Einstein's Theory of Relativity
versus
Classical
Mechanics
by
Paul
Marmet
Chapter Three
Demonstration of the Lorentz Equations
without Einstein's Relativity Principles.
3.1 - Fundamental Physical
Principle.
In this chapter, we will show that the Lorentz equations can be
demonstrated using the principle of mass-energy conservation and
quantum mechanics. The equations obtained are mathematically identical
to the usual Lorentz transformations. There is no need for Einstein's
relativity principles or for the hypothesis of the constancy of the
velocity of light. In fact, no new physical principle is required and
the constancy of the velocity of light appears as a consequence to
mass-energy conservation.
We have
seen in chapter one that the principle of mass-energy conservation
implies that the mass of a particle changes with the gravitational
potential. In this chapter, we will consider particles with kinetic
energy. We will take into account that masses increase with kinetic
energy, using Einstein's relativistic relationship mv[rest]
= gms[rest].
This relationship shows that a moving particle has a larger mass than
the same particle at rest (using rest mass units). This relationship has been
demonstrated previously (see Web). However, as expected, when
observed within the moving frame (using proper values), the mass does
not appear to change.
In order to demonstrate the Lorentz equations using physical
considerations instead of a mathematical transformation of coordinates,
we must define accurately the physical meaning of the quantities used.
We have seen that Einstein considered that time is what clocks display.
We know that clocks run more slowly when they are located in a
gravitational potential. However, time does not flow more slowly
because clocks run at a slower rate.
Consequently, even if the equations that we will find are
mathematically the same as the Lorentz equations, because of Einstein's
interpretation, the parameter representing the time t in the equation
will actually be a clock display CD. Therefore due to Einstein's
confusion between clock display and time, the units (second)
characterizing time t in Lorentz's equations should not exist because t
is actually a clock display (which is a pure number).
When we compare Einstein's model of time dilation with the natural
explanation in which the clock rate is simply slower, we are obliged to
compare clock displays, which have no units, with real time, which
needs to be expressed in seconds. In this chapter, since we wish to
establish a comparison between Einstein's model and mass-energy
conservation, it is impossible to avoid momentarily giving Einstein's
units of time to quantities that represent only clock displays.
Furthermore, we see that the relationship in which length l
equals velocity times a time interval (l = vDt),
leads
to
an erroneous length because Einstein's definition of time is
not time but a clock display. Therefore the length found is not a
length but a pure number (of local meters). The length of a rod is a
reality independent of the observer and does not depend on the rate at
which a measuring clock is running. There is no change of length of a
rod when the observer uses a clock running more slowly. Consequently,
comparing our calculations with Einstein’s theory is very subtle
because Einstein confused the slowing down of clocks with time dilation.
3.2 - Change of Energy and
Bohr Radius Due to Kinetic Energy.
We have explained that the Bohr equation (equation 1.12) gives a
relationship between the parameters that describe the rate at which an
atomic clock runs. The energy levels in the Bohr atom for each of the n
quantum levels are:
 |
3.1 |
where the subscript o means that the atom is at rest. When the hydrogen
atom is given a velocity, the energy of each of the n levels changes as
seen by an observer remaining at rest and using rest units.
We must notice that the frame in which the observer is actually located
has no physical relevance. However, a description of the units (of
mass, length and clock rate) used by the observer is necessary. Of
course, one generally assumes that the observer uses the units that
exist in his own frame. However, the description will be complete only
when we specify the frame of origin of the units instead of assuming
every time that the observer uses the units of his own frame.
The energy levels of the moving atom (using rest frame units) are given
by putting equations 2.22 and 2.23 in equation 3.1. The Bohr equation
becomes:
 |
3.2 |
Furthermore,
since the Bohr radius ao of an atom at rest is:
 |
3.3 |
using
equations 2.22, 2.23 and 3.3, the Bohr radius of a moving atom will be:
 |
3.4 |
This means
that the Bohr radius ao increases linearly with g.
This will be discussed in section 3.4. From equation 3.2, we see that
the energy between atomic transitions of a moving atom (which
determines the clock rate) decreases linearly as g
increases (using the units of the rest frame). We conclude that
according to quantum mechanics, the rate of a moving clock slows down
when its velocity increases.
This is
compatible with the slower clock rate of moving atoms as observed
experimentally and interpreted erroneously as time dilation. The
popular phrase "time dilation" should be interpreted as meaning that
the rate of the moving clock has slowed down and not that time has
dilated. Combining the Bohr equation (equation 3.2) with solely the
mass relationship (equation 2.23) and neglecting equation 2.22 would
lead to a rate increase of the moving clock. This is contrary to
observations and to mass-energy conservation, as seen in chapter two.
