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.

Written 2001

Formatted 2010

  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, w1 and w2, and a variation in the wavelength, Bd, as follows:

F = cos(w1t - Bdx) +

cos(w2t + Bdx)

F = cos[1/2( w 1+ w 2)t] *

cos[Bd 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.

 

clock and ruler

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/Bd

time 2 for ruler

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)

time 3 for ruler

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.

time 4 for ruler

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.

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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[Bd x - (w1- w2)t]

Ruler

1/B

1/ Bd

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/Bd = 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? - You Decide.

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/Bd = 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

 
 

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