## Einstein's Theory of Relativity

## versus

## Classical Mechanics

by: Paul Marmet

Where to get a Hard Copy of this BookChapter Two

Transformation of Excitation Energy between Frames.Upgraded version.

We consider now the kinetic energy given to masses when there is no gravitational potential. The principle of mass-energy conservation requires that masses increase when given kinetic energy. This has been demonstrated previously (see Web). This is expressed by the relationship:

m_{v}[rest] = gm_{s}[rest] |
2.1 |

2.2 |

Since masses can be excited particles containing internal potential energy, we must study how to transform that potential energy between frames. The mass-equivalent of this internal potential energy has always been ignored in relativity. In order to be coherent, it must be taken into account. Let us show how this correction restores physical reality in relativity. To calculate the relationship between masses in different frames we use the principle of mass-energy conservation (equation 2.1). Let us find an equivalent relationship for the case of energy released by an excited atom.

**2.2 - Difference
between Time and What Clocks Display.**

It has
been suggested that time is what clocks measure. This definition
is incomplete and misleading. We have seen in chapter one that
due to mass-energy conservation, clocks in different
gravitational potentials run at different rates. We must realize
that "time" is not elapsed more slowly because a clock functions
at a slower rate or because the atoms and molecules in our body
function at a slower rate.

We have
seen in equation 1.22 that in the case of a change of
gravitational potential, the Bohr radius is larger when the
electron mass is smaller. We also know that according to quantum
mechanics, atomic clocks run more slowly when the electron mass
is smaller. When we say that an atomic clock runs more slowly,
we mean that for that atomic clock, it takes more "time" to
complete one full cycle than for an atomic clock in the initial
frame, where the electron has a larger mass. That slower rate
can only be measured by comparing the duration of a cycle in the
initial frame with the duration of a cycle in the new frame. It
is the time rate measured in the initial frame at rest that is
considered the "reference time rate". We will see that all
observations are compatible with this unchanging "reference time
rate".

The
change of clock rate is not unique to atomic clocks. We recall
that quantum mechanics shows that the intermolecular distances
in molecules and in crystals are proportional to the Bohr radius
(see appendix I). Consequently,
due to velocity, the length of a mechanical pendulum will
change. Therefore it can be shown that the period of oscillation
of all clocks (electronic or mechanical) will also change with
velocity.

We
cannot say that "time" flows at the rate at which all clocks run
because not all clocks run at the same rate. However, a coherent
measure of time must always refer to the reference rate. That
reference rate corresponds to the one given by a reference clock
for which all conditions are fully described. It never changes.
However, all matter around us (including our own body) is
influenced by a change of electron mass (see appendix I) so that we are deeply
tied to the rate of clocks running in our frame. Since our body
and all experiments in our frame are closely synchronized with
local clocks, it is much more convenient to describe the results
of experiments as a function of the clock rate in our own frame.
This is what we call the "apparent time".

We
generally refer to the clock rate of our organism believing that
we are referring to the "real time". What appears as a "time
interval" for our organism is in fact the difference between two
"clock displays" on a clock located in our own frame.
"Difference of clock displays" (DCD)
is a heavier phrase than "time interval" but it is necessary for
an accurate description of nature. Of course, clocks are
instruments measuring time but during the same time interval
there is a difference by a factor of proportionality between the
"differences of clock displays" of different frames. In order to
avoid any misinterpretation, we must use the word "time" with
great caution when we want to shorten the description. In that
case, "time" is an ** apparent time interval**
corresponding to the difference of clock displays in a given
frame when no correction has been made to compare it with the
reference time. Since all our clocks and biological mechanisms
depend on the electron's mass and energy, humans feel nothing
unusual when going to a new frame. However, the time measured by
the observer in that new frame is an

**2.3 - Description of
the Reference Time Rate.**

We do
not know how to build a clock whose rate will not change when
brought to a different gravitational potential or to a different
velocity. However, using the mass-energy conservation principle,
we have seen in equation 1.22 how to calculate the difference of
clock rate between clocks without relative velocity and located
in different gravitational potentials. This means that we can
calculate the clock rate in one frame as a function of the clock
rate in a different frame, as long as the gravitational
potential and kinetic energies are fully described in both
frames.

An
absolute "reference time rate" can be defined using a clock
located in a frame in which the velocity and the gravitational
potential are well described. For example this could be a clock
at rest with respect to the Sun and far enough from it so that
the residual gravitational potential would be negligible. We
could then arbitrarily define the "reference time rate" as the
rate at which that clock operates in these particular
conditions. Everywhere in the universe we would refer to that
rate as the "reference time rate". If such a reference clock
were brought from outer space to a location near the Sun, we
have found in chapter one that due to mass-energy conservation,
it would run more slowly because the electron would lose mass
into energy that would escape away from its initial frame.