The correction due to mass-energy must be applied to the Planck
parameter h as given by equation 2.22. Consequently, the observed
slowing down of the clock rate of moving clocks, which is implied by
equation 3.2, is an experimental confirmation of equation 2.22. This
also solves the apparent contradiction presented in section 2.7.
3.3 - The Lorentz Equation
for Time.
From the relativistic Bohr equation presented above, let us calculate
the energy of an atom located on a stationary frame. From equation 3.1
we see that the energy states of a stationary atom (using rest frame
units) are:
 |
3.5 |
where hono[rest]
is the internal energy of excitation in the atom, using rest frame
units. Due to its velocity, the atom located on the moving frame has a
different internal energy. Equation 3.2 gives (using rest frame units):
 |
3.6 |
where honv[rest] is the
internal energy of excitation of the moving atom (using rest frame
units) that can possibly be received on a frame at rest
in order to be compatible with mass-energy conservation. Consequently,
the radiation emitted from such an atom has a lower absolute energy and
frequency. This can be seen from equations 3.5 and 3.6:
 |
3.7 |
From
equation 3.7, we see that using rest units, there is less internal
energy Ev[rest] in the moving atom (due to equation 2.22)
than in the atom at rest (Eo[rest]).
The middle term of equation 3.6 represents the internal excitation
energy of the moving atom in rest units while the right hand side term
represents the same internal energy available that can be received by
an observer at rest (also in rest units). Since the energy states of
the moving atom have less energy (always in rest units), the observer
at rest will detect a lower frequency (as measured using rest frame
units) if that energy is emitted. We must notice that in both cases
(equations 3.5 and 3.6), the constant h refers to a measurement done in
the stationary frame (meaning that the measurement is made from a frame
having zero velocity and using rest units) so that the parameter h must
have the subscript o.
One must
notice a fundamental physical mechanism implied in the decrease of
internal energy in the hydrogen atom as given in equation 3.7 (using
rest units). The internal potential energy in a hydrogen atom is given
by equation 1.12. When the hydrogen atom is moving, equation 1.12 shows
that due to the increase of velocity, the electron mass me
and therefore the energy En increases by a factor g.
However, at the same time, the Planck parameter which is squared and
located at the denominator also increases. The overall effect is that
the internal energy En in the atom decreases when the
velocity increases. One must then realize that when the velocity
increases, the electron mass becomes larger but the decrease of the
Planck parameter corresponds to a decrease of the force between the
electron and the proton.
From
equations 3.5, 3.6 and 3.7 we obtain that the ratio between the clock
rates of the moving clock and the clock at rest is:
 |
3.8 |
The last
term nv[rest]/no[rest]
of equation 3.8 gives the ratio between the frequencies (in rest units)
of oscillation of two independent clocks having different velocities
according to the Bohr equation. This relationship has nothing to do
with the relative values of the frequencies of an electromagnetic wave
as given in equation 2.21. In equation 3.8, there are two different
frequencies emitted by two different clocks observed in a single frame.
However, in the case of equation 2.21, we have a single clock emitting
a single frequency observed by two independent observers located in
different frames.
Let us
consider
figure 3.1 on which a moving clock M travels in front of a station (at
rest) from A to B. Let us measure the difference of clock displays DCDo
recorded on a clock located on the station at rest between the instants
the moving clock M passes from A to B. We will also measure the
difference of clock displays DCDv
recorded on the moving clock while it passes from A to B. It is clear
that the absolute time (as defined in section 2.3) is the same for M to
pass from A to be B in both observations.
Figure 3.1
However the two clocks will not display the same difference because
they do not run at the same rate. The ratio between those two
differences of clock displays DCDo
and DCDv is proportional to the
ratio of the clock rates no[rest] and nv[rest]. Therefore:
 |
3.9 |
Combining
equation 3.9 with equation 3.8 gives:
 |
3.10 |
which is
mathematically identical to:
 |
3.11 |
From the
usual definition of g, equation 2.2:
 |
3.12 |
we find
that, using equation 3.11:
 |
3.13 |
Einstein
made the hypothesis that "time is what clocks are measuring". This
means that the Dt
in Einstein's relativity and in the Lorentz equations is only a
difference of clock displays on a clock at rest to which the units of
time were given:
In reality,
since Dt is nothing more than a DCD, the units of DCDo
(which is a pure number) must be given to Dt.
Let
us
give an example. It is believed that in Einstein's relativity
and in the Lorentz equations, when an excited atomic state of a moving
atom has not become de-excited after a classical time interval, it is
because the time interval was shorter within the moving frame than in
the rest frame. We have seen above that this explanation is incorrect
and that the reason is that the principle of mass-energy conservation
requires a change in the atom parameters and consequently, a slower
internal motion inside atoms. This slower internal motion makes moving
clocks function more slowly. Therefore, the Dt
measured
by
Einstein's and Lorentz's clocks is not a time interval at
all, but a difference of clock displays (DCD)
of
a
clock running more slowly. The correct explanation is that when,
in the Lorentz equation, we find that the Dt'
is
different
from Dt
during the same time interval, we are fooled by clocks running at
different rates in different frames. It is an error of interpretation
to give time units to Dt and Dt' in the Lorentz equations while they are no
more than differences of clock displays as admitted by Einstein. Since
the DCD is a pure number, the Dt in equation 3.14 is also a pure number.
