Toronto, Ontario, Canada
© J.L. Gaasenbeek, B.Sc., P.Eng. 1990
Revised: July 24, 1998
Equations which define the observed parameters of a moving body aregiven; the variable speed of light is defined in terms of movingelectromagnetic frames of reference (EFORS).
This paper will discuss two kinds of reference frames:
(a) The reference frame of the "stationary observer", and
(b) The reference frame of the light which conveys the informationto the observer.
THE OBSERVATION OF A MOVING BODY BY A STATIONARY OBSERVER
First let us consider the elementary case where a stationary observerlooks at an object that travels either directly away or towards him.
What will he or she see?
Figure l: An object travels either directly away or towards the observer
Since it takes time for the image of a moving object to reach theobserver he only perceives the object where it used to be and what itused to look like at some time in the past, i.e., the actual object willhave travelled an additional distance during the time it took for itshistoric image to reach the observer, even though the image travelledtowards the observer at the speed of light. Consequently, its observedparameters, such as velocity and aging rate, will differ from its actualvelocity and aging rate.
Providing the velocity of the observed object is a fraction of the speedof light, the following "Newtonian" formulas apply (see Figure 1).
XACT = tV 
XACT = the actual distance travelled by the moving object at time t,
t = the elapsed time since the object left the observer, and
V = the actual velocity of the body away from the observer.
Next let us derive:
XOBS = the observed distance travelled by the moving object attime t.
In order to simplify matters we shall call
XACT = Y and XOBS = X.
As per Figure l we get:
When we substitute we get:
from which follows:
The observed velocity (VOBS) of the moving object is equal to:
The observed or historic age of the moving object is equal to:
AOBS = X/V = t  or
AOBS = t which is the moving objects real time t.
The observed aging rate of the moving object is equal to:
A'OBS = XOBS/XACT or
A'OBS = C/(V+C) 
The observed length of time (tOBS) it takes the object to travel theactual distance is:
tOBS = XOBS/VOBS  or
tOBS is the observer's real time t.
Finally the Doppler frequency of the light emitted by the moving object
where is the actual light frequency radiated by the body.
Even though the above Equations l to 8 only cover the most elementarysituation, they prove quite enlightening when applied to the TwinParadox(1).
THE TWIN PARADOX
In order to illustrate the phenomenon of observed reality versus actualreality let us examine the famous example of the travelling twin who, onreturning from a fast round trip, allegedly aged less than his twinbrother who stayed at home.
Example: Suppose an astronaut left the (stationary) earth at l/2 C for aperiod of one year. Next he remains at his destination for one year.
Finally he returns to earth at a speed of l/4 C.
NOTE: See also a more straight forward worked example.
Question: How fast and how much will the astronaut appear to have agedas compared to the earthbound observer and how much will the astronautand the observer actually have aged over the duration of the voyage?
First let us consider what the observer sees after the astronaut leaves.
Since the historic image the observer sees of the astronaut getsprogressively older the further away he travels from the observer, as ittakes progressively longer for the image of the spaceship to reach him,it appears to the observer as if the astronaut is aging at a slower ratethan he actually is.
That is to say the apparent aging rate of the observed historic image ofthe astronaut as he travels away from the observer at 1/2 C is:
On the other hand, after the astronaut begins his journey home at 1/4 Cthe astronaut appears to age faster than normal to the observer:
However, the thing to remember is that, not only does the astronautappear to age at an abnormal rate on his way out and on his way back butalso that his observed velocity differs from his actual velocity. As aresult it will appear to the observer as if it took the astronaut longerthan the actual time it took him to get to his destination and less timethat it actually took him to return home.
