狹義相對論中的加速度

狹義相對論中的加速度類似於牛頓力學中的概念，乃速度對於時間的微分。因為相對論中的勞侖茲轉換及時間膨脹，時間與距離的概念變為複雜，因此「加速度」的定義也變得複雜。狹義相對論為平直閔考斯基時空的理論，即使加速度存在依然有效，前提是能量動量張量所造成的重力場效應可以忽略。否則，則需用到廣義相對論以及彎曲時空來詮釋。在地球表面附近，時空彎曲程度不明顯，因此實務上採用狹義相對論來詮釋物理現象仍是合宜作法，比如粒子加速器實驗。^[1]

如同在外界慣性座標系中的測量，三維空間中的普通加速度（稱為「三維加速度」或「座標加速度」）的轉換式可以推導得出。此外作為一特例，也可用共動（comoving）的加速規來測量固有加速度。另一種有用的形式是四維加速度，其分量可透過勞侖茲轉換在不同參考系中做連結。連結加速度與力的運動方程式也可得到。幾種特殊形式的加速物體運動方程式以及它們的彎曲世界線可以透過對上述方程式的積分求得。知名的特例如雙曲運動（英语：hyperbolic motion (relativity)），適用於常數值縱向固有加速度的例子，以及等速率圓周運動。最後，在狹義相對論的架構下，描述加速參考系中的物理現象亦為可行。

歷史演進上，在相對論發展的早年即已出現包含加速度的相對論性方程式，在早年的教科書中有整理，如馬克斯·馮·勞厄（1911年、1921年）^[2]或沃夫岡·包立（1921年）。^[3]舉例來說，運動方程式以及加速度轉換式於以下學者的論文中建立起來：亨德里克·勞侖茲（1899年、1904年）、儒勒·昂利·龐加萊（1905年）、阿爾伯特·愛因斯坦（1905年）、馬克斯·普朗克（1906年）；四維加速度、固有加速度與雙曲運動的分析參見赫爾曼·閔考斯基 （1908年）、馬克斯·玻恩（1909年）、古斯塔夫·赫格洛茨（英语：Gustav Herglotz）（1909年）、阿諾·索末菲（1910年）、馮·勞厄（1911年）。

在牛頓力學與狹義相對論中，三維加速度或座標加速度的定義保持一致。${\displaystyle \mathbf {a} =\left(a_{x},\ a_{y},\ a_{z}\right)}$是速度${\displaystyle \mathbf {u} =\left(u_{x},\ u_{y},\ u_{z}\right)}$對座標時間的一階導數，亦即是位置${\displaystyle \mathbf {r} =\left(x,\ y,\ z\right)}$對座標時間的二階導數：

${\displaystyle \mathbf {a} ={\frac {d\mathbf {u} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}}$。

然而在另一相異的慣性參考系中做三維加速度測量時，兩項理論的預測就出現重大歧異。牛頓力學中，時間是絕對的（${\displaystyle t'=t}$），採用的慣性系轉換式為伽利略轉換。因此，從伽利略轉換推導而得的三維加速度在所有慣性系中皆相同：^[4]

${\displaystyle \mathbf {a} =\mathbf {a} '}$。

相反地，在狹義相對論中，${\displaystyle \mathbf {r} }$與${\displaystyle t}$兩者皆與勞侖茲轉換相依，因此三維加速度${\displaystyle \mathbf {a} }$及其分量在不同慣性系也各不相同。當慣性系間的相對速度是沿著x軸，即${\displaystyle v=v_{x}}$（${\displaystyle \gamma _{v}=1/{\sqrt {1-v^{2}/c^{2}}}}$為相對應的勞侖茲因子），勞侖茲轉換式為：

${\displaystyle {\begin{array}{c|c}{\begin{aligned}x'&=\gamma _{v}(x-vt)\\y'&=y\\z'&=z\\t^{\prime }&=\gamma _{v}\left(t-{\frac {v}{c^{2}}}x\right)\end{aligned}}&{\begin{aligned}x&=\gamma _{v}(x'+vt')\\y&=y'\\z&=z'\\t&=\gamma _{v}\left(t'+{\frac {v}{c^{2}}}x'\right)\end{aligned}}\end{array}}}$

