狹義相對論中的加速度
加速度與力

更多：相對論力學 § 力和狹義相對論中的質量
四維力 \mathbf {F} 可寫為三維力 \mathbf {f} 的函數： \mathbf {F} =\gamma \left((\mathbf {f} \cdot \mathbf {u} )/c,\ \mathbf {f} \right)。四維力、四維加速度（式(2)）以及不變質量 m則具有如下關係式： \mathbf {F} =m\mathbf {A} ；因此可得
速度沿任意方向的情形下，三維力與三維加速度的關係式則可寫成：
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|  {\begin{aligned}\mathbf {f} &=m\gamma ^{3}\left({\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}\right)+m\gamma \mathbf {a} \\\mathbf {a} &={\frac {1}{m\gamma }}\left(\mathbf {f} -{\frac {(\mathbf {f} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}\right)\end{aligned}} |  | 5a |
當速度沿著x方向，即 u=u_{x}，且僅考慮平行（x方向）或垂直（y、z方向）於速度方向的加速度與力，則三維力與三維加速度的關係式為：
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|  {\begin{array}{c|c|cc}{\begin{aligned}f_{x}&={\frac {ma_{x}}{\left(1-{\frac {u^{2}}{c^{2}}}\right)^{3/2}}}\\f_{y}&={\frac {ma_{y}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\\f_{z}&={\frac {ma_{z}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\end{aligned}}&{\begin{aligned}a_{x}&={\frac {f_{x}}{m}}\left(1-{\frac {u^{2}}{c^{2}}}\right)^{3/2}\\a_{y}&={\frac {f_{y}}{m}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}\\a_{z}&={\frac {f_{z}}{m}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}\end{aligned}}&{\begin{aligned}\mathbf {f} &=m\mathbf {a} \left(\gamma ^{3},\ \gamma ,\ \gamma \right)\\\mathbf {a} &={\frac {\mathbf {f} }{m}}\left({\frac {1}{\gamma ^{3}}},\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)\end{aligned}}\end{array}} |  | 5b |
牛頓力學中，將質量簡單定義為三維力與三維加速度的比值；此想法在狹義相對論中變為拙劣，因為這樣定義的質量將與速度的大小及方向相依。是故如下曾出現在舊版教科書中的質量定義在當代已捨棄不用：
式(5a)中三維加速度與三維力的關係式也可透過運動方程式求得：
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|  \mathbf {f} ={\frac {d\mathbf {p} }{dt}}={\frac {d(m\gamma \mathbf {u} )}{dt}}={\frac {d(m\gamma )}{dt}}\mathbf {u} +m\gamma {\frac {d\mathbf {u} }{dt}}=m\gamma ^{3}\left({\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}\right)+m\gamma \mathbf {a}  |  | 5c |
其中 \mathbf {p} 是三維動量。若慣性參考系 S與 S'間的相對速度為沿著x方向，即 v=v_{x}，且僅考慮平行（x方向）或垂直（y、z方向）於速度方向的情形時， S中的三維力 \mathbf {f} 與 S'中的三維力 \mathbf {f} '之間的轉換關係式可透過對 \mathbf {u} 、 \mathbf {a} 、 m\gamma 、 d(m\gamma )/dt等相關轉換式做代換，或透過四維力進行勞侖茲轉換後取其分量，而得到以下結果：
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|  {\begin{array}{c|c}{\begin{aligned}f_{x}^{\prime }&={\frac {f_{x}-{\frac {v}{c^{2}}}(\mathbf {f} \cdot \mathbf {u} )}{1-{\frac {u_{x}v}{c^{2}}}}}\\f_{y}^{\prime }&={\frac {f_{y}}{\gamma _{v}\left(1-{\frac {u_{x}v}{c^{2}}}\right)}}\\f_{z}^{\prime }&={\frac {f_{z}}{\gamma _{v}\left(1-{\frac {u_{x}v}{c^{2}}}\right)}}\end{aligned}}&{\begin{aligned}f_{x}&={\frac {f_{x}^{\prime }+{\frac {v}{c^{2}}}(\mathbf {f} ^{\prime }\cdot \mathbf {u} ^{\prime })}{1+{\frac {u_{x}^{\prime }v}{c^{2}}}}}\\f_{y}&={\frac {f_{y}^{\prime }}{\gamma _{v}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)}}\\f_{z}&={\frac {f_{z}^{\prime }}{\gamma _{v}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)}}\end{aligned}}\end{array}} |  | 6a |
又或將之推廣到任意方向的 \mathbf {u} 及 \mathbf {v} （其中 |\mathbf {v} |=v）：
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|  {\begin{aligned}\mathbf {f} '&={\frac {{\frac {\mathbf {f} }{\gamma _{v}}}-\left\{(\mathbf {f\cdot u} ){\frac {v^{2}}{c^{2}}}-(\mathbf {f\cdot v} )\left(1-{\frac {1}{\gamma _{v}}}\right)\right\}{\frac {\mathbf {v} }{v^{2}}}}{1-{\frac {\mathbf {v\cdot u} }{c^{2}}}}}\\\mathbf {f} &={\frac {{\frac {\mathbf {f} '}{\gamma _{v}}}+\left\{(\mathbf {f'\cdot u} '){\frac {v^{2}}{c^{2}}}+(\mathbf {f'\cdot v} )\left(1-{\frac {1}{\gamma _{v}}}\right)\right\}{\frac {\mathbf {v} }{v^{2}}}}{1+{\frac {\mathbf {v\cdot u'} }{c^{2}}}}}\end{aligned}} |  | 6b |