Distance and the moving frames V2
edit
The two reference systems are moving with a relative constant velocity v in a direction e (given by a unit vector). So at,
x
=
(
0
,
0
,
0
)
{\displaystyle \mathbf {x} =(0,0,0)}
,
x
′
=
v
t
′
e
{\displaystyle \mathbf {x} '=vt'\mathbf {e} }
. This equation may be written,
x
ν
′
=
v
t
′
e
ν
{\displaystyle x'_{\nu }=vt'e_{\nu }}
Note: There is an assumption here that
e
{\displaystyle \mathbf {e} }
and
e
′
{\displaystyle \mathbf {e} '}
are the same. This assumes that the co-ordinate axis are parallel.
x
μ
=
e
μ
(
p
μ
,
0
t
′
+
∑
ν
p
μ
,
ν
x
ν
′
)
+
(
1
−
e
μ
)
x
μ
′
{\displaystyle x_{\mu }=e_{\mu }(p_{\mu ,0}t'+\sum _{\nu }p_{\mu ,\nu }x'_{\nu })+(1-e_{\mu })x'_{\mu }}
(equation 1)
(Equation 1) which applies for any
x
μ
′
{\displaystyle x'_{\mu }}
. The general formula may be applied to describe the origin of the co-ordinate systems. Substituting,
x
ν
{\displaystyle x_{\nu }}
for
0
{\displaystyle 0}
x
ν
′
{\displaystyle x'_{\nu }}
for
v
t
′
e
ν
{\displaystyle vt'e_{\nu }}
in (equation 1) gives,
0
=
e
μ
(
p
μ
,
0
t
′
+
∑
ν
p
μ
,
ν
v
t
′
e
ν
)
+
(
1
−
e
μ
)
v
t
′
e
μ
{\displaystyle 0=e_{\mu }(p_{\mu ,0}t'+\sum _{\nu }p_{\mu ,\nu }vt'e_{\nu })+(1-e_{\mu })vt'e_{\mu }}
which simplifies to,
0
=
p
μ
,
0
+
v
(
(
1
−
e
μ
)
+
∑
ν
p
μ
,
ν
e
ν
)
{\displaystyle 0=p_{\mu ,0}+v((1-e_{\mu })+\sum _{\nu }p_{\mu ,\nu }e_{\nu })}
and then,
−
p
μ
,
0
v
=
(
1
−
e
μ
)
−
∑
ν
m
μ
,
ν
e
ν
{\displaystyle -{\frac {p_{\mu ,0}}{v}}=(1-e_{\mu })-\sum _{\nu }m_{\mu ,\nu }e_{\nu }}
This suggests a more natural constant for
(
μ
,
0
)
{\displaystyle (\mu ,0)}
. Define,
n
μ
,
0
{\displaystyle n_{\mu ,0}}
as,
n
μ
,
0
=
−
p
μ
,
0
v
{\displaystyle n_{\mu ,0}=-{\frac {p_{\mu ,0}}{v}}}
(definition 1)
then,
n
μ
,
0
=
(
1
−
e
μ
)
+
∑
ν
p
μ
,
ν
e
ν
{\displaystyle n_{\mu ,0}=(1-e_{\mu })+\sum _{\nu }p_{\mu ,\nu }e_{\nu }}
Then define
n
μ
,
ν
{\displaystyle n_{\mu ,\nu }}
as,
p
μ
,
ν
=
n
μ
,
0
n
μ
,
ν
{\displaystyle p_{\mu ,\nu }=n_{\mu ,0}n_{\mu ,\nu }}
(definition 2)
then,
n
μ
,
0
=
(
1
−
e
μ
)
+
∑
ν
n
μ
,
0
n
μ
,
ν
e
ν
{\displaystyle n_{\mu ,0}=(1-e_{\mu })+\sum _{\nu }n_{\mu ,0}n_{\mu ,\nu }e_{\nu }}
1
−
1
−
e
μ
n
μ
,
0
=
∑
ν
n
μ
,
ν
e
ν
{\displaystyle 1-{\frac {1-e_{\mu }}{n_{\mu ,0}}}=\sum _{\nu }n_{\mu ,\nu }e_{\nu }}
(equation 3)
Substituting the (definition 1) and (definition2) into (equation 1) gives,
x
μ
=
e
μ
n
μ
,
0
(
−
v
t
′
+
∑
ν
n
μ
,
ν
x
ν
′
)
+
(
1
−
e
μ
)
x
μ
′
