Natural Science Forum / Physics / Relativity / February 2006

                       THE DUAL VELOCITY THEORY OF RELATIVITY

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 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".

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 cent/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 than 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