Poniżej przedstawiamy wzory ogólne na transformacie prędkości, czasu, zmiany położenia, przy wielkościach wektorowych prostopadłych i równoległych do prędkości nowego układu odniesienia
V
→
{\displaystyle {\vec {V}}\;}
Podamy tutaj transformacje dla układów globalnie (lokalnie) płaskich transformujące wielkości pomiędzy układami globalnie (lokalnie) płaskimi.
Transformacja prędkości ciała ze starego układu współrzędnych do nowego względem prędkości jest napisana:
v
→
′
=
C
p
1
−
(
v
→
,
V
→
)
c
2
{
V
→
(
V
→
,
v
→
)
V
2
−
V
→
+
1
−
V
2
c
2
[
v
→
−
V
→
(
v
→
,
V
→
)
V
2
]
}
=
C
p
1
−
(
v
→
,
V
→
)
c
2
[
v
→
|
|
−
V
→
+
1
−
V
2
c
2
v
→
⊥
]
=
{\displaystyle {\vec {v}}^{'}={{C_{p}} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}\left\{{\vec {V}}{{({\vec {V}},{\vec {v}})} \over {V^{2}}}-{\vec {V}}+{\sqrt {1-{{V^{2}} \over {c^{2}}}}}\left[{{\vec {v}}-{\vec {V}}{{({\vec {v}},{\vec {V}})} \over {V^{2}}}}\right]\right\}={{C_{p}} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}\left[{\vec {v}}_{||}-{\vec {V}}+{\sqrt {1-{{V^{2}} \over {c^{2}}}}}{\vec {v}}_{\perp }\right]=\;}
=
C
p
{
v
→
|
|
−
V
→
1
−
(
v
→
,
V
→
)
c
2
+
1
−
V
2
c
2
v
→
⊥
1
−
(
v
→
,
V
→
)
c
2
}
{\displaystyle =C_{p}\left\{{{{\vec {v}}_{||}-{\vec {V}}} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}+{{{\sqrt {1-{{V^{2}} \over {c^{2}}}}}{\vec {v}}_{\perp }} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}\right\}\;}
(14.1)
Wzór 14.1
Transformacje nieskończenie małej zmiany położenia ciała ze starego układu współrzędnych do nowego względem różniczki zmiany położenia i czasu w starym układzie odniesienia wyrażamy:
d
r
′
→
=
C
p
{
V
→
(
V
→
,
d
r
→
)
V
2
−
V
→
d
t
1
−
V
2
c
2
+
d
r
→
−
V
→
(
V
→
,
d
r
→
)
V
2
}
=
C
p
{
d
r
→
|
|
−
V
→
d
t
1
−
V
2
c
2
+
d
r
→
⊥
}
{\displaystyle d{\vec {r'}}=C_{p}\left\{{{{\vec {V}}{{({\vec {V}},d{\vec {r}})} \over {V^{2}}}-{\vec {V}}dt} \over {\sqrt {1-{{V^{2}} \over {c^{2}}}}}}+d{\vec {r}}-{\vec {V}}{{({\vec {V}},d{\vec {r}})} \over {V^{2}}}\right\}=C_{p}\left\{{{d{\vec {r}}_{||}-{\vec {V}}dt} \over {\sqrt {1-{{V^{2}} \over {c^{2}}}}}}+d{\vec {r}}_{\perp }\right\}\;}
(14.2)
Równanie 14.2
Gdy założymy, że dla
t
′
=
t
=
0
{\displaystyle t^{'}=t=0\;}
r
→
′
=
r
→
=
0
{\displaystyle {\vec {r}}^{'}={\vec {r}}=0\;}
14.2
r
′
→
=
C
p
{
V
→
(
V
→
,
r
→
)
V
2
−
V
→
t
1
−
V
2
c
2
+
r
→
−
V
→
(
V
→
,
r
→
)
V
2
}
=
C
p
{
r
→
|
|
−
V
→
t
1
−
V
2
c
2
+
r
→
⊥
}
