Einstein's Theory of Relativity## versus

## Classical Mechanics

by Paul Marmet

Where to get a Hard Copy of this BookChapter Eleven

Internal Phenomena inside Atoms.

In this chapter, we will give a physical description of the absolute changes that happen inside an hydrogen atom when it is accelerated to a high velocity. We have seen that this acceleration produces an increase of the electron mass and in the Bohr radius. We also notice from chapter three that the principle of mass-energy conservation is respected inside the hydrogen atom without having to involve any change of electric charge when the hydrogen atom is brought to high velocity. We will now show how the absolute parameters of the hydrogen atom change when it acquires kinetic energy. We present some considerations to the problem of internal potentials inside the nucleus of atoms. Finally, we will see how the nature of the interactions taking place inside nuclei can be predicted using these considerations.

**11.2 - Transformations
inside Fast Moving Atoms.**

We have
seen in equation 3.4 that when the velocity of the hydrogen atom
increases, the absolute value of the Bohr radius *a*
increases according to:

a_{v}[rest] = ga_{o}[rest] |
11.1 |

a_{v} = 2a_{o} |
11.2 |

m_{v} = gm_{o} |
11.3 |

m_{v} = 2m_{o} |
11.4 |

__Figure 11.1__

Figure 11.1, illustrates the simultaneous increase of the Bohr radius and of the electron mass when g = 2. Let us examine how these results are compatible with the Bohr model, the de Broglie wavelength of particles and quantum mechanics.

**11.3 - Electric
Potentials.**

Let us
examine first the compatibility between the description given
above with the laws regulating the electron and the proton in
the hydrogen atom. We recall that when an atom is accelerated to
a high velocity, the electric charges and the absolute electric
field around those charges do not have to change in order to
remain compatible with the principle of mass-energy
conservation. The electric energy E_{o} of the electron
in the electric field of the proton is:

11.5 |

11.6 |

e^{-}_{o
}[rest] = e^{-}_{v
}[rest] |
11.7 |

In the case of kinetic energy, in order to be able to establish comparisons, all parameters are calculated using rest units. According to the Bohr model of the atom, when the electron of the hydrogen atom moves in an electric field (i.e. the field of the proton), one must have an equilibrium between the attracting electric force and the centrifugal force. The electric and the centrifugal forces are defined as:

11.8 |

11.9 |

11.10 |

11.11 |

11.12 |

11.13 |

In order to be compatible with the Bohr equation and quantum mechanics, the length of the circumference of the orbit of an electron around a proton must be equal to an integer number of the wavelength of the electron. In the case of the hydrogen ground state, the electron wavelength must be equal to the length of one circular orbit. The de Broglie wavelength l is given by:

11.14 |

11.15 |

This satisfies the wave condition of the constructive interference of the electron wave after each translation since the radius of the orbit (therefore its circumference) of the moving atom is twice as large as the radius for the atom at rest as illustrated on figure 11.1.

Furthermore, when all these fundamental conditions are perfectly satisfied, the frequency of emission of light between electronic transitions is reduced by two, since the energy between the states is reduced by two when the atom is moving, exactly as observed experimentally from the red shift of spectral lines and from the slowdown of moving atomic clocks. It is this absolute reduction of frequency of a moving clock located in a moving frame that has been erroneously interpreted by Einstein as time dilation.

We must then conclude that the predicted absolute change of parameters inside a moving frame, resulting from mass-energy conservation is coherent inside moving atoms. We must also note that all the transformations given above are in perfect agreement with constant absolute electric charges in all frames when

**11.4 - Sommerfeld Fine
Structure.**

The
prediction of the advance of the perihelion of Mercury seen in
chapter five is not the sole example of the success of the
principle of mass-energy conservation and classical mechanics.
There is also a well documented example in atomic and molecular
physics in which it is clearly observed that the principle of
mass-energy conservation influences the electronic structure
inside atoms. There are many similarities between Mercury moving
inside the gravitational field of the Sun and the electrons of
atoms orbiting inside the electric potential of the proton.
However, an important difference is that the electron mass is
not concentrated into a relatively small location with respect
to the size of the atom contrary to the case of Mercury and the
Sun.