Similarly, the difference of cloc k displays DCDv
is called Dt' in the Lorentz equations:
A comparison with the Lorentz equations, as given with equations 3.14
and 3.15, is useful to examine some mathematical properties common to
both interpretations. Equations 3.14 and 3.15 in equation 3.13 give:
 |
3.16 |
By
definition, the number of units x representing the distance traveled
during Dt (for Einstein corresponding to
the time while a clock shows DCDo)
is:
| x = vDt or x = vDCDo |
3.17 |
Of course, x
is not a real distance, as explained in section 3.1. Let us substitute Dt from equation 3.17 to the second term Dt of equation 3.16. We get:
 |
3.18 |
Equation
3.18 gives the relationship between Dt'
(which is a difference of clock displays) displayed by a clock located
at a distance x from the origin and moving at a velocity v and Dt
displayed by a stationary clock. We observe that equation 3.18 is
exactly the Lorentz equation for time and that it is compatible with
Einstein's hypothesis that time is what clocks display. This equation
is simply an exact mathematical description of mass-energy conservation
in agreement with equations 2.22 and 2.23 and with the physical
mechanism implied by equation 3.2 We notice finally that the Lorentz
transformation for time has been demonstrated here without using the
hypothesis of the constancy of the velocity of light nor any new
hypothesis. We have used only the mass-energy relationship E = Km from
equation 2.3. In fact, we have obtained the Lorentz equation for time
without the use of any of Einstein's relativity principles.
One must conclude that the Lorentz transformation derived above is in
reality a transformation of relative clock displays between frames.
Then Dt and Dt'
(when related to this Lorentz equation) represent differences of clock
displays DCD.
3.4 - Length Dilation Due
to Kinetic Energy.
Length dilation and contraction have been demonstrated in chapter one
for matter placed in a gravitational potential. Using equation 3.4, we
will now show that the Bohr equation also gives a change of length when
matter acquires a velocity v. This will be done without involving the
constancy of the velocity of light. According to equation 3.4, we have:
 |
3.19 |
Therefore,
the relative size of the Bohr radius as a function of velocity is:
 |
3.20 |
Let us consider a reference meter made of ordinary classical atoms. We
see from equation 3.20 that the size of atoms, which is proportional to
the Bohr radius or to the interatomic distance (see Appendix I),
increases as a function of velocity. This means that the size of all
material matter increases with velocity.
We know that
the number of atoms Na
making up the length of a rod does not change with velocity.
Furthermore, it is well established in modern physics that the
interatomic distance jo is
proportional to the Bohr radius ao so that jv[rest] = gjo[rest].
The
length
lo of a rod is:
| lo[rest] = (Na-1)jo[rest] |
3.21 |
At velocity
v, the length lv is:
| lv[rest] = (Na-1)jv[rest] = (Na-1)gjo[rest] |
3.22 |
We note that
the number of atoms Na is much larger than unity.
Therefore, using equations 3.21 and 3.22 we have:
 |
3.23 |
Equation 3.23 shows that there is length dilation of matter when its
velocity increases (in a constant gravitational potential). Length
dilation is a real physical phenomenon involving no stress nor any
pressure, similar to length dilation and length contraction in a
gravitational field, as shown in chapter one. It is just the natural
equilibrium of matter given by quantum mechanics that makes it dilate
at relativistic velocities. Space dilation or space contraction is
meaningless.
The fact
that we are
led from our reasoning to length dilation instead of length contraction
does not represent a problem since the assumed phenomenon of length
contraction has never been observed experimentally in special
relativity. On the contrary, we need length dilation to be compatible
with the slowing down of clocks, which is also required by quantum
mechanics and has been observed experimentally. In order to be coherent
with quantum mechanics and mass-energy conservation, one must
understand that there exists no length (nor space) contraction in
special relativity because g is always
equal to or larger than one (equation 3.23). Only length dilation can
be produced when there is an increase of velocity.
3.5 - The Lorentz
Transformation for Lengths.
Let us consider two identical frames O-X at rest. The axis of those
frames are constructed with many rods in series each having a length
exactly equal to one reference meter (defined in section 2.4). A mass M
is located at a distance x[rest] from the origin O[rest]. For a
stationary observer using the reference meters located on the frame at
rest, the coordinate of the mass M is:
| x[rest] = nometer[rest] |
3.24 |
where no
is the number of times the meter rod, when defined at rest
(meter[rest]) must be used to form the length x[rest]. The symbol no
is a pure number measured in the stationary (subscript o) frame. We
must recall that contrary to Newtonian physics, the simple use of the
number no is not sufficient to represent a length. A length
must necessarily be represented by a pure number multiplied by the
length of the reference meter.