{\displaystyle {\vec {r'}}=C_{p}\left\{{{{\vec {V}}{{({\vec {V}},{\vec {r}})} \over {V^{2}}}-{\vec {V}}t} \over {\sqrt {1-{{V^{2}} \over {c^{2}}}}}}+{\vec {r}}-{\vec {V}}{{({\vec {V}},{\vec {r}})} \over {V^{2}}}\right\}=C_{p}\left\{{{{\vec {r}}_{||}-{\vec {V}}t} \over {\sqrt {1-{{V^{2}} \over {c^{2}}}}}}+{\vec {r}}_{\perp }\right\}\;}
(14.3)
Transformacja nieskończenie małego upływu czasu ze starego układu współrzędnych do nowego względem zmiany infinitezymalnego czasu i infinitezymalnej zmiany położenia ciała wyrażamy w sposób:
d
t
′
=
d
t
−
(
V
→
,
d
r
→
)
c
2
1
−
V
2
c
2
{\displaystyle dt'={dt-{{({\vec {V}},d{\vec {r}})} \over {c^{2}}} \over {\sqrt {1-{{V^{2}} \over {c^{2}}}}}}\;}
(14.4)
Równanie 14.4
Podobnie jak dla 14.3
t
′
=
t
−
(
V
→
,
r
→
)
c
2
1
−
V
2
c
2
{\displaystyle t'={t-{{({\vec {V}},{\vec {r}})} \over {c^{2}}} \over {\sqrt {1-{{V^{2}} \over {c^{2}}}}}}\;}
(14.5)
Policzmy kwadrat wyrażenia
v
′
2
=
(
v
→
′
,
v
→
′
)
=
v
→
′
T
A
′
v
→
′
{\displaystyle {v'}^{2}=\left({\vec {v}}^{'},{\vec {v}}^{'}\right)={{\vec {v}}^{'}}^{T}A^{'}{{\vec {v}}^{'}}\;}
4.1 14.1
v
′
2
=
v
′
→
T
A
′
v
′
→
=
{
v
→
|
|
−
V
→
1
−
(
v
→
,
V
→
)
c
2
+
1
−
V
2
c
2
v
→
⊥
1
−
(
v
→
,
V
→
)
c
2
}
T
C
p
T
A
′
C
p
⏟
A
{
v
→
|
|
−
V
→
1
−
(
v
→
,
V
→
)
c
2
+
1
−
V
2
c
2
v
→
⊥
1
−
(
v
→
,
V
→
)
c
2
}
=
{\displaystyle {v^{'}}^{2}={\vec {v^{'}}}^{T}A^{'}{\vec {v^{'}}}=\left\{{{{\vec {v}}_{||}-{\vec {V}}} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}+{{{\sqrt {1-{{V^{2}} \over {c^{2}}}}}{\vec {v}}_{\perp }} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}\right\}^{T}\underbrace {C_{p}^{T}A^{'}C_{p}} _{A}\left\{{{{\vec {v}}_{||}-{\vec {V}}} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}+{{{\sqrt {1-{{V^{2}} \over {c^{2}}}}}{\vec {v}}_{\perp }} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}\right\}=\;}
=
v
|
|
2
+
V
2
+
(
1
−
V
2
c
2
)
v
⊥
2
−
2
(
V
→
,
v
→
)
(
1
−
(
V
→
,
v
→
)
c
2
)
2
=
v
2
+
V
2
−
2
(
V
→
,
v
→
)
−
V
2
c
2
v
⊥
2
(
1
−
(
V
→
,
v
→
)
c
2
)
2
=
v
2
+
V
2
−
2
(
V
→
,
v
→
)
−
V
2
c
2
(
v
2
−
v
|
|
2
)
(
1
−
(
V
→
,
v
→
)
c
2
)
2
=
{\displaystyle ={{v_{||}^{2}+V^{2}+\left(1-{{V^{2}} \over {c^{2}}}\right)v_{\perp }^{2}-2({\vec {V}},{\vec {v}})} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}={{v^{2}+V^{2}-2({\vec {V}},{\vec {v}})-{{V^{2}} \over {c^{2}}}v_{\perp }^{2}} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}={{v^{2}+V^{2}-2({\vec {V}},{\vec {v}})-{{V^{2}} \over {c^{2}}}\left(v^{2}-v_{||}^{2}\right)} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}=\;}
=
v
2
+
V
2
−
2
(
V
→
,
v
→
)
−
V
2
c
2
(
v
2
−
(
V
→
,
v
→
→
)
2
V
2
)
(
