Einstein's Theory of Relativity## versus

## Classical Mechanics

by Paul Marmet

Where to get a Hard Copy of this BookChapter Five

Calculation of the Advance

of the Perihelion of Mercury.

In this chapter we will deal with two kinds of transformations. The first kind is a mathematical transformation of units which brings no physical change to the quantities being described. In such a transformation, there is no physics, just mathematics. For example, let us suppose that we measure a rod on Mercury and find that it is 100 times longer than the local Mercury meter. Then we say that the length of the rod is 100 Mercury meters. However, if we know that on Mercury, the local meter is 1% longer than the local reference meter in outer space, we know that the same rod is actually equal to 101 times the outer space reference meter. These two descriptions by units of different frames are perfectly identical. The rod has not changed.

The observer on Mercury can also use his clock to measure a time interval. If the Mercury observer measures 100 units on his clock (i.e. 100 Mercury seconds), knowing that clocks on Mercury run at a rate which is 1% slower than clocks in outer space, we can calculate that during that absolute time interval the difference of clock displays on a clock in outer space will be 101 outer space units. No physics is involved in that transformation, only mathematics. The same physical phenomenon is described using different units.

Other units must also be transformed. For example, the absolute mass of the Sun does not change because we observe it from Mercury location near the Sun. However, measuring the same solar mass using the smaller Mercury unit of mass will lead to a larger number of Mercury units. Similarly, the physical amplitude of the absolute gravitational constant G does not change because the phenomenon takes place near the Sun. We have seen in chapter four that the absolute constant G is represented by different numbers of Mercury and outer space units. Again, no physics is involved.

**5.1.1 - Consequence
of a Simple Change of Units.**

Let us
suppose that using Newton's relationships, we want to calculate
the period of Mercury using Mercury units. We must then compare
this answer with the one obtained with the same relationships
using outer space units. If we do so, we find that the numbers
of units found for the period are different. However, when we
take into account that the Mercury clock runs at a slower rate,
we see that the absolute times obtained from either frame are
the same.

In the
next section we will see that in order to be compatible with the
principle of mass-energy conservation, one must add another kind
of transformations which are physical transformations. Contrary
to the identical consequences resulting from the mathematical
transformation explained above, different absolute results are
found when Newton's laws are applied with the proper values
belonging to different frames.

**5.2 - Physical
Transformations Due to Mass-Energy Conservation.**

The
second kind of transformations consists of real physical
changes. We have seen in chapters one and three that when an
object in outer space is moved to Mercury location, its absolute
mass changes because of the change of gravitational potential
and kinetic energies. (In the case of gravitational energy, the
difference of mass is transformed into work). The object that
remains at Mercury location is physically different from the
object that existed in outer space because the dimensions of its
atoms, their mass and clock rate have changed. This physical
change of mass is quite different from the mathematical change
of units mentioned above.

Here is
an example. An observer on Mercury measures that a mass on his
frame is 100 times larger that the unit of mass on Mercury.
Another observer in outer space measures the mass of the same
object after it has been carried out to outer space. In that new
frame, the outer space observer finds the same number (100) of
new units of mass. Both observers measure 100 __local__
kilograms. However, the absolute mass of this object has changed
when moved from Mercury location to outer space. The Mercury
kilogram is not equal to the outer space kilogram. To realize
this, we need to know the mass at Mercury location using outer
space units. Applying the principle of mass-energy conservation,
we find that using the same outer space units, the mass of the
object is reduced to only 99 outer space kilograms when brought
to Mercury location (since the Mercury kilogram is 1% lighter
than the outer space kilogram). This is a real physical change.
It is not a simple mathematical transformation of units like the
one explained in section 5.1.

We will
see in section 5.3 that these physical changes lead to results
that are physically different when calculated using proper
values in different frames. Using Newton's classical mechanics,
we will find that the results obtained using the proper
parameters in one frame are not coherent with the results
obtained using parameters proper to another frame.

In
order to clarify this description, in this chapter we will use
the expression ** transformation of units** to
designate only a pure mathematical transformation of units. When
a physical change is involved as a consequence of mass-energy
conservation, we will speak of a

We consider that the interactions between physical elements (like fields, masses, lengths and clock rates) existing on Mercury, using Mercury parameters, must be the same as the ones that we calculate in outer space using outer space parameters. This means that the mathematical relationships so well-known in physics are the only ones that are valid but it is required that on Mercury we use the physical quantities (mass, length and clock rate) existing on Mercury while in outer space, we use the physical quantities (which are different) existing in outer space. In other words, we must always use proper values. It is

**5.3 - Incoherence
between Outer Space and Mercury Predictions Using Newton's
Physics.**

In this
book, we use Newton's equations which are always perfectly valid
in all frames. However, there is a difference between Newton's
equations and Newton's physics. Newton's physics is different
from the physics described in this book because it is not
compatible with the principle of mass-energy conservation. In
Newton's physics, there is no place for changes of mass, length
and clock rate. According to that physics, the mass of an object
in outer space does not change if it is transported to Mercury
location or to anywhere in the universe.

