Einstein’s Simple Mathematical Trick –and the Illusion of a Constant Speed of Light

As Einstein regarded the situation, the [aether] experiments, seemed to indicate a "conspiracy" on the part of nature against man's efforts to obtain knowledge of the physical world,
–Science writer Lillian R. Lieber (1945)

1. Introduction
2. Basic Motion
3. How Einstein achieves the illusion of lightspeed INVARIANCE
4. How Nature Achieves the Illusion of Lightspeed Invariance
5. In the Real World
6. Conclusion

1.   Introduction

In the real world aether exists and serves as the conducting medium of light waves.

Pay no attention to the contrary assertion in your high school and college texts which invariably state that there is no such thing as aether. The aether is described as just an old discarded concept; strictly 19^th century voodoo science. Do, however, pay attention to what is written in the research articles of the various physics journals. It is now commonly accepted that the Universe is permeated by aether (a generic term for the space medium). While the existence of aether is not at issue, the properties of aether are subject to considerable debate.

Also understand that most professionals are embarrassed to actually use the term “aether” with its negative connotations. They prefer to use more exotic terms such as quantum foam, cosmic substratum, dynamic three-space, background substructure, and others. … Can’t really blame them. I understand and I sympathize. Imagine having to acknowledge that you had it all wrong; and that your profession had it wrong for almost 100 years!  Ouch!

2.  Basic Motion

The most fundamental equation of motion is

(distance traveled) = (speed) × (interval of time).

And when rearranged,

(interval of time) = (distance) ÷ (speed),  or

.                              (2-1)

Consider an airplane making a two-way crossing of a lake on a windy day. (See Fig. 1.) The time-of-travel going with the wind will obviously be shorter than the travel time for the crossing going against the wind. It is simply because the plane is flying with a constant speed with respect to the air, but the air itself is moving with respect to the Earth’s surface, including the lake.

	Fig. 1.   In the presence of the atmospheric wind, the plane's flight time going from A to B is less than the time going from B to A. (up is the air-speed of the plane.)

During the outbound crossing the speed is, in our example, (u_PLANE + u_AIR). And, during the return flight, assuming the wind has not changed, the speed must be (u_PLANE − u_AIR). Here u_PLANE is the plane’s speed with respect to the surrounding air. (And u_AIR is the atmospheric wind with respect to the Earth reference frame.)

All you need to understand is that the plane moves at some FIXED speed through the air, while the air itself also moves (as “wind”).

Here are the equations of motion for the flights between runways A and B (as labeled in Fig. 1):

.               (2-2)

Simple enough. Anyway, the propagation of light waves through the aether works in the same way. … Yes, I know what you were taught; but just follow along with me.

Light waves pass through the aether —that is, they are conducted by the aether— with a fixed speed of about 300,000 kilometers per second. In shorthand, c = 300,000 km/s. (Note that symbol “c” is not a variable, it represents a constant of Nature.) Meanwhile the aether itself may be in motion —a motion historically called the aether wind.

We now consider a light pulse beaming outward across the lake, and then being reflected back to the starting point. A lightspeed round trip from A to B, then back across to A. See Fig. 2.

	Fig. 2.   As a consequence of the "aether wind", the travel time of the light pulse going from A to B is less than the time going from B to A.

Obviously the two time intervals will be different:

  .                         (2-3)

This means that the speed of the light pulse is not the same for the two directions. Speed (c + u_a) is not the same as speed (c – u_a).

Now here comes the good part. I will show how Einstein makes the speed of light the same (constant) for both directions. Then later, I will detail how Nature actually makes it appear that the speed of light is constant!

3.  How Einstein Achieves the Illusion of Lightspeed INVARIANCE

Einstein uses a simple procedure to make the speed of light constant.

First he deems the two travel times to be identical. That’s right, as if by magic.

In his own words:

He simply declares,  (∆time_{A to B} ) = (∆time_{B to A} ).

Which is the same as declaring:

 .                           (3-1)

Time A to B = ½ (∆time_{A to B}  + ∆time_{B to A} ).                 (3-2)

But from the basic motion equation (see above) a pulse’s travel time is

 .                                                     (3-3)

Therefore, combining (3-2) and (3-3)

(Distance) ÷ (pulse speed) = ½ (∆t_{A to B} + ∆t_{B to A} ) .     (3-4)

Substituting expressions (for ∆t_{A to B} and ∆t_{B to A}) from (2-3)

.

Apply Einstein’s definition —the one that makes u_a equal to zero, and:

,

Which simplifies to
 

pulse speed = c .

The same argument, of course, applies to the travel time in the opposite direction, from B to A. And so regardless of the magnitude of the aether wind, the observed speed of light should always be c.

This remarkable mathematical illusion became enshrined in Einstein’s second postulate, making the speed of light invariant for any non-accelerating observer.

But what about the real world?

4.  How Nature Achieves the Illusion of Lightspeed Invariance

In the real world ∆t_AB does not equal ∆t_BA !  And yet amazingly the speed of a light pulse still appears to be invariant!