1
−
(
V
→
,
v
→
)
c
2
)
2
=
c
2
[
v
2
c
2
+
V
2
c
2
−
2
(
V
→
,
v
→
)
c
2
−
V
2
c
4
v
2
+
(
V
→
,
v
→
)
2
c
4
(
1
−
(
V
→
,
v
→
)
c
2
)
2
]
=
{\displaystyle ={{v^{2}+V^{2}-2({\vec {V}},{\vec {v}})-{{V^{2}} \over {c^{2}}}\left(v^{2}-{{{({\vec {V}},{\vec {\vec {v}}})}^{2}} \over {V^{2}}}\right)} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}=c^{2}\left[{{{{v^{2}} \over {c^{2}}}+{{V^{2}} \over {c^{2}}}-2{{({\vec {V}},{\vec {v}})} \over {c^{2}}}-{{V^{2}} \over {c^{4}}}v^{2}+{{\left({\vec {V}},{\vec {v}}\right)^{2}} \over {c^{4}}}} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]=\;}
=
c
2
[
v
2
c
2
+
V
2
c
2
(
1
−
v
2
c
2
)
−
2
(
V
→
,
v
→
)
c
2
+
(
V
→
,
v
→
)
2
c
4
(
1
−
(
V
→
,
v
→
)
c
2
)
2
]
=
c
2
[
v
2
c
2
+
V
2
c
2
(
1
−
v
2
c
2
)
−
1
+
1
−
2
(
V
→
,
v
→
)
c
2
+
(
V
→
,
v
→
)
2
c
4
(
1
−
(
V
→
,
v
→
)
c
2
)
2
]
=
{\displaystyle =c^{2}\left[{{{{v^{2}} \over {c^{2}}}+{{V^{2}} \over {c^{2}}}\left(1-{{v^{2}} \over {c^{2}}}\right)-2{{({\vec {V}},{\vec {v}})} \over {c^{2}}}+{{\left({\vec {V}},{\vec {v}}\right)^{2}} \over {c^{4}}}} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]=c^{2}\left[{{{{v^{2}} \over {c^{2}}}+{{V^{2}} \over {c^{2}}}\left(1-{{v^{2}} \over {c^{2}}}\right)-1+1-2{{({\vec {V}},{\vec {v}})} \over {c^{2}}}+{{\left({\vec {V}},{\vec {v}}\right)^{2}} \over {c^{4}}}} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]=\;}
=
c
2
[
v
2
c
2
+
V
2
c
2
(
1
−
v
2
c
2
)
−
1
+
(
1
−
(
V
→
,
v
→
)
2
c
4
)
2
(
1
−
(
V
→
,
v
→
)
c
2
)
2
]
=
c
2
[
1
−
1
−
v
2
c
2
−
V
2
c
2
(
1
−
v
2
c
2
)
(
1
−
(
V
→
,
v
→
)
c
2
)
2
]
=
{\displaystyle =c^{2}\left[{{{{v^{2}} \over {c^{2}}}+{{V^{2}} \over {c^{2}}}\left(1-{{v^{2}} \over {c^{2}}}\right)-1+\left(1-{{\left({\vec {V}},{\vec {v}}\right)^{2}} \over {c^{4}}}\right)^{2}} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]=c^{2}\left[1-{{1-{{v^{2}} \over {c^{2}}}-{{V^{2}} \over {c^{2}}}\left(1-{{v^{2}} \over {c^{2}}}\right)} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]=\;}
=
c
2
[
1
−
(
1
−
V
2
c
2
)
(
1
−
v
2
c
2
)
(
1
−
(
V
→
,
v
→
)
c
2
)
2
]
⇒
v
′
2
=
c
2
[
1
−
(
1
−
V
2
c
2
)
(
1
−
v
2
c
2
)
(
1
−
(
V
→
,
v
→
)
c
2
)
2
]
{\displaystyle =c^{2}\left[1-{{\left(1-{{V^{2}} \over {c^{2}}}\right)\left(1-{{v^{2}} \over {c^{2}}}\right)} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]\Rightarrow {v^{'}}^{2}=c^{2}\left[1-{{\left(1-{{V^{2}} \over {c^{2}}}\right)\left(1-{{v^{2}} \over {c^{2}}}\right)} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]}
(14.6)
Wzór 14.6
v
→
{\displaystyle {\vec {v}}\;}