Since
electrons exist as waves, the electric potential between the
electron cloud and the proton can be calculated using the wave
distribution given in quantum
mechanics. This leads to the same average energy and distance *a*_{o}
that we would find if all the electron was concentrated at a
distance equal to the Bohr radius from the proton. Consequently,
one can calculate the potential of that electron cloud using
quantum mechanics, as if it were located at a distance from the
proton, equal to the Bohr radius. That electron cloud can either
oscillate through the proton if the angular momentum is zero or
around it, if the angular momentum is not zero.

When
the electron cloud is trapped into the electric field of a
proton, an hydrogen atom is formed. During its formation, energy
is given up as emitted radiation. This is similar to the energy
that Mercury must release when it is trapped into the Sun's
gravitational potential. The electron cloud can be distributed
according to many configurations having different energies
corresponding to different quantum states. Consequently, during
the formation of each of those states, the electron loses mass
the same way Mercury does when it is trapped in the Sun's
gravitational potential.

Let us
use the Bohr model in which an electron moves on an orbit around
the nucleus. We know that the Rydberg states of hydrogen
correspond to electrons traveling on an orbit whose
circumference is exactly equal to an integer number of the
wavelength of the electron. Then, there is a constructive
interference of the electron wave when moving to the next orbit
around the nucleus. The number of wavelengths forming the orbit
is equal to the principal quantum number. This model is
compatible with the energies calculated by quantum mechanics.

Experimentally, after the Rydberg states were measured, it was
noticed that the transitions between these states are not as
simple as originally expected. It was discovered that the
transitions between each pair of states are generally made of
several very close spectral lines. Sommerfeld carried out
calculations using general relativity and he discovered that
instead of simple transitions between quantum states, there
should be multiple transitions due to the fine structure.

Due to
the change of electron mass as a function of its distance from
the proton, the wavelength of the electron changes.
Consequently, the radius of the orbit changes because it is
necessary to have an integer number of wavelengths in a
circumference. Due to that change of the distance from the
proton, the electrostatic potential changes so that the electron
energy becomes different. Consequently, the ** force**
(not the field) between the electron and the proton does not
follow exactly a quadratic function. Therefore the electron
orbit around the proton precesses as in the case of Mercury
around the Sun as given in equation 5.52. Due to this
precession, the transitions between different quantum states
have slightly different energies depending on the relative
direction of the velocity of the electron around the nucleus
involved in the quantum transition.

Experimentally, the fine structure is well known. The Sommerfeld fine structure constant is equal to:

11.16 |

This fine structure term is observed between all quantum states as long as transitions are allowed by the selection rules. Sommerfeld's fine structure is explained in many textbooks [1]. It is often illustrated by precessing ellipses forming rosettes identical to the path of Mercury on figure 6.2.

The Sommerfeld fine structure constant can be explained more accurately using the principle of mass-energy conservation as done in the case of the orbit of Mercury. However, this is beyond the scope of this book. We will limit our explanations to this qualitative description. We understand now that the fine structure inside atoms is due to the principle of mass-energy conservation. Of course, Sommerfeld's calculations do not lead to a complete agreement in the case of an electron, because one must consider the electron spin. However, this last correction is irrelevant in the case of Mercury.

**11.5 - Atomic Structure
inside Free Falling Atoms.**

Let us
study a hydrogen atom falling freely in a gravitational field.
We can assume that the atom was initially located in outer space
before it slowly started to drift and accelerate gradually
toward the Sun. Then, gradually, the hydrogen atom acquires a
high velocity. An observer accompanying the falling mass would
not ** feel** any internal acceleration. We will now
calculate the absolute rate of the falling clock.

