Length of The Cycle
All Observers agree they see is a Cycle; what they disagree on is the Cycle-Length
Maxwell’s Electromagnetic Wave Equation is traditionally read as a three-dimensional wave equation containing a fixed speed of light. An alternative way to read the mathematics is that
Maxwell’s 3D Wave Equation contains a Fixed 3D Speed.
NeoClassical Relativity (NCR) is based on this interpretation of Maxwell’s mathematics.
In the previous post, I introduced the first of two postulates of NeoClassical Relativity.
The first postulate is that:
The Speed of Light is a three dimensional combination of the Speed of Oscillation and the Speed of Linear Travel.
Mathematically, this postulate can be written as the “3D Speed Equation”
Equation (1):_______________ c² = u² + v²
{Where u is the speed of oscillation in two spatial dimensions, and, v is the speed of travel in the third.}
In this post, I will introduce the second postulate of NCR.
A Fixed and Variable Perspective
In the previous post, I addressed Einstein’s thought experiment on “chasing a light beam”. In this post, I want to address another of Einstein’s thought experiments, and that’s the one about the train and the “light-clock”.
{Note 1: To quote Wikipedia, “The light-clock is a simple way of showing a basic feature of Special Relativity. A clock is designed to work by bouncing a flash of light off a distant mirror and using its return to trigger another flash of light, meanwhile counting how many flashes have occurred along the way.”}
In this thought experiment, Einstein has an individual seated on a train next to a light-clock. The light in the light-clock bounces up and down in a vertical direction. The train moves horizontally…
Einstein says that this individual sees the bouncing light move at the speed of light (c). He further states that an external observer, seeing the same bouncing light, would also say that the light is moving at (c) — and that’s despite the fact that the light-clock itself moves with the train.
{Note 2: The fact that the light-clock is moving means that the external observer is seeing the bouncing light travel a greater distance in the same amount of time.}
So obviously there is a conflict here which Einstein goes on to neatly resolve by mathematically showing that “Time” is different for the different observers. This idea is known as “Time Dilation” and has been experimentally tested and found to be true. So all’s well and good on that front…
But, curiously, time dilation is not the only thing that is interesting about this thought experiment. Something everyone seems to have overlooked — including Einstein himself — is that this thought experiment is actually describing two different perspectives on a single oscillation.
One perspective sees only fixed up and down motion, while the other sees both fixed up and down motion and variable sideways motion…
The Speed of a “Wave”
Oscillations are, as we all know, ubiquitous in the real world and consequently so too in physics.
Similarly, moving oscillations are also ubiquitous; but moving oscillations are usually referred to as “waves”.
Obviously, the speed of travel is fundamental to any such wave motion; and this speed is given by
Equation (5) _________________ v = f λ
{Where lambda (λ) is the wavelength, and, the frequency (f) is the number of waves per second.}
Virtual Oscillation Length
Now, traditionally a “wavelength” and the associated “wave speed” relate only to linear motion. But, actually, equation (5) can equally be used to describe the motion of an oscillation.
Like waves, oscillations also have a fixed frequency; but unlike waves, oscillations have a variable velocity that oscillates between zero and some maximum. This oscillating velocity is characteristic of its “Simple Harmonic Motion”.
Four hundred years ago Galileo realized that simple harmonic motion is actually a one-dimensional version of uniform circular motion; consequently, thanks to Galileo’s observation, we can use equation (5) to describe the “uniform” speed of any oscillation.
To do so we simply treat the amplitude of an oscillation as if it is the radius of a circle. By doing this, a single back and forth oscillation can be converted into uniform travel around a circle (whose circumference equals 2π times the amplitude of the oscillation).
Thus, 2π times the amplitude of an oscillation is the “virtual” distance that we can use as the lambda of the oscillation. We will call this lambda the “Oscillation-Length”.
Lambdas of the 3D cycle
We will assign the oscillation-length the symbol “lambda-u”, and so now we can use equation (5) to define the uniform oscillation-speed in terms of the virtual oscillation-length
Equation (6) _______________ u= f (λ-u)
Likewise, we can also use equation (5) to define the other two speeds associated with any three-dimensional cycle.
Equation (7) _______________ v = f (λ-v)
Equation (8) _______________ c= f (λ-c)
To differentiate between these two new lambdas, I will call lambda-v the “Wave-Length”, because it represents the traditional idea of a linear distance from crest to crest; and I will call lambda-c the “Cycle-Length” because it represents the total 3D length of the cycle.