v
=
c
{\displaystyle v=c\;}
v
′
=
c
{\displaystyle v^{'}=c\;}
Obliczamy różniczkę obustronną równości 14.1
d
v
→
′
=
C
p
(
a
→
|
|
+
1
−
V
2
c
2
a
⊥
)
(
1
−
(
v
→
,
V
→
)
c
2
)
−
(
v
|
|
−
V
→
+
1
−
V
2
c
2
v
→
⊥
)
(
−
(
a
→
,
V
→
)
c
2
)
(
1
−
(
v
→
,
V
→
)
c
2
)
2
d
t
=
{\displaystyle d{\vec {v}}^{'}=C_{p}{{\left({\vec {a}}_{||}+{\sqrt {1-{{V^{2}} \over {c^{2}}}}}a_{\perp }\right)\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)-\left(v_{||}-{\vec {V}}+{\sqrt {1-{{V^{2}} \over {c^{2}}}}}{\vec {v}}_{\perp }\right)\left(-{{({\vec {a}},{\vec {V}})} \over {c^{2}}}\right)} \over {\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)^{2}}}dt=\;}
=
C
p
(
a
→
|
|
+
1
−
V
2
c
2
a
⊥
)
(
1
−
(
v
→
,
V
→
)
c
2
)
+
(
v
|
|
−
V
→
+
1
−
V
2
c
2
v
→
⊥
)
(
a
→
,
V
→
)
c
2
(
1
−
(
v
→
,
V
→
)
c
2
)
2
d
t
=
{\displaystyle =C_{p}{{\left({\vec {a}}_{||}+{\sqrt {1-{{V^{2}} \over {c^{2}}}}}a_{\perp }\right)\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)+\left(v_{||}-{\vec {V}}+{\sqrt {1-{{V^{2}} \over {c^{2}}}}}{\vec {v}}_{\perp }\right){{({\vec {a}},{\vec {V}})} \over {c^{2}}}} \over {\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)^{2}}}dt=\;}
=
C
p
a
→
|
|
+
a
→
⊥
1
−
V
2
c
2
−
a
→
|
|
(
v
→
,
V
→
)
c
2
−
a
→
⊥
1
−
V
2
c
2
(
v
→
,
V
→
)
c
2
+
a
→
|
|
(
v
→
,
V
→
)
c
2
−
a
→
|
|
V
2
c
2
+
1
−
V
2
c
2
v
→
⊥
(
a
→
,
V
→
)
c
2
(
1
−
(
v
→
,
V
→
)
c
2
)
2
d
t
=
{\displaystyle =C_{p}{{{\vec {a}}_{||}+{\vec {a}}_{\perp }{\sqrt {1-{{V^{2}} \over {c^{2}}}}}-{\vec {a}}_{||}{{({\vec {v}},{\vec {V}})} \over {c^{2}}}-{\vec {a}}_{\perp }{\sqrt {1-{{V^{2}} \over {c^{2}}}}}{{({\vec {v}},{\vec {V}})} \over {c^{2}}}+{\vec {a}}_{||}{{({\vec {v}},{\vec {V}})} \over {c^{2}}}-{\vec {a}}_{||}{{V^{2}} \over {c^{2}}}+{\sqrt {1-{{V^{2}} \over {c^{2}}}}}{\vec {v}}_{\perp }{{({\vec {a}},{\vec {V}})} \over {c^{2}}}} \over {\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)^{2}}}dt=\;}
=
C
p
a
→
|
|
(
1
−
V
2
c
2
)
2
+
[
(
1
−
(
v
→
,
V
→
)
c
2
)
a
→
⊥
+
v
→
⊥
(
a
→
,
V
→
)
c
2
]
1
−
V
2
c
2
(
1
−
(
v
→
,
V
→
)
c
2
)
2
d
t
{\displaystyle =C_{p}{{{\vec {a}}_{||}{\left({\sqrt {1-{{V^{2}} \over {c^{2}}}}}\right)}^{2}+\left[\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right){\vec {a}}_{\perp }+{\vec {v}}_{\perp }{{({\vec {a}},{\vec {V}})} \over {c^{2}}}\right]{\sqrt {1-{{V^{2}} \over {c^{2}}}}}} \over {\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)^{2}}}dt}
(14.7)
Równość 14.7 14.4
a
→
′
=
C
p
{
a
→
|
|
(
1
−
V
2
c
2
)
2
+
[
(
1
−
(
v
→
,
V
→
)
c