1a

或是對於一長度${\displaystyle v}$及任意方向的速度向量${\displaystyle \mathbf {v} =\left(v_{x},\ v_{y},\ v_{z}\right)}$（其中${\displaystyle |\mathbf {v} |=v={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}$），勞侖茲轉換式為：^[5]

${\displaystyle {\begin{array}{c|c}{\begin{aligned}\mathbf {r} '&=\mathbf {r} +\mathbf {v} \left[{\frac {\left(\mathbf {r\cdot v} \right)}{v^{2}}}\left(\gamma _{v}-1\right)-t\gamma _{v}\right]\\t^{\prime }&=\gamma _{v}\left(t-{\frac {\mathbf {r\cdot v} }{c^{2}}}\right)\end{aligned}}&{\begin{aligned}\mathbf {r} &=\mathbf {r} '+\mathbf {v} \left[{\frac {\left(\mathbf {r'\cdot v} \right)}{v^{2}}}\left(\gamma _{v}-1\right)+t'\gamma _{v}\right]\\t&=\gamma _{v}\left(t'+{\frac {\mathbf {r'\cdot v} }{c^{2}}}\right)\end{aligned}}\end{array}}}$

1b

為了求得三維加速度的轉換式，必須分別對勞侖茲轉換式中的空間座標${\displaystyle \mathbf {r} }$及${\displaystyle \mathbf {r} '}$做時間${\displaystyle t}$與${\displaystyle t'}$的微分。首先是得到三維速度${\displaystyle \mathbf {u} }$及${\displaystyle \mathbf {u} '}$的轉換式（亦稱為速度加成式）；爾後再次做時間${\displaystyle t}$與${\displaystyle t'}$的微分運算而得到三維加速度${\displaystyle \mathbf {a} }$及${\displaystyle \mathbf {a} '}$的轉換式。從式(1a)出發，所得到的轉換式為平行（x方向）與垂直（y、z方向）於速度${\displaystyle v=v_{x}}$之加速度：^{[6][7][8][9][H 1][H 2]}

${\displaystyle {\begin{array}{c|c}{\begin{aligned}a_{x}^{\prime }&={\frac {a_{x}}{\gamma _{v}^{3}\left(1-{\frac {u_{x}v}{c^{2}}}\right)^{3}}}\\a_{y}^{\prime }&={\frac {a_{y}}{\gamma _{v}^{2}\left(1-{\frac {u_{x}v}{c^{2}}}\right)^{2}}}+{\frac {a_{x}{\frac {u_{y}v}{c^{2}}}}{\gamma _{v}^{2}\left(1-{\frac {u_{x}v}{c^{2}}}\right)^{3}}}\\a_{z}^{\prime }&={\frac {a_{z}}{\gamma _{v}^{2}\left(1-{\frac {u_{x}v}{c^{2}}}\right)^{2}}}+{\frac {a_{x}{\frac {u_{z}v}{c^{2}}}}{\gamma _{v}^{2}\left(1-{\frac {u_{x}v}{c^{2}}}\right)^{3}}}\end{aligned}}&{\begin{aligned}a_{x}&={\frac {a_{x}^{\prime }}{\gamma _{v}^{3}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)^{3}}}\\a_{y}&={\frac {a_{y}^{\prime }}{\gamma _{v}^{2}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)^{2}}}-{\frac {a_{x}^{\prime }{\frac {u_{y}^{\prime }v}{c^{2}}}}{\gamma _{v}^{2}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)^{3}}}\\a_{z}&={\frac {a_{z}^{\prime }}{\gamma _{v}^{2}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)^{2}}}-{\frac {a_{x}^{\prime }{\frac {u_{z}^{\prime }v}{c^{2}}}}{\gamma _{v}^{2}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)^{3}}}\end{aligned}}\end{array}}}$

1c

若從式(1b)出發，則得到通解，速度與加速度可以是任意方向：^[10][11]

