A Simple Refutation of Special Relativity [ edit ] 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.
S
(
x
,
t
)
{\displaystyle S(x,t)}
x
{\displaystyle x}
x
+
=
c
t
{\displaystyle x_{+}=ct}
and the other in the negative
x
{\displaystyle x}
x
−
=
−
c
t
{\displaystyle x_{-}=-ct}
where
x
+
{\displaystyle x_{+}}
x
−
{\displaystyle x_{-}}
t
{\displaystyle t}
S
(
x
,
t
)
{\displaystyle S(x,t)}
c
{\displaystyle c}
x
′
{\displaystyle x'}
t
′
{\displaystyle t'}
S
′
(
x
′
,
t
′
)
{\displaystyle S'(x',t')}
v
{\displaystyle v}
S
(
x
,
t
)
{\displaystyle S(x,t)}
x
{\displaystyle x}
x
′
=
γ
(
x
−
v
t
)
{\displaystyle x'=\gamma \left(x-vt\right)}
t
′
=
γ
(
t
−
x
v
c
2
)
{\displaystyle t'=\gamma \left(t-{\frac {xv}{c^{2}}}\right)}
γ
=
1
1
−
v
2
c
2
{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
where
x
′
{\displaystyle x'}
t
′
{\displaystyle t'}
S
′
(
x
′
,
t
′
)
{\displaystyle S'(x',t')}
x
{\displaystyle x}
t
{\displaystyle t}
S
(
x
,
t
)
{\displaystyle S(x,t)}
γ
{\displaystyle \gamma }
x
=
x
+
=
c
t
{\displaystyle x=x_{+}=ct}
x
+
′
=
γ
(
x
+
−
v
t
)
=
γ
(
c
t
−
v
t
)
=
γ
(
1
−
v
c
)
c
t
=
γ
(
1
−
v
c
)
x
+
{\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_{+}}
t
′
=
γ
(
t
−
v
x
+
c
2
)
=
γ
(
t
−
v
c
t
c
2
)
=
γ
(
t
−
v
t
c
)
=
γ
(
1
−
v
c
)
t
{\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
γ
(
1
−
v
c
)
=
1
−
v
c
1
−
v
2
c
2
=
1
−
v
c
(
1
+
v
c
)
(
1
−
v
c
)
=
1
−
v
c
1
+
v
c
<
1
{\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
t
{\displaystyle t}
S
(
x
,
t
)
{\displaystyle S(x,t)}
Δ
x
/
Δ
t
=
Δ
x
′
/
Δ
t
′
=
c
{\displaystyle \Delta x/\Delta t=\Delta x'/\Delta t'=c}
x
=
x
−
=
−
c
t
{\displaystyle x=x_{-}=-ct}
x
−
′
=
γ
(
x
−
−
v
t
)
=
γ
(
−
c
t
−
v
t
)
=
−
γ
(
1
+
v
c
)
c
t
=
γ
(
1
+
v
c
)
x
−
{\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_{-}}
t
′
=
γ
(
t
−
v
x
−
c
2
)
=
γ
(
t
+
v
c
t
c
2
)
=
γ
(
t
+
v
t
c
)
=
γ
(
1
+
v
c
)
t
{\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
γ
(
1
+
v
c
)
=
1
+
v
c
1
−
v
2
c
2
=
1
+
v
c
(
1
+
v
c
)
(
1
−
v
c
)
=
1
+
v
c
1
−
v
c
>
1
{\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
t
{\displaystyle t}
S
(
x
,
t
)
{\displaystyle S(x,t)}
Δ
x
/
Δ
t
=
Δ
x
′
/
Δ
t
′
=
−
c
{\displaystyle \Delta x/\Delta t=\Delta x'/\Delta t'=-c}
Time cannot run at different rates at the same time
(
t
′
=
0
)
{\displaystyle (t'=0)}
(
x
′
=
0
)
{\displaystyle (x'=0)}
