4.1 Definition of the
Absolute Standard Units [o.s.].
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. 4.2
- The Absolute Reference Meter.
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 absolute reference meter (metero.s.)
as the distance traveled by light during 1/299 792 458 of a
second given by a clock located at rest in outer space away from
any mass. The subscript o.s. defines where the meter is located.
This unit of length is equal to a number Bo.s. times
the length of the Bohr radius ao.s. in outer
space. An absolute reference meter must have the same absolute
physical length, independently of the frame where it is located
(and of the frame where the observer is located). Consequently,
an observer must make relevant corrections to his local meter to
reproduce the absolute reference meter. The definition of the
absolute reference meter is then:
| metero.s. = Bo.s.ao.s. |
4.1 |
The
absolute meter can be reproduced in any frame but it is defined
with respect to a length in outer space. The constant Bo.s.
(the inverse of the Bohr radius) is about 1.8897263×1010. Since the Bohr radius a varies with the
electron mass (which changes with potential and kinetic energies),
the constant number Bo.s. times the outer space Bohr
radius a is not an absolute standard when the meter is not
located in outer space. The Earth meter (meterE) is different from the absolute reference meter (metero.s.)
because the Bohr radius is longer on Earth. The length of the
Earth meter is:
We see
that the length of a meter at a Mercury distance from the Sun is
also different from the length of a meter in outer space or on
Earth. Let us study the example of Mercury since we wish to
predict a phenomenon taking place at the distance from the Sun
where Mercury is orbiting. The length of the Mercury meter (meterM) is:
In order
to avoid useless lengthy repetitions, we will shorten some of the
descriptions. Instead of repeating that we refer to a location at
the Mercury distance from the Sun which has zero orbital velocity,
we will simply say "Mercury location" and the context will provide
the supplementary information. The velocity component of Mercury
will be considered separately later. All other parameters will be
taken into account only later because they are not relevant in
this chapter and would bring confusion. An absolute standard of
reference will sometimes be called in short "absolute meter",
"absolute time rate" or "absolute mass" when it corresponds to the
standard established in outer space.
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 (Bo.s.) the Bohr radius in
outer space (ao.s.). However, the Bohr radius
does not change solely with the gravitational potential. It also
changes with velocity. We define the absolute outer space meter as
being a meter in outer space with zero velocity. From equation
1.22, the relationship giving the Bohr radius a when there
is no change of velocity is (using outer space units):
 |
4.4 |
which
gives:
 |
4.5 |
where mgDh is the change of potential energy (Pot.)
of a mass m in a gravitational field across height Dh. In the case of a central force,
Newton's laws say that the gravitational potential (Pot.) of a
body decreases when the distance (R) from the central body
increases. The gravitational potential of a body of mass M(M)
(in the case of Mercury) at a distance RM from the Sun of mass M(S) with respect to outer
space is:
 |
4.6 |
where G
is the Cavendish gravitational constant and g is the gravitational
acceleration where the mass is located (here in the solar
gravitational field).
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 (meterM) compared with the
absolute reference meter (metero.s.) is:
 |
4.7 |
We recall
that the length of the meter (metero.s.) in outer space
is the absolute standard reference. However, we know that when an
observer is located on a different frame to measure a given
length, he finds a different answer because his unit of comparison
(his local meter) is different.
It is
useless here to specify the units of GM(S)/c2RM. Logically, they
should be coherent i.e. either [M] or [o.s.]. Physically, it makes
no difference whether the units of G, M(S) or R are the
same or not since the error brought in this way is of the order of
10-9 on GM(S)/c2RM which is itself of
the order of 10-9
with respect to the meter.
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 clocko.s.). |
4.8 |
During
one absolute second, a cesium clock in outer space emitting N(S)
cycles shows a difference of clock displays labeled DCDo.s.(S). We must emphasize
that DCDo.s.(S) does not
correspond to any value of DCD, it
corresponds only to the number of counts on the outer space clock
leading to the absolute second. This is shown by (S) following the
DCD. Consequently, DCDo.s.(S)
representing the absolute reference second must not be confused
with a simple value of DCDframe (without (S)) which can be any number of seconds. We
have:
| 1 abs. sec. º DCDo.s.(S) º N(S) Oscillations(cesium clocko.s.) |
4.9 |
When an
observer on Mercury observes that his cesium clock has emitted the
same number N(S) of cycles, the absolute time interval elapsed is
not the absolute second since the Mercury clock is slower. That
time interval is called the Mercury second. We have:
| 1 Mercury sec. º DCDM(S) º N(S)
Oscillations(cesium clockM) |
4.10 |
Therefore
we define one "local second" as the time elapsed when the
numerical value shown on a local frame is equal to DCDframe(S).