${\displaystyle {\begin{aligned}\mathbf {a} '&={\frac {\mathbf {a} }{\gamma _{v}^{2}\left(1-{\frac {\mathbf {v\cdot u} }{c^{2}}}\right)^{2}}}-{\frac {\mathbf {(a\cdot v)v} \left(\gamma _{v}-1\right)}{v^{2}\gamma _{v}^{3}\left(1-{\frac {\mathbf {v\cdot u} }{c^{2}}}\right)^{3}}}+{\frac {\mathbf {(a\cdot v)u} }{c^{2}\gamma _{v}^{2}\left(1-{\frac {\mathbf {v\cdot u} }{c^{2}}}\right)^{3}}}\\\mathbf {a} &={\frac {\mathbf {a} '}{\gamma _{v}^{2}\left(1+{\frac {\mathbf {v\cdot u} '}{c^{2}}}\right)^{2}}}-{\frac {\mathbf {(a'\cdot v)v} \left(\gamma _{v}-1\right)}{v^{2}\gamma _{v}^{3}\left(1+{\frac {\mathbf {v\cdot u} '}{c^{2}}}\right)^{3}}}-{\frac {\mathbf {(a'\cdot v)u} '}{c^{2}\gamma _{v}^{2}\left(1+{\frac {\mathbf {v\cdot u} '}{c^{2}}}\right)^{3}}}\end{aligned}}}$

1d

此轉換式表示：若有兩慣性系${\displaystyle S}$與${\displaystyle S'}$，兩者相對速度${\displaystyle \mathbf {v} }$，則${\displaystyle S}$系中測到一物體瞬時速度為${\displaystyle \mathbf {u} }$、加速度為${\displaystyle \mathbf {a} }$，該物體在${\displaystyle S'}$系中則具有瞬時速度${\displaystyle \mathbf {u} '}$、加速度${\displaystyle \mathbf {a} '}$。一如速度加成式，這些加速度轉換式可保證一物體無法加速到光速，遑論超過光速。

若改採用四維向量，即${\displaystyle \mathbf {R} }$乃四維位置，${\displaystyle \mathbf {U} }$乃四維速度，則一物體的四維加速度${\displaystyle \mathbf {A} =\left(A_{t},\ A_{x},\ A_{y},\ A_{z}\right)=\left(A_{t},\ \mathbf {A} _{r}\right)}$可透過對原時${\displaystyle \mathbf {\tau } }$的微分求得：^[12][13][14]

${\displaystyle {\begin{aligned}\mathbf {A} &={\frac {d\mathbf {U} }{d\tau }}={\frac {d^{2}\mathbf {R} }{d\tau ^{2}}}=\left(c{\frac {d^{2}t}{d\tau ^{2}}},\ {\frac {d^{2}\mathbf {r} }{d\tau ^{2}}}\right)\\&=\left(\gamma ^{4}{\frac {\mathbf {u} \cdot \mathbf {a} }{c}},\ \gamma ^{4}{\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}+\gamma ^{2}\mathbf {a} \right)\end{aligned}}}$

2

其中${\displaystyle \mathbf {a} }$為物體的三維加速度；${\displaystyle \mathbf {u} }$為物體的瞬時三維速度，長度為${\displaystyle |\mathbf {u} |=u}$，所對應的勞侖茲因子為${\displaystyle \gamma =1/{\sqrt {1-u^{2}/c^{2}}}}$。若只考慮空間分量且速度是沿著x方向（即${\displaystyle u=u_{x}}$），且只考慮與速度平行（x方向）或垂直（y、z方向）的加速度，則關係式可簡化為：^[15][16]

${\displaystyle \mathbf {A} _{r}=\mathbf {a} \left(\gamma ^{4},\ \gamma ^{2},\ \gamma ^{2}\right)}$

與前述的三維加速度不同，四維加速度不需要推導新的轉換關係式，因為所有四維向量（包括四維加速度）在兩個具有相對速度${\displaystyle v}$的慣性系之間都呈現勞侖茲協變性。因此只要將式(1a)中的${\displaystyle x,\ y,\ z,\ ct}$代換為${\displaystyle A_{x},\ A_{y},\ A_{z},\ A_{t}}$，即為四維加速度在兩慣性系之間的轉換式：^[17]