The observed velocity of the astronaut on the way out is:
(as compared to his actual velocity of 1/2 C)
The observed length of time it takes the astronaut to reach hisdestination is:
tOBS = XOBS/VOBS 
tOBS = (1/2 C)/(1/3 C) = 1 1/2 years(as compared to his actual travelling time of one year)
The number of years the astronaut appears to have aged during theobserved l l/2 years it took him to reach his destination is:
AOBS = X/V 
AOBS = (1/2 C)/(1/2 C) = 1 year
AOBS = A'OBS tOBS
AOBS = (2/3)(3/2)
AOBS = 1 year
While at his destination of course, the astronaut appears to be aging atthe normal rate as seen by the observer i.e., during his stay of one yearhe ages one year. Of course, the historic image of the astronaut theobserver sees is always 1/2 year old as this is the time required for theastronaut's image to reach the observer.
Next the astronaut appears to take off for home. (In reality, of coursehe took off half a year earlier). However, now his observed velocity is:
(as compared to his actual return speed of -l/4 C)
The observed length of time it takes the astronaut to return home is:
tOBS = XOBS/VOBS 
tOBS = (-1/2 C)/(-1/3 C) = 1 1/2 years, as compared to his actual travelling time of:
t = XACT/V or 
t = (-1/2 C)/(-1/4 C) = 2 years.
Since the observed aging rate of the astronaut on his way home is 4/3 times normal (see above), he will appear to have aged A'OBS times tOBS = 4/3 times 3/2 = 2 years.
Alternatively the number of years the astronaut appears to have aged during the observed 1.5 years it took him to return home is:
AOBS = X/V or 
AOBS = (-1/2 C)/(-1/4 C) = 2 years
In summary, it appeared to the earthbound observer that the astronautaged slower than normal on his way out and faster than normal on his wayback because he appeared to travel slower than actual on his way out andfaster than actual on his way home.
In reality, of course, both the astronaut and the observer aged one yearduring the actual year it took for the astronaut to reach hisdestination. Similarly, both the astronaut and the observer aged l yearduring the year the astronaut remained at his destination and a further2 years for the actual length of time it took the astronaut to get backto earth, i.e., both the astronaut and the observer aged a total of
l + l + 2 = 4 years for the duration of the trip which lasted 4 years.
THE OBSERVATION OF A MOVING BODY WHICH PASSES A STATIONARY OBSERVER AT A DISTANCE (d)
Whereas in the previous instance time zero is when the object reaches orleaves the observer, in case of a fly-by, time zero is when the object isclosest to the observer i.e., at right angles to the flight path, adistance (d) from the observer. This is comparable to our calendar whichstarts at the birthday of Christ, i.e., we measure time in years beforeand after Christ.
Figure 2 shows a moving object which passes the observer at a distance d
at a velocity V.
Figure 2: An object passes the observer at a distance d
The following formulas apply (The derivation of the formulas have been omitted for the sake of brevity. (See Frames of Reference: Part Two; Appendix)).
The above formula [l0] accounts for stellar aberration.
tOBS = XOBS/VOBS 
The observed age of the moving body is:
AOBS = X/V 
Alternatively, the observed age of the moving body is:
AOBS = A'OBS tOBS
A'OBS = XOBS/XACT 
The actual distance between the observer and the moving body is equal to:
SACT = (d2 + V2t2)1/2 
The observed distance between the observer and the moving body is equalto:
SOBS = (d2 + X2OBS)1/2 
The Doppler frequency is equal to:
The case where a moving observer observes a moving body and where astationary observer watches the fly-by of a superluminal star, will bediscussed in a future paper.
THE VARIABLE SPEED OF LIGHT MODEL
In the forgoing analysis it was assumed that in every instance the imageof the moving object travelled towards the observer at the speed of lightin relation to the observer, regardless of the velocity and size of themoving object or body.
Even though all earthbound measurements have shown that the velocity oflight is independent of its source velocity, I have always felt thatthese experiments only told half the story.
Accordingly it is further proposed that each and every body radiates itsown electromagnetic field (frame) of reference (EFOR) which causes anyelectromagnetic radiation within its sphere of influence to travel at thespeed of light (C) in relation to that body (disregarding the refractiveindex of its atmosphere). Moreover the strength of a body's EFOR isproportional to the total amount of electromagnetic radiation it emits,that is to say, on its luminosity. The luminosity of a body in turndepends on its size and temperature.
