, we use the Bohr model of the atom which is so familiar everywhere
in physics. We also find that using proper values, the physical
relationships
are valid in all frames as in Einstein's relativity. At the same time,
a rational explanation is given. No time nor space distortion is
required
and the new interpretation is compatible with classical logic. There is
certainly an extremely strong preference in favor of this new model
when
we apply Occam's razor.
Important Note:
After the writing of that book,
a complete detailed description of the advance of the perihelion of
Mercury
entitled:
"A Detailed Classical Description of the Advance
of the Perihelion of Mercury".
This new paper appears: At
this location
6.7 - References.
[1] A. Hall, A
Suggestion
in the Theory of Mercury, Astr. J. 14, 49-51, 1894.
[2] H. Goldstein, Classical
Physics, Addison-Wesley, Reading, Mass., second Edition, p.
123,
1980.
[3] E. T. Whittaker, A
Treatise on the Analytical Dynamics of Particles and Rigid Bodies,
Cambridge University Press, Fourth Edition, Chapter 4, 1937. (also
Dover,
New York, 1944).
6.8 - Symbols and
Variables.
| FM(o.s.) |
number of outer space newtons for the
gravitational force
on Mercury |
| G(o.s.) |
number of outer space units for the
gravitational constant |
| kgframe |
mass of the local kilogram in absolute units |
| M(M)M(o.s.) |
number of outer space kilograms for Mercury
at Mercury
location |
| M(M)o.s.(o.s.) |
number of outer space kilograms for Mercury
in outer
space |
| m(M)M[o.s.] |
mass of Mercury in outer space units at
Mercury location |
| m(M)o.s.[o.s.] |
mass of Mercury in outer space units in
outer space |
| M(S)(o.s.) |
number of outer space units for the mass of
the Sun |
| RM(o.s.) |
number of outer space units for the
distance of Mercury
from the Sun |
| VM(o.s.) |
number of outer space units for the
gravitational potential
on Mercury |
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