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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