Let us
suppose a Newtonian observer wants to measure the period of
Mercury. He wishes to know its mass. To do this, he imagines the
following thought experiment. He takes Mercury out of its orbit
to outer space and puts the planet on a balance to measure its
mass. Then he puts Mercury back on its orbit. Being a Newtonian
observer using Newton's physics, the mass he will use in his
calculations of Mercury's period will be the mass he just
measured in outer space. However, we know this mass is wrong
because of mass-energy conservation. We also know that other
parameters (like length and clock rate) at Mercury location are
modified due to the change of mass. Therefore this observer's
Newtonian calculation of the orbit of Mercury will be wrong even
when he uses the correct equations.

We will
see that when the orbit of a planet moving around the Sun is
calculated, using outer space physical parameters, we find a
perfect ellipse. However, when we use the proper parameters
existing on Mercury, we find a different orbit which is a
precessing ellipse. This explains the advance of the perihelion
of Mercury. When neglecting the changes of mass, length and
clock rate on Mercury with respect to outer space, we find an
erroneous prediction because we use outer space physical
parameters instead of proper parameters.

**5.4 - Incoherence of
the Gravitational Force Using Newton's Physics.**

Let us
give an example that shows that the calculated force of gravity
is different depending on what the physical parameters are used
(outer space or Mercury). For the Newtonian observer, the
gravitational force is:

5.1 |

5.2 |

M(M)_{M}(M) = M(M)_{o.s.}(o.s.) |
5.3 |

5.4 |

5.5 |

5.6 |

5.7 |

5.8 |

5.9 |

Using the physical parameters existing on Mercury and outer space units, equation 5.9 shows that the absolute gravitational force on Mercury is different from the one calculated using the physical parameters existing in outer space and given in equation 5.1. The two results are not compatible. They predict different orbits. As explained above, the logical choice requires that we choose the equation obtained using the proper physical parameters existing at the location where the interaction of Mercury takes place with the gravitational field. We must reject the calculation obtained using outer space parameters when the experiment is taking place on Mercury. Finally, we now realize that equations 5.1 is the limit of equations 5.9 when R

There is another direct consequence of mass-energy conservation. Contrary to equation 5.1, we see in equation 5.9 that using the physical parameters existing on Mercury, the decrease of the gravitational force is no longer perfectly quadratic. We will see in chapter six that in classical mechanics the orbits of an object submitted to a non quadratic gravitational force must have a precession.

**5.5 - Relevant Physical
Parameters.**

Let us
assume that an object on Mercury has a length of 100 Mercury
meters. This means that independently of the units used to
describe it, this is the relevant physical length. If we find
that the meter on Mercury is 1% longer that the outer space
meter, that length will be represented by 101 outer space
meters. However, a Newtonian observer in outer space would
predict 100 outer space meters from his own (incorrect)
calculation.

In the
case of time, if the Mercury observer measures that a phenomenon
lasts 100 Mercury seconds, this means that the outer space
observer measuring the same time interval on his clock (that
runs 1% faster) will measure 101 outer space seconds. For the
outer space observer, this means that the physics taking place
on Mercury is such that the phenomenon takes place more slowly.
We must remember that this is not a simple transformation of
mathematical units. The difference is due to the slowing down of
the processes on Mercury in order to maintain the internal
coherence within the Mercury frame. One must recall that if the
phenomenon takes place in outer space, the outer space observer
will also measure 100 of his seconds which are different from
100 Mercury seconds. However, since the phenomenon is taking
place on Mercury, it takes one extra outer space second before
being completed.

If one
could observe a physical phenomenon from outer space taking
place in a very deep gravitational potential, one would see that
objects are bigger and react more slowly. Furthermore if the
outer space observer calculates quite independently the
phenomena taking place on Mercury using outer space parameters,
he would find that the observations reveal that everything
functions at an unexpected slower rate with respect to his frame
since the physics at Mercury location must be compatible with
Newton's laws when using proper physical parameters.

**5.6 - Fundamental
Phenomena Responsible for the Advance of the Perihelion of
Mercury.**

This
section is very important to understand the phenomena
responsible for the advance of the perihelion of Mercury. Let us
consider that the orbit calculated by the Mercury observer has a
length equal to 1000 kilometers as determined with Newton's
equations using proper parameters on Mercury. Of course, an
observer located in outer space, also using Newton's equations
and proper values existing in outer space will calculate that
the length of the orbit is 1000 outer space kilometers.