Let us
assume that an observer near the Sun wants to measure the period
of variation of light coming from a remote variable star. He
uses his clock and records a clock display every time the star
is at its maximum of brightness. The difference between two
maxima will give him the period of variation of the star, using
his clock rate. Let us represent by DCD_{s}
(where s stands for Sun) the difference of clock displays for
the clock near the Sun. In Einstein's relativity, since time is
what clocks measure, DCD_{s}
is interpreted as a time interval. However, we know that a
difference of clock displays simply gives a pure number without
any information on what the absolute time is. The subscript of DCD_{s} refers only to the
location of the clock and not to an absolute time unit. We know
however that another clock far away from the Sun (in a higher
gravitational potential) will give a different difference of
clock displays called DCD_{o.s.}
(where o.s. stands for outer space) between each maxima because
it runs at a different rate (that is equal to the "reference
outer space clock rate"). Consequently, the DCD_{s}
recorded near the Sun will not be the same as the DCD_{o.s.} recorded in outer
space. The observer near the Sun will have the illusion of a
"time interval" (that he might call Dt)
that is different from the one measured by the observer located
in outer space simply because the clock rate at his location is
different due to a different electron mass. One must understand
that the real time interval for a star to complete a cycle does
not vary because the observe r has moved somewhere else or
because his clock runs at a different rate. Consequently, when
we refer to DCD, we must always
specify (with a subscript) in which frame the clock is located.
Then a correction needs to be made to that number if we want to
calculate the corresponding DCD
given by a reference clock in outer space. We must remember that
the DCD given by a local clock is a
pure number that must be multiplied by a unit of time to give a
"real time" interval. Therefore, an absolute reference of "time
unit" must be defined. Furthermore, the absolute standard of
unit of time will appear different in different frames since we
have seen that local clocks run at different rates in different
gravitational potentials.

We see
that there is no time dilation nor time contraction. There is no
magic. In order to be able to make a comparison between systems,
it is absolutely necessary to compare the differences of clock
displays (which are not time but ** numbers of units**
of time) instead of the time intervals.

This problem cannot be discussed properly using directly the parameter "time" because of the psychological impression on humans that time is the rate at which our own organism runs. This last rate depends on the electron mass in the frame in which we are located. Consequently, we must get familiar with the phrase "difference of clock displays" (DCD

We have seen above that two clocks located in different gravitational potentials will not show the same difference of clock displays during the same real time interval. We will see now that quantum mechanics also predicts that clock rates are different when these clocks are carried in frames having different kinetic energy. We might assume that the relativistic correction could be made simply by taking into account the increase of electron mass due to the addition of kinetic energy, but this correction is too simple and incomplete (as we will see in sections 2.8 and 2.9) and disregards the need to consider the transfer of internal excitation energy between systems. In order to be able to calculate relative clock rates, we must first find the relationship between the excitation energy of atoms in frames having different velocities.

**2.4 - Description of
the Reference Meter.**

The
standard definition of length uses a unit called the "meter". In
order to be coherent, we must define the meter in a way that can
be reproduced in any frame. It is generally believed in physics
that one can transfer, without any change of length, a standard
meter from the rest frame to the moving frame. This is wrong
because this is not compatible with the principle of mass-energy
conservation and with quantum mechanics. When kinetic energy (or
potential energy) is added to or removed from a rod, the
electron mass and the Bohr radius change as required by the
principle of mass-energy conservation. Consequently, the length
of a rod will not be the same in frames having different
velocities. The change of length of a standard rod which is one
meter long in an initial frame can be calculated considering its
kinetic and potential energies.

Even
the most fundamental definition of the meter (which is 1/299 792
458 of the distance traveled by light in one second) suffers
from the same error since it requires the use of the unit of
time and since the "apparent second" in the moving frame (DCD(S)[mov]) is different from the
"apparent second" in the rest frame (DCD(S)[rest])
due to the change of mass of the electrons in the atomic clock
carried by the moving system. Consequently, to be able to
compare lengths in different frames, we must complete the
international definition of the reference meter and state its
potential and kinetic energies.

