## Classical Mechanics

by: Paul Marmet
Chapter Two

Transformation of Excitation Energy between Frames. Upgraded version.

2.1 - Introduction.
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:
 mv[rest] = gms[rest] 2.1
where:
 2.2
The index [rest] means that the measurement is made using the units of the rest frame. The subscripts v and s refer to masses having respectively a velocity v and no velocity (stationary). These indices will be explained in detail in section 2.6.
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 apparent time and it must be corrected to be compared with a time interval on the fundamental reference frame.

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 DCDs (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, DCDs 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 DCDs 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 DCDo.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 DCDs recorded near the Sun will not be the same as the DCDo.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" (DCDframe) remembering that it corresponds to the "time interval" believed to be felt by an observer in that particular frame.
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 definehere that the length of the reference meter corresponds to 1/299 792 458 of the distance traveled by light during one second on a clock located at rest in outer space, far away from the Sun.

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
However, it has been observed experimentally that the value of K is equal to the square of what is interpreted to be the velocity of light. Whatever c is, for practical reasons, we define it as:
 2.4
Everywhere in this book, the meaning of c is fundamentally bound to equation 2.4. We believe that the fact that the velocity of light is equal to the square root of the constant K in the mass-energy relationship is not just a coincidence and results from a fundamental mechanism. However, it is very likely that the best method of measuring the mass-energy constant K is through the measurement of c.

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 ms[rest] and the mass of a moving particle (using units of the rest frame), by mv[rest]. According to relativity, we must write:

 mv[rest] = gms[rest] 2.5
Similarly, the mass of a moving particle measured using moving units is represented by mv[mov] and the mass of a stationary particle measured using moving units is represented by ms[mov]. Consequently, the number of kilograms in ms[rest] is identical to the number in mv[mov] because they are both measured using proper parameters. However, the mass ms[rest] is different from mv[rest] as seen in equation 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 ns equals nv. However, the distance between the atoms changes. Since the interatomic distance a changes when a physical body is moved to another frame, the number of atoms Ns along a length of one meter[rest] in a stationary rod is different from the number of atoms Nv along the same length (one meter[rest]) when the rod is in motion at velocity v. Therefore when measuring the same absolute constant length in two frames we find:
 2.6
Of course, the indexes [rest] and [mov] are irrelevant with the numbers ns, nv, Ns and Nv because they are pure numbers.
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 does not appear to be compatible with the slower rate of disintegration of fast moving muons. This apparent contradiction is a very serious problem that requires a more careful study. Using the principle of mass-energy conservation, we will solve that problem by showing that one important parameter has been ignored.
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 Mso 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:

 Mvo[rest] = gMso[rest] 2.7
where the index v means that the particle has a velocity v.
Let us consider that an internal energy of excitation Exs[rest] is given to that particle before its acceleration. The index x refers to internal excitation energy. The total mass Msxt[rest] of the stationary excited atom is then:
 2.8
where the index t refers to the total mass-energy which includes rest mass, internal and kinetic energies when relevant. From equation 2.8, we calculate that the internal excitation energy Exs[rest] alone has a mass-equivalent Mxs[rest] given by:
 2.9
where hns[rest] is the energy Exs measured using the units of time and length of the rest frame. Equations 2.8 and 2.9 give:
 Msxt = Mso[rest]+Mxs[rest] 2.1
The particle of mass Msxt can emit its energy of excitation according to equation 2.9. When that particle (Msxt) is accelerated to a velocity v, its mass becomes Mvxt which is g times its mass at rest as given by equation 2.5. This gives:
 Mvxt[rest] = gMsxt[rest] 2.11
Putting 2.10 in 2.11 gives:
 Mvxt[rest] = gMso[rest]+gMxs[rest] 2.12
If the particle does not possess any internal energy, then the second term of equation 2.12 vanishes and we get equation 2.7. Putting equation 2.7 in 2.12, we have:
 Mvxt[rest] = Mvo[rest]+gMxs[rest] 2.13
Equations 2.13 and 2.9 give:
 2.14
Equation 2.13 shows that the velocity of the excited particle leads to the mass component Mvo[rest]. The second term gMxs[rest] gives the mass-energy equivalent of the excitation energy of the moving particle. This term is composed of the mass equivalent of the excitation energy of the particle (which is hns/c2[rest]) and of the energy required to accelerate it (given by g). From equations 2.13 and 2.14, we see that the principle of mass-energy conservation requires that the total energy of excitation combined with the energy necessary to accelerate that energy of excitation (or its mass equivalent) give:
 En(Excit.+acceleration of excit.) = gMxsc2[rest] = ghns[rest] 2.15
Equation 2.15 gives the total energy [rest] that the excited moving atom must lose (by emission of a photon) to go to its ground state.
However, when the observer moves with the excited atom and uses rest units, he will deduce from his measurements a frequency nv[rest] from which he will naturally decide that the energy of internal excitation is hnv[rest]. Therefore:
 En[rest](emitted) = hnv[rest] 2.16
The energy that was required to accelerate the mass-equivalent of that excitation energy may appear irrelevant to the moving observer. However, due to mass-energy conservation, that energy cannot disappear and be ignored. According to the principle of mass-energy conservation, since no other photon is emitted during the transition, the emitted photon must possess all the energy available which includes the energy of excitation plus the kinetic energy of the mass equivalent of that excitation energy.
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:
 gMxsc2[rest] = ghns[rest] = hnv[rest] 2.17
In equation 2.17, we have the Planck parameter h that comes from the measurement of hns in a stationary frame. We also have the Planck parameter h that comes from a measurement of hnv in the moving frame (always using the same common units [rest]). In order to be coherent and since the Planck parameter comes from measurements from different frames, we must individually label each Planck parameter. Equation 2.17 becomes:
 ghsns[rest] = hvnv[rest] 2.18
Equation 2.18 is an important relationship that must be applied when the energy of excitation is given a new velocity.