2
)
a
→
⊥
+
v
→
⊥
(
a
→
,
V
→
)
c
2
]
1
−
V
2
c
2
}
1
−
V
2
c
2
(
1
−
(
v
→
,
V
→
)
c
2
)
3
=
{\displaystyle {\vec {a}}^{'}=C_{p}{{\left\{{\vec {a}}_{||}{\left({\sqrt {1-{{V^{2}} \over {c^{2}}}}}\right)}^{2}+\left[\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right){\vec {a}}_{\perp }+{\vec {v}}_{\perp }{{({\vec {a}},{\vec {V}})} \over {c^{2}}}\right]{\sqrt {1-{{V^{2}} \over {c^{2}}}}}\right\}{\sqrt {1-{{V^{2}} \over {c^{2}}}}}} \over {\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)^{3}}}=\;}
=
C
p
a
→
|
|
(
1
−
V
2
c
2
)
3
+
[
(
1
−
(
v
→
,
V
→
)
c
2
)
a
→
⊥
+
v
→
⊥
(
a
→
,
V
→
)
c
2
]
(
1
−
V
2
c
2
)
(
1
−
(
v
→
,
V
→
)
c
2
)
3
⇒
{\displaystyle =C_{p}{{{\vec {a}}_{||}{\left({\sqrt {1-{{V^{2}} \over {c^{2}}}}}\right)}^{3}+\left[\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right){\vec {a}}_{\perp }+{\vec {v}}_{\perp }{{({\vec {a}},{\vec {V}})} \over {c^{2}}}\right]\left(1-{{V^{2}} \over {c^{2}}}\right)} \over {\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)^{3}}}\Rightarrow \;}
⇒
a
→
=
C
p
{
a
→
|
|
(
1
−
V
2
c
2
1
−
(
v
→
,
V
→
)
c
2
)
3
+
[
(
1
−
(
v
→
,
V
→
)
c
2
)
a
→
⊥
+
v
→
⊥
(
a
→
,
V
→
)
c
2
]
1
−
V
2
c
2
(
1
−
(
v
→
,
V
→
)
c
2
)
3
}
{\displaystyle \Rightarrow {\vec {a}}=C_{p}\left\{{\vec {a}}_{||}\left({{\sqrt {1-{{V^{2}} \over {c^{2}}}}} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}\right)^{3}+\left[\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right){\vec {a}}_{\perp }+{\vec {v}}_{\perp }{{({\vec {a}},{\vec {V}})} \over {c^{2}}}\right]{{1-{{V^{2}} \over {c^{2}}}} \over {\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)^{3}}}\right\}}
(14.8)
Stąd policzmy przyśpieszenie równoległe i prostopadłe do prędkości nowego układu odniesienia:
a
→
|
|
′
=
C
p
a
→
|
|
(
1
−
V
2
c
2
1
−
(
V
→
,
v
→
)
c
2
)
3
{\displaystyle {\vec {a}}_{||}^{'}=C_{p}{\vec {a}}_{||}\left({{\sqrt {1-{{V^{2}} \over {c^{2}}}}} \over {1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}}}\right)^{3}\;}
(14.9)
a
→
⊥
′
=
C
p
[
(
1
−
(
v
→
,
V
→
)
c
2
)
a
→
⊥
+
v
→
⊥
(
V
→
,
a
→
)
c
2
]
1
−
V
2
c
2
(
1
−
(
v
→
,
V
→
)
c
2
)
3
{\displaystyle {\vec {a}}_{\perp }^{'}=C_{p}\left[\left(1-{{\left({\vec {v}},{\vec {V}}\right)} \over {c^{2}}}\right){\vec {a}}_{\perp }+{\vec {v}}_{\perp }{{\left({\vec {V}},{\vec {a}}\right)} \over {c^{2}}}\right]{{1-{{V^{2}} \over {c^{2}}}} \over {\left(1-{{\left({\vec {v}},{\vec {V}}\right)} \over {c^{2}}}\right)^{3}}}}
(14.10)
Transformacja 14.1 14.2 14.3 14.6 14.8 14.9 14.10 10.6
C
p
{\displaystyle C_{p}\;}