Of course, the Mercury second represented by DCDM(S) lasts longer than the outer space second represented
by DCDo.s.(S) because even
if the differences of clock displays DCDo.s.(S)
and DCDM(S) are equal, the Mercury clock is slower.
Consequently, during one local second, we have for the outer space
clock the same DCD than for the
Mercury clock:
| 1 local second º DCDframe(S) |
4.11 |
Since the
principle of mass-energy conservation and Bohr equation teach us
by how much the rates of two clocks located in outer space and on
Mercury differ, an observer on Mercury can calculate the absolute
time using his Mercury clock and making suitable corrections due
to the gravitational potential at Mercury location (we will
consider the velocity of Mercury later).
Let us
consider that a clock in outer space records a difference of clock
displays equal to the number DCDo.s..
The corresponding absolute time interval elapsed is called Dto.s.[o.s.]. That absolute time
interval can be measured on different locations like Mercury or
outer space. For a phenomenon taking place in outer space, a time
interval can be written:
| Dto.s.[o.s.]
=
DCDo.s.(o.s.)DCDo.s.(S) |
4.12 |
where Dto.s.[o.s.] is the absolute
time interval, DCDo.s.(o.s.)
is the number of seconds shown by the outer space
clock and DCDo.s.(S) is the
absolute unit of time in outer space given by the o.s. clock.
In
equation 4.12, the symbol [o.s.] after Dto.s.
is due to the units of time DCDo.s.(S).
The parentheses in DCDo.s.(o.s.)
indicate the units used for the measurement. The subscript o.s. of
Dto.s.[o.s.] andDCDo.s.(o.s.) refers to the
location where the phenomenon takes place (this is different from
what we did in chapter three). When an outer space phenomenon is
observed using a Mercury clock, the absolute time interval Dto.s. [M] measured on a clock
on Mercury is given by the relationship:
| Dto.s.[M]
=
DCDo.s.(M)DCDM(S) |
4.13 |
where
DCDo.s.(M) is the number
of Mercury seconds and DCDM(S) is the unit of time of the clock located on Mercury,
as described in equation 4.10.
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 Dto.s.[o.s.]
(equation 4.12). If this same phenomenon is measured using the
Mercury clock, the same absolute time interval will be represented
by Dto.s.[M] (equation
4.13). Of course, one single phenomenon does not last a longer
absolute time because it is observed from a different location
using a different clock. The real absolute duration is the same in
any frame. This gives:
| Dto.s.[o.s.]
=
Dto.s.[M] |
4.14 |
Using
equations
4.12
and
4.13
in
4.14, we find:
| DCDo.s.(o.s.)DCDo.s.(S) = DCDo.s.(M)DCDM(S) |
4.15 |
4.3.1 - Example.
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 DCDo.s.(S)
has elapsed and this is repeated DCDo.s.(o.s.)
times (with DCDo.s.(o.s.) =
20). Consequently, the corresponding time interval Dto.s.[o.s.] elapsed is 20
absolute (or outer space) seconds, as given in equation 4.12. That
same clock is moved to a stationary location (for example Mercury)
near a very massive star so that the relativistic electron mass
decreases by 1.0% due to the change of gravitational potential.
Quantum mechanics shows that the atomic clock will then run at a
rate which is 1.0% slower (as explained in chapter one).
Consequently, since the atomic clock on that planet is slower than
when it was in outer space, it will take a longer absolute time to
make the same number N(S) of oscillations. Since the Mercury
second is defined (in equation 4.10) as the time required for the
clock on Mercury to emit N(S) cycles, it is longer than the outer
space second. This gives:
| 1 Mercury second = 1.01 Absolute
second |
4.16 |
Consequently,
during
the
time
interval
in
which the outer space clock will record an absolute time interval
Dto.s.[o.s.] equal to 20
outer space seconds (DCDo.s.(o.s.)),
the Mercury clock will record a smaller DCDo.s.(M)
because
it
runs
at a slower rate. The DCDo.s.(M)
recorded
on
Mercury
will
be
1.0% smaller:
 |
4.17 |
giving
the numerical value:
 |
4.18 |
Therefore,
in
agreement
with
equation
4.14,
since the Mercury second lasts longer, as seen in equation 4.16,
the total absolute time elapsed on Mercury (Dto.s.[M])
is the same as the total absolute time in outer space. We find in
equation 4.12:
| Dto.s.[o.s.]
