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**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 m_{v}[rest] = gm_{s}[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 |

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 |

3.3 |

3.4 |

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 |

3.6 |

3.7 |

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 m

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 |

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 DCD

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 DCD_{o}
and DCD_{v} is proportional
to the ratio of the clock rates n_{o}[rest] and n_{v}[rest]. Therefore:

3.9 |

3.10 |

3.11 |

3.12 |

3.13 |

DCD_{o} = Dt |
3.14 |

DCD_{v} = Dt' |
3.15 |

3.16 |

x = vDt or x = vDCD_{o} |
3.17 |

3.18 |

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 |

3.20 |

We know that the number of atoms N

l_{o}[rest] = (N_{a}-1)j_{o}[rest] |
3.21 |

l_{v}[rest] = (N_{a}-1)j_{v}[rest] = (N_{a}-1)gj_{o}[rest] |
3.22 |

3.23 |

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] = n_{o}meter[rest] |
3.24 |

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 n

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] = n_{v}meter[mov] = n_{o}meter[mov] |
3.25 |

n_{v} = n_{o} |
3.26 |

x'[rest] = gx[rest] or Dx'[rest] = gDx[rest] | 3.27 |

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:

t ¹ t' | 3.28 |

3.29 |

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 S

DCD(S_{o})[rest]
= DCD(S_{v})[mov] |
3.30 |

3.31 |

n_{o}[rest]
= gn_{v}[rest] |
3.32 |

3.33 |

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 S

t[rest] = xS_{o}[rest] |
3.34 |

t'[rest] = xS_{v}[rest] |
3.35 |

t'[rest] = gt[rest] or Dt'[rest] = gDt[rest] | 3.36 |

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 |

**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 |

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

3.39 |

3.40 |

l_{o}[mov] = gl_{o}[rest] or
meter[mov] = gmeter[rest] |
3.41 |

3.42 |

Dt_{o}[mov]
= gDt_{o}[rest] or
sec[mov] = gsec[rest] |
3.43 |

3.44 |

3.45 |

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:

V = -V' | 3.46 |

V_{o}[rest] = -V_{v}[mov] |
3.47 |

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:

x' = x_{o}'+V't' |
3.48 |

x_{v}[mov] = x_{ov}[mov]+V_{v}[mov]t_{v}[mov] |
3.49 |

From equation 3.44, we have:

t_{v}[rest] = gt_{v}[mov] |
3.50 |

x_{v}[rest] = gx_{v}[mov] |
3.51 |

x_{ov}[rest] = gx_{ov}[mov] |
3.52 |

x_{v}[rest] = x_{ov}[rest]+V_{v}[mov]t_{v}[rest] |
3.53 |

x_{v}[rest] = g(x_{oo}[rest]-V_{o}[rest]t_{o}[rest]) |
3.54 |

x' = g(x-Vt) | 3.55 |

**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:

c = c' | 3.56 |

c_{v}[mov] = c_{o}[rest] |
3.57 |

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:

Dx = vDt | 3.58 |

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 m_{o}[rest]. Its total energy
is:

E_{o}[rest] = m_{o}[rest]c^{2} |
3.59 |

3.60 |

m_{v}[rest] > m_{o}[rest] |
3.61 |

M_{o}[rest] º
m_{v}[rest] = gm_{o}[rest] |
3.62 |

Now, the moving observer takes the object of mass M

M_{v}[rest] = gM_{o}[rest] |
3.63 |

M_{v}[rest] > M_{o}[rest] |
3.64 |

M_{v}[rest] º
m_{o}[rest] |
3.65 |

m_{o}[rest] º
M_{v}[rest] = g^{2}m_{o}[rest] |
3.66 |

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.

[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.

a_{o}[rest] |
Bohr radius at rest in rest units |

a_{v}[rest] |
Bohr radius in motion in rest units |

DCD_{o} |
difference of clock displays on a clock at rest |

DCD(S_{o})[frame] |
DCD corresponding to an apparent second in any frame |

DCD_{v} |
difference of clock displays on a clock in motion |

E_{n,o}[rest] = E_{o}[rest] |
energy of the Bohr atom at rest in state n in rest units |

E_{n,v}[rest] = E_{v}[rest] |
energy of the Bohr atom in motion in state n in rest units |

h_{o}[rest] |
Planck parameter on the rest frame in rest units |

h_{v}[rest] |
Planck parameter on the frame in motion in rest units |

l_{o}[rest] |
length of a rod at rest in rest units |

l_{v}[rest] |
length of a rod in motion in rest units |

n_{o}[rest] |
clock rate of a clock at rest in rest units |

N_{s} |
number of clock oscillations in an apparent second |

n_{v}[rest] |
clock rate of a clock in motion in rest units |

(S_{o})[rest] |
definition of the absolute second in rest units |

(S_{v})[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 = V_{o}[rest] |
velocity of M with respect to the moving frame in rest units |

V'= V_{v}[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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