استخراج

برای مولفه x

{\displaystyle {\begin{aligned}N_{x}'&={\frac {E'}{c^{2}}}x'-t'p_{x}'\\&={\frac {\ گاما }{c^{2}}}(E-vp_{x})\گاما (x-vt)-\gamma \left(t-{\frac {xv}{c^{2}}}\راست) \gamma \left(p_{x}-{\frac {vE}{c^{2}}}\right)\\&=\gamma ^{2}\left[{\frac {1}{c^{ 2}}}\left(E-vp_{x}\right)(x-vt)-\left(t-{\frac {xv}{c^{2}}}\right)\left(p_{x }-{\frac {vE}{c^{2}}}\right)\right]\\&=\gamma ^{2}\left[{\frac {Ex}{c^{2}}}- {\frac {Evt}{c^{2}}}-{\frac {vp_{x}x}{c^{2}}}+{\frac {vp_{x}vt}{c^{2} }}-tp_{x}+{\frac {xv}{c^{2}}}p_{x}+t{\frac {vE}{c^{2}}}-{\frac {xv}{ c^{2}}}{\frac {vE}{c^{2}}}\right]\\&=\gamma ^{2}\left[{\frac {Ex}{c^{2}}}{\cancel {-{\frac {Evt}{c^{2}}}}}{\cancel {- {\frac {vp_{x}x}{c^{2}}}}}+{\frac {v^{2}}{c^{2}}}p_{x}t-tp_{x}{ \cancel {+{\frac {xv}{c^{2}}}p_{x}}}{\cancel {+t{\frac {vE}{c^{2}}}}}-{\frac {v^{2}}{c^{2}}}{\frac {Ex}{c^{2}}}\right]\\&=\gamma ^{2}\left[\left({\ frac {Ex}{c^{2}}}-tp_{x}\right)+{\frac {v^{2}}{c^{2}}}\left(p_{x}t-{\ frac {Ex}{c^{2}}}\right)\right]\\&=\gamma ^{2}\left[1-{\frac {v^{2}}{c^{2}} }\right]N_{x}\\&=\گاما ^{2}{\frac {1}{\gamma ^{2}}}N_{x}\end{تراز شده}}}

جزء y

{\displaystyle {\begin{aligned}N_{y}'&={\frac {E'}{c^{2}}}y'-t'p_{y}'\\&={\frac {1 }{c^{2}}}\gamma (E-vp_{x})y-\gamma \left(t-{\frac {xv}{c^{2}}}\right)p_{y}\ \&=\gamma \left[{\frac {1}{c^{2}}}(E-vp_{x})y-\left(t-{\frac {xv}{c^{2}} }\right)p_{y}\right]\\&=\gamma \left[{\frac {1}{c^{2}}}Ey-{\frac {1}{c^{2}}} vp_{x}y-tp_{y}+{\frac {xv}{c^{2}}}p_{y}\right]\\&=\gamma \left[\left({\frac {1} {c^{2}}}Ey-tp_{y}\right)+{\frac {v}{c^{2}}}(xp_{y}-yp_{x})\right]\\&= \gamma \left(N_{y}+{\frac {v}{c^{2}}}L_{z}\right)\end{تراز شده}}}

و مولفه z

{\displaystyle {\begin{aligned}N_{z}'&={\frac {E'}{c^{2}}}z'-t'p_{z}'\\&={\frac {1 }{c^{2}}}\gamma (E-vp_{x})z-\gamma \left(t-{\frac {xv}{c^{2}}}\right)p_{z}\ \&=\gamma \left[{\frac {1}{c^{2}}}(E-vp_{x})z-\left(t-{\frac {xv}{c^{2}} }\right)p_{z}\right]\\&=\gamma \left[{\frac {1}{c^{2}}}Ez-{\frac {1}{c^{2}}} vp_{z}z-tp_{z}+{\frac {xv}{c^{2}}}p_{z}\right]\\&=\gamma \left[\left({\frac {1} {c^{2}}}Ez-tp_{z}\right)+{\frac {v}{c^{2}}}(xp_{z}-zp_{x})\right]\\&= \gamma \left(N_{z}-{\frac {v}{c^{2}}}L_{y}\right)\end{تراز شده}}}

جمع آوری اجزای موازی و عمود بر هم مانند قبل

{\displaystyle {\begin{aligned}\mathbf {N} _{\parallel }'&=\mathbf {N} _{\parallel }\\\mathbf {N} _{\perp }'&=\gamma ( \mathbf {v} )\left(\mathbf {N} _{\perp }-{\frac {1}{c^{2}}}\mathbf {v} \times \mathbf {L} \راست)\ \\پایان{تراز شده}}}

باز هم، مولفه های موازی با جهت حرکت نسبی تغییر نمی کنند، آنهایی که عمود هستند تغییر می کنند.