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Planar hyperspace waves

This article briefly describes concept of hyperspace electro-magnetics waves by the "simple way" .  No knowledge of quaternions is required. Planar electromagnetic waves are used for derivation.

STL case (slower than light moving frames of reference)

First let's consider 2 inertial frames each moving by velocity |v|<c (STL) with respect to each other. We  can simply apply Lorentz transformation, and  transform different quantities between the 2 inertial frames. This is regular special theory of relativity. For x and t coordinates the transformation writes (assume  that frames are moving only in x-x' direction):


Note: frame F(t,x,y,z) can be considered stationary, whereas frame F'(t',x',y',z') is considered as moving along x-x' direction at speed v "to right" (in direction of increasing x). 

Now let's consider wave of unit  amplitude (i. e. 1)  propagating in x-x' direction written in complex notation:


After substitutions

and simplifications we will get finally:



The above formulas are formulas for relativistic Doppler effect  (see also redshift and blueshift). Up to now it was the physics, which is known from the school benches.


FTL case (faster than light moving frames of reference)
Now let's have 2 inertial frames each moving by velocity |v|>c with respect to each other. There is nothing wrong with this according to special theory of relativity, because no real object with real mass is moving at this velocity, but only a though object called "inertial frame" or "frame of reference".  Therefore we can simply apply Lorentz transformation, and thus transform different quantities between the 2 frames. Let's start with the coordinate system itself (assume now that frames are moving only in x-x' direction).


For FTL case (|v|>c) coordinate system is transformed between frames using the following formula (please note uncertainty in signature) .



Now let's consider wave of unit amplitude propagating in x-x' direction written in complex notation:


After substitutions


and simplifications we get:




The solution can be further split to 2 cases:

1st case


2nd case


By itemization, we will get 4 different wave (for each +/- sign) with purely imaginary frequency and wave numbers.


Please note that the total number of complex waves will be 8 and number of real waves will 4. To see this let's look again on general notation of wave using complex exponential function:

 
Plus or minus sign determines the direction of rotation of the arrow or point, if we draw so the wave in the complex plane. Thus any real measurable wave can be expressed as sum of two complex exponentials (we ignore phase here) (2.15):
                                                                          

If we sum all the waves together, we will realize, that each wave is contained twice, therefore we will finally get  2 real waves amplified by factor 2, if we base our notation on hyperbolic cosine function or alternatively we get 4 real waves if we base our notation on exponential function (2.16) :



Please note, that these waves can be no more represented as rotation of arrow in complex plane left or right, but instead they are regular exponential functions, and only the length of arrow changes as exponential function of time t (for constant x). Let' call these exponential pulses "hyperspace waves".

The conclusion of the above derivation is therefore, that if there exists a harmonic electromagnetic wave propagating in inertial frame F'( t',x',y',z'), which is moving at speed v (|v|>c) with the respect to another inertial frame F(t,x,y,z) along the x-x' direction, then electromagnetic wave can be received in the frame F as superposition of 4 exponential infinite pulses. The parameters of the pulses are given by the above equation, and depend on parameters (omega',k') of the wave in it's original frame F' as well as on the speed v and it's direction. Also note that the above case is a very special case because we expected, that the wave is propagating in the same direction as the frame moves (i.e. direction x-x'). 

Please note that these 4 waves are only subset of all possible hyperspace waves, as we considered only special case of direction of wave propagation. The general case of propagation should be also investigated. Also note, that the above 4 exponential pulses differ in direction of propagation, monotony and dumping coefficient.   




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