Lorentz Transformation and Waves

The Lorentz Transformation as a Wave Function

We will demonstrate that translation-dependent transformations, of which the Lorentz Transformation is a special case, arise naturally out of wave systems. As a result, the Lorentz Transformation might be considered a natural consequence of the wave characteristics of matter. Does this concept form the link between quantum theory's wave-particle duality and relativity's Lorentz-Transformation? You decide.

Last updated September 2012

Part 1

Imagine an entity composed of standing waves in one dimension. This system can be characterized by equations in various ways:

1: Two waves traveling in opposite directions

F = cos(wt0-Bx) + cos(wt+Bx)

2: One system with nodes and antinodes.

F = 2 * cos(wt) * cos(Bx)

where

w => frequency
B => wavelength
t => time
x => distance

This second form gives us a natural ruler - the wavelength, or distance between nodes

Ruler = 1/B

and a natural clock - the oscillation time of the antinodes:

Clock = 1/ w

see the simulation below

Set 2: We can alter this form by allowing for two frequencies, w₁and w₂, and a variation in the wavelength, B_d, as follows:

F = cos(w₁t - B_dx) +

cos(w₂t + B_dx)

F = cos[1/2( w ₁+ w ₂)t] *

cos[B_d x - (w₁- w ₂)t]

The perceptive will already see the implications in the second equation, but we will forgo the discussion until after a few simulation snap shots.

Two Standing waves, the green representing the first set of equations above, the purple representing the second set of equations.		At the beginning both"standing" waves are in phase at X=0. The arrows above show how we will use our standing waves as rulers measuring from peak to peak.
We can see that our rulers are different: 1/B and 1/B_d		Above the wave we show our simulation time. Below the wave we show the time implied by the wave. The green wave has just passed 1/4 cycle, the purple wave is approaching 1/4 cycle.
We can see that our clocks are different: 1/w and 2/( w₁+ w₂)		By this point, it is clear that our green clock is proceeding faster than our purple clock, and our purple ruler, the two peaks, is moving to the right.
Since we noticed that the ruler, and clock showed up in our equations we can look back and see that the motion: v = (w₁- w ₂) showed up there also.		This simulation has demonstrated that the same things that make a standing wave translate (move) will alter the wavelength and frequency of the waves. Thus, if our rulers and clocks depend on standing waves, then motion will alter them. The idea that motion alters rulers and clocks is the basis of the Lorentz Transformation which is the basis of Relativity.

Part 1:

Concieved 1983

Posted 2001

Formatted 2010

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To sum:

Standing Wave

Accelerated "Standing" Wave

Equation

**F = 2 * cos(**wt) * cos(Bx)

F = cos[1/2( w ₁+ w ₂)t] *

cos[B_d x - (w₁- w₂)t]

Ruler

1/B

1/ B_d

Clock

1/ w

1/(w₁+ w ₂)

velocity

none

(w₁- w ₂)

Lorentz & Standing Waves

In the Lorentz Transformation accelerating an object will contract its ruler, and dilate its clock. In standing waves changing the coefficients to contract the ruler and dilate the clock means changing the same variables that will produce acceleration.

Using a little algebra, the wave coefficients will relate to the velocity in the Lorentz Transformation:

2/( w ₁+ w ₂) = 1/Sqrt(1-v²/c²) to transform the clock
1/B_d = Sqrt(1-v²/c²) to transform the ruler
w ₁ = Sqrt(1-v²/c²) - 1/2 v required change in frequencies
w ₂ = Sqrt(1-v²/c²) + 1/2 v

With out the use of higher level multi-dimensional mathematics and special considerations from physical requirements, we can not determine for sure if this result is significant.

Evidence that the result is physically significant is above: standing wave systems transform intrinsically when accelerated.

Evidence that the result is mathematically trivial: whenever a system is transformed, all waves within that system must transform accordingly. Imagine a violin on a space ship, it should sound correct to the astronauts, regardless of the acceleration of the ship.

Physically Significant or Mathematically Trivial?

Newtonian Transformation - ie. translation without transformation:

Standing waves may produce other transformation including the Newtonian Transformation of classical physics

2/( w ₁+ w ₂) = 1/ w to keep the same clock
1/B_d = 1/B to keep the same ruler
v = (w₁- w ₂) to produce to translation (velocity)
w ₁ = w + 1/2 v required change in frequencies
w ₂ = w - 1/2 v

Part 2:

In 1983 I recognized (above) that the Lorentz Translation could result from a wave equation. I was unable to determine the significance of the result. One reason may have been that I assumed that a standing wave system required both the forward and reverse parts to have the same wavelength.
However, around the same time that recognized this possibility, a Russian scientist named Yuri Ivanov determined that a standing wave can form even with unmatched wavelengths. Please proceed to a beautiful demonstration of his work at this site.

Considerations and Special Cases
1: We could do the opposite of above and interfere two waves which have different wavelengths and the same frequency.

2 Interfering Waves	Interference Pattern
1/2 cos(v t + k1x) + 1/2 cos (v t - k2x)	cos [1/2(k1+k2)x] * cos[1/2(2v t + (k1-K2)x)]
Backwards + forwards	Still * moving

This pattern is interesting, because it gives us a standing wave overlapping a moving wave. This may be a good general description when we see ripples between rocks in a flowing stream. But this form probably does not apply to the Lorentz Transformation.

2: We could approach the problem from an energy and momentum perspective.

2 Interfering Waves	Interference Pattern
1/2 cos(h/P t + E1/hx) + 1/2 cos (h/P t - E2/hx)	cos [1/2((h/P1+h/P2)x + (E2/h-E1/h)t)] * cos [1/2((h/P1-h/P2)x + (E1/h+E2/h)t)]
Backwards + forwards	moving * moving

It's hard to see whether the interference pattern can be made to conform to a moving "standing wave." But we can see some interesting considerations in the equation. The total energy of the system should be E1+E2. The total momentum of the system should be P1 - P2. From the perspective of matter as a standing wave, we would expect momentum to correlate to two de Broglie wavelengths: = 1 - 2 = h/P.

3: We could look at the general form and ask what special conditions must the general form meet to conform to the measured characteristics of relativity and quantum theory.

2 Interfering Waves	Interference Pattern
2 cos(v1 t + k1x) + 2 cos (v2 t - k2x)	cos [1/2(( v1+ v2)t +(k1-k2)x)] * cos [1/2(( v1- v2)t +(k1+k2)x)]
Backwards + forwards	moving * moving

This is the challenge. Did Ivanov take this form far enough to show that it could fit the Lorentz Transformation? If not, what wave characteristics will make it do so?