Using
mass-energy conservation, let us assume that due to a different
gravitational potential, the unit meter on Mercury is 1% longer
than the unit meter in outer space. Consequently, in order to be
coherent, we calculate that clocks in outer space will run at a
rate which is 1% faster than the rate on Mercury.

From
the above information, let us calculate the clock display
measured on the outer space clock DCD(o.s.)
while Mercury travels the distance of 1000 km_{M}. Since
the distance traveled is 1000 km_{M}, equation 4.34
shows that due to the longer Mercury meter, the outer space
observer will measure 1010 km_{o.s.}. The circumference
of the orbit is:

Circ[M] = 1000 km_{M} = 1010 km_{o.s.} |
5.10 |

Consequently, in order to respect the physical laws existing on Mercury where the interaction with the gravitational potential takes place, we see that we must take into account two phenomena slowing down the completion of the ellipse in the frame where Mercury interacts with the gravitational potential. One is due to the increase of length of the Mercury meter and the second is due to the slowing down of the physical mechanisms on Mercury. We will calculate these two quantities in detail in the next sections of this chapter.

Let us note that in the above description, we have seen that the exact distance 1000 km

**5.7 - Change of Length
from Outer Space to Mercury Location.**

We have
seen that the relevant parameters responsible for the physical
interaction with the solar gravitational field are the ones at
Mercury location even though the final results are observed by
the outer space observer. Let us calculate the physical length
observed in outer space corresponding to the length calculated
using Mercury parameters where the interaction takes place.
There are two physical phenomena that make the Mercury meter
longer than the outer space meter. The first one is due to the
gravitational potential as explained in chapter one. The second
phenomena is due to the velocity of Mercury on its orbit as
calculated in chapter three.

Let a_{M}(o.s.)
and a_{M}(M) be the numbers representing the semi-major
axis of Mercury. Using equation 4.34, we get the relationship:

5.11 |

5.12 |

Mercury is not only located in a gravitational potential, it also has a velocity. Because of this velocity v, there is a difference between the length of the moving meter and the length of the meter at rest, both at Mercury distance from the Sun (see equation 3.23). The moving Mercury meter is also the one that is relevant here since it is the one involved in the physics taking place on Mercury. The rest meter being shorter, the number of rest meters needed to describe the length of the orbit will be larger than the number of moving Mercury meters.

Let us call N

L[rest] = N_{o} meter[rest] = N_{v}
meter[mov] |
5.13 |

5.14 |

5.15 |

5.16 |

5.17 |

5.18 |

5.19 |

5.20 |

Equation 5.12 gives the relative increase of the number of outer space meters with respect to the number of Mercury meters due to mass-energy conservation in the static gravitational potential of the Sun. Equation 5.20 gives another relative increase of the number of meters at rest with respect to the number of moving meters as explained in chapter three. From these two causes, the total relative number

5.21 |

5.22 |

**5.8 - Change of Clock
Rate from Outer Space to Mercury Location.**

There
are two independent phenomena that slow down the clocks on
Mercury's orbit. One is due to its gravitational potential, the
other is due to its velocity. On the Mercury clock, during the
period required to complete one full revolution, the difference
of clock displays called DCD_{M}(M) is smaller than the difference of clock displays DCD_{M}(o.s.) in outer space
since the physical mechanisms and clocks in outer space run at a
faster rate. Let us calculate DCD_{M}(o.s.) with respect to DCD_{M}(M) during the same absolute time interval. From
equation 4.49 we have:

5.23 |

5.24 |

5.25 |

5.26 |

5.27 |

5.28 |

5.29 |

We have seen in equation 5.23 that clocks (and therefore the absolute physical mechanisms) slow down on Mercury as a consequence of the gravitational potential at that location. Equation 5.29 also shows a slowing down of the clocks due to the velocity of Mercury on its orbit. Let us calculate the total slowing down of clocks on Mercury due to both the gravitational potential and the velocity of Mercury on its orbit. The total difference of clock displays DCD

5.30 |

5.31 |

We have seen in sections 5.7 and 5.8 how the changes of length and clock rate modify the period of translation of Mercury around the Sun. The first phenomenon given by equation 5.22 gives the relative length of the orbit as measured in outer space when the phenomenon is calculated using the parameters existing on Mercury where the interaction with the gravitational field of the Sun takes place. The circumference of the orbit

Each phenomenon makes an independent contribution to modify lengths and clock rates on moving Mercury with respect to the ones at rest in outer space. Consequently both phenomena will contribute to the larger number of units for the period of Mercury as measured by an outer space observer.

