Natural Science Forum / Physics / Particle Physics / May 2006

TO Moc
Re your post "The Lorentz Transformation for Velocity" . you are sorely
confused. Take a look at the following:

  THE DUAL VELOCITY THEORY OF RELATIVITY

           Vertner Vergon
Email: avergon@verizon.net

The special theory of relativity enjoys its unusual status because
certain conclusions were made
by Einstein that a rational mind has difficulties digesting. There is a
saying that many physics
theoreticians are correct in their mathematics but err in their
interpretations.

The object here is to correct Einstein's misinterpretations and clear
up some of the
misunderstandings. Rather than analyze the said misinterpretations, we
simply present the
proper interpretations and let the reader draw his own inferences.

First, we must recognize that what happens to length, mass, energy,
momentum and time is
simply an observation. These parameters in their own co-ordinate system
do not change. There
they are known as "proper".

LENGTH

Let us consider length, say the length of a rod. As its co-ordinate
system approaches the speed
of light, the length of the rod appears to foreshorten. Next we
recognize that length and distance
are synonymous. We also recognize that velocity is distance/time.
Therefore, we are drawn to the
conclusion that if the observation of a length contracts, so does the
observation of a velocity.

So what we have, then, is the dual situation of proper length and
observed length -- and likewise,
we have proper velocity and observed velocity, thus, the dual velocity
that is the subject of this
writing.

We shall refer to the proper velocity as "Newtonian" for that is what
it is -- and the observed
velocity as "relativistic" because that is what it is.

Next, we note (after proper investigation) that as the velocity of the
moving co-ordinate system
approaches infinity, the observed relativistic velocity approaches c.
The relationship can be
written as

V x sqrt(1-v^2/c^2) = v, where V = Newtonian velocity, v = relative
velocity, and sqrt(1-v^2/c^2) is
the Lorentz transformation -- for which, henceforward, we shall use the
letter, R.
Thus we have V R = v.

MASS AND MOMENTUM

Amongst other things, this posit clears up the old bugaboo of
relativistic mass which was inferred
from the expression for relativistic momentum, p:   p = mv/R

In this expression, as v goes to c, p goes to infinity. And it was
concluded that since v had the
limit c, the only way p could go to inifinty was if the mass, m,
increased. So Einstein concluded.

But in the light of dual velocities, we see another explanation.
Instead of R modifying m, (m/R) it
modifies v, (v/R) -- and we see that v/R = V. So it is the Newtonian
velocity that goes to infinity.

TIME

We start our examination of the time dilation concept by going to its
source -- Einstein's paper,
On the Electrodynamics of Moving Bodies. We refer to his gedanken
experiment of moving
clocks. One of two identical clocks remains at rest while the other
moves away and returns. When
Einstein perceived the difference of the clock readings in his
calculations, he stated the moving
clock "was slow by ...". The immediate perception by the public was
that he meant if a clock was
"slow by" - it had to have run slower. He also said the moving clock
was "behind" the inertial
clock by ... . These two statements do not mean the same thing. If one
clock is running slower,
then it is running slower, and that has only one meaning.

On the other hand to say one clock is behind the other is open to
alternative explanations, eg, the
moving clock could have traveled a shorter world line -- or may have
traveled faster than
observed. In either case the clock would maintain its normal (proper)
rate but for a shorter
duration than the inertial clock and thus be behind. At any rate the
accepted version is that the
clock ran slower and thus was born the concept of time dilation which
led to the famous Twin
Paradox.

We see here the answer to the problem is that the moving clock traveled
faster than observed.
The clock in the moving system kept proper time but did so for a
shorter duration than
experienced by the stationary clock because its velocity (Newtonian)
was faster than the
observed relative velocity. To conclude that the moving clock kept
proper time (which it does) and
dilation (slower) time simultaneously is a reductio ad absurdum.

We shall illustrate -- and in doing so, we shall also illustrate the
existence of super c velocities.
We assume a co-ordinate system that travels at the rate of 1.732 c. By
VR = v we see the relative
(observed) velocity to be .866c. {R can be obtained from V by
sqrt{1/(1+V^2)}.

We set the conditions as follows: The traveling system ,S, will travel
a distance of 1.732 light
seconds (LS). Since the velocity is 1.732 LS/sec, the clock in the
system will record the transit as
taking one second. One second up and one second back equals 2 seconds
for the round trip.
The observation in the inertial system is different. There, the
observed velocity (relative) is .866
LS/sec, and the round trip will occupy 4 seconds -- two seconds each
way.

However, that is not the way the inertial system actually observes. One
has to take into account
that in observing a moving co-ordinate system, the time for radiation
to transmit the record of it
has to be included. So the actual subjective description can be
mathematically described as
follows:

( where c=1) v_ diverging = v/(1+v), v_converging = v/(1-v).
Thus in our illustration the subjective observed velocity in recession
is .4641c -- and the transit
time is 1.731 LS/.4641c = 3.732 sec.

The subjective observed velocity in approach is 6.4641c -- and the
transit time is
1.732 LS/ 6.4641c = .26795 sec.

Adding the recession time to the approach time gives us a 4 second
round trip, for the inertial
observer whereas it is 2 seconds for the moving observer.

One last consideration. Now that we have established super c
velocities, what are the
complications?

Let us, as a means of clearer illustration, use the Twins example. We
shall call them Astronaut
and Astronomer.

We see that to the Astronaut the round trip is two seconds, whereas to
the Astronomer it is four
seconds. (Let us transpose seconds to years).

