Einstein's Theory of Relativity## versus

## Classical Mechanics

by: Paul Marmet

Where to get a Hard Copy of this BookChapter Four

Fundamental Nature of the Mechanism Responsible for the Advance of the Perihelion of Mercury.

In order to understand the mechanism responsible for the advance of the perihelion of Mercury, we need to explain the meaning of quantities such as an absolute standard of mass, time or length. The meaning of absolute standards is such that each of them must always represent the same and unique physical quantity in any frame. This condition is necessary since the absolute length of a rod does not change because it is measured from a different frame. This also applies to an absolute time interval and an absolute mass: they do not change when measured in different frames. However, an absolute length, time interval or mass can be described using different parameters (e.g. different units). One must conclude that lengths, time intervals and masses are absolute and exist independently of the observer. They never change as long as they remain within one constant frame. However, they appear to change with respect to an observer who moves to a different frame because they are then compared with new units located in a different frame.

In relativity, we always read an expression with respect to a frame "of reference". The phrase "of reference" gives the illusion that masses, lengths and clock rates really change as a function of the "reference" used to measure them. That there could be a real physical change of mass, length and clock rate because the observer uses a different "reference" does not make sense. This apparent change of length, clock rate or mass is simply due to the observer using different units of comparison. In this book, we avoid the words "of reference" because they are clearly misleading.

We have seen that when a rod changes frames, its absolute length changes. However, when an observer carrying his reference meter changes frames, the length of the rod that remains at rest corresponds to a different number of the observer's new reference meter. When a rod changes frames, the change of its length is real as seen in chapters one and three. However, when the observer changes frames (with his reference meter) and the rod does not, there is only a change in the number of measured meters; the rod does not change. Consequently, the change of frame of the rod and the change of frame of the observer (carrying his reference meter) are not symmetrical.

The usual definition of the meter is 1/299 792 458 of the distance traveled by light during one second. The local clock is used to determine the second. We recall from section 2.4 that this definition is not absolute because it depends on the definition of the second which is a function of the local clock rate which changes from frame to frame.

Unfortunately, there is no direct way to reproduce an absolute meter within a randomly chosen frame. We have seen that carrying a piece of solid matter from one frame to another one (in which the potential or kinetic energy is different) leads to a change of the Bohr radius and consequently to a change in the dimensions of the piece of matter. However, a local meter can apparently be reproduced in any other frame using a solid meter previously calibrated in outer space and brought to the local frame. Of course, the absolute length of that local meter in the new frame will not be equal to its absolute length when it was in outer space because the potential and kinetic energies may change from frame to frame.

One can
also reproduce a local meter in any frame by calculating 1/299
792 458 of the distance traveled by light in ** one local
second**. However, the duration of the local second
must be corrected with respect to the reference clock-rate
existing in outer space (with v = 0). It is illusionary to
believe that absolute time and absolute length can be obtained
in any frame just by carrying a reference atomic clock and a
reference meter to the new frame.

We define the

meter_{o.s.} = B_{o.s.}a_{o.s.} |
4.1 |

meter_{E} = B_{o.s.}a_{E} |
4.2 |

meter_{M} = B_{o.s.}a_{M} |
4.3 |

In the problems considered in these first chapters, the relative changes of length, time rate and mass will always be extremely small. In the case of Mercury, which is the closest planet to the Sun, these changes will be as small as about one part per billion. Consequently we will regularly simplify the calculations by using only the first order. This will be an excellent approximation. The derivative of the function will then become equal to the finite difference as used in chapter one. This does not change the fundamental understanding of the phenomenon as we will see below.

We have seen in equation 4.1 that the absolute reference meter is a constant number of times (B

4.4 |

4.5 |

4.6 |

In previous chapters, we have used the brackets [rest] and [mov] to indicate the units. From now on, depending on whether we refer to the units of length, mass, clock rate, etc., located in outer space (free from a gravitational potential) or units in the gravitational potential of Mercury, we will use the indices [o.s.] or [M]. The units will always be "translated" in absolute units (e.g. Mercury second = 1.01 absolute seconds). Using equations 4.1, 4.3, 4.5 and 4.6, we find that the length of the Mercury meter (meter

4.7 |

It is useless here to specify the units of GM(

**4.3 - The Absolute
Reference Second.**

An
equivalent transformation must be taken into account when time
is defined. We can evaluate time on different frames using a
local cesium clock. However, one must recall that the rate of
such a clock (or of any other clock) changes with the electron
mass and therefore with the potential and kinetic energies where
the clock is located. Therefore a correction must be made if we
want to know the absolute time.