Let
us give the velocity V to one of the frames that we now call O'-X'. At
time t = 0, the origin O' of the moving frame coincides with the origin
O of the rest frame. The axis O'-X' is arbitrarily displaced on figure
3.2 in order to avoid confusion. Before the frame O'-X' acquired its
velocity, the distance between the origin O and the mass M was
identical in both systems. After the frame O'-X' has reached velocity
V, we have seen that the Bohr radius and all physical material on the
moving frame are dilated as given by equation 3.23. Therefore the
reference meters used to form the axis are longer. The mass M' on the
moving frame is fixed with respect to that frame and does not move with
respect to the particular segment of meter where it is fixed. Therefore
the number nv of those standard moving rods between M' and
the origin O' is necessarily the same and no = nv.
Figure 3.2
However, the absolute distance x'[mov] between M' and O' will increase
because the length of the standard meter has increased due to the
increase of the Bohr radius. The distance x'[mov] between M'[mov] and
the origin O' is given by:
| x'[mov] = nvmeter[mov] = nometer[mov] |
3.25 |
with
Using the
notation x[rest] = lo[rest] and x'[rest] = lv[rest]
equation
3.23
gives:
| x'[rest] = gx[rest] or Dx'[rest] = gDx[rest] |
3.27 |
Equation
3.27 means that using rest frame units, the distance x' (which is
O'-M') is g times longer than the distance
x (which is O-M) also using rest frame units even if the numbers of
local meters no and nv are the same. 3.5.1 - Apparent and Absolute
Time.
In order to predict the consequences of the change of "clock" rate
between systems, we must be able to compare predictions between
different frames. Let us examine the relationship between the "apparent
time" in different frames. In Einstein's relativity, the "time" is
defined as what is perceived by each observer. It is equal to what a
clock measures in its own frame. It is called t in the rest frame and
t' in the moving frame.
Consequently, each frame has its own
"time" but we know that it is only apparent. Real physical time does
not flow faster because the local clock runs faster. For an observer at
rest, Einstein's interpretation assumes that his "time" t is the one
shown by his clock at rest. Similarly, the "time" t' is the apparent
time in the moving frame. Since the moving clock runs at a different
rate than the clock at rest (see equation 3.8), the time on the moving
frame "appears" (as seen by an observer at rest) to elapse at a
different rate giving:
We define
the "absolute second" So[rest]
as the time interval t taken by the atomic clock at rest (located away
from any gravitational potential) to record a constant number Ns
of oscillations. Since that clock at rest runs at a frequency no[rest], the apparent rest second
(called absolute second) will be elapsed when So equals
unity. This gives:
 |
3.29 |
On a moving
frame, the "apparent second" Sv[mov] is equal to the time
taken by the local clock moving at velocity V to record the same number
of oscillations Ns. Therefore during one "apparent second" (Sv)
on
the
moving frame (at velocity V), by definition, the clock must
record the same number of oscillations as the clock on the rest frame
does during one "absolute second" (So). This means that
during one "apparent second" inside any frame, the local DCD
is always the same number. Then, since clocks have different rates, in
different frames, the "absolute duration" of the "apparent second"
varies with the velocity of the frame carrying the clock.
It is arbitrarily decided that the rest second (in zero gravitational
potential) is called the "absolute second of reference". Since the
number of oscillations is the same for any local second, we have, for
the case of apparent second Sv in a frame moving at velocity:
| DCD(So)[rest] = DCD(Sv)[mov] |
3.30 |
From the definition of apparent seconds in a frame moving at velocity
V, with equations 3.29 and 3.30, we find that the duration of one
moving second is:
 |
3.31 |
In order to be able to compare "apparent seconds" generated in
different frames, we must be able to express the "apparent time"
duration using common units. We have from equation 3.8:
| no[rest] = gnv[rest] |
3.32 |
Equation
3.32 in equations 3.31 and 3.29 gives:
 |
3.33 |
Equation
3.33 shows that the unit of time Sv in the moving frame is g times longer than the unit of time So
in the rest frame.
Let us consider the "real time intervals" corresponding to the same
numerical value of local apparent "x" seconds elapsed in both the rest
frame and the moving frame. The DCD shown
by either clock is the same in both frames. In Einstein's relativity,
this was erroneously interpreted as the same time interval in both
frames. In the rest frame, the real time t[rest] is equal to the number
of seconds "x" times the duration of the apparent second So
at rest. This gives:
In the moving frame, the real time (in rest units) is called t'[rest].