C
¯
p
{\displaystyle {\overline {C}}_{p}\;}
10.4 10.4 10.8 10.9
S i M
edytuj
Macierzowo transformacje n+1 wymiarowego tensora położenia w czasoprzestrzeni
d
x
μ
=
[
c
d
t
,
d
r
→
]
T
{\displaystyle dx^{\mu }=[cdt,d{\vec {r}}]^{T}\;}
d
x
′
μ
=
[
c
d
t
′
,
d
r
→
′
]
T
{\displaystyle d{x^{'}}^{\mu }=[cdt',d{\vec {r}}']^{T}\;}
14.2 14.4
[
c
d
t
′
d
r
→
′
]
=
S
[
c
d
t
d
r
→
]
=
[
γ
−
γ
V
→
T
c
A
−
γ
C
p
V
→
c
C
p
(
γ
V
→
V
→
T
V
2
A
+
I
−
V
→
V
→
T
V
2
A
)
]
⏟
S
[
c
d
t
d
r
→
]
=
[
γ
−
γ
V
→
T
c
A
−
M
p
V
→
c
C
p
(
γ
P
|
|
+
I
−
P
|
|
⏟
P
⊥
)
]
[
c
d
t
d
r
→
]
=
{\displaystyle {\begin{bmatrix}cdt'\\d{\vec {r}}'\end{bmatrix}}=S{\begin{bmatrix}cdt\\d{\vec {r}}\end{bmatrix}}=\underbrace {\begin{bmatrix}\gamma &-\gamma {{{\vec {V}}^{T}} \over {c}}A\\-\gamma C_{p}{{\vec {V}} \over {c}}&C_{p}\left(\gamma {{{\vec {V}}{\vec {V}}^{T}} \over {V^{2}}}A+I-{{{\vec {V}}{\vec {V}}^{T}} \over {V^{2}}}A\right)\end{bmatrix}} _{S}{\begin{bmatrix}cdt\\d{\vec {r}}\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma {{{\vec {V}}^{T}} \over {c}}A\\-M_{p}{{\vec {V}} \over {c}}&C_{p}\left(\gamma P_{||}+\underbrace {I-P_{||}} _{P_{\perp }}\right)\end{bmatrix}}{\begin{bmatrix}cdt\\d{\vec {r}}\end{bmatrix}}=\;}
=
[
γ
−
γ
V
→
T
c
A
−
M
p
V
→
c
C
p
(
γ
P
|
|
+
P
⊥
)
]
[
c
d
t
d
r
→
]
=
[
γ
−
γ
V
→
T
c
A
−
M
p
V
→
c
γ
C
p
|
|
+
C
p
⊥
]
[
c
d
t
d
r
→
]
=
[
γ
−
γ
V
→
T
c
A
−
M
p
V
→
c
M
p
]
⏟
M
[
c
d
t
d
r
→
]
=
M
[
c
d
t
d
r
→
]
{\displaystyle ={\begin{bmatrix}\gamma &-\gamma {{{\vec {V}}^{T}} \over {c}}A\\-M_{p}{{\vec {V}} \over {c}}&C_{p}\left(\gamma P_{||}+P_{\perp }\right)\end{bmatrix}}{\begin{bmatrix}cdt\\d{\vec {r}}\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma {{{\vec {V}}^{T}} \over {c}}A\\-M_{p}{{\vec {V}} \over {c}}&\gamma C_{p||}+C_{p\perp }\end{bmatrix}}{\begin{bmatrix}cdt\\d{\vec {r}}\end{bmatrix}}=\underbrace {\begin{bmatrix}\gamma &-\gamma {{{\vec {V}}^{T}} \over {c}}A\\-M_{p}{{\vec {V}} \over {c}}&M_{p}\end{bmatrix}} _{M}{\begin{bmatrix}cdt\\d{\vec {r}}\end{bmatrix}}=M{\begin{bmatrix}cdt\\d{\vec {r}}\end{bmatrix}}}
(14.11)
Stąd macierz transformacji
S
{\displaystyle S\;}
M
{\displaystyle M\;}
S
=
M
{\displaystyle S=M\;}
![{\displaystyle {\vec {v}}^{'}={{C_{p}} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}\left\{{\vec {V}}{{({\vec {V}},{\vec {v}})} \over {V^{2}}}-{\vec {V}}+{\sqrt {1-{{V^{2}} \over {c^{2}}}}}\left[{{\vec {v}}-{\vec {V}}{{({\vec {v}},{\vec {V}})} \over {V^{2}}}}\right]\right\}={{C_{p}} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}\left[{\vec {v}}_{||}-{\vec {V}}+{\sqrt {1-{{V^{2}} \over {c^{2}}}}}{\vec {v}}_{\perp }\right]=\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0613362bbdbd5aa59f8d108b5dd2e8582c7c2785)