2.9 - Relative Frequencies between Systems.
In order to solve equation 2.18, we need to find a relationship between ns[rest] and nv[rest]. Let us consider an electromagnetic wave of frequency nv[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 ns[rest], which is identical to the frequency emitted nv[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 ns[rest] received in the [rest] frame is identical to the frequency emitted nv[rest] in the moving frame. We have:

 nv[rest] = ns[rest] 2.19
The second phenomenon is due to the slower rate of emission of light on the moving frame due to the increase of electron mass with kinetic energy. We explain now this physical phenomenon in more detail.

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 Msxt [rest], as a function of its mass "in the ground state" plus its "excitation energy". The excitation energy in the stationary frame is

 2.2
After the acceleration of the internally excited particle, we find equation 2.14   We see in equation 2.14, that the mass term Mvo[rest] has increased g times (see eq. 2.7), and also the new energy term g(hsns/c2)[rest] is g times larger.  This is certainly expected from the principle of mass-energy conservation.  The g term appears equally in the energy term, because we have taken into account the energy required to accelerate the energy of the excited state. This last energy term predicts the extra energy of an eventual re-emitted photon.  Furthermore, we know that in the moving frame, particles always emit at a slower rate, just as all clocks in the moving frame run at a slower rate, because of the increase of the electron mass in atoms.
Consequently, the excitation energy (gExs/c2)[rest] of the atom which has now been carried into the moving frame, will be emitted at a rate which is g times slower than on the rest frame.
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 ghsns/c2[rest]  (eq. 2.14).  When that same excitation energy is transformed into a photon, the frequency of the emitted photon is given by the relationship (hvnv/c2 [rest]) (eq. 2.21) due to the slower clock rate in that moving frame.  Therefore, depending on its frame, the excitation energy of the same atom appears as:
 2.21
Combining equation 2.21 and 2.19, we get

 hv[rest] = ghs[rest] 2.22
Equation 2.22 means that when we use the Planck parameter h to determine the energy in a moving system, we must make a correction (g) because of the kinetic energy of the equivalent mass of the excitation energy hvnv[rest]. This is the relationship necessary to transform excitation energies between frames.
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:
 mv[rest] = gms[rest] 2.23
Equation 2.22 is a relationship previously ignored. However this equation, which is required by the principle of mass-energy conservation, is absolutely necessary when treating problems dealing with a change of kinetic energy.  We will see in chapter three how equation 2.22 allows us to solve the apparent contradiction described in section 2.7.

2.10 - Cases of Relevance of the Relationship hv=ghs
We must notice that equation 2.22 (hv[rest] = ghs[rest]) results from the fact that the internal excitation energy of particles (that has a mass equivalent) acquires a velocity v that produces an increase of mass-energy equivalent. However, in the case of a change of gravitational potential energy, as seen in chapter one, the mass-equivalent of the internal excitation energy has no kinetic energy since it has no velocity. Therefore in the case of potential energy, the relationships  (hv[rest] = ghs[rest]) and mv[rest] = gms[rest] are irrelevant since g = 1 when v = 0. In the case of gravitational potential, the changes of energy and length are given by equation 1.22 in chapter one. Let us finally note that the relationship hv[rest] = ghs[rest] is absolutely necessary to satisfy the principle of invariance of physical laws in any frame of reference as will be seen in the rest of this book.

2.11 - Symbols and Variables.

 DCDframe 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 Exs[rest] energy of excitation given at rest in rest units hs[rest] Planck parameter on the rest frame in rest units hv[rest] Planck parameter on the frame in motion in rest units ms[rest] mass of an object at rest in rest units Mso[rest] mass of a particle at rest in its ground state in rest units Mxs[rest] mass of the excitation energy of a particle at rest in rest units Msxt[rest] total mass of a particle at rest in its excited state in rest units mv[rest] mass of an object moving at velocity v in rest units Mvo[rest] mass of a particle in motion in its ground state in rest units Mvxt[rest] total mass of a particle in motion in its excited state [rest units] ns[rest] frequency of light measured by an observer at rest in rest units nv[rest] frequency of light measured by a moving observer in rest units