It is equal to the number "x" of seconds times the duration of the
apparent moving second Sv:
| t'[rest] = xSv[rest] |
3.35 |
Combining
equations 3.33, 3.34 and 3.35 gives:
| t'[rest] = gt[rest] or Dt'[rest] = gDt[rest] |
3.36 |
Equation 3.36 shows that when we consider the same number of local
"apparent seconds" (i.e. the same difference of clock displays) in two
different frames, the real absolute time spent on the moving frame is g times longer that the absolute time spent on
the rest frame.
Equation 3.36 is equivalent to equation 3.18 when time is measured at
the same location (x = 0). However, one must understand that the change
of time between systems suggested by Einstein is only apparent because
clocks in different frames run at different rates. This has erroneously
been interpreted as time dilation in the past, but we see now that it
is nothing else than clocks running at different rates in different
frames.
3.5.2 - Relationship
between Velocities V and V'.
On figure 3.2, the right hand side direction of the axes O-X and O'-X'
is positive in both frames. When the moving frame O'-X' has a velocity
toward the right hand side, the coordinate of the location M' increases
(in time) with respect to the rest frame O-X. Therefore location M' has
a positive velocity with respect to the rest frame O-X. However, figure
3.2 shows that when the moving frame (with origin O') travels to the
right hand side, location M moves to the left hand side with respect to
the frame O'-X'. The coordinate of location M is getting more and more
negative (in time) with respect to the frame O'-X', while the
coordinate of location M' is getting more positive in time with respect
to the frame O-X. This means that the velocity V' of point M' (with
respect to O-X) has the opposite sign of the velocity V of point M with
respect to O'-X'. This result comes out of pure geometrical
considerations illustrated on figure 3.2. Therefore:
 |
3.37 |
Equation 3.37 signifies that the velocities have opposite directions.
We will show now that the velocities V and V' have the same magnitude.
3.5.3 - Relative
Velocities within Systems.
Let us consider a rest frame and a moving frame. Both frames were
identical before the moving frame started to move at velocity V[rest].
Inside both frames, we consider rods that had the same length when they
were initially in the same frame at rest. This can be verified later if
we count the same number of atoms in both frames for the length of
either rods. The rod at rest extends from O to M and the moving rod
extends from O' to M'.
There are at
least two different ways to compare velocities between frames. One way
consists of measuring directly the velocity in each frame using proper
values and comparing numbers. Another way, the one we will use here, is
to use a definition of velocity in each frame and to compare the
corresponding elements of the definitions. The velocity u of a moving
object across O-M with respect to the rest frame is defined as:
 |
3.38 |
With equation 3.38, we start dealing with a series of equations related
to velocities. These velocities can have any direction in space and
might be described by vectors. However, such a description would lead
to a very heavy notation that could be confusing and would require
useless efforts. This is avoided by defining that in every equation
between 3.38 and 3.46, we consider that u and u' represent the
magnitudes |u| and |u'| of these parameters. The appropriate
mathematical sign of the velocities will be considered starting with
equation 3.46.
Inside the
moving frame, a similar slowly moving object moves from O' to M'
(distance Dx'). During the time Dt' the slow moving object crosses the distance Dx' from O' to M'. The velocity of the slow
moving object with respect to the moving frame is defined as:
 |
3.39 |
We have seen that, before the moving rod (O'-M') started to move, it
was similar to the rod in the rest frame (O-M) and that both clock
rates were similar. Consequently, we can use equations 3.27 and 3.36.
Let us put the transformation of coordinates given by equations 3.27
and 3.36 into the equation 3.38. We get:
 |
3.40 |
Let us use
equation 3.23 to calculate the ratio between the units of length. If
the length lo is a unit of length equal to one meter
using rest units, we see that this unit of length becomes glo on the moving frame.
Therefore the relationship between the units of length is:
| lo[mov] = glo[rest]
or
meter[mov]
= gmeter[rest] |
3.41 |
This means
that when we move from the rest frame to the moving frame, the unit of
length becomes g times longer. Therefore,
in order to represent the same physical length using longer units of
length, the number of units Dx'[mov] must
be smaller. This gives:
 |
3.42 |
In the case
of time, a corresponding phenomenon takes place. Let us consider
equation 3.36. We see that a time interval Dto
equal to one unit of time in the rest frame becomes g
times larger in the moving frame because it takes more time for the
slower clock to show the same DCD. In that
case, we see from equation 3.36 that the change of local units of time Dto between frames gives:
| Dto[mov] = gDto[rest] or sec[mov] = gsec[rest] |
3.43 |
This means
that when we move from the rest frame to the moving frame, the local
unit of time becomes g times larger.