=
20×1 absolute second = 20 absolute seconds |
4.19 |
From
equations 4.13, 4.16 and 4.18 we have:
| Dto.s.[M]
=
19.80198×(1.01 abs. seconds) = 20 abs. seconds |
4.20 |
Therefore, Dt is a real absolute time
interval in all frames.
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 |
Equation
4.21 gives the relative frequencies of clocks located in different
frames. Obviously, it does not matter whether the phenomenon
measured is in outer space or on Mercury, as long as both clocks
measure the same phenomenon. This means that the subscripts of the
left hand side of 4.21 could both be M instead of o.s.. If there
is a difference of kinetic energy between the frames, equation 3.9
must be applied. Any difference of clock rate is caused by the
difference of gravitational potential and/or kinetic energy
between an outer space location and the orbit of Mercury. In the
case of pure potential energy, using equations 1.22 and 4.6, the
relative clock rate is given by the relationship:
 |
4.22 |
which
gives:
 |
4.23 |
Using
equation 4.21 with equation 4.23, we see that during the same
absolute time interval, the relative difference of clock displays
is:
 |
4.24 |
Let us
note that these equations do not take into account a second order
that might exist when the particle moves down in the gravitational
potential. Since that second order effect is quite negligible in
the first chapters of this book, we will consider it only if it
becomes significant.
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 (kgo.s.)
in outer space contains a different amount of mass after it is
carried to Mercury. When we carry a mass of one kilogram (kgo.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 (kgM) 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 |
Equation
4.25 gives the mass of the outer space kilogram with respect to
the Mercury kilogram.
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 |
With a =
$10.00, and b = $0.50 each, we find
Now, if
we also find that an orange costs 50 cents, how many oranges will
we have for $10.00? Using again equation 4.26 with a = $10.00 and
b = $0.50 each, we find:
We also
want to buy peas. They cost 1 cent each. How many peas do we get
for $10.00? Using again equation 4.26, we find that the number of
peas is:
Equations
4.27, 4.28 and 4.29 illustrate that the mathematical parameter N
does not represent apples, oranges or peas. It represents only the
numerical value of the unit. The unit must be specified
separately. One must know that the units also follow a separate
mathematical relationships. This is called a dimensional analysis
which requires an analysis separate from the numerical analysis.
Therefore, "a" represents the number of units of
length. The same remark must be applied to all physical quantities
that are pure numbers obtained from a previous definition of other
standard units. Furthermore, in order to be compatible with the
principle of causality given above, the units of length, mass and
clock rate must necessarily be the ones existing on Mercury where
the phenomenon takes place. We will see below how this description
leads to a perfect coherence.
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 number of seconds giving the
period P is a function of the parameters a, G, M(S) and M(M).
However, due to mass-energy conservation we have seen that the
units of length, time and mass are different at Mercury distance
from the Sun than in outer space. In section 4.5, we have also
seen that the action of the gravitational potential on Mercury
must be calculated using the number of units of mass (and all
other parameters) that Mercury has at that location.
4.7 - Transformations
of Units.
4.7.1 - aM(o.s.) versus aM(M).
When we
measure the number of meters that constitute a
given length, we find that this number depends on the length of
the unit used in conjunction with it. We call aM(o.s.), the number of outer space
meters that represents the length of the semi-major axis of the
orbit of Mercury when we use outer space meters. The absolute
physical length LM[o.s.] being
measured using outer space meters is then:
| LM[o.s.] = aM(o.s.)metero.s. |
4.30 |
The value
of the absolute length LM[o.s.]
of the semi-major axis of the orbit of Mercury corresponds to
measuring the number aM(o.s.) of
meters in the orbit times the outer space meter (metero.s.). We
now have to determine the number aM(M) of Mercury meters (meterM) found in conjunction with Mercury units. aM(M) represents the corresponding number
of Mercury meters to measure the same length when we use Mercury
meters. We find that the absolute physical length LM[M] of the semi-major axis, is given by:
Since a
physical length does not change because we use a different
reference meter to measure it, we must understand that the
absolute physical length of the semi-major axis is the same
whether it is measured using outer space or Mercury units.