For the
case of zero gravitational potential, we now define an absolute
time interval called the absolute reference second just as in
section 3.5.1 where the second was defined for the case of zero
velocity. During one absolute second, a cesium clock makes N(S)
(where the index (S) refers to the definition of a second)
oscillations that are counted from the number of cycles of
electromagnetic radiation emitted. That cesium clock must be
located outside the gravitational potential of the Sun and have
zero velocity. By definition, that absolute time interval will
be called the "outer space second". We have:

absolute ref. second º
N(S) Oscillations (cesium clock_{o.s.}). |
4.8 |

1 abs. sec. º DCD_{o.s.}(S) º N(S) Oscillations(cesium clock_{o.s.}) |
4.9 |

1 Mercury sec. º DCD_{M}(S) º N(S)
Oscillations(cesium clock_{M}) |
4.10 |

1 local second º DCD_{frame}(S) |
4.11 |

Let us consider that a clock in outer space records a difference of clock displays equal to the number DCD

Dt_{o.s.}[o.s.]
=
DCD_{o.s.}(o.s.)DCD_{o.s.}(S) |
4.12 |

In equation 4.12, the symbol [o.s.] after Dt

Dt_{o.s.}[M]
=
DCD_{o.s.}(M)DCD_{M}(S) |
4.13 |

Of course, a Mercury second is not equal to one real outer space second. The absolute second is defined in outer space. Therefore a Mercury second is not a real time interval. It corresponds to a difference of clock displays which can be described as an apparent time on Mercury.

If a phenomenon taking place in outer space is measured using a clock located in outer space, its duration will be represented by the absolute time interval Dt

Dt_{o.s.}[o.s.]
=
Dt_{o.s.}[M] |
4.14 |

DCD_{o.s.}(o.s.)DCD_{o.s.}(S) = DCD_{o.s.}(M)DCD_{M}(S) |
4.15 |

In order to clarify this description, let us give a numerical example. Let us assume that an atomic clock located in outer space has emitted 20 times N(S) cycles of E-M radiation. After N(S) cycles, one more absolute second DCD

_{1 Mercury second = 1.01 Absolute
second } |
_{4.16} |

4.17 |

_{4.18} |

Dt_{o.s.}[o.s.]
=
20×1 absolute second = 20 absolute seconds |
4.19 |

Dt_{o.s.}[M]
=
19.80198×(1.01 abs. seconds) = 20 abs. seconds |
4.20 |

**4.3.2 - Relative
Clock Displays between Frames.**

We have
seen that the clock used in each frame simply counts the number
of cycles emitted by the local atomic clock. In all frames, the
local second is equal to the count of N(S) cycles on the local
clock. During one absolute time interval, the number of cycles
is then proportional to the absolute clock rate which is its
absolute frequency as given by equation 1.22 (when v = 0).
Therefore, during one absolute time interval, the ratio of the
differences of clock displays between frames is directly
proportional to the ratio of the natural frequency of each
clock. This gives:

4.21 |

4.22 |

4.23 |

4.24 |

**4.4 - The Absolute
Reference Kilogram.**

The
absolute unit of mass is also defined in outer space. We have
seen in chapter one that one absolute kilogram (kg_{o.s.})
in outer space contains a different amount of mass after it is
carried to Mercury. When we carry a mass of one kilogram (kg_{o.s.})
from outer space to Mercury location (at rest), the amount of
mass decreases (because it gives up energy during the transfer).
However, the observer on Mercury will still call it one Mercury
kilogram (kg_{M}) since the
number of atoms has not changed. In fact, nothing ** appears**
to change for an observer moving with the kilogram and observing
a physical phenomenon on Mercury. The relationship between two
kilograms located in different potentials is given in equation
1.5. Using equations 1.5 and 4.6, we find:

4.25 |

**4.5 - Space and Time
Corollaries within the Action-Reaction Principle.**

Let us
discuss what happens inside a frame located at the position
where Mercury interacts with the Sun's gravitational field. What
is the behavior of Newton's laws at that location?

We
believe in the principle of causality. The cause is the reason
for the action. Newton applied this principle and stated that an
action is always accompanied by a reaction. However, even if
this has not been stated specifically, it becomes obvious that
there are two corollaries to that principle. The first corollary
is that both the action and the reaction take place at exactly
the same location where the interaction takes place. The second
corollary is that both the action and the reaction take place at
exactly the same time the interaction takes place. The principle
of causality implies that it is illogical and indefensible to
believe that the cause of a phenomenon does not take place at
the same location and at the same time that the effect does.