Therefore in order to represent the same absolute time interval using
longer units of time, the number of units Dt'[mov]
must
be
smaller. This gives:
 |
3.44 |
Equations
3.39, 3.40, 3.42 and 3.44 give:
 |
3.45 |
Equation 3.45 shows that the velocity u measured using the rest frame
units is the same as the velocity u' using the moving frame units.
Among the values of velocities which can be given to u, we can choose
the velocity V which is the velocity of the moving frame with respect
to the rest frame (rest frame units). Symmetrically, let us call V' the
velocity u' of the rest frame with respect to the moving frame (using
moving frame units). Using equations 3.37 and 3.45 gives:
or
Equation 3.46 shows that the proper value of the velocity of the moving
frame with respect to the rest frame is the same (negative) as the
proper value of the velocity of the rest frame with respect to the
moving frame.
Let us add
that a
velocity appears as a physical concept for a physicist. However, we
have seen above that a comparison of velocities in two different frames
having a relative velocity leads to the same numbers. We have seen that
when we are in a moving frame, the ratio between the distance traveled
and the time taken to travel it changes with respect to the rest frame.
Both the numerator (the distance) and the denominator (time interval)
change by the same ratio. Consequently, a constant velocity is nothing
more that a constant ratio between two fundamental physical quantities.
On can say that the constant velocity in different frames means the
same thing as three oranges out of six is the same thing as four apples
out of eight. Velocities are just ratios of physical quantities.
3.5.4 - Lorentz's Second
Relationship.
In order to find the dynamical relationship between the coordinates x'
and x, let us now combine the quantities x, V and t calculated above.
In classical mechanics inside the moving frame we have:
where xo'
is
the
coordinate x at t = 0 and V' is the velocity between frames. In
order to be more specific, in complete notation, equation 3.48 should
be:
| xv[mov] = xov[mov]+Vv[mov]tv[mov] |
3.49 |
Let us
consider first in equation 3.49 the expression tv[mov]. The
term tv
represents the number of units that is multiplied by the length of the
unit [mov]. Let us calculate what would be the quantity tv[mov]
using
the
[rest] units of length instead of the [mov] units of length.
From
equation 3.44, we have:
In the case
of the units of distance (xv or xov) we use again
the same method. With the help of equation 3.42 we find:
and
| xov[rest] = gxov[mov] |
3.52 |
From
equation 3.49, transforming xv[mov] with 3.51, xov[mov]
with
3.52,
and tv[mov] with 3.50, we get after multiplying
both sides by g:
| xv[rest] = xov[rest]+Vv[mov]tv[rest] |
3.53 |
From
equation 3.53, transforming xov[rest] with 3.27, Vv[mov]
with
3.47,
and tv[rest] with 3.36, we get:
| xv[rest] = g(xoo[rest]-Vo[rest]to[rest]) |
3.54 |
Using a more
conventional notation this is:
Equation 3.55 gives the relationship between the coordinate x' on the
moving frame and the coordinate x, the velocity V and the time t on the
rest frame. This relationship results solely from mass-energy
conservation and quantum mechanics without using any of Einstein's
relativity principles. However, equation 3.55 is exactly identical to
the Lorentz equation related to lengths. The demonstrations leading to
equations 3.18 and 3.55 show the uselessness of Einstein's special
relativity principles. Most importantly, this demonstration provides a
way to give a logical interpretation to experiments without space or
time contraction or dilation.
3.6 - Constant Velocity of
Light within Any Frame of Reference.
We must notice that c is also a velocity obtained from the quotient of
a distance by time within any frame. Let us consider that the internal
velocity u is the velocity of light c. In the moving frame, the
velocity u' equals c'. Therefore when the velocities u and u'
considered are applied to light, equation 3.45 gives:
When we use
the complete notation, we get:
This means that following equations 3.45 and 3.56, one must conclude
that the physical mechanism resulting from mass-energy conservation and
quantum mechanics leads to the conclusion (not the hypothesis) that any
velocity, including the velocity of light, is constant as measured
within any frame (using proper values). Contrary to Einstein and
Lorentz, we do not have to make the arbitrary hypothesis that the
velocity of light is constant inside all frames. We have found that the
constancy of the velocity of light is a necessary conclusion to
mass-energy conservation and the quantum mechanical equations.
From another point of view, the value of c, called the velocity of
light, has been defined in section 2.4 as the square root of K (the
quotient between energy and mass) which is the fundamental basis of
mass-energy equivalence. Any theory or experiment not compatible with
the constancy of the velocity of light (using proper values) is
therefore necessarily not compatible with quantum mechanics and
mass-energy conservation. However, since the velocity of light is given
as the quotient of two quantities (length and DCD)
that
are
different in different frames, the physical meaning of that
constant ratio is subtle.
3.7 - Non-Reality of Space
Dilation, Contraction or Distortion.
