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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