${\displaystyle {\begin{array}{c|c}{\begin{aligned}A_{x}^{\prime }&=\gamma _{v}\left(A_{x}-{\frac {v}{c}}A_{t}\right)\\A_{y}^{\prime }&=A_{y}\\A_{z}^{\prime }&=A_{z}\\A_{t}^{\prime }&=\gamma _{v}\left(A_{t}-{\frac {v}{c}}A_{x}\right)\end{aligned}}&{\begin{aligned}A_{x}&=\gamma _{v}\left(A_{x}^{\prime }+{\frac {v}{c}}A_{t}^{\prime }\right)\\A_{y}&=A_{y}^{\prime }\\A_{z}&=A_{z}^{\prime }\\A_{t}&=\gamma _{v}\left(A_{t}^{\prime }+{\frac {v}{c}}A_{x}^{\prime }\right)\end{aligned}}\end{array}}}$

又或將式(1b)中的${\displaystyle \mathbf {r} ,\ ct}$代換為${\displaystyle \mathbf {A} _{r},\ A_{t}}$，亦可得到任意相對速度${\displaystyle \mathbf {v} }$情形的轉換式：

${\displaystyle {\begin{array}{c|c}{\begin{aligned}\mathbf {A} {}_{r}^{\prime }&=\mathbf {A} _{r}+{\frac {\mathbf {v} }{c}}\left[{\frac {\left(\mathbf {A} _{r}\cdot \mathbf {v} \right)c}{v^{2}}}\left(\gamma _{v}-1\right)-A_{t}\gamma _{v}\right]\\A_{t}^{\prime }&=\gamma _{v}\left(A_{t}-{\frac {\mathbf {A} _{r}\cdot \mathbf {v} }{c}}\right)\end{aligned}}&{\begin{aligned}\mathbf {A} _{r}&=\mathbf {A} {}_{r}^{\prime }+{\frac {\mathbf {v} }{c}}\left[{\frac {\left(\mathbf {A} {}_{r}^{\prime }\cdot \mathbf {v} \right)c}{v^{2}}}\left(\gamma _{v}-1\right)+A_{t}^{\prime }\gamma _{v}\right]\\A_{t}&=\gamma _{v}\left(A_{t}^{\prime }+{\frac {\mathbf {A} {}_{r}^{\prime }\cdot \mathbf {v} }{c}}\right)\end{aligned}}\end{array}}}$,

此外，四維加速度自身內積${\displaystyle \mathbf {A} ^{2}=-A_{t}^{2}+\mathbf {A} _{r}^{2}}$及向量大小${\displaystyle |\mathbf {A} |={\sqrt {\mathbf {A} ^{2}}}}$為不變量（這裡所採用的度規標記（metric signature）為(−,+,+,+)），因此：^[16][13][18]

${\displaystyle |\mathbf {A} '|=|\mathbf {A} |={\sqrt {\gamma ^{4}\left[\mathbf {a} ^{2}+\gamma ^{2}\left({\frac {\mathbf {u} \cdot \mathbf {a} }{c}}\right)^{2}\right]}}}$。

3

在無限小的瞬間，總有一慣性系${\displaystyle S'}$與一加速物體（加速參考系）相對靜止，即兩者相對速度為0。在這樣的慣性系中，勞侖茲轉換成立。相對應的三維加速度${\displaystyle \mathbf {a} ^{0}=\left(a_{x}^{0},\ a_{y}^{0},\ a_{z}^{0}\right)}$可透過加速規直接測量，稱之為固有加速度^{[19][H 3]}或靜止加速度。^{[20][H 4]}此瞬時慣性系${\displaystyle S'}$中的${\displaystyle \mathbf {a} ^{0}}$與外界另一慣性系${\displaystyle S}$所測到的${\displaystyle \mathbf {a} }$之間的關係式為(1c)與(1d)，其中${\displaystyle \mathbf {a} '=\mathbf {a} ^{0}}$，${\displaystyle \mathbf {u} '=0}$，${\displaystyle \mathbf {u} =\mathbf {v} }$，而${\displaystyle \gamma =\gamma _{v}}$。因此如同(1c)中的情形，速度沿x方向（${\displaystyle u=u_{x}=v=v_{x}}$），且只考慮與速度平行（x方向）或垂直（y、z方向）的加速度，則關係式為：^{[12][20][19][H 5][H 6][H 3][H 4]}

