在前面计算推迟势的过程中,我们得到了t与t’的导数关系:
\(f’(t’)=1-v(x-vt’)/cR\),也就是:
\(\frac{\Delta(t)}{\Delta(t’)} => 1-v(x-vt’)/cR \)
\( =(c^2(t-t’)-v(x-vt’))/cR \)
\( =(c^2t-vx-(c^2-v^2)t’)/cR \)
\( =(t-vx/c^2-\frac{c^2-v^2}{c^2}t’)/(t-t’) \)
\( =\frac{\sqrt{(x-vt)^2+(y^2+z^2)/\gamma^2}}{\sqrt{(x-vt’)^2+y^2+z^2}} \)
\( =\frac{\sqrt{(x-vt)^2\gamma^2+(y^2+z^2)}}{\gamma\sqrt{(x-vt’)^2+y^2+z^2}} \)
\( =\frac{R’}{\gamma R} \)
其中: \(t’=\gamma^2 (t-vx/c^2-\frac{R’}{c}) \)
假设x对应vt,则:
\( => c^2(t-t’)-v(vt-vt’)/cR \)
\( =(c^2t-v^2-(c^2t’-v^2t’))/cR \)
\( =(c^2-v^2)(t-t’)/c^2(t-t’) \)
\( =(c^2-v^2)/c^2 \)