The distance
Dx traveled in a time interval Dt is defined as:
Let us assume an observer traveling between the ends of
a long stationary rod having a length Dx.
That length Dx is calculated from the
velocity v times the time interval Dt
necessary to travel between the ends of the rod. We know that the
velocity v is the same on any frame. However, the difference of clock
displays DCDo (which is
interpreted as time Dt by Einstein) on the
rest frame is different from DCDv
(interpreted by Einstein as time interval Dt')
on
the
moving frame. Consequently, according to Einstein's
interpretation, the length Dx' measured by
the moving observer is different from the length Dx
of
the
same rod measured by the observer at rest. At the velocity of
light, the DCDc decreases to
zero so that the (apparent Einstein's) length Dx'
becomes
zero
for the moving observer because his moving clock has
stopped running.
It is irrational to claim that the length of the stationary rod changes
and even becomes zero just because the observer changes his velocity.
How can the length of a rod logically change because a non interacting
observer looks at it? The rod would become longer or shorter depending
on the observer's own velocity. The length (and other properties) of
the rod would not be a property that would belong to matter. It is the
observer that would set the length of the rod and different observers
would simultaneously find different lengths for the same rod depending
on their observing conditions. Then, what would be the length of the
rod if there were no observer? It is just like the statement that the
moon is not there when nobody is looking at it. We believe that this is
nonsense and that the length of matter is independent of the observer.
This is the same irrationality that appears in quantum mechanics and
which has already been discussed [1].
We have not yet defined how to measure space. This is because space is
not measurable unless we fill it up at least partially with matter.
Then, it is that matter that we measure, not space. Whether space is
empty or full of matter, we generally refer to it as "space". We know
several methods of measuring lengths of objects but there does not
exist any method of measuring space without using matter as a
reference. In relativity, space is often referred to as being
contracted or dilated. How can it be contracted or dilated when there
is no method of measuring it without assuming some matter in it? The
properties of matter are then inadvertently attributed to or confused
with space. The same comment applies to the belief of space distortion.
How can there be space distortion when we cannot measure space directly
in the absence of matter? The interpretation of space distortion is
nothing more than a change of the Bohr radius in the measuring
instrument or in the matter filling the space.
This problem is easily solved logically when we consider that the
internal atomic mechanism of the observer runs at a different rate
since electrons in motion have a different mass. This has nothing to do
with the illusion of space dilation or distortion. One must conclude
that the expressions "space contraction" and "space distortion" are
irrational. They bring confusion and must be eliminated.
3.8 - Transformation of
Units in Different Frames.
There are many other consequences to the relativistic changes of
lengths and masses. For example, in chapter one we have seen that the
mass of particles decreases when located at rest in a lower
gravitational potential. In chapter three we have seen that masses
increase with velocity due to the absorption of kinetic energy. This
means that if we take an object of one kilogram on Earth and move it to
a location at rest on the solar surface, about one millionth of its
mass will disappear and be carried away by the energy generated during
the slowing down of the object falling into the Sun. Even if there is
exactly the same number of atoms in one Earth kilogram after it is
carried on the Sun's surface, we see that the solar kilogram has less
mass than the Earth kilogram using any common frame of comparison of
mass units. Consequently, there is more energy (in Earth joules) in one
Earth kilogram than in one solar kilogram. This is required by the
principle of mass-energy conservation.
Similar considerations must be applied to most physical constants.
Because of the principle of mass-energy conservation, the units must
always be specified (kg[Earth], meter[Earth], joule[Earth],
second[Earth]). However, the electric charge appears to be constant in
any frame. This means that the ratio of the electron charge divided by
the electron mass (e/m) is different in different frames. For example,
e/m is smaller on Earth (when using Earth units) than on the solar
surface (using Earth units). In order to be able to compare those
quantities with the ones calculated in different frames, we must take
into account the difference of gravitational potential or the
difference of kinetic energy. To define accurately the reference
kilogram, the reference meter, etc., we must know the exact altitude on
Earth at which these units have been defined.
3.9 - Failure of the
Reciprocity Principle.
We have studied above some of the differences existing between a frame
in motion and a frame at rest. In a moving frame, clocks run at a
slower rate, the Bohr radius is larger and so are masses because of
their kinetic energy. Let us consider a body on the rest frame having a
mass mo[rest]. Its total energy is:
| Eo[rest] = mo[rest]c2 |
3.59 |
When mo[rest]
is
accelerated
to velocity vo[rest] with respect to the rest
frame, its mass becomes mv[rest]. We get:
 |
3.60 |
Equation
3.60 shows that the moving mass mv[rest] is larger than the
rest mass mo[rest]:
Let us
consider now a train moving at velocity vo[rest]
carrying an observer and the mass mentioned above. The mass of the
train, of the observer and of the body described above becomes g times larger than when at rest. However, since
the units in the moving train have been modified by the same ratio g,
the changes of mass, clock rate and length are undetectable to the
moving observer, even if they are real. Inside the moving train, an
observer using Einstein's reciprocity principle will claim that the
object of mass mv[rest] is at rest with respect to him. He
will thus call it Mo[rest]. Therefore:
| Mo[rest] º mv[rest]
=
gmo[rest] |
3.62 |
It is
because we use Einstein's hypothesis of reciprocity that we write
[rest] after Mo
in equation 3.62, since Einstein's hypothesis assumes that the mass
that has been transferred to the train is now at rest for the observer
moving with the train. Furthermore, the symbol º
used in equation 3.62 does not mean that we are defining a new
quantity. The symbol º means that Mo
is the same object in the same physical condition as mv[rest].