${\displaystyle {\begin{array}{c|c|cc}{\begin{aligned}a_{x}^{0}&={\frac {a_{x}}{\left(1-{\frac {u^{2}}{c^{2}}}\right)^{3/2}}}\\a_{y}^{0}&={\frac {a_{y}}{1-{\frac {u^{2}}{c^{2}}}}}\\a_{z}^{0}&={\frac {a_{z}}{1-{\frac {u^{2}}{c^{2}}}}}\end{aligned}}&{\begin{aligned}a_{x}&=a_{x}^{0}\left(1-{\frac {u^{2}}{c^{2}}}\right)^{3/2}\\a_{y}&=a_{y}^{0}\left(1-{\frac {u^{2}}{c^{2}}}\right)\\a_{z}&=a_{z}^{0}\left(1-{\frac {u^{2}}{c^{2}}}\right)\end{aligned}}&{\begin{aligned}\mathbf {a} ^{0}&=\mathbf {a} \left(\gamma ^{3},\ \gamma ^{2},\ \gamma ^{2}\right)\\\mathbf {a} &=\mathbf {\mathbf {a} } ^{0}\left({\frac {1}{\gamma ^{3}}},\ {\frac {1}{\gamma ^{2}}},\ {\frac {1}{\gamma ^{2}}}\right)\end{aligned}}\end{array}}}$

4a

或將固有加速度此一特例條件代入通式(1d)，其中任意方向的速度${\displaystyle \mathbf {u} }$其長度為${\displaystyle |\mathbf {u} |=u}$：^[21][22][18]

${\displaystyle {\begin{aligned}\mathbf {a} ^{0}&=\gamma ^{2}\left[\mathbf {a} +{\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{u^{2}}}\left(\gamma -1\right)\right]\\\mathbf {a} &={\frac {1}{\gamma ^{2}}}\left[\mathbf {a} ^{0}-{\frac {(\mathbf {a} ^{0}\cdot \mathbf {u} )\mathbf {u} }{u^{2}}}\left(1-{\frac {1}{\gamma }}\right)\right]\end{aligned}}}$

另外固有加速度與四維加速度的長度也有密切關係：如前述，四維加速度的長度為不變量，可在瞬間共動慣性系${\displaystyle S'}$中被測定，其中${\displaystyle \mathbf {A} _{r}^{\prime }=\mathbf {a} ^{0}}$，且因${\displaystyle dt'/d\tau =1}$，${\displaystyle d^{2}t'/d\tau ^{2}=A_{t}^{\prime }=0}$：^{[20][12][23][H 7]}

${\displaystyle |\mathbf {A} '|={\sqrt {0+\left.\mathbf {a} ^{0}\right.^{2}}}=|\mathbf {a} ^{0}|}$。

4b

因此四維加速度長度對應到固有加速度長度。將此結果與式(3)結合，可得到將${\displaystyle S'}$系中${\displaystyle \mathbf {a} ^{0}}$與${\displaystyle S}$系中${\displaystyle \mathbf {a} }$連結的另一種關係式求法，亦即：^[13][18]

${\displaystyle |\mathbf {a} ^{0}|=|\mathbf {A} |={\sqrt {\gamma ^{4}\left[\mathbf {a} ^{2}+\gamma ^{2}\left({\frac {\mathbf {u} \cdot \mathbf {a} }{c}}\right)^{2}\right]}}}$

從這裡可得到式(4a)，只要再次採用如下條件：速度沿著x方向（${\displaystyle u=u_{x}}$），只考慮與速度平行（x方向）或垂直（y、z方向）的加速度。

四維力${\displaystyle \mathbf {F} }$可寫為三維力${\displaystyle \mathbf {f} }$的函數：${\displaystyle \mathbf {F} =\gamma \left((\mathbf {f} \cdot \mathbf {u} )/c,\ \mathbf {f} \right)}$。四維力、四維加速度（式(2)）以及不變質量${\displaystyle m}$則具有如下關係式：${\displaystyle \mathbf {F} =m\mathbf {A} }$^[24]；因此可得^[25]