What happens when the Astronaut lands and strolls over to stand
shoulder to shoulder with the
Astronomer? He must necessarily see the same as does the Astronomer.
What would that be?
And would that violate any laws of physics? He realizes that although
only two years have
elapsed for him, it has been four years for his brother, the
astronomer.
To illustrate further, we can imagine a similar situation. We
contemplate a galaxy a million light
years away. It is agreed that as we observe it, we observe it as it was
a million years ago - we
are looking into our past.

Should a creature from a planet in that galaxy suddenly appear, we
would conclude that he came
from over a million years in our past. There is no conflict with known
physical laws.

The question arises, what about the accepted concept that one can never
chase a light beam and
catch up to it? *

Consider the following: As one increases their velocity in this
pursuit, the beam gradually reduces
in frequency - until at the speed of light, there is no frequency at
all. Note that the reduction in
frequency does not alter the fact of the beam always preceding the
observer at c until the
frequency reaches zero. That would be c on the relativistic scale and
infinitely great on the
Newtonian. Thus we conclude it would take an infinitely great Newtonian
velocity to catch a
photon - which disappears at that velocity because the observer is
keeping pace with the EM
transmission.
------------------------------------------------

* CHASING A LIGHT BEAM

For an observer to chase a light beam means to chase photons that have
recorded his existence
and are proceeding in the same direction as he. Since c is a constant,
we conclude that the
photons will always precede the observer at that speed and he can never
overtake them. We
therefore conclude that superluminal velocities are impossible.
However, we recognize that
super c velocities are. We distinguish one from the other. A super c
velocity is a Newtonian
velocity greater than 3x10^10 cm/sec whereas a superluminal velocity is
greater than light -
which is potentially infinitely great.
--------------------------------------------------
Above, we mentioned time dilation - another misconception by
Einstein. He asserted that
whether receding or approaching, the moving clock ran slower. The
nightly observations of
astronomers belies this.

Any known constant frequency is a clock (A certain vibration of the
excited cesium atom is the
new standard for the second.). Constellations and stars contain excited
atoms of known, constant
frequency. They are in effect clocks.

We note that, in effect, when these clocks are receding, the observed
frequency slows. That
means the clock is observed to keep slower time. Conversely, when the
clock is approaching, it is
observed to run fast. Thus, we are drawn to the conclusion that time
dilation as Einstein proposed
it is in error for in that concept the clock is running slower.

Not only that but in its own co-ordinate system it would have to keep
slower time - and
simultaneously keep proper time, as we said before, a reductio ad
absurdum.

Another fact that unseats the time dilation concept is that the rate
for the two systems are
different. The frequency system just described is also known as the
Doppler effect. Now
"Doppler" is just the name describing the mechanics of it. This
does not alter the fact that Doppler
rate is observed time rate. I call it "Doppler time". If one
examines the figures given in the
illustration above, they will discover that the transit times are of
the Doppler rate. When used in
the Twin Paradox situation, the paradox never appears.

ENERGY

We write generically E = m a d = mv^2 (m = mass, a = acceleration, d =
distance)
For kinetic energy we write E_k = mv^2/2

This is the Newtonian expression and valid for very low velocities.
By considerations not displayed here the factor 1/2 is replaced by
1 /R + R^2

Thus we write the expression for kinetic energy as

            mv^2
E_k = ---------------
         (R + R^2)

It will be found this is exactly equal to Einstein's
E_k = (1/R - 1)mc^2 and good for all velocities.
We note that the expression for the kinetic energy of radiation is

E_k = h nu = m_photon c^2, which is of the form mc^2.
The mass of the photon is derived from Einstein's m = E/c^2.
Total energy is E_t = mc^2/R

We also note that in the form, E = mc^2, there is no modifying factor
as there is for ponderous
bodies. We take this to mean that the Lorentz transformation is not
applicable to radiation (except
where there is an interaction between radiation and matter in motion).

In the Newtonian case, we also note that the factor ½ is only
applicable at very low velocities. In
the kinetic energy equation above --- good for all velocities --- we
have the factor 1/(R + R^2)
replacing ½.

Note that at very low velocities, R is 1 --- and the factor 1/ (R +
R^2) becomes ½. As we go up the
scale of velocities, this factor goes to infinity as does the velocity.

As stated before every Newtonian velocity has a corresponding relative
velocity. It should be
noted the parameters of momentum, kinetic energy and transit time found
in the Newtonian
velocity are associated, unchanged, with the corresponding relative
velocity. In short, they do not
undergo the Lorentz/Fitzgerald transformation.

It is this phenomenon that accounts for much of the mystique of special
relativity --- to wit --- at c
velocity the energy and momentum go to infinity, while transit time
goes to zero. Recall that the
Newtonian velocity corresponding to c is infinitely great - and we
would expect the momentum
and energy to go to infinity. At infinite velocity, the transit time to
anywhere is zero, so the clock in
the moving co-ordinate system is assumed to have stopped.

As to length, that does undergo the transformation - which, with
respect to inertial time, shows up
as a reduced Newtonian velocity which is our relative velocity. (As a
seeming contradiction, we
note that inertial time times R equals the transit time in the
Newtonian velocity, but the fact
remains that transit time with the Newtonian velocity is the time
attributed to the relativistic
velocity.)

For more information and monographs see the General Science Journal at
http://www.wbabin.com. Go to "List of Authors" and click on Vertner
Vergon.

V. Vergon

Feb. 2006