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The Theory of Special Relativity with its mathematical reprensentation, the Lorentz Transformation, is easily refuted by examining its results for the propagation of light signals.

Consider two light fronts travelling in opposite directions in an inertial frame ${\displaystyle S(x,t)}$. One light front is moving in the positive ${\displaystyle x}$-direction

${\displaystyle x_{+}=ct}$

and the other in the negative ${\displaystyle x}$-direction

${\displaystyle x_{-}=-ct}$

where ${\displaystyle x_{+}}$ and ${\displaystyle x_{-}}$ denote the coordinates of the light fronts at time ${\displaystyle t}$ in the system ${\displaystyle S(x,t)}$, and ${\displaystyle c}$ denotes the speed of light.

Following special relativity, the spatial coordinate ${\displaystyle x'}$ and the time ${\displaystyle t'}$ in an inertial frame ${\displaystyle S'(x',t')}$, which is moving with velocity ${\displaystyle v}$ relative to ${\displaystyle S(x,t)}$ in the positive ${\displaystyle x}$-direction, are given by the Lorentz transformation

${\displaystyle x'=\gamma \left(x-vt\right)}$

${\displaystyle t'=\gamma \left(t-{\frac {xv}{c^{2}}}\right)}$

${\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$

where ${\displaystyle x'}$ is the space coordinate and ${\displaystyle t'}$ is the time in the system ${\displaystyle S'(x',t')}$, which correspond to the space coordinate ${\displaystyle x}$ and the time ${\displaystyle t}$ in the system ${\displaystyle S(x,t)}$, respectively. ${\displaystyle \gamma }$ is also known as Lorentz factor.

Substituting ${\displaystyle x=x_{+}=ct}$ for the first light front we get

${\displaystyle x_{+}'=\gamma \left(x_{+}-vt\right)=\gamma \left(ct-vt\right)=\gamma \left(1-{\frac {v}{c}}\right)ct=\gamma \left(1-{\frac {v}{c}}\right)x_{+}}$

${\displaystyle t'=\gamma \left(t-{\frac {vx_{+}}{c^{2}}}\right)=\gamma \left(t-{\frac {vct}{c^{2}}}\right)=\gamma \left(t-{\frac {vt}{c}}\right)=\gamma \left(1-{\frac {v}{c}}\right)t}$

with the common factor

${\displaystyle \gamma \left(1-{\frac {v}{c}}\right)={\frac {1-{\frac {v}{c}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1-{\frac {v}{c}}}{\sqrt {(1+{\frac {v}{c}})(1-{\frac {v}{c}})}}}={\frac {\sqrt {1-{\frac {v}{c}}}}{\sqrt {1+{\frac {v}{c}}}}}<1}$

From these equations special relativity tells us that, for a given time ${\displaystyle t}$, the distance the light front travels is shorter than in the system ${\displaystyle S(x,t)}$, and time runs slower by the same factor, resulting in a constant speed of light ${\displaystyle \Delta x/\Delta t=\Delta x'/\Delta t'=c}$ in both systems.

Now substituting ${\displaystyle x=x_{-}=-ct}$ for the second light front we get

${\displaystyle x_{-}'=\gamma \left(x_{-}-vt\right)=\gamma \left(-ct-vt\right)=-\gamma \left(1+{\frac {v}{c}}\right)ct=\gamma \left(1+{\frac {v}{c}}\right)x_{-}}$

${\displaystyle t'=\gamma \left(t-{\frac {vx_{-}}{c^{2}}}\right)=\gamma \left(t+{\frac {vct}{c^{2}}}\right)=\gamma \left(t+{\frac {vt}{c}}\right)=\gamma \left(1+{\frac {v}{c}}\right)t}$

with the common factor

${\displaystyle \gamma \left(1+{\frac {v}{c}}\right)={\frac {1+{\frac {v}{c}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1+{\frac {v}{c}}}{\sqrt {(1+{\frac {v}{c}})(1-{\frac {v}{c}})}}}={\frac {\sqrt {1+{\frac {v}{c}}}}{\sqrt {1-{\frac {v}{c}}}}}>1}$

From these equations special relativity tells us now that, for a given time ${\displaystyle t}$, the distance the light front travels is longer than in the system ${\displaystyle S(x,t)}$, and time runs faster by the same factor, resulting in a constant speed of light ${\displaystyle \Delta x/\Delta t=\Delta x'/\Delta t'=-c}$ in both systems.

This leads to a basic contradiction:

Time cannot run at different rates at the same time ${\displaystyle (t'=0)}$ and at the same place ${\displaystyle (x'=0)}$ in the same system ${\displaystyle (S'(x',t'))}$.

The Theory of Special Relativity, applied to the simple case of two light fronts moving in opposite directions, leads a contradiction and is thus refuted.

${\displaystyle y(x,t)=A*cos(2\pi (t/T-x/\lambda ))}$

Period: ${\displaystyle T}$

Frequency: ${\displaystyle f=1/T}$

Propagation speed: ${\displaystyle c}$

Wavelength: ${\displaystyle \lambda =cT}$

Wave function maxima (wave crests):

${\displaystyle 2\pi (t/T-x/\lambda )=k*2\pi }$, ${\displaystyle k}$ integer
${\displaystyle t/T-x/\lambda =k}$

${\displaystyle k^{th}}$ wave crest:
${\displaystyle x_{k}=ct-k\lambda =c(t-kT)}$

${\displaystyle x^{o}=x-vt}$
${\displaystyle t^{o}=t}$

Results for ${\displaystyle x_{k}=ct-k\lambda }$:
${\displaystyle x_{k}^{o}=(c-v)t^{o}-k\lambda }$

Result summary:
${\displaystyle T^{o}={\frac {T}{(1-v/c)}}}$
${\displaystyle f^{o}={\frac {1}{T^{\circ }}}=(1-v/c)f}$
${\displaystyle \lambda ^{o}=\lambda }$
${\displaystyle c^{o}=c-v}$

${\displaystyle x'=\gamma (x-vt)}$
${\displaystyle t'=\gamma (t-vx/c^{2})}$
${\displaystyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}$

Results for ${\displaystyle x_{k}=ct-k\lambda }$:
${\displaystyle x_{k}'=\gamma (x_{k}-vt)=\gamma (ct-k\lambda -vt)=\gamma ((c-v)t-k\lambda )}$
${\displaystyle t'=\gamma (t-vx_{k}/c^{2})=\gamma (t-v(ct-k\lambda )/c^{2})=\gamma (1-v/c)t+\gamma kTv/c=\gamma (1-v/c)t+\gamma kTv/c}$

Substituting ${\displaystyle t}$ to get ${\displaystyle x_{k}'}$ as function of ${\displaystyle t'}$(using ${\displaystyle \gamma (1+v/c)=1/\gamma (1-v/c)}$):
${\displaystyle x_{k}'=ct'-{\frac {k\lambda }{\gamma (1-v/c)}}=c(t'-{\frac {kT}{\gamma (1-v/c)}})}$

Result summary:
${\displaystyle T'={\frac {T}{\gamma (1-v/c)}}}$
${\displaystyle f'=1/T'=\gamma (1-v/c)f}$
${\displaystyle \lambda '={\frac {\lambda }{\gamma (1-v/c)}}}$
${\displaystyle c'=c}$

Gerd Termathe