${\displaystyle \mathbf {F} =\left(\gamma {\frac {\mathbf {f} \cdot \mathbf {u} }{c}},\ \gamma \mathbf {f} \right)=m\mathbf {A} =m\left(\gamma ^{4}\left({\frac {\mathbf {u} \cdot \mathbf {a} }{c}}\right),\ \gamma ^{4}\left({\frac {\mathbf {u} \cdot \mathbf {a} }{c^{2}}}\right)\mathbf {u} +\gamma ^{2}\mathbf {a} \right)}$。

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：^[26][27][24]

${\displaystyle {\begin{aligned}\mathbf {f} &=m\gamma ^{3}\left({\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}\right)+m\gamma \mathbf {a} \\\mathbf {a} &={\frac {1}{m\gamma }}\left(\mathbf {f} -{\frac {(\mathbf {f} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}\right)\end{aligned}}}$

5a

當速度沿著x方向，即${\displaystyle u=u_{x}}$，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：^{[28][27][24][H 6][H 8]}

${\displaystyle {\begin{array}{c|c|cc}{\begin{aligned}f_{x}&={\frac {ma_{x}}{\left(1-{\frac {u^{2}}{c^{2}}}\right)^{3/2}}}\\f_{y}&={\frac {ma_{y}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\\f_{z}&={\frac {ma_{z}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\end{aligned}}&{\begin{aligned}a_{x}&={\frac {f_{x}}{m}}\left(1-{\frac {u^{2}}{c^{2}}}\right)^{3/2}\\a_{y}&={\frac {f_{y}}{m}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}\\a_{z}&={\frac {f_{z}}{m}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}\end{aligned}}&{\begin{aligned}\mathbf {f} &=m\mathbf {a} \left(\gamma ^{3},\ \gamma ,\ \gamma \right)\\\mathbf {a} &={\frac {\mathbf {f} }{m}}\left({\frac {1}{\gamma ^{3}}},\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)\end{aligned}}\end{array}}}$

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：^{[28][29][H 6]}

${\displaystyle m_{\Vert }={\frac {f_{x}}{a_{x}}}=m\gamma ^{3}}$，稱為「縱向質量」；

${\displaystyle m_{\perp }={\frac {f_{y}}{a_{y}}}={\frac {f_{z}}{a_{z}}}=m\gamma }$，稱為「橫向質量」。

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：^{[30][26][H 6][H 8]}

${\displaystyle \mathbf {f} ={\frac {d\mathbf {p} }{dt}}={\frac {d(m\gamma \mathbf {u} )}{dt}}={\frac {d(m\gamma )}{dt}}\mathbf {u} +m\gamma {\frac {d\mathbf {u} }{dt}}=m\gamma ^{3}\left({\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}\right)+m\gamma \mathbf {a} }$

5c

其中${\displaystyle \mathbf {p} }$是三維動量。若慣性參考系${\displaystyle S}$與${\displaystyle S'}$間的相對速度為沿著x方向，即${\displaystyle v=v_{x}}$，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，${\displaystyle S}$中的三維力${\displaystyle \mathbf {f} }$與${\displaystyle S'}$中的三維力${\displaystyle \mathbf {f} '}$之間的轉換關係式可透過對${\displaystyle \mathbf {u} }$、${\displaystyle \mathbf {a} }$、${\displaystyle m\gamma }$、${\displaystyle d(m\gamma )/dt}$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：^{[30][31][25][H 9][H 2]}

${\displaystyle {\begin{array}{c|c}{\begin{aligned}f_{x}^{\prime }&={\frac {f_{x}-{\frac {v}{c^{2}}}(\mathbf {f} \cdot \mathbf {u} )}{1-{\frac {u_{x}v}{c^{2}}}}}\\f_{y}^{\prime }&={\frac {f_{y}}{\gamma _{v}\left(1-{\frac {u_{x}v}{c^{2}}}\right)}}\\f_{z}^{\prime }&={\frac {f_{z}}{\gamma _{v}\left(1-{\frac {u_{x}v}{c^{2}}}\right)}}\end{aligned}}&{\begin{aligned}f_{x}&={\frac {f_{x}^{\prime }+{\frac {v}{c^{2}}}(\mathbf {f} ^{\prime }\cdot \mathbf {u} ^{\prime })}{1+{\frac {u_{x}^{\prime }v}{c^{2}}}}}\\f_{y}&={\frac {f_{y}^{\prime }}{\gamma _{v}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)}}\\f_{z}&={\frac {f_{z}^{\prime }}{\gamma _{v}\left(1+{\frac {u_{x}^{\prime }v}{c^{2}}}\right)}}\end{aligned}}\end{array}}}$

