Einstein's Theory of Relativity
versus
Classical Mechanics
by Paul Marmet
Chapter Eight
The Doppler Effect.
8.1 - Fundamental
Principles of the Doppler Effect.
In
chapter two, we considered the special case of zero Doppler
effect. This means that the source was moving in a direction
perpendicular to the direction of propagation of light. The change
of frequency due to the Doppler effect was zero because the radial
velocity between the source and the detector was equal to zero.
When there is a relative radial velocity between the source and
the detector, the Doppler effect must be taken into account.
Unfortunately, this phenomenon does not seem to be completely well
understood in physics.
There
have been many discussions about the question of the conservation
of energy in the Doppler effect. For example, Weiss and Baez wrote
an article [1] entitled:
"Is Energy Conserved in General Relativity?"
They
consider the case of the cosmic radiation that has been redshifted
over billions of years. "Each photon gets redder and redder. What
happens to this energy?" They report that:"... the energy is
simply lost".
Such an
answer is not acceptable since we believe in mass-energy
conservation. We do not believe that any kind of energy can ever
be lost whatever the circumstances are. If this were possible,
energy would be created from nothing when an emitter moves toward
an observer because of the Doppler effect.
Of
course, an increasing radial velocity necessarily produces a
reddening but one sees that a reddening is not a proof of a
Doppler effect since it can be produced by other ways. It has been
shown [2] that the
reddening of the cosmic radiation can be better explained by a
different phenomenon in which mass-energy is conserved. The
reddening results from the energy lost following numerous
interactions of photons on interstellar gases during billions of
years. In that case, the residual energy is scattered elsewhere so
that there is no difficulty to be compatible with the principle of
mass-energy conservation.
8.2 - Mass-Energy
Conservation in the Context of the Doppler Effect.
Doppler
reddening is a real phenomenon which can occur in some cases and
which is always compatible with mass-energy conservation. For
example, let us consider the case of a hydrogen atom excited to
10.2 eV (the Lyman state) moving away from a stationary source.
If the hydrogen atom moves at half the velocity of light, the
theory of the Doppler effect (using the wave property of light)
teaches us that we will receive only half of the frequency of
the excited state. This means that the photon received from the
moving particle will have only half the energy of excitation.
The question is: Where does the difference of energy (5.1 eV)
go? It has been claimed in several papers that the energy is
missing.
The
demonstration using the change of frequency of a wave due to the
relative velocity does not take into account all the energy
available in the experiment. Let us calculate the Doppler effect
without using waves but using only the principle of mass-energy
conservation.
8.3 - The Doppler
Effect without Using Waves.
Let us
consider a mass mo (at rest) moving away at velocity
V with respect to an observer at rest. Let us assume that the
mass is a hydrogen atom. This moving atom has a total energy of:
 |
8.1 |
Let us
consider the case when that hydrogen atom is excited at the Lyman
a atomic state with an energy hno of 10.2 eV. The total energy
(potential plus kinetic) of that excited atom (neglecting the
higher order terms) is:
 |
8.2 |
Let us
use the moving frame of the particle from which the photon is
emitted. To be detected in the rest frame, the photon must be
emitted backward (-x axis) from the moving atom, in the direction
of the rest frame where the observer is located. When the photon
is emitted, the atom gets a recoil in the forward (+x axis)
direction giving it an increase of velocity Dv.
Of
course,
the
total
change
of momentum DP of the moving system
(photon plus atom) is zero. At the moment of emission, considering
the photon's momentum, we have:
 |
8.3 |
or
 |
8.4 |
With
respect to the rest frame, the velocity of the hydrogen atom was V
before the emission of the photon. After the emission of the
photon, the final velocity Vf of the atom with respect
to the rest frame becomes:
Equation
8.4 in 8.5 gives:
 |
8.6 |
The total
(mass plus kinetic) energy of the de-excited hydrogen atom after
the emission of the photons is (neglecting the higher order
terms):
 |
8.7 |
Using
equation 8.6 gives:
 |
8.8 |
The
change of kinetic energy of the hydrogen atom due to the recoil of
the photon is:
 |
8.9 |
From
equations 8.8 and 8.1, neglecting the second order, we have:
 |
8.10 |
 |
8.11 |
Equation
8.11 gives the increase of kinetic energy of the atom due to its
recoil. According to the mass-energy conservation principle, the
increase of kinetic energy of the atom must come from the photon
energy. Since the excitation energy initially available was hno, and since equation 8.11
gives the energy transferred to the atom (as kinetic energy), the
residual photon energy hnf
is:
 |
8.12 |
which is:
 |
8.13 |
Equation 8.13 is exactly identical to the Doppler equation.
We
have demonstrated the Doppler equation using no wave model but
only mass-energy conservation. The energy apparently lost in the
Doppler phenomenon is simply transferred as kinetic energy to
the emitting atom whose velocity has increased due to the recoil
momentum. It is also important to notice that the amount of
kinetic energy lost in equation 8.11 is independent of the mass
of the particle.
The
above demonstration solves the problem discussed by Weiss and
Baez and others. We conclude that the energy redshifted by the
Doppler mechanism is not lost. It is simply transmitted as
kinetic energy to the emitting atom due to recoil at the moment
of emission. We must notice that this explanation has nothing to
do with relativity.
8.4 - References.
[1] http://www-hpcc.astro.washington.edu/mirrors/physicsfaq/energy_gr.html
[2] P.
Marmet, A New Non-Doppler Redshift,
Physics Essays,
1, 1, P. 24-32, 1988.
8.5 - Symbols and
Variables.
EV |
energy of a mass mo moving at
velocity V |
E*V |
energy of a mass mo moving at
velocity V and excited to 10.2 eV |
E'V |
energy of a mass mo moving at
velocity V after losing its energy of excitation |
no |
frequency corresponding to the
excitation energy |
nf |
frequency emitted by the atom |
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