光走的时间分析:
根据\( x’+y’=(ct’)^2 \),\( x’=\gamma(x-vt)\), \(y’=y\), \(t’=\gamma(t-xv/c^2)\),我们得到:
\( (\gamma(x-vt))^2 + y^2 = c^2 \gamma(t-xv/c^2)^2 \)
在x轴上,/( y=0 /),化简上面的式子,得到:
(x-vt)^2 =c^2 (t-xv/c^2)^2,于是有:
\( x-vt =c(t-xv/c^2) =ct-xv/c \),即:
\(x(1+v/c) =ct(1+v/c)\),即:
\(x=ct\) 于是我们反推上述式子,根据 \(x=ct\),可以反推得到:
\(x-vt = c(t-xv/c^2)\),
可见,当从x变成\(x-vt\)时,发射的光走的距离变成了\(c(t-xv/c^2)\),也就是相当于:
\(t_x=(t-xv/c^2)\)
这是我们假设在O坐标系内沿着x方向发射时,相对于O’坐标系原点走的时间 假设\(t=x/v\)时,即\(x=vt\),那么:
根据\(x^2+y^2=(ct)^2\), 有\(y^2=(ct)^2-(vt)^2 \)
\( (\gamma (x-vt))^2 +y^2 =c^2 (\gamma (t-xv/c^2))^2 \),可得:
\( y^2=c^2(\gamma(t-xv/c^2))^2 \),化简可正好得:
\( y^2 = (ct)^2-(vt)^2 \), 与前面相同,所以:
\( t_y=\gamma(t-xv/c^2) \),化简可得 (\t_y=t/\gamma \)
此时因为\(x’=0\),所以有\((ct’)^2 = y^2 =c^2(\gamma(t-xv/c^2))^2\),
于是有 \(t’=\gamma(t-xv/c^2)=t/\gamma \)
也就是用\(t=x/v\)计时的话,根据\(y^2=(ct)^2-(vt)^2 \),就会得到上述变换关系。