6a

又或將之推廣到任意方向的${\displaystyle \mathbf {u} }$及${\displaystyle \mathbf {v} }$（其中${\displaystyle |\mathbf {v} |=v}$）：^[32][33]

${\displaystyle {\begin{aligned}\mathbf {f} '&={\frac {{\frac {\mathbf {f} }{\gamma _{v}}}-\left\{(\mathbf {f\cdot u} ){\frac {v^{2}}{c^{2}}}-(\mathbf {f\cdot v} )\left(1-{\frac {1}{\gamma _{v}}}\right)\right\}{\frac {\mathbf {v} }{v^{2}}}}{1-{\frac {\mathbf {v\cdot u} }{c^{2}}}}}\\\mathbf {f} &={\frac {{\frac {\mathbf {f} '}{\gamma _{v}}}+\left\{(\mathbf {f'\cdot u} '){\frac {v^{2}}{c^{2}}}+(\mathbf {f'\cdot v} )\left(1-{\frac {1}{\gamma _{v}}}\right)\right\}{\frac {\mathbf {v} }{v^{2}}}}{1+{\frac {\mathbf {v\cdot u'} }{c^{2}}}}}\end{aligned}}}$

6b

透過一共動彈簧秤來測量一瞬時慣性系中的力${\displaystyle \mathbf {f} ^{0}}$可稱為固有力。^[34][35]從式(6a)與式(6b)，設定${\displaystyle \mathbf {f} '=\mathbf {f} ^{0}}$、${\displaystyle \mathbf {u} '=0}$、${\displaystyle \mathbf {u} =\mathbf {v} }$、${\displaystyle \gamma =\gamma _{v}}$等條件，可得到固有力的關係式。當速度沿x軸，${\displaystyle u=u_{x}=v=v_{x}}$，且僅考慮平行（x方向）或垂直（y、z方向）之加速度，可採用式(6a)：^[36][34][35]

${\displaystyle {\begin{array}{c|c|cc}{\begin{aligned}f_{x}^{0}&=f_{x}\\f_{y}^{0}&={\frac {f_{y}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\\f_{z}^{0}&={\frac {f_{z}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\end{aligned}}&{\begin{aligned}f_{x}&=f_{x}^{0}\\f_{y}&=f_{y}^{0}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}\\f_{z}&=f_{z}^{0}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}\end{aligned}}&{\begin{aligned}\mathbf {f} ^{0}&=\mathbf {f} \left(1,\ \gamma ,\ \gamma \right)\\\mathbf {f} &=\mathbf {f} ^{0}\left(1,\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)\end{aligned}}\end{array}}}$

7a

任意方向、大小為${\displaystyle |\mathbf {u} |=u}$的速度${\displaystyle \mathbf {u} }$之通則採用式(6b)：^[36][37]

${\displaystyle {\begin{aligned}\mathbf {f} ^{0}&=\mathbf {f} \gamma -{\frac {(\mathbf {f} \cdot \mathbf {u} )\mathbf {u} }{u^{2}}}(\gamma -1)\\\mathbf {f} &={\frac {\mathbf {f} ^{0}}{\gamma }}+{\frac {(\mathbf {f} ^{0}\cdot \mathbf {u} )\mathbf {u} }{u^{2}}}\left(1-{\frac {1}{\gamma }}\right)\end{aligned}}}$

因為${\displaystyle \gamma =1}$，牛頓力學關係式${\displaystyle \mathbf {f} ^{0}=m\mathbf {a} ^{0}}$在瞬時慣性系中成立，是故式(4a)、式(5b)、式(7a)可歸結為：^[38]

${\displaystyle {\begin{aligned}\mathbf {f} ^{0}&=\mathbf {f} \left(1,\ \gamma ,\ \gamma \right)=m\mathbf {a} ^{0}=m\mathbf {a} \left(\gamma ^{3},\ \gamma ^{2},\ \gamma ^{2}\right)\\\mathbf {f} &=\mathbf {f} ^{0}\left(1,\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)=m\mathbf {a} ^{0}\left(1,\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)=m\mathbf {a} \left(\gamma ^{3},\ \gamma ,\ \gamma \right)\end{aligned}}}$