Now, the
moving observer takes the object of mass Mo[rest] (that is
stationary with respect to him) and throws it at velocity vo[rest]
with
respect
to his moving train (considered at rest in his frame) in
the direction opposite to the direction of motion of the train.
According to Einstein's principle of reciprocity, the mass projected at
velocity vo[rest] with respect to the moving frame acquires
velocity and energy with respect to the moving frame (now considered at
rest). Einstein's principle of reciprocity says that all frames are
identical which means that mass Mo[rest] increases when
accelerated with respect to the train to become Mv[rest]. In
fact, the reciprocity principle implies that the passage of the object
of mass Mo[rest] from zero velocityo[rest] to vo[rest]
(with
respect
to the train) increases its mass by g
times, independently of the direction of the velocity of the
mass with respect to the train. This gives:
| Mv[rest] = gMo[rest] |
3.63 |
As expected
from the relativity principle, equation 3.63 shows that mass Mv[rest]
is
larger
than Mo[rest] giving:
A physical
representation of these changes of velocity shows that the mass Mv[rest]
now
has
zero velocity with respect to the rest frame. It is back at
rest on the rest frame. Mass Mv[rest] is then physically
undistinguishable from mass mo[rest]
since it is the very same object having the same zero velocity with
respect to the same rest frame. Therefore physically, we must have:
Combining
equations 3.62, 3.63 and 3.65 gives:
| mo[rest] º Mv[rest]
=
g2mo[rest] |
3.66 |
Obviously,
equation 3.66 is correct only if g
equals unity so that the velocity must always be zero. This shows that
the principle of reciprocity cannot be valid when we apply the
principle of mass-energy conservation. We must conclude that Einstein's
reciprocity principle is not coherent.
Contrary to Einstein's claim, the energy given to a mass accelerated
with respect to the train must depend on the direction of its velocity
with respect to the direction of the velocity of the train. When the
directions are opposite, the two velocities (whose magnitudes are
equal) cancel out and the mass of the body must come back to its
original value in the rest frame. Otherwise we would discover that
atoms of matter having traveled to another frame would have a different
mass after their return to the initial frame. We must conclude that two
frames cannot be equivalent when there exists a relative motion between
them.
3.10 - References.
[1] P. Marmet, Absurdities in
Modern Physics: A
Solution, ISBN 0-921272-15-4, Les Éditions du
Nordir, c/o R.
Yergeau, 165 Waller, Ottawa, Ontario K1N 6N5, 144p. 1993.
3.11 - Symbols and
Variables.
| ao[rest] |
Bohr radius at rest in rest units |
| av[rest] |
Bohr radius in motion in rest units |
| DCDo |
difference of clock displays on a clock at
rest |
| DCD(So)[frame] |
DCD corresponding
to an apparent second in any frame |
| DCDv |
difference of clock displays on a clock in
motion |
| En,o[rest] = Eo[rest] |
energy of the Bohr atom at rest in state n in
rest units |
| En,v[rest] = Ev[rest] |
energy of the Bohr atom in motion in state n
in rest units |
| ho[rest] |
Planck parameter on the rest frame in rest
units |
| hv[rest] |
Planck parameter on the frame in motion in
rest units |
| lo[rest] |
length of a rod at rest in rest units |
| lv[rest] |
length of a rod in motion in rest units |
| no[rest] |
clock rate of a clock at rest in rest units |
| Ns |
number of clock oscillations in an apparent
second |
| nv[rest] |
clock rate of a clock in motion in rest units |
| (So)[rest] |
definition of the absolute second in rest
units |
| (Sv)[rest] |
duration of one moving second in rest units |
| u[rest] |
definition of the velocity in the rest frame
in rest units |
| u'[rest] |
definition of the velocity in the moving
frame in rest units |
| V = Vo[rest] |
velocity of M with respect to the moving
frame in rest units |
| V'= Vv[mov] |
velocity of M' with
respect to the rest frame in motion units |
| x[rest] |
distance between O and M in rest units |
| x'[mov] |
distance between O'
and M' in motion units |
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