{\displaystyle x_{\mu }=e_{\mu }n_{\mu ,0}(-vt'+\sum _{\nu }n_{\mu ,\nu }x'_{\nu })+(1-e_{\mu })x'_{\mu }}
Assuming that displacements in one direction in one co-ordinate system, result only in displacements in the same direction in the primed system,
μ
≠
ν
⟹
n
μ
,
ν
=
0
{\displaystyle \mu \neq \nu \implies n_{\mu ,\nu }=0}
Then equation 3 gives,
1
−
1
−
e
μ
n
μ
,
0
=
n
μ
,
μ
e
μ
{\displaystyle 1-{\frac {1-e_{\mu }}{n_{\mu ,0}}}=n_{\mu ,\mu }e_{\mu }}
Solve for
n
μ
,
μ
{\displaystyle n_{\mu ,\mu }}
gives,
n
μ
,
μ
=
1
−
1
−
e
μ
n
μ
,
0
e
μ
=
1
+
(
1
−
1
n
μ
,
0
)
(
1
e
μ
−
1
)
{\displaystyle n_{\mu ,\mu }={\frac {1-{\frac {1-e_{\mu }}{n_{\mu ,0}}}}{e_{\mu }}}=1+(1-{\frac {1}{n_{\mu ,0}}})({\frac {1}{e_{\mu }}}-1)}
gives,
x
μ
=
e
μ
n
μ
,
0
(
−
v
t
′
+
1
−
1
−
e
μ
n
μ
,
0
e
μ
x
μ
′
)
+
(
1
−
e
μ
)
x
μ
′
{\displaystyle x_{\mu }=e_{\mu }n_{\mu ,0}(-vt'+{\frac {1-{\frac {1-e_{\mu }}{n_{\mu ,0}}}}{e_{\mu }}}x'_{\mu })+(1-e_{\mu })x'_{\mu }}
x
μ
=
e
μ
n
μ
,
0
(
−
v
t
′
+
(
1
+
(
1
−
1
n
μ
,
0
)
(
1
e
μ
−
1
)
)
x
μ
′
)
+
(
1
−
e
μ
)
x
μ
′
{\displaystyle x_{\mu }=e_{\mu }n_{\mu ,0}(-vt'+(1+(1-{\frac {1}{n_{\mu ,0}}})({\frac {1}{e_{\mu }}}-1))x'_{\mu })+(1-e_{\mu })x'_{\mu }}
x
μ
=
e
μ
n
μ
,
0
(
−
v
t
′
+
x
μ
′
+
(
1
−
1
n
μ
,
0
)
(
1
e
μ
−
1
)
x
μ
′
)
+
(
1
−
e
μ
)
x
μ
′
{\displaystyle x_{\mu }=e_{\mu }n_{\mu ,0}(-vt'+x'_{\mu }+(1-{\frac {1}{n_{\mu ,0}}})({\frac {1}{e_{\mu }}}-1)x'_{\mu })+(1-e_{\mu })x'_{\mu }}
x
μ
=
e
μ
n
μ
,
0
(
−
v
t
′
+
x
μ
′
)
+
e
μ
n
μ
,
0
(
1
−
1
n
μ
,
0
)
(
1
e
μ
−
1
)
x
μ
′
+
(
1
−
e
μ
)
x
μ
′
{\displaystyle x_{\mu }=e_{\mu }n_{\mu ,0}(-vt'+x'_{\mu })+e_{\mu }n_{\mu ,0}(1-{\frac {1}{n_{\mu ,0}}})({\frac {1}{e_{\mu }}}-1)x'_{\mu }+(1-e_{\mu })x'_{\mu }}
x
μ
=
e
μ
n
μ
,
0
(
−
v
t
′
+
x
μ
′
)
+
(
n
μ
,
0
−
1
)
(
1
−
e
μ
)
x
μ
′
+
(
1
−
e
μ
)
x
μ
′
{\displaystyle x_{\mu }=e_{\mu }n_{\mu ,0}(-vt'+x'_{\mu })+(n_{\mu ,0}-1)(1-e_{\mu })x'_{\mu }+(1-e_{\mu })x'_{\mu }}
x
μ
=
e
μ
n
μ
,
0
(
−
v
t
′
+
x
μ
′
)
+
(
n
μ
,
0
−
1
+
1
)
(
1
−
e
μ
)
x
μ
′
{\displaystyle x_{\mu }=e_{\mu }n_{\mu ,0}(-vt'+x'_{\mu })+(n_{\mu ,0}-1+1)(1-e_{\mu })x'_{\mu }}
x
μ
=
e
μ
n
μ
,
0
(
−
v
t
′
+
x
μ
′
)
+
n
μ
,
0
(
1
−
e
μ
)
x
μ
′
{\displaystyle x_{\mu }=e_{\mu }n_{\mu ,0}(-vt'+x'_{\mu })+n_{\mu ,0}(1-e_{\mu })x'_{\mu }}
x
μ
=
e
μ
n
μ
,
0
(
−
v
t
′
)
+
n
μ
,
0
x
μ
′
{\displaystyle x_{\mu }=e_{\mu }n_{\mu ,0}(-vt')+n_{\mu ,0}x'_{\mu }}
x
μ
=
n
μ
,
0
(
x
μ
′
−
v
t
′
e
μ
)
{\displaystyle x_{\mu }=n_{\mu ,0}(x'_{\mu }-vt'e_{\mu })}
???? same result - seems broken ????