Let us call P

Let us add the contribution of the two phenomena described above. The correction on the period will be the product of the contributions given by equations 5.22 and 5.31. This gives:

5.32 |

5.33 |

5.34 |

5.35 |

5.36 |

Before completing this section, we must notice that following Newton's law, the advance of the perihelion of Mercury given by equation 5.36 can be written in a more simple form. Let us consider the gravitational potential "Pot" as a function of the distance R

5.37 |

5.38 |

**5.10 - Correction for
an Elliptical Orbit.**

There
is one more term that needs to be taken into account to get a
better accuracy. We know that Mercury travels on an elliptical
orbit. However, in our calculation we have always considered the
distance of Mercury from the Sun (R_{M}) as a constant. In an elliptical motion, the distance
from the Sun is not constant but varies according to a
relationship characteristic of an ellipse. From geometrical
considerations, it is demonstrated [1] that the distance R_{M}
of the orbiting body (Mercury) from the occupied focus (where
the Sun is located) of an ellipse is given by the relationship:

5.39 |

The orbiting body is sometimes at a closer distance from the Sun where the gravitational potential is larger. At those times, the velocity of the planet is larger. Of course, there are other parts of the orbit where the planet moves more slowly in a shallower gravitational potential. However, we can see that the smaller gravitational potential does not compensate completely for the larger one. The eccentricity must be taken into account. The clock rate and the unit of length must be taken into account at every point of the elliptical orbit. We have calculated above that the change of gravitational potential and of velocity produce an average effect represented mathematically by an "effective potential" Pot/M(

5.40 |

5.41 |

5.42 |

5.43 |

5.44 |

5.45 |

5.46 |

**5.11 - Mathematical
Identity with Einstein's Equation.**

Einstein presented a mathematical relationship for the advance
of the perihelion of Mercury. Many books report that result.
Straumann's [2]
equations 3.1.11 and 3.3.7 give:

5.47 |

[1] Kenneth R. Lang, ** Astrophysical
Formulae**, Springer-Verlag, ISBN 3-540-09933-6.
second corrected and enlarged edition, p. 541, 1980.

[2] Norbert Straumann, ** General
Relativity and Relativistic Astrophysics**,
Springler-Verlag, second printing, 1991.

[3] U. J. J. Le Verrier,
** Théorie du mouvement de Mercure**, Ann.
Observ. imp. Paris (Mémoires) 5, p. 1 to 196, 1859.

[4] N. T. Roseveare, ** Mercury's
Perihelion
from
Le
Verrier
to
Einstein**, Clarendon Press, Oxford, 208 p. 1982.

a_{M}(M) |
number of Mercury meters for the semi-major axis |

a_{M}(o.s.) |
number of outer space meters for the semi-major axis |

DCD_{M}(M) |
DCD for the period of Mercury measured by a Mercury clock |

DCD_{M}(o.s.) |
DCD for the period of Mercury measured by an outer space clock |

DCD_{M,v} |
DCD for the period of Mercury measured by a moving Mercury clock |

DCD_{o.s.,o} |
DCD for the period of Mercury measured by an outer space clock at rest |

DCD_{o} |
DCD for the period of Mercury on a clock at rest |

DCD_{v} |
DCD for the period of Mercury on a clock in motion |

DPDCD(o.s.,rest)_{l}, |
relative increase of the number of absolute seconds for the period of Mercury |

Df | advance of the perihelion of Mercury in radians |

F_{G}(M) |
number of Mercury newtons for the gravitational force on Mercury |

F_{G}(o.s.) |
number of outer space newtons for the gravitational force on Mercury |

G(M) | number of Mercury units for the gravitational constant |

G(o.s.) | number of outer space units for the gravitational constant |

km_{frame} |
length of the local kilometer in a frame |

l_{M}(M) |
number of Mercury meters for the orbit of Mercury |

l_{M}(o.s.) |
number of outer space meters for the orbit of Mercury |

l_{M,v} |
number of Mercury moving meters for the orbit of Mercury |

l_{o.s.,o} |
number of outer space rest meters for the orbit of Mercury |

L[rest] | length of the orbit of Mercury in rest units |

meter[frame] | length of the local meter in a frame |

M(M)_{M}(M) |
number of Mercury kilograms for Mercury at Mercury location |

M(M)_{o.s.}(o.s.) |
number of outer space kilograms for Mercury in outer space |

M(S)(M) |
number of Mercury units for the mass of the Sun |

M(S)(o.s.) |
number of outer space units for the mass of the Sun |

N_{o} |
number of rest meters for the orbit of Mercury |

N_{v} |
number of moving meters for the orbit of Mercury |

PDCD(o.s.,rest)_{l}, |
number of outer space (rest) seconds for the period of Mercury taking into account the gravitational potential and the velocity of Mercury |

PDCD(M,mov)_{l}, |
number of Mercury (motion) seconds for the period of Mercury taking into account the gravitational potential and the velocity of Mercury |

R_{M}(M) |
distance of Mercury from the Sun in Mercury units |

R_{M}(o.s.) |
distance of Mercury from the Sun in outer space units |

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