Brace yourself for some real magic. As for the mathematics, I promise nothing more complicated than basic algebra.

 We again start with the basic equation of motion

.

 The “measured speed” that we are interested in is the apparent speed of the light pulse:

  .                              (4-1)

 Consider the set-up shown in Fig. 3. Ignore the impracticality of trying to use a stopwatch to time the round-trip motion of a light pulse; simply focus on the equation for the apparent speed of light  and the two effects that influence it.

	Fig. 3.   Method for measuring the speed of light. We already know that the speed of the light pulse is constant c with respect to aether. But what is the measured (the apparent) speed of the light pulse?

Now if we are not careful here we might be tempted to replace ∆t with the sum of the two expressions from eqn (2-3). But we would get the wrong answer —meaning disagreement with actual experimental results.

The reality is that we must take into consideration the affect that the aether wind has on the rigid platform and on the clock.

The conversion expression we need for length is (from textbox, above)

D_app = D_con γ .                          (4-2)

Where g is the gamma factor, g = 1 / (1− u_a²/c² )^1/2, also known as the Lorentz factor. In the form 1/g it is known as the contraction factor. Notice that it contains the aether-referenced speed u_a. For the derivation, see The Physical Nature of Length Contraction.^[4]  What is quite interesting, and not well known, is that clock slowing (due to absolute motion) is caused by length contraction (see textbox, below) and for this reason the gamma factor also appears in the time-interval conversion. 
 

And the conversion expression ^[5] we need for time is

Dt_real = γ Dt_app.slow , or 

Dt_app.slow = Dt_real /γ .            (4-3)
 

(The “apparent slow” time is always less than, or equal to, the “real” absolute-rest time. Note that γ is always greater than, or equal to, one. But don’t worry about γ, it cancels out in the end.) Next, we apply the conversions (4-2) and (4-3) to expression (4-1), now properly subscripted,

                        (4-4)

.        (4-5)

The real round-trip time, when expressed with the real velocities shown in Fig. 3, is,

,

.                        (4-6)

Which then allows eqn (4-5) to be simplified to:

.             (4-7)

The apparent speed of light equals c which is a constant (and has the well-known value of about 300,000 km/s).

Thus, it is the contraction of object length and the slowing of clocks that gives us the remarkable illusion of the constancy of the speed of light under the conditions just described.

For an alternate and more technical proof of the “remarkable illusion” of invariance, the reader may wish to check into Professor Stephan Gift’s article in Physics Essays.^[6]

So what then is the secret behind the magic of this illusion? …Well, the illusion only works when a two-way light-path method is used. Note, however, that the two-way light-path method is not always obvious.  Imagine, for example, a one-way light-path set-up. The light pulse is timed from the moment it departs from distant point B to the split-second instant it is received by the observer at A. For this to work the clocks at A and B must obviously be synchronized. And there’s the catch. You see, in order to synchronize those clocks, one has to follow Einstein’s defined procedure (see textbox, above, Einstein’s Averaging Procedure). And doing so requires a two-way light-path method. There really is no other way. It’s a catch 22, if you will.

The result is a built-in, cleverly hidden, synchronization error; and the illusion of invariance still holds. It still fools the audience.

But in the real world …
 

5.  In the Real World

The astute reader, at this point, might claim that the above analysis of Fig. 3 could also be interpreted as proof that Einstein was right; that u_a equals zero, or even that there is no aether and no aether wind.

But no, that interpretation is not an option. It is not an option because the one-way speed of light has been shown, by actual experiments, to be VARIANT. That is, the propagation speed of the pulse from A to B is NOT the same as the speed from B to A.

This difference can only arise in the presence of a light-conducting aether which is in motion!

In the real world the actual one-way speed of light
 is not constant.

Notwithstanding the observable illusion described in the previous section, the real world speed of light is NOT INVARIANT. In the real world the actual one-way speed of light VARIES with the aether wind!
 

Here is a selection of the experimental evidence and supporting theory. The experiments detailed by the various authors somehow manage to overcome Einstein’s clock-synchronization problem which usually contaminates one-way lightspeed measurements. ^{[7] [8] [9] [10] [11] [12] [13] [14]}  

One such experiment, the Marinov (1974, 1977a, 1980b) coupled-mirrors experiment, has been described as “one of the most brilliant and ingenious experiments of all time. It measures the very small quantity u/c, where u is the absolute velocity of the observer, by using very clever stratagems.” The coupled-mirrors experiment demonstrated that the absolute velocity of the solar system is of the order of 300 ±20 km/sec, and that the speed of light is c − u in the direction of motion of the solar system, and c + u in the opposite direction. ^[15]

6.  Conclusion

Whether you are a student or a teacher or a researcher, pay close attention.

Here’s the bottom line: Your text books are wrong. The fact is, lightspeed invariance is merely an illusion. The illusion depends on the measuring method.

What you need to know is that the ONE-WAY speed of light has long been proven to be NON-constant. (References: [7] to [15] )

The one-way speed of light is not what Einstein’s postulate says it is.

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