![{\displaystyle ={{v^{2}+V^{2}-2({\vec {V}},{\vec {v}})-{{V^{2}} \over {c^{2}}}\left(v^{2}-{{{({\vec {V}},{\vec {\vec {v}}})}^{2}} \over {V^{2}}}\right)} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}=c^{2}\left[{{{{v^{2}} \over {c^{2}}}+{{V^{2}} \over {c^{2}}}-2{{({\vec {V}},{\vec {v}})} \over {c^{2}}}-{{V^{2}} \over {c^{4}}}v^{2}+{{\left({\vec {V}},{\vec {v}}\right)^{2}} \over {c^{4}}}} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]=\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/717b6c1877ea382698a71864aad5a0ba01303e65)
![{\displaystyle =c^{2}\left[{{{{v^{2}} \over {c^{2}}}+{{V^{2}} \over {c^{2}}}\left(1-{{v^{2}} \over {c^{2}}}\right)-2{{({\vec {V}},{\vec {v}})} \over {c^{2}}}+{{\left({\vec {V}},{\vec {v}}\right)^{2}} \over {c^{4}}}} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]=c^{2}\left[{{{{v^{2}} \over {c^{2}}}+{{V^{2}} \over {c^{2}}}\left(1-{{v^{2}} \over {c^{2}}}\right)-1+1-2{{({\vec {V}},{\vec {v}})} \over {c^{2}}}+{{\left({\vec {V}},{\vec {v}}\right)^{2}} \over {c^{4}}}} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]=\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e96617f84a8ab185be47ed6779dad968f5cd69)
![{\displaystyle =c^{2}\left[{{{{v^{2}} \over {c^{2}}}+{{V^{2}} \over {c^{2}}}\left(1-{{v^{2}} \over {c^{2}}}\right)-1+\left(1-{{\left({\vec {V}},{\vec {v}}\right)^{2}} \over {c^{4}}}\right)^{2}} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]=c^{2}\left[1-{{1-{{v^{2}} \over {c^{2}}}-{{V^{2}} \over {c^{2}}}\left(1-{{v^{2}} \over {c^{2}}}\right)} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]=\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dbf0a23cd646ad1955ff9c4835e24e654fb576f)
![{\displaystyle =c^{2}\left[1-{{\left(1-{{V^{2}} \over {c^{2}}}\right)\left(1-{{v^{2}} \over {c^{2}}}\right)} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]\Rightarrow {v^{'}}^{2}=c^{2}\left[1-{{\left(1-{{V^{2}} \over {c^{2}}}\right)\left(1-{{v^{2}} \over {c^{2}}}\right)} \over {\left(1-{{\left({\vec {V}},{\vec {v}}\right)} \over {c^{2}}}\right)^{2}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00f0d013b22bb01a7e49636ef9ac63411f815872)
