GHT153

A Simple Refutation of Special Relativity

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 $S(x,t)$. One light front is moving in the positive $x$-direction

$x_{+}=ct$

and the other in the negative $x$-direction

$x_{-}=-ct$

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

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

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

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

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

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

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

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

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

with the common factor

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

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

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

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

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

with the common factor

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

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

This leads to a basic contradiction:

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

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.

Relativistic Doppler Effect

Electromagnetic Wave

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

Period: $T$

Frequency: $f=1/T$

Propagation speed: $c$

Wavelength: $\lambda =cT$

Wave function maxima (wave crests):

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

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

Diagram 1. Some successive wavecrests propagating at speed c in the reference frame of the source (v=0.25c)

Galilean transformation

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

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

Diagram 2. Result of Galilean transformation, based on the scenario given in Diagram 1

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

Lorentz transformation

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

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

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

Diagram 3. Results of the Lorentz transformation, based on the scenario given in Diagram 1

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

Gerd Termathe