Therefore, the absolute length LM[frame]
of
the
semi-major
axis
of
the orbit of Mercury is the same independently of the units used
to measure it. Therefore, equations 4.30 and 4.31 are identical:
|
LM[M]
= LM[o.s.] = aM(o.s.)metero.s. = aM(M)meterM.
|
4.32 |
Equation
4.32 gives us the relationship between the number aM(o.s.) of outer space meters and the number aM(M) of Mercury meters to measure the same length. This
gives:
 |
4.33 |
Combining
equations 4.7 and 4.33 gives:
 |
4.34 |
Equation
4.34 shows that the number aM(M) of Mercury meters required to equal the semi-major
axis of Mercury is smaller than the number aM(o.s.) of outer space meters since the outer space meter
is shorter. Therefore the outer space observer will record a
larger number aM(o.s.)
of meters than the Mercury observer even if both observers are
measuring the very same semi-major axis.
4.7.2 - M(S)(o.s.)
and M(M)M(o.s.) versus M(S)(M) and M(M)M(M).
The
symbols (S) and (M) represent respectively the Sun
and Mercury. M(S)(o.s.) and M(M)M(o.s.) represent the numbers of
absolute outer space kilograms (kgo.s.) for the Sun and Mercury respectively. The subscript
M of M(M)M(o.s.)
indicates that the planet is at Mercury location. The numbers of
Mercury units that give the same masses are represented by M(S)(M)
and M(M)M(M). The
absolute solar mass m(S)[o.s.]
using
outer
space units is:
| m(S)[o.s.] =
M(S)(o.s.)kgo.s. |
4.35 |
Using
Mercury units, the same absolute solar mass is given by:
| m(S)[M] = M(S)(M)kgM |
4.36 |
Since the
solar mass does not change because we measure it using Mercury
units instead of outer space units, we have:
| m(S)[o.s.] =
m(S)[M] |
4.37 |
Similarly, the mass of Mercury measured with outer space units is:
| m(M)M[o.s.] = M(M)M(o.s.)kgo.s. |
4.38 |
When the
measurement is done with Mercury units, the same mass is given by:
| m(M)M[M] = M(M)M(M)kgM |
4.39 |
Since it
is the same absolute mass of Mercury described using different
units, we have:
| m(M)M[o.s.]
=
m(M)M[M] |
4.40 |
Due to
mass-energy conservation, the amount of mass contained in one
local Mercury kilogram is different from the one in one outer
space kilogram. From equations 4.35, 4.36 and 4.37 we have:
 |
4.41 |
The left
hand side of equation 4.41 gives the ratio between the number
of outer space kilograms and the number of Mercury
kilograms needed to measure the same solar mass. From equation
4.25, we get:
 |
4.42 |
Combining
equations 4.41 and 4.42 gives:
 |
4.43 |
Equation
4.43 shows that the number of kilograms M(S)(o.s.)
found
in
the measurement of the solar mass is smaller when measured in
conjunction with the outer space kilogram than when measured in
conjunction with the Mercury kilogram. Combining equations 4.38,
4.39 and 4.40 with 4.42, we get for the case of the mass of
Mercury:
 |
4.44 |
Consequently, the number M(M)M of kilograms giving the mass of Mercury is smaller
using outer space kilograms than using Mercury kilograms.
4.7.3 - PM(o.s.) versus PM(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 number of seconds PM(o.s.)
giving the period of Mercury when measured with an outer space
clock is given by the relationship:
| DtM[o.s.] = PM(o.s.)
DCDo.s.(S) |
4.45 |
and the
period PM(M) measured on Mercury
using a Mercury clock (with Mercury units) refers to the
relationship:
| DtM[M] = PM(M)
DCDM(S) |
4.46 |
The time
intervals DtM[o.s.] and DtM[M] in equations 4.45 and 4.46 represent the absolute
time interval for the period P of translation of Mercury around
the Sun. An absolute time interval is not different because it is
measured with a Mercury clock instead of an outer space clock:
| DtM[o.s.] = DtM[M] = PM(M)
DCDM(S) = PM(o.s.) DCDo.s.(S) |
4.47 |
We have
seen in equation 4.24 the ratio of the numbers DCDM(o.s.) and DCDM(M) between two frames in different gravitational
potentials. We see that the numbers PM(o.s.) and PM(M)
displayed by the clocks correspond to DCDM(o.s.) and DCDM(M) during one period of translation. Therefore,
 |
4.48 |
Combining
equation 4.48 with 4.24 gives:
 |
4.49 |
Equation
4.49 shows that even if the absolute time interval Dt for the period is the same in both
frames, the differences of clock displays are different because
the clocks run at different rates.