We ** define**here
that
the
length of the reference meter corresponds to 1/299 792 458 of
the distance traveled by light during one second

**2.5 - Definition of the
Velocity of Light.**

We want
to point out that none of the above definitions depends on the
experimental measurement of the velocity of light. The value of
the parameter c is defined in equation 1.3 from the fundamental
concept requiring an absolute constant K of proportionality
between mass and energy:

E = Km | 2.3 |

2.4 |

**2.6 - Need of
Parameters with a Double Index.**

From
the above description, we realize that the observer's frame is
submitted to several particular conditions like its
gravitational potential and kinetic energies. However, an
observer moving with his clock cannot measure the change of
clock rate because all phenomena in the moving frame, including
the clock rate, change in the same proportion.

The
same can be said of masses. When an observer and some masses
move at an identical velocity, the values of the masses (as
measured by the observer inside the moving system) are
indistinguishable from the values obtained before the common
change of velocity. After claiming that a mass increases with
velocity with respect to an observer at rest, it would be
incoherent to claim that the same mass does not increase when
the observer moves with it.

In
order to make a clear and coherent description, one must use a
suitable notation which gives a complete description of the
units used. To do this, two independent indexes are necessary.
The first index indicates the units used for the measurement.
For example, we can measure the length of an object either with
respect to a reference meter at rest or with respect to a moving
meter. It must be realized that the reference meter at rest is a
unit that has a different length than the same reference meter
in motion. It is almost like using inches instead of
centimeters. When we measure a length *l* and a mass m
using the units of length and mass issued from the system at
rest, the length is represented by *l*[rest] and the mass
is represented by m[rest]. When we measure lengths and masses
using the units of the system in motion, we represent the length
by *l*[mov] and the mass by m[mov]. The indexes [rest] and
[mov] do not tell us whether the mass is moving or not. They
only tell us what units are used.

The
second index indicates the state of motion of the system on
which parameters (like length or mass) are measured. We describe
the frame in which the particle is located using the subscript
"v" when the particle is moving and the subscript "s" when the
particle is stationary. For example, the mass of a stationary
particle (using units of the rest frame) is represented by m_{s}[rest]
and the mass of a moving particle (using units of the rest
frame), by m_{v}[rest]. According to relativity, we must
write:

m_{v}[rest] = gm_{s}[rest] |
2.5 |

The number "n" of meters of a rod does not change when the rod is moved to another frame as long as we measure proper values (number of proper meters). Then n

2.6 |

The fundamental importance of the necessity of using a double index must not be underestimated because relativity cannot be explained properly without it. This is a consequence of having different units of mass and length in different frames. These double indices are irrelevant in Newtonian mechanics. In principle, a third index could be added giving the information about the gravitational potential energy. This third parameter will be considered separately.

**2.7 - Apparent Lack of
Compatibility for Fast Moving Particles.**

When a
body is accelerated, its mass increases according to the
relationship given by equation 2.5. Therefore fast moving atoms
possess more massive electrons. Using the Bohr equation, let us
calculate the consequences of a heavier electron in the case of
the hydrogen atom.

When
the electron mass is larger ** and no other parameter is
taken into account**, then according to the Bohr
equation (equation 1.12), all the atomic energy levels should
have more energy (equation 1.13). Consequently, since E = hn, the atoms formed with those heavier
electrons should emit electromagnetic radiation at a higher
frequency n. This means that an
atomic clock located in the moving frame should run at a higher
rate. However, we know from experiments that fast moving
particles disintegrate at a slower rate and atoms emit a lower
frequency. This has been clearly observed in the muon's and
spectroscopic experiments. We conclude that the increase of
electron mass that causes atoms to disintegrate at a higher rate
in a gravitational potential

In the next section, we will consider solely experiments in which the gravitational potential energy is always constant. This corresponds to the study of special relativity. Only the velocity (and therefore the kinetic energy) will change. The problem of combining gravitational potential energy with kinetic energy will be studied in chapters five and six.

**2.8 - Demonstration of
the Energy Relationship between Systems.**

Let us
consider a stationary particle M_{so} where the index s
stands for stationary and the index o means that the particle is
in its ground state of internal excitation. That particle can be
a single hydrogen atom. When accelerated to a velocity v, its
mass becomes:

M_{vo}[rest] = gM_{so}[rest] |
2.7 |

Let us consider that an internal energy of excitation E

2.8 |

2.9 |

M_{sxt} = M_{so}[rest]+M_{xs}[rest] |
2.10 |

M_{vxt}[rest] = gM_{sxt}[rest] |
2.11 |

M_{vxt}[rest] = gM_{so}[rest]+gM_{xs}[rest] |
2.12 |

M_{vxt}[rest] = M_{vo}[rest]+gM_{xs}[rest] |
2.13 |

2.14 |

E_{n}(Excit.+acceleration of excit.) =
gM_{xs}c^{2}[rest]
= ghn_{s}[rest] |
2.15 |