Let us
apply those corollaries to relativity. When a mass moves in a
gravitational field, its trajectory is modified by the action of
the gravitational field. The interaction between a mass and a
gravitational field takes place at the location of the mass and
at the moment the mass is interacting with the field.
Consequently, the relevant parameters during the interaction are
the amount of mass and the intensity of the gravitational field
at the location of the interaction. It would be absurd to
calculate an interaction using quantities that exist somewhere
else than where the interaction takes place. When we study the
behavior of Mercury interacting with the solar gravitational
potential, we must logically use the physical quantities
existing where Mercury is located. This means that when we
calculate the behavior of planet Mercury, we must use the units
of length, clock rate and mass existing at Mercury location.
This is the only logical way to be compatible with the principle
of causality and with its natural corollaries leading to the
principle of action-reaction. It would not make sense for the
mass of Mercury involved in the interaction with the solar
gravitational field to be the mass it has in outer space rather
than its real mass where it is located at the moment it is
interacting near the Sun.

Therefore the amount of mass, length and clock rate that must be
used in the equations are the ones that appear at Mercury
location, since they are the only relevant parameters logically
compatible with the physics taking place on Mercury. At Mercury
location, there is no other physics than the one using the local
mass, length and clock rate. Logically, it must be so everywhere
within any frame in the universe. This point is extremely
important and is fundamental in the calculations below because
it is the basic phenomenon that explains the advance of the
Mercury perihelion around the Sun.

**4.6 - Fundamental
Mechanism Taking Place in Planetary Orbits.**

In
classical mechanics, it is demonstrated that planets revolve
around the Sun in a circular or elliptical orbit. The complete
period of an orbit can be defined as the time taken to complete
a full translation of 2p radians
around the Sun or as the time interval taken by the planet to
complete its ellipse between the passages of a pair of
perihelions. It is usually considered that these two definitions
of a period of an orbit are identical. However, if the ellipse
is precessing, the angle spanned between the two passages of a
pair of perihelion is larger than for a non precessing ellipse
i.e. larger than 2p radians. This
means that the full translation of 2p
radians is completed before the ellipse reaches the next
perihelion. Therefore we expect the period of that precessing
ellipse to be larger.

One of
the fundamental phenomena implied in such an orbital motion is
the gravitational potential decreasing as the inverse of the
distance from the Sun where the planet is orbiting. When the
orbit is circular, it is difficult to determine at what instant
one full orbit is completed other than measuring a translation
of 2p radians with respect to masses
seen in outer space. However, in an elliptical orbit (as in the
case of Mercury around the Sun), the direction of the major axis
can be easily located in space from the instant Mercury is at
its perihelion, i.e. its closest distance from the Sun.

**4.6.1 - Significance
of Units in an Equation.**

In
Galilean mechanics, when the units are identical in all frames,
the pure number that multiplies the unit is undistinguishable
from the quantity that includes the unit. For example, when
someone reports that a rod is ten meters long, we can assume
that either he has in mind that the rod is ten times the length
of the standard meter (in which ten is a pure number separated
from the unit of length), or he means a single global quantity
with unit, corresponding to one single quantity ten times longer
than the unit meter. Of course, the difference brings no
consequence at all when we always use the same standard meter.
However, the correct interpretation must be understood and
specified here because the size of the reference meter (and all
other units) changes from frame to frame.

If "a"
represents the semi-major axis of the elliptical orbit of
Mercury, we have to find whether "a" represents a pure number
(to which a unit is added and considered separately) or a single
global quantity (with units included). This can be answered if
we study the fundamental role of a mathematical equation. In
mathematics, we learn that an equation is a fundamental
relationship between numerical quantities. The same mathematical
equation can relate numbers (or concepts) having different
units. This can be illustrated in the following way.

If an
apple costs 50 cents, how many apples (N) will we buy with
$10.00? We use the following equation:

4.26 |

N = 20 apples | 4.27 |

N = 20 oranges | 4.28 |

N = 1000 peas | 4.29 |

Therefore, "a" represents the

In the solar system, the orbit of Mercury is very elongated and is an excellent example to study Kepler's laws. However, since there are several other planets moving around the Sun, there are other classical corrections due to the interactions between these other planets that need to be taken into account. Extensive classical calculations show that the interaction of the other planets of the solar system also produces an important advance of the perihelion of Mercury. After accurate calculations, data show that the advance of the perihelion of Mercury is larger than the value predicted by classical mechanics. The advance of the perihelion is observed to be 43 arcsec per century larger than expected from all classical interactions by all planets.

