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

Conrad Ranzan
2011

Abridged version of an article published in Applied Physics Research Vol.5, No.4, 2013
(websites: www.ccsenet.org/apr & www.ccsenet.org/journal/index.php/apr/issue/view/852 )

Contents:

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

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

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 historical 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. It is entirely understandable. Imagine having to acknowledge that you had it all wrong; and that your profession had it wrong for almost 100 years! This is embarrassing to say the least.

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. (υ_pis the air-speed of the plane.)

During the outbound crossing the speed is, in our example, (υ_PLANE + υ_AIR). And, during the return flight, assuming the wind has not changed, the speed must be (υ_PLANE − υ_AIR). Here υ_PLANE is the plane’s speed with respect to the surrounding air. (And υ_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. We have 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 + υ_a) is not the same as speed (c – υ_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:

“… we establish by definition that the “time” required by light to travel from A to B equals the “time” it requires the light to travel from B to A.”

–(From Einstein’s famous 1905 paper, page 40)

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

Which is the same as setting:

. (3-1)

Essentially, this makes υ_a equal to zero. (Just solve the equation for υ_a and you will find that υ_a = 0.) It is in the application of his definition that Einstein disconnects from the real world and makes any speed with respect to aether equal to zero! (In effect, he declares aether to be non-existent!)

Now comes the math part of the illusion. Einstein has a core definition for the fundamentally important concepts of time interval and simultaneity. See the adjacent textbox.

And Einstein uses it in the simple mathematical trick to achieve the illusion of a constant speed of light:

From his core definition of time interval and simultaneity, a pulse’s travel time from A to B is,

Einstein’s Core Definition and Averaging Procedure

Einstein recognized that, in a discussion at this fundamental level, the intuitive notion of time intervals (say, between events at different places) is inadequate. And so, he devised an operational definition of simultaneity and time-interval at different places as follows: Suppose time-intervals at different points (points A and B) of a given coordinate system are measured by clocks of similar construction; we may then synchronize these clocks by means of light signals. A emits a light ray at time t_A by A’s clock, it is received and reflected by B at time t_B by B’s clock, and returns to A at t′_A by A’s clock. Then B’s time t_B is defined to be simultaneous with A’s time ½(t′_A + t_A). ^[1]

Essentially, this says a light pulse’s travel time ∆t_AB , from A to B, is:

∆t_AB = ½ (∆time_{A to B} + ∆time_{B to A} );

And, similarly

∆t_BA = ½ (∆time_{B to A} + ∆time_{A to B} ).

This is deviously clever: ∆t_AB is made equal to ∆t_BA —even for cases when ∆time_A to B does NOT equal ∆time_{B to A} !! Check it for yourself. Try it with some test values. It's foolproof !

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 υ_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 Eq. (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 pulse being measured has an absolute speed c with respect to aether. The speed υ_a is the flow of the aether itself (this is the same as saying that the apparatus frame, in Fig. 3, is moving through the aether with absolute speed υ_a). Now, to be consistent, object-length D and clock-time ∆t must likewise be expressed in terms of their motion through aether. In other words, they must be expressed in their aether-altered form.

To be consistent with “absolute” c and “absolute” υ_a, everything on the equation’s right side, the apparent length and apparent time-interval, must be converted into intrinsic terms. This means that both length contraction and clock slowing must be taken into account.

The problem of measuring the motion-induced contraction of rigid objects: The problem is that the length of objects (such as that of the platform in Fig. 3) is not something that can be measured directly. If one attempts a carpenter’s method of using a standardized meter stick, one would find that the particular length being checked will always measure the same value —regardless of motion: the numeric value will be the same whether performed while at absolute rest with respect to the aether or while moving at some significant fraction of the speed of light. Keep in mind there is NO Einsteinean relative motion here; the carpenter-experimenter, the meter-stick, and the apparatus-platform are all moving together.

Incidentally, only object dimensions in the direction of motion are actually contracted and NOT spatial lengths.^[2]

If one wants to know the actual contracted length, one must do the calculation. The formula is^[3]

D_contracted = D_apparent / γ ,

D_app = (D_con) γ . (4-2)

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

D_app = (D_con) γ . (4-2)

Where γ is the gamma factor, γ = 1 / (1− υ_a²/c² )^1/2, also known as the Lorentz factor. In the form 1/γ it is known as the contraction factor. Notice that it contains the aether-referenced speed υ_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

∆t_real = γ ∆t_app.slow , or

∆t_app.slow = ∆t_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 Eq. (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.