However, when the observer moves with the excited atom and uses rest units, he will deduce from his measurements a frequency n

E_{n}[rest](emitted) = hn_{v}[rest] |
2.16 |

Using the same units, it is clear that the total energy of equation 2.15 (excitation plus the energy required to accelerate the mass-equivalent of the energy of excitation) is equal to the energy of the photon received during the de-excitation by the observer at rest (equation 2.16). This gives:

gM_{xs}c^{2}[rest] = ghn_{s}[rest] = hn_{v}[rest] |
2.17 |

gh_{s}n_{s}[rest] = h_{v}n_{v}[rest] |
2.18 |

**2.9 - Relative
Frequencies between Systems.**

In
order to solve equation 2.18, we need to find a relationship
between n_{s}[rest] and n_{v}[rest]. Let us consider an electromagnetic wave of
frequency n_{v}[rest] emitted by an atom having a velocity v. That
electromagnetic wave is measured by an observer in the rest
frame. When the measurement of the frequency is made, he must
consider two different phenomena that might change the frequency
due to the velocity of the emitting atom. The first one is the
change of clock rate of the emitter and the second is the
classical Doppler effect due to the radial velocity between the
stationary source of radiation and the moving observer.

Let
us study those two effects separately starting with the
classical Doppler effect. In order to avoid the problem,
let us suppose that the moving emitter of radiation is traveling
at a velocity v, in a direction perpendicular to the direction
of emission of light. The observer at rest, receives the
radiation at a frequency n_{s}[rest], which is identical to
the frequency emitted n_{v}[rest] when using
the same units. Consequently, the Doppler effect can be
eliminated, and there is then no change of frequency due to
the light emitted from a moving frame. Since we use
constant units [rest] and there
is no Doppler correction, the frequency n_{s}[rest]
received
in the [rest] frame is identical to the frequency emitted n_{v}[rest] in the
moving frame. We have:

n_{v}[rest]
= n_{s}[rest] |
2.19 |

**Physical Phenomenon Explained. **

The
physical
phenomenon involved can be seen when we compare equations 2.8
and 2.14 Equation 2.8 gives the total amount of
mass-energy of the stationary excited particle M_{sxt}
[rest], as a function of its mass "in the ground
state" plus its "excitation energy". The excitation energy in
the stationary frame is

2.20 |

Consequently, the excitation energy (gE

Let us find a relationship to calculate the frequency of a photon emitted from an excited state of an atom in a moving frame, with respect to the frequency of the same atom, when it is located in a rest frame. We have seen that after the atom is accelerated to the moving frame, its excitation energy becomes gh

2.21 |

h_{v}[rest] = gh_{s}[rest] |
2.22 |

We must notice that the change of the Planck constant by g in equation 2.22 is not apparent to the observers inside the moving frame or inside the rest frame when the measure internal values. This phenomenon appears only when an observer calculates the photon energy of atoms in an external frame, with respect to the excitations energy of atoms that has moved from the rest frame to that external frame. The reason for that phenomenon is because the internal structure of the atom carried to a moving frame, becomes modified due to the change of kinetic energy. The atom retransmits all its internal excitation energy using a lower frequency, but of course, using a longer time of coherence for the re-emitted radiation. In fact, the re-emitted photon is different because the clock rate of that emitting atom is different.

Equation 2.22 is the relationship we were looking for in section 2.1. It is, for energy, the relationship equivalent to the mass-energy conservation principle:

m_{v}[rest] = gm_{s}[rest] |
2.23 |

**2.10 - Cases of Relevance of the Relationship
h**_{v}**=gh _{s}. **

We must notice that equation 2.22 (h

DCD_{frame } |
difference of clock displays on a clock located in a frame |

DCD(S)[mov] | DCD for the apparent second in the moving frame |

DCD(S)[rest] | DCD corresponding to the apparent second in the rest frame |

E_{xs}[rest] |
energy of excitation given at rest in rest units |

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

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

m_{s}[rest] |
mass of an object at rest in rest units |

M_{so}[rest] |
mass of a particle at rest in its ground state in rest units |

M_{xs}[rest] |
mass of the excitation energy of a particle at rest in rest units |

M_{sxt}[rest] |
total mass of a particle at rest in its excited state in rest units |

m_{v}[rest] |
mass of an object moving at velocity v in rest units |

M_{vo}[rest] |
mass of a particle in motion in its ground state in rest units |

M_{vxt}[rest] |
total mass of a particle in motion in its excited state [rest units] |

n_{s}[rest] |
frequency of light measured by an observer at rest in rest units |

n_{v}[rest] |
frequency of light measured by a moving observer in rest units |

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