Let us examine this problem separating mathematically the two components of energy acting on the falling mass. With respect to a rest frame in outer space, the speeding hydrogen atom is now at a location where there exists a gravitational potential. We have seen that to calculate the exact mass of the particle, this potential must be taken into account. Furthermore, the falling hydrogen has acquired a velocity which must also be taken into account.

We have seen in equation 1.22 that the gravitational potential, where the atom is now located, is such that the mass of the particle has decreased and is now different from its mass in outer space. We also know that the kinetic energy increases the mass of the particle by an amount, which must be equal to the mass lost due to the potential energy.

This can be easily calculated and we see that the decrease of mass due to the gravitational potential compensates exactly the increase of mass due to kinetic energy. Consequently, the absolute mass of the particle (proton and electron) does not change while it is falling.

**11.6 - High Potentials
and Higher Order Terms.**

Contrary to Einstein, in this book we have not arbitrarily
postulated that physical quantities are invariant in all frames.
We have used only the principle of mass-energy conservation.
However, we have found that when we consider the zero and first
orders of v/c (or the gravitational potential), the physical
laws appear (almost) invariant in all frames as arbitrarily
assumed by Einstein. In that particular case, the physical
consequences are almost all identical to what Einstein found
with his arbitrary postulate. However, our results are obtained
using solely the principle of mass-energy conservation. The
physical laws derived from the use of the first order of v/c are
invariant, up to the point we reach higher order terms in (v/c)^{2} and other still higher terms (however small) that
have been neglected.

One
could repeat all the above calculations without neglecting the
higher order terms. Then, one could have an exact answer to the
problem of extreme energies. We can foresee that if we dealt
with physical phenomena in which the higher terms were not
negligible (correction due to velocity), the physical laws
observed might be slightly different. Within those physical
limiting conditions, at high energy, the behavior of matter
would not correspond to the description we are used to see in a
rest frame and in a frame in which the ratio v/c is not too
high.

We have
to realize that the experimental conditions that correspond to
such high energies are quite common in physics. It is clear that
when the nucleus of an atom emits particles having energies of
millions of electron volts, the second and third order terms of
the potential involved are not negligible. Consequently, we
expect that the internal phenomena taking place in the nucleus
of atoms at such high potential taking into account the higher
order terms lead to physics we are not accustomed to see. It is
for that reason that nuclear forces are not familiar to us and
to classical mechanics. We believe that the principle of
mass-energy conservation is one of the ultimate principles in
physics that possesses the wonderful power of informing us, in a
logical way, on the correct physical nature of the forces
involved in nuclear and particle physics. Mass-energy
conservation is relevant ** everywhere** in physics
and can be applied everywhere in nature, especially when
enormous potentials are involved as in the nucleus of atoms and
at the center of the stars.

A general study of physics in which the principle of mass-energy conservation is fully applied is beyond the scope of this book. However, we are convinced that a physical and realistic description of our physical world can be logically achieved without having to involve the non-realistic, the non-conservation of mass-energy and the contradictory hypotheses used in modern physics [2].

**11.7 - References.**

[1] H. Semat, ** Introduction
to
Atomic
and
Nuclear Physics,** Holt, Rinehart and Winston, Forth
edition, P. 245, 1962.

[2] P. Marmet,

a_{o} |
Bohr radius of the atom at rest |

a_{v} |
Bohr radius of the moving atom |

E_{o} |
Electric energy of the atom at rest |

E_{v} |
Electric energy of the moving atom |

l_{o} |
de Broglie wavelength of the atom at rest |

l_{v} |
de Broglie wavelength of the moving atom |

m_{o} |
mass of the atom at rest |

m_{v} |
mass of the moving atom |

v_{o} |
velocity of the electron relative to the proton of the atom at rest |

v_{v} |
velocity of the electron relative to the proton of the moving atom |

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