Part II (A)


Toronto, Ontario, Canada
© J.L. Gaasenbeek, B.Sc., P.Eng. 1990

Revised: April 14, 1998


Equations which describe the observation of a moving body by a moving observer and of a superluminal body by a stationary observer are given. Stellar aberration is explained.


In my second paper "Frames of Reference"(1), I discussed:

(a) The reference frame of the "stationary observer", and

(b) The reference frame of the light which conveys the information to the observer.

In this paper I propose to expand on what is meant by a "stationary observer" as well as discuss the case where:

(a) a moving observer observes a moving body,

(b) where a stationary observer watches the fly-by of a superluminal star, and

(c) stellar aberration.


For the sake of clarity the equations in this paper will be numbered in continuation to those in the previous paper: "Frames of Reference"(1),

i.e., the numbering will start at [26].


In my paper "Frames of Reference"(1) the question whether a given event occurs in one or another frame of reference is avoided since all observations are always related to the frame of reference of the observer which considers himself to be at rest, even though he may be in motion relative to the earth or the solar system, etc.

For example, let us suppose that the observer, as well as the moving body he is observing, each have a different velocity in relation to the "stationary" sun. In this instance, in order to be able to apply equations [l] to [25], he must first vectorially subtract his velocity from the object he is observing, as explained below.

This is the case because the observer can only relate his observations to himself who, as far as he is concerned, is at rest even though he may be travelling in a speeding spaceship.

The alternative, of not having the observer consider himself stationary is unworkable since the moving object he is studying will appear different to any other observer who is not travelling at his velocity and who is not located in exactly the same position he is, at a given point in time.

The above observed reality should not be confused with actual reality. In actual reality each body is positioned at a given position in relation to all other bodies in the universe and is travelling at a given velocity (although it may be accelerating due to gravitational forces etc.) at a given point in time.

However, since it takes time for the image of an object to reach the observer, what the observer sees at any given time depends on where he is positioned at the time the historic image of the observed object reaches him. That is to say, even though what he sees is reality to the observer, what he is looking at is something that happened some time ago.

Or, to use a simile, no two people in a concert hall hear exactly the same thing while listening to a symphony orchestra perform. Not only will the people on the right side hear the cellos more clearly than the first violins, because they are closer to the cellos than the violins, but the acoustics throughout the concert hall vary, depending on where one is seated. Moreover should the orchestra be in motion in relation to the listener, its sound will undergo a Doppler frequency shift. Finally, a listener in the back of the hall will, for example, see the kettle drum being struck before he can hear it, etc.

Of course, an observer could elect to consider any moving object he observes to be stationary and himself to be moving. However, if this were the case each observation would call for a different frame of reference, depending on the velocity of the observed object. The consequence of such an approach would be chaotic since no two separate observations would be comparable.

A further reason why an observer should consider himself at rest is that the average speed at which the historic image of an observed object travels towards him depends on the strength and velocity of both the effective EFOR of the object and the effective EFOR of the observer. However, for the purpose of this discussion, we shall disregard this added complication.


One advantage of the foregoing approach is that it is applicable to the most general of cases such as when both the observer and the observed entity each travel through space at their own separate velocity as measured in a common frame of reference.

Figure l shows a three-dimensional view of two moving entities A and B at a given point in time in a given frame of reference. Let us suppose that A observes B. What will he see?

First, for A to consider himself at rest he vectorially subtracts his velocity from B's velocity. The resultant velocity VAB and A now define a new reference plane in which equations [9] to [17] apply. (See previous paper: "Frames of Reference").

Next the new time t, which corresponds to the present, is determined by calculating the time it takes for B to move back in time and space at velocity -VAB until B is closest to A who remains stationary. This is the distance d which runs from A perpendicular to the line of flight as defined by VAB to point D, at which t = 0.

Finally, all observed parameters can now be calculated by substituting the new values for VAB, a, d, and t in equations [9] to [17], etc.

Next follow the derivation of the relevant equations and a numerical example.

Figure 1
Two moving entities A and B

Next we will calculate what A sees when he observes B.

When we substitute ac for VAB in Equation [10] we get:

Finally, the historic image of B, as observed by A, is where it was located and as it looked like C4.7051/C = 4.7051 seconds ago.

AB OBS = 6.5109 sec, as compared to the present which is 11.216 seconds.

It is realized, of course, that the above analyses would apply equally if B was selected as the observer of A in which case B's velocity would be vectorially subtracted from A's velocity, etc.

This example clearly illustrates that the apparent reality as seen by a given observer is unique to that observer in that a second observer, who is positioned in a different location, will see a different set of events even though both are observing the same actual set of events.

This is why each observer must consider himself stationary while observing a distant moving object, quite beside the fact that that is what it feels like when we look out of the window of a moving plane.

Finally the above also illustrates that time dilation is a perceived rather than an actual phenomenon.

It should be noted that by comparison the Lorentz-Einstein transformation equations(2) do not clearly specify what frame of reference is selected and why, nor what the position of the observer is in relation to the object he observes, nor, for that matter, what the age is of the observed historic image of the object etc.

Finally Einstein's theory expects one to embrace the illogical concept that the flow of time varies throughout the universe. In so doing his explanation raises more questions than it answers. Besides, "time waits for no man", no matter how fast or how far he travels.


About once a day a burst of intense gamma radiation emanates from a different part of the sky, none of which have been positively identified with a previously known object. Not only is the burst of radiation intense but it is of high frequency and of short duration(3).

As outlined in the preceding paper(1), it was proposed that objects can travel at a superluminal linear velocity providing their EFOR dominates. Accordingly it is further proposed that gamma-ray bursters are in reality superluminal stars which travel past the observer. To this end the appropriate equations will be derived below.

Suppose a superluminal star travels from left to right past the observer. Since the star travels faster than its image, it will actually have passed the observer before he will first see it. For example suppose the star's velocity V = 2C km/sec. We can now draw a light front which travels from the star at an angle = arc sin C/V. (See Figure 2 below). Since the light front has not yet reached the observer he will be completely unaware that the star has past him, similar to a plane which travels at a speed in excess of the speed of sound cannot be heard until it has past the observer.

However the instant the light front reaches the observer he will first see the star at 1 although the real star is actually at 2 (see Figure 3).

Figure 2

Superluminal star travels past an observer

Figure 3

Schematic diagram of the fly-by of a superluminal star

Since the first appearance of the star is such a crucial event all distances and times are measured from this point l rather than point 4. That is to say at t = 0 the actual star is at 1 but invisible. However at time to the actual star has reached 2 as it becomes first visible at 1.

Next follows the derivation of XOBS and VOBS.

VACT = V = ac as V is a multiple of C.

Y = act

Now the actual star will have travelled from 6 to 3 during the time it took for its historic image to travel from 6 to the observer at 5. Similarly and at the same time, the actual star travelled from 7 to 3 during the time it took for its historic image to travel from 7 to the observer at 5, i.e. the observed image of the star splits in two.


After solving the quadratic equation we get:

When we differentiate X1,2 OBS we get

to can be calculated by taking equation [39] and equating it to zero which gives:

which is the time at which the observer first sees the star. Finally the chronological age of the observed historic images of the star is equal to:

It should be noted that it is assumed in the above equations that the image of the star travels towards the observer at the speed of light C. In reality of course CAV should be used instead.

Continue with Frames of Reference Part II (B).