The illusion depends entirely on length contraction:
The illusion depends on the aether wind via the length-contraction factor. But what about the time measured by the clock(s), you ask? . . . It turns out that the slowing of clocks is the direct result of length contraction within the components of the particular clock —no matter what components are involved (even the atomic-scale parts of atomic clocks, even bio-tissue of a beating heart).*

Thus, the illusion of the constant speed of light (in our example of non-relative motion, with everything being in the same reference frame) is caused solely by the real phenomenon of length contraction.

* There is one exception whereby it is possible to have a clock that “slows” without invoking length contraction: a so-called light-clock in which the light pulses travel perpendicular to the direction of motion. No length contraction is involved. Nevertheless, the “ticking” rate will agree perfectly with an identical light-clock oriented so that the path of the light-pulses is aligned with the direction of motion.

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 υ_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. ^[6] ^[7] ^[8] ^[9] ^[10] ^[11] ^[12] ^[13]

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 υ/c, where υ 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 − υ) in the direction of motion of the solar system, and (c + υ) in the opposite direction.^[14]

6. Conclusion

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

Here’s the bottom line: Your textbooks are wrong. The glaring factual error in most textbooks is the claim, “It is impossible to detect our motion relative to the aether.” Also in error is the claim that the famous Michelson-Morley experiment had a “null result”. The truth is, Albert Michelson and Edward Morley back in 1887 did measure an aether wind —it was merely less than had been expected. They had simply failed to take into account the physical length contraction of their apparatus, (a phenomenon of which they were unaware at the time).

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: [6] to [14] )

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

* * * *

LINKS to the full-length printable research paper (which addresses the question of How Nature Achieves the Illusion Even Without Rigid-platform Length-contraction, when there is no length contraction between measuring point A and reflection point B):

Journal published:
C. Ranzan, Einstein’s Simple Mathematical Trick –and the Illusion of a Constant Speed of Light,
Applied Physics Research Vol.5, No.4, p85, (2013 Aug); Doi: http://dx.doi.org/10.5539/apr.v5n4p85

(Download PDF)

This article is an open-access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

References and Notes

^ Mary B. Hesse, Forces and Fields (a study of action at a distance in the history of physics) (Littlefield, Adams & Co., Totown, New Jersey, 1965) p231
^ Joseph Lévy, From Galileo to Lorentz... and Beyond, Principles of a Fundamental Theory of Space and Time (Published by C. Roy Keys Inc., Montreal, Canada, 2003) “It should be pointed out that according to the Lorentz assumptions, it is matter that is contracted, not space.” –Lévy p41
^ C. Ranzan, DSSU Relativity –The Lorentz Transformations Applied to Aether-Space. Physics Essays Vol.23, No.3, p520 (2010) Posted at: http://www.CellularUniverse.org/
^ C. Ranzan, The Physical Nature of Length Contraction –the DSSU Theory of Length Contraction Induced by Absolute Motion. (2012) Posted at: http://www.CellularUniverse.org/
^ C. Ranzan, DSSU Relativity –The Lorentz Transformations Applied to Aether-Space. Physics Essays Vol.23, No.3, p520 (2010) Posted at: http://www.CellularUniverse.org/
^ F. Selleri, Noninvariant one-way velocity of light, Found. Phys 26, 641-664, 1996. Posted at: http://adsabs.harvard.edu/abs/1996FoPh...26..641S
^ S.J.G. Gift, One-Way Light Speed Determination Using the Range Measurement Equation of the GPS.
Posted at: http://vixra.org/pdf/1008.0035v1.pdf
^ S.J.G. Gift, One-Way Light Speed Measurement Using the Synchronized Clocks of the Global Positioning System (GPS), Physics Essays, Vol.23, No.2, pp271-275, June 2010
^ S.J.G. Gift, Light Speed Invariance is a Remarkable Illusion, Physics Essays Vol.23, 1, 2010
^ S.J.G. Gift, Doppler Shift Reveals Light Speed Variation, Apeiron, Vol.17, 13, 2010
^ S.J.G. Gift, The Relative Motion of the Earth and the Ether Detected, Journal of Scientific Exploration Vol.20, 201, 2006
^ A. Kelly, Challenging Modern Physics (BrownWalker Press, Florida, 2005)
^ M. Sato, The Velocity of Electromagnetic Wave is Observed Differently Depending on the Observer’s Velocity, Physics Essays, Vol.23, 405, 2010
^ J. Lévy, From Galileo to Lorentz... and Beyond, Principles of a Fundamental Theory of Space and Time (C. Roy Keys Inc., Montreal, Canada, 2003) p49

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