(
S
′
(
x
′
,
t
′
)
)
{\displaystyle (S'(x',t'))}
Relativistic Doppler Effect [ edit ] Electromagnetic Wave [ edit ]
y
(
x
,
t
)
=
A
∗
c
o
s
(
2
π
(
t
/
T
−
x
/
λ
)
)
{\displaystyle y(x,t)=A*cos(2\pi (t/T-x/\lambda ))}
Period:
T
{\displaystyle T}
Frequency:
f
=
1
/
T
{\displaystyle f=1/T}
Propagation speed:
c
{\displaystyle c}
Wavelength:
λ
=
c
T
{\displaystyle \lambda =cT}
2
π
(
t
/
T
−
x
/
λ
)
=
k
∗
2
π
{\displaystyle 2\pi (t/T-x/\lambda )=k*2\pi }
k
{\displaystyle k}
t
/
T
−
x
/
λ
=
k
{\displaystyle t/T-x/\lambda =k}
k
t
h
{\displaystyle k^{th}}
x
k
=
c
t
−
k
λ
=
c
(
t
−
k
T
)
{\displaystyle 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 [ edit ]
x
o
=
x
−
v
t
{\displaystyle x^{o}=x-vt}
t
o
=
t
{\displaystyle t^{o}=t}
Results for
x
k
=
c
t
−
k
λ
{\displaystyle x_{k}=ct-k\lambda }
x
k
o
=
(
c
−
v
)
t
o
−
k
λ
{\displaystyle 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
=
T
(
1
−
v
/
c
)
{\displaystyle T^{o}={\frac {T}{(1-v/c)}}}
f
o
=
1
T
∘
=
(
1
−
v
/
c
)
f
{\displaystyle f^{o}={\frac {1}{T^{\circ }}}=(1-v/c)f}
λ
o
=
λ
{\displaystyle \lambda ^{o}=\lambda }
c
o
=
c
−
v
{\displaystyle c^{o}=c-v}
Lorentz transformation [ edit ]
x
′
=
γ
(
x
−
v
t
)
{\displaystyle x'=\gamma (x-vt)}
t
′
=
γ
(
t
−
v
x
/
c
2
)
{\displaystyle t'=\gamma (t-vx/c^{2})}
γ
=
1
/
1
−
v
2
/
c
2
{\displaystyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}
Results for
x
k
=
c
t
−
k
λ
{\displaystyle x_{k}=ct-k\lambda }
x
k
′
=
γ
(
x
k
−
v
t
)
=
γ
(
c
t
−
k
λ
−
v
t
)
=
γ
(
(
c
−
v
)
t
−
k
λ
)
{\displaystyle x_{k}'=\gamma (x_{k}-vt)=\gamma (ct-k\lambda -vt)=\gamma ((c-v)t-k\lambda )}
t
′
=
γ
(
t
−
v
x
k
/
c
2
)
=
γ
(
t
−
v
(
c
t
−
k
λ
)
/
c
2
)
=
γ
(
1
−
v
/
c
)
t
+
γ
k
T
v
/
c
=
γ
(
1
−
v
/
c
)
t
+
γ
k
T
v
/
c
{\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}
t
{\displaystyle t}
x
k
′
{\displaystyle x_{k}'}
t
′
{\displaystyle t'}
γ
(
1
+
v
/
c
)
=
1
/
γ
(
1
−
v
/
c
)
{\displaystyle \gamma (1+v/c)=1/\gamma (1-v/c)}
x
k
′
=
c
t
′
−
k
λ
γ
(
1
−
v
/
c
)
=
c
(
t
′
−
k
T
γ
(
1
−
v
/
c
)
)
{\displaystyle 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
′
=
T
γ
(
1
−
v
/
c
)
{\displaystyle T'={\frac {T}{\gamma (1-v/c)}}}
f
′
=
1
/
T
′
=
γ
(
1
−
v
/
c
)
f
{\displaystyle f'=1/T'=\gamma (1-v/c)f}
λ
′
=
λ
γ
(
1
−
v
/
c
)
{\displaystyle \lambda '={\frac {\lambda }{\gamma (1-v/c)}}}
c
′
=
c
{\displaystyle c'=c}
Gerd Termathe