For example, when an airplane turns on its landing lights, the light willleave the plane at the speed of light (C) in relation to the movingplane. However, the EFOR of the moving plane is so weak compared to theEFOR of the "spaceship earth" that almost immediately after leaving theplane the light will fall in step with the electromagnetic radiation theearth radiates i.e., the earth's EFOR overpowers the airplane's EFOR socompletely that it becomes the Dominant EFOR or DEFOR.
However, when we consider the preceding phenomenon on a cosmologicalscale matters become more complex since one moving star's EFOR may be asstrong as that of a neighbouring star. Moreover, they may be many lightyears apart. Accordingly, it follows that at some point between thestars both their combined or Resultant EFOR or REFOR will determine atwhat speed the light will travel as related to one or the other star.
In order to explore the phenomenon in greater depth let us return to theproposed helical photon wave light model. As stated previously, unlike asound wave which travels through the stationary air, a helical photonwave does not require a medium to travel through. One interestingconsequence is that once the light leaves its source it has no way ofknowing at what speed it is travelling since, unlike a soundwave, itsspeed is not controlled by the medium through which it travels. However,light can judge its speed by comparing it to other light waves it meetson its journey as follows:
Suppose that a star travels away from the sun at a velocity of 1/4 C.
Now we know that the light from the sun will start its journey towardsthe star at C as related to the sun. Similarly, the light from the starwill start its journey towards the sun at C as related to its source, thestar. This is the case because once we conclude that the sun radiatesits light at C it logically follows that the star will also radiate itslight at C since, for the sake of argument, we could have called the sunthe star and the star the sun as they are both stars.
However, we also know that by the time the star's light reaches the sunit will be travelling at the speed of light (C) in relation to the sun,not as related to its source the star. Similarly, by the time the sun'slight reaches the star it will be travelling at C in relation to thestar, not as related to its source the sun. Again, what applies to onemust apply to the other.
What happens is that the large number of photons emitted by the sun overpower the few photons that arrive from the star, that is to say, the sun's EFOR dominates, close to the sun. Of course the opposite is true for the few photons from the sun that happen to reach the star. Here the star's outgoing photons outnumber the few incoming photons from the sun and force them to fall in step with them.
It follows from the above that the degree to which helical photon wavesfall in step with opposing helical photon waves will depend on thedensity of the oncoming waves as compared to their own photon wavedensity. Now the number of photons a star emits per second isproportional to its luminosity. Suppose the luminosity of the star inour example is equal to the luminosity of the sun, then halfway betweenthe sun and the star the density of the star's photons and that of thesun will be the same. Since neither one can overpower the other theysplit the difference and each increases its equivalent linear velocitysuch that their relative velocity once more adds up to 2 C. That is tosay, the sun's photons increase their equivalent linear velocity by 1/8 C to l 1/8 C and the star's photons increase their equivalent linearvelocity by 1/8 C to 7/8 C (both velocities as related to the"stationary" sun). In reality of course, this is a gradual process.
Since the photon density is proportional to the luminosity (or K value)of its source and inversely proportional to the square of the distancefrom its source, formula [l8] gives the speed at which a star's lighttravels toward the earth, measured at a distance X from the earth.
S = the distance between the star and the earth, the K value of the sunis one and the velocity of the star away from the earth is V.
The average velocity at which a moving star's light travels toward theearthbound observer is:
THE OBSERVATION OF FAST MOVING BODIES
Should the observed body travel at a relativistic velocity, CAV, as performula  should be substituted for C in the above formula  to .
For example, the observed or apparent aging rate  of the relativistictwin (of course the K value of the spaceship is very small as compared tothe K value of the earth) becomes:
The substitution of CAV for C is a simple matter for the case where theobserved object travels directly away or towards the observer, i.e., formulas  to .
In the case where the object passes the observer at a distance d(see formulas  to ) the calculation of CAV becomes morecomplex. First XOBS is calculated based on the speed of light beingequal to C. This value of XOBS can now be used to calculate a valuefor CAV.