In order to solve this problem, we have to examine in more detail the conditions in which the equations must be applied. As we will see in chapter five, the

**4.7 - Transformations
of Units.**

**4.7.1 - a _{M}(o.s.) versus a_{M}(M).**

When we measure the

L_{M}[o.s.] = a_{M}(o.s.)metero.s. |
4.30 |

L_{M}[M] = a_{M}(M)meter_{M} |
4.31 |

L |
4.32 |

4.33 |

4.34 |

**4.7.2 - M( S)(o.s.)
and M(M)_{M}(o.s.) versus M(S)(M) and M(M)_{M}(M).**

The symbols (

m(S)[o.s.] =
M(S)(o.s.)kg_{o.s.} |
4.35 |

m(S)[M] = M(S)(M)kg_{M} |
4.36 |

m(S)[o.s.] =
m(S)[M] |
4.37 |

m(M)_{M}[o.s.] = M(M)_{M}(o.s.)kg_{o.s.} |
4.38 |

m(M)_{M}[M] = M(M)_{M}(M)kg_{M} |
4.39 |

m(M)_{M}[o.s.]
=
m(M)_{M}[M] |
4.40 |

4.41 |

4.42 |

4.43 |

4.44 |

**4.7.3 - P _{M}(o.s.) versus P_{M}(M).**

In equations 4.12 and 4.13, we have calculated absolute time intervals Dt as measured from outer space location (Dto.s.[o.s.]) and Mercury location (Dto.s.[M]). Let us consider now that the time interval Dt is the period of translation of Mercury to complete an ellipse around the Sun. The

Dt_{M}[o.s.] = P_{M}(o.s.)
DCDo.s.(S) |
4.45 |

Dt_{M}[M] = P_{M}(M)
DCDM(S) |
4.46 |

Dt_{M}[o.s.] = Dt_{M}[M] = P_{M}(M)
DCDM(S) = P_{M}(o.s.) DCDo.s.(S) |
4.47 |

4.48 |

4.49 |

**4.7.4 - G(o.s.)
versus G(M) .**

Since
lengths, clock rates and masses are not the same in different
frames, we see now that the gravitational constant G is
different when measured using Mercury units. The ** number**
of outer space units of the gravitational constant is called
G(o.s.) and the

J[o.s.] = G(o.s.)U_{o.s.} |
4.50 |

J[M] = G(M)U_{M}. |
4.51 |

J[o.s.] = J[M] | 4.52 |

4.53 |

4.54 |

F = ma | 4.55 |

4.56 |

4.57 |

4.58 |

4.59 |

vo.s. = v_{M} |
4.60 |

4.61 |

4.62 |

4.63 |

4.64 |

4.65 |

**4.7.5 - F(o.s.)
versus F(M).**

From
equation 4.56 we have:

4.66 |

4.67 |

4.68 |

4.69 |

4.70 |

a_{frame}[o.s.] |
length of the local Bohr radius in absolute units |

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

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

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}(S) |
apparent second on Mercury |

DCD_{o.s.}(M) |
DCD in outer space measured by a Mercury clock |

DCD_{o.s.}(o.s.) |
DCD in outer space measured by an outer space clock |

DCD_{o.s.}(S) |
absolute second in outer space |

Dt_{M}[M] |
period of Mercury in Mercury units |

Dt_{M}[o.s.] |
period of Mercury in outer space units |

Dt_{o.s.}[M] |
time interval in outer space in Mercury units |

Dt_{o.s.}[o.s.] |
time interval in outer space in outer space units |

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

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

J[M] | gravitational constant in Mercury units |

J[o.s.] | gravitational constant in outer space units |

kg_{frame} |
mass of the local kilogram in absolute units |

L_{M}[M] |
length of the semi-major axis of the orbit of Mercury in Mercury units |

L_{M}[o.s.] |
length of the semi-major axis of the orbit of Mercury in outer space units |

meter_{frame} |
length of the local meter in absolute units |

M(M)_{M}(M) |
number of Mercury units for the mass of Mercury at Mercury location |

m(M)_{M}[M] |
mass of Mercury in Mercury units at Mercury location |

M(M)_{M}(o.s.) |
number of outer space units for the mass of Mercury at Mercury location |

m(M)_{M}[o.s.] |
mass of Mercury in outer space units at Mercury location |

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 |

m(S)[M] |
mass of the Sun in Mercury units |

m(S)[o.s.] |
mass of the Sun in outer space units |

N(S) | number of oscillations of an atomic clock for one local second |

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

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

R_{M} |
distance between Mercury and the Sun |

U_{frame} |
unit of the gravitational constant in the local frame |

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