![{\displaystyle =C_{p}{{{\vec {a}}_{||}{\left({\sqrt {1-{{V^{2}} \over {c^{2}}}}}\right)}^{2}+\left[\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right){\vec {a}}_{\perp }+{\vec {v}}_{\perp }{{({\vec {a}},{\vec {V}})} \over {c^{2}}}\right]{\sqrt {1-{{V^{2}} \over {c^{2}}}}}} \over {\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)^{2}}}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9dc6a5011adaa6d4f8627aff081f9a294dc9cb)
![{\displaystyle {\vec {a}}^{'}=C_{p}{{\left\{{\vec {a}}_{||}{\left({\sqrt {1-{{V^{2}} \over {c^{2}}}}}\right)}^{2}+\left[\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right){\vec {a}}_{\perp }+{\vec {v}}_{\perp }{{({\vec {a}},{\vec {V}})} \over {c^{2}}}\right]{\sqrt {1-{{V^{2}} \over {c^{2}}}}}\right\}{\sqrt {1-{{V^{2}} \over {c^{2}}}}}} \over {\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)^{3}}}=\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4818af596bbf92b60882b5dd05632eb695df4f12)
![{\displaystyle =C_{p}{{{\vec {a}}_{||}{\left({\sqrt {1-{{V^{2}} \over {c^{2}}}}}\right)}^{3}+\left[\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right){\vec {a}}_{\perp }+{\vec {v}}_{\perp }{{({\vec {a}},{\vec {V}})} \over {c^{2}}}\right]\left(1-{{V^{2}} \over {c^{2}}}\right)} \over {\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)^{3}}}\Rightarrow \;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c6f8c91ad31b768ea72ce8d7ebcd97c7f340fa2)
![{\displaystyle \Rightarrow {\vec {a}}=C_{p}\left\{{\vec {a}}_{||}\left({{\sqrt {1-{{V^{2}} \over {c^{2}}}}} \over {1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}}}\right)^{3}+\left[\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right){\vec {a}}_{\perp }+{\vec {v}}_{\perp }{{({\vec {a}},{\vec {V}})} \over {c^{2}}}\right]{{1-{{V^{2}} \over {c^{2}}}} \over {\left(1-{{({\vec {v}},{\vec {V}})} \over {c^{2}}}\right)^{3}}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b37d7367f6bebc07e00c46f50acda02aec5f449a)
![{\displaystyle {\vec {a}}_{\perp }^{'}=C_{p}\left[\left(1-{{\left({\vec {v}},{\vec {V}}\right)} \over {c^{2}}}\right){\vec {a}}_{\perp }+{\vec {v}}_{\perp }{{\left({\vec {V}},{\vec {a}}\right)} \over {c^{2}}}\right]{{1-{{V^{2}} \over {c^{2}}}} \over {\left(1-{{\left({\vec {v}},{\vec {V}}\right)} \over {c^{2}}}\right)^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e303cf097e6d10400d773bca4aef1224e37c041a)
![{\displaystyle dx^{\mu }=[cdt,d{\vec {r}}]^{T}\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32c49ea3990cbc17f88f172ab3c5312c769367fc)
![{\displaystyle d{x^{'}}^{\mu }=[cdt',d{\vec {r}}']^{T}\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d48886aaa876d8c4e88cac5627ddb1b01822dfa)