So now, having established these relationships (between the speeds and the lambdas of the cycle), we can now proceed to the second postulate of NeoClassical Relativity…
Second Postulate of NCR
Using these equations (6, 7 & 8) we can now rewrite equation (1) as
Equation (9) ___________ (f λ-c)² = (f λ-u)² + (f λ-v)²
And by eliminating frequency squared from both sides, we arrive at
Equation (10) ___________ (λ-c)² = (λ-u)² + (λ-v)²
This is the “3D Length Equation” and it contains the three individual lengths that are characteristic of any three-dimensional cycle.
And so this brings me to the second of the two postulates that I am putting forward, in my revision of Special Relativity.
The second postulate of NCR states that:
The Oscillation-Length of a 3D Cycle is a Fixed Quantity.
There are two significant consequences of this statement
- All three-dimensional cycles must be a nonlinear combination of a Fixed Oscillation-Length and a Variable Wave-Length.
- The fixed oscillation-length must also be the “Minimum Cycle-Length” (which going forward I will refer to as “Lambda-Min”…).
Mathematically, the second postulate of NCR can be summarized as
Equation (11) ___________ (λ-u) = (λ-min)
And this means we that equation (6) can be rewritten as
Equation (12) ____________ u = (f)(λ-min)
It also means we can now write an alternative version of the 3D Length Equation
Equation (13)___________ (λ-c)² = (λ-min)² + (λ-v)²
{Where, λ-c is the cycle-length, λ-v is the wave-length, and λ-min equals 2π times the Planck Length — which equates to a length of 1.0155 x 10⁻³⁴ meters.}
[Note 3: I will show at a later date what this distance physically represents and why it should have this value…]
Everything in Terms of Lambda-Min
Now, given that lambda-min is the minimum distance involved in any cycle, it simplifies things a lot to express both (λ-v) and (λ-c) in terms of a quantity of (λ-min).
To do so we need to, once again, start with the 3D Speed Equation
Equation (1) _______________ c² = v² + u²
Next, we divide across both sides by (u) squared
_____________________ (c/u)² = (u/u)² + (v/u)²
Then, we multiply across both sides by (λ-min) squared
____________ (c/u (λ-min))² = (u/u (λ-min))² + (v/u (λ-min))²
Finally, we compare with equation (13) and we see that
Equation (14) ___________ (λ-v) = v/u (λ-min)
Equation (15) ___________ (λ-c) = c/u (λ-min)
Now that we have all lengths of the cycle expressed in terms of (λ-min), it becomes a lot easier to see why the speed of light is always associated with the linear speed of travel.
As the oscillation-length becomes a smaller and smaller fraction of the overall cycle-length, the wave-length and cycle-length effectively converge — as the table below shows…
Furthermore, it is worth noting that equation (15) describes lambda-c as being equal to “gamma” times lambda-min. In other words, the cycle-length (i.e. the quantity traditionally viewed as the wavelength) is proportional to the ubiquitous quantity of Special Relativity (γ), and the constant of proportionality is a universal constant of length — Lambda-min.
Change of Perspective
So now finally, getting back to Einstein’s thought experiment. We can see that what Einstein was actually describing (with his hypothetical light-clock) is the motion of light in two different directions — an up and down motion over a fixed distance that is equivalent to (λ-min), and a sideways motion through a variable distance that is equivalent to (v/u)(λ-min).
Moreover, he calculated that the combination of these two independent motions must be a variable distance that we can clearly see is equivalent to (gamma)(λ-min). So, although this was not his intention,
Einstein’s light-clock thought experiment perfectly matches the set-up associated with a single 3D cycle of light.
And the interesting thing about the same 3D cycle of light seen from two different frames of reference is that:
While both observers might agree that they are both observing a single cycle, they will disagree on the cycle-length (and by extension, the frequency)…
This, admittedly, is quite a subtle change in perspective on one of the most famous thought experiments in the history of science. But it does, nonetheless, reveal a lot about how Einstein was able to be wrong about the linear dynamics of light, and yet still manage to be right about the concept of “Time Dilation” - which I will address in my next post…
© Kieran D. Kelly
This is Post #2 in the series on NeoClassical Relativity Theory