This calculated value of CAV can now in turn be used to recalculateXOBS and so on. By repeating the procedure any required degree ofaccuracy can be achieved.
It was noted that the apparent aging rate of the observed historic imageof a moving light source (A'OBS) can be used to calculate the observedor Doppler frequency of the light source .
That is to say, in the case where the light source travels eitherdirectly away or towards the observer the Doppler frequency is equal to:
In the case where the light source is massive and travels at arelativistic speed:
Alternatively, in case of a fly-by where the light source passes theobserver at a distance (d) the Doppler frequency is equal to:
Similarly, should the light source be massive and travel at arelativistic speed, CAV  should be substituted for C in the aboveformula  as discussed previously.
INTERACTION BETWEEN MOVING BODIES
It follows from the above that if an observer can only see a moving bodywhere it used to be and what it used to look like at some time in thepast, any electromagnetic, electrostatic or gravitational force etc.,between two moving bodies will also be a function of the observeddistance between the bodies rather than the actual distance, since it isgenerally assumed that all force fields advance through space at thespeed of light.
For example, the gravitational attraction between two stationary bodiesor masses is equal to(2):
However, should the second body travel away from the first body at avelocity V, we can substitute the observed distance XOBS for thedistance r in formula .
the formula becomes:
Should their relative velocity approach the speed of light (C) theaverage value of the speed of light CAV should be used to calculateXOBS etc.
H.L. Fizeau first showed in 1851, by means of a specially equippedoptical interferometer, that instead of the full speed, about half thespeed of the water is added to the speed of light, when the light beamand the water travel in the same direction, whereas half the speed of thewater is subtracted from the speed at which the light beam normallytravels through water, when the light beam travels in the oppositedirection(3).It took me quite a while to understand what was happening since thisresult implied that the flowing water was somehow capable of moving theEFOR of the light at about half its speed.
Eventually the following thought experiment occurred to me.
Suppose a man (A) was sitting towards the front of a fast movingspaceship and a second man (B) towards the back of the same spaceship and(A) flashed a beam of light at (B). It surely would take just as longfor the lightpulse to travel from (A) to (B) as it would take for alightpulse to travel from (B) to (A), regardless of the speed of thespaceship, since all the electromagnetic radiation inside the shiptravels at C in relation to the ships interior. In other words, theinside of the spaceship constitutes a moving EFOR!
The next step was obvious. What happens if the spaceship is providedwith a rear and front window through which a lightbeam can enter andleave the moving spaceship? It follows logically that the speed of theship will be added to the speed of the light during the time it travelsthrough the spaceship from the rear to the front window.
But if this is the case why did Fizeau's optical interferometer only addor subtract 1/2 its water velocity to or from the speed of light whereasthe spaceship should add or subtract all its velocity to or from thespeed of the light that travels through it? (Assuming that the inside ofthe spaceship is a vacuum).
Finally, it dawned on me, why this was so. Inside the spaceship allsources of electromagnetic radiation move at the velocity of thespaceship. In Fizeau's experimental apparatus on the other hand, theinternal sources that emit electromagnetic radiation into the flowingwater do not all move at the velocity of the water since the tube, through which the water flows is stationary. Consequently, the Resultant EFOR or REFOR which effects the light is a combination of the EFOR of the movingwater and the EFOR of the stationary tube through which it flows, whichis why only about half the water's velocity is algebraically added to thevelocity of the light that travels through it.
My previous paper: "Helical Particle Waves" and this paper: "Frames ofReference" together provide an alternative explanation for Einstein'stheories of Special and General Relativity. In my next paper I proposeto combine such concepts as the conversion of matter into energy and thegravitational force, with the strong and weak nuclear forces and myearlier theorems, into a single Unified Field Theory.
1. A.P. French, Special Relativity, Pages 154-159, The Twin Paradox.
2. Halliday and Resnick, Second Edition, Fundamentals of Physics, Page 243.
3. A.P. French, Special Relativity, Pages 46-49, Fizeau's Measurement of the Drag Coefficient.
End of Paper 2 of 5
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