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 number of Mercury units of the
same gravitational constant is called G(M). The fundamental
units corresponding to the gravitational constant G are called
respectively Uo.s. and UM. The total gravitational constant G is called J[o.s.]
when measured from outer space and J[M] when measured from
Mercury orbit. Therefore we have:
| J[o.s.] = G(o.s.)Uo.s. |
4.50 |
and
Since the
absolute gravitational constant does not change because we measure
it from a different location, we have:
The
relative number of units between G(o.s.) and G(M) is found using a
dimensional analysis. The units of G can be obtained from Newton's
well known gravitational law:
 |
4.53 |
where the
force F is in newtons, M and m are in kilograms and the radius R
is in meters. From equation 4.53 and recalling that the units of
G(o.s.) are called Uo.s., we
find:
 |
4.54 |
From the
relationship
where a is the acceleration, we find that the
units of F are:
 |
4.56 |
Combining
4.54 with 4.56 we get:
 |
4.57 |
From the
definition of velocity, the units of v are:
 |
4.58 |
Equation
4.58 in 4.57 gives:
 |
4.59 |
We have
seen in sections 3.5.3 and 3.6 that a velocity is represented by
the same number within any frame. This means that the number
representing a velocity is the same within any frame when it is
measured using any coherent system of local units. Since a
velocity is the quotient between a length and a time interval,
this quotient stays constant even when switching between frames
because the same correction is made on both lengths and clock
displays. Consequently, we have:
Equations
4.7, 4.42 and 4.60 in equation 4.59 give:
 |
4.61 |
The first
order expansion of equation 4.61 gives:
 |
4.62 |
By
analogy with 4.59 for UM, we
have:
 |
4.63 |
Equation
4.63 in 4.62 gives:
 |
4.64 |
Equations
4.50, 4.51, 4.52 and 4.64 give the relationship between the number
of units of G:
 |
4.65 |
Equation
4.65 shows that the gravitational constant G is represented by
different numbers when measured with the units existing on Mercury
and in outer space.
4.7.5 - F(o.s.)
versus F(M).
From
equation 4.56 we have:
 |
4.66 |
Using
equations 4.7, 4.15, 4.24 and 4.25, we find:
 |
4.67 |
To the
first order, this is equal to:
 |
4.68 |
and:
 |
4.69 |
Consequently, the relationship between the number of Mercury
newtons and the number of outer space newtons is given by:
 |
4.70 |
4.8 - Symbols and
Variables.
| aframe[o.s.] |
length of the local Bohr radius in
absolute units |
| aM(M) |
number of Mercury meters for the
semi-major axis of Mercury |
| aM(o.s.) |
number of outer space meters for the
semi-major axis of Mercury |
| DCDM(M) |
DCD for the
period of Mercury measured by a Mercury clock |
| DCDM(o.s.) |
DCD for the
period of Mercury measured by an outer space clock |
| DCDM(S) |
apparent second on Mercury |
| DCDo.s.(M) |
DCD in outer
space measured by a Mercury clock |
| DCDo.s.(o.s.) |
DCD in outer
space measured by an outer space clock |
| DCDo.s.(S) |
absolute second in outer space |
| DtM[M] |
period of Mercury in Mercury units |
| DtM[o.s.] |
period of Mercury in outer space units |
| Dto.s.[M] |
time interval in outer space in Mercury
units |
| Dto.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 |
| kgframe |
mass of the local kilogram in absolute
units |
| LM[M] |
length of the semi-major axis of the orbit
of Mercury in Mercury units |
| LM[o.s.] |
length of the semi-major axis of the orbit
of Mercury in outer space units |
| meterframe |
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 |
| PM(M) |
DCD for the
period of Mercury measured by a Mercury clock |
| PM(o.s.) |
DCD for the
period of Mercury measured by an outer space clock |
| RM |
distance between Mercury and the Sun |
| Uframe |
unit of the gravitational constant in the
local frame |
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Chapter3
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