7b

透過這些式子，歷史上對橫向質量${\displaystyle m_{\perp }}$定義中的明顯矛盾可以得到解釋。^[39]愛因斯坦（1905年）的定義是固有力與三維加速度的比值：^{[H 10]}

${\displaystyle m_{\perp \ \mathrm {Einstein} }={\frac {f_{y}^{0}}{a_{y}}}={\frac {f_{z}^{0}}{a_{z}}}=m\gamma ^{2}}$，

而勞侖茲（1899年與1904年）、普朗克（1906年）的定義則是三維力與三維加速度的比值^{[H 6]}

${\displaystyle m_{\perp \ \mathrm {Lorentz} }={\frac {f_{y}}{a_{y}}}={\frac {f_{z}}{a_{z}}}=m\gamma }$。

對運動方程式做積分，可得到一加速物體的世界線，對應到一連串的瞬時慣性系。如此則需考慮相關的「時鐘假設」：^[40][41]共動時鐘的原時與加速度無關。也就是說，對外部慣性系而言，這些時鐘的時間膨脹只相依於和外部慣性系之間的相對速度。以下是兩個簡單的彎曲世界線範例，透過對式(4a)中固有加速度的積分而得：

a) 雙曲運動（英语：Hyperbolic motion (relativity)）：式(4a)中為恆定的縱向固有加速度${\displaystyle \alpha =a_{x}^{0}=a_{x}\gamma ^{3}}$，造成世界線^{[12][19][20][26][42][43][H 11][H 2]}

${\displaystyle {\begin{aligned}&t(\tau )={\frac {c}{\alpha }}\sinh {\frac {\alpha \tau }{c}},\quad x(\tau )={\frac {c^{2}}{\alpha }}\left(\cosh {\frac {\alpha \tau }{c}}-1\right),\quad y=0,\quad z=0,\\&\tau (t)={\frac {c}{\alpha }}\ln \left({\sqrt {1+\left({\frac {\alpha t}{c}}\right)^{2}}}+{\frac {\alpha t}{c}}\right),\quad x(t)={\frac {c^{2}}{\alpha }}\left({\sqrt {1+\left({\frac {\alpha t}{c}}\right)^{2}}}-1\right)\end{aligned}}}$

8

此世界線對應到雙曲方程式${\displaystyle c^{4}/\alpha ^{2}=\left(x+c^{2}/\alpha \right)^{2}-c^{2}t^{2}}$，因此這樣的移動物體世界線被稱作雙曲運動。此方程組常用來計算孿生子悖論或貝爾太空船悖論的不同版本案例，亦與等加速度太空旅行（英语：space travel using constant acceleration）有關。

b) 式(4a)中為恆定的橫向固有加速度${\displaystyle a_{y}^{0}=a_{y}\gamma ^{2}}$，可視為向心加速度，^[13]造成一勻速旋轉物體的世界線：^[44][45]

${\displaystyle {\begin{aligned}x&=r\cos \Omega _{0}t=r\cos \Omega \tau \\y&=r\sin \Omega _{0}t=r\cos \Omega \tau \\z&=z\\t&=\gamma \tau ={\frac {\tau }{\sqrt {1-\left({\frac {r\Omega _{0}}{c}}\right)^{2}}}}=\tau {\sqrt {1+\left({\frac {r\Omega }{c}}\right)^{2}}}\end{aligned}}}$

其中${\displaystyle v=r\Omega _{0}}$為切線速率，${\displaystyle r}$是軌道半徑；角速度${\displaystyle \Omega _{0}}$為座標時間的函數，另外${\displaystyle \Omega =\gamma \Omega _{0}}$為固有角速度。

加速運動亦可透過加速座標系或曲線座標系（英语：curvilinear coordinates）來描述。以此方式建立的固有參考系與費米座標（英语：Fermi coordinates）密切相關。^[46][47]舉例而言，一雙曲加速參考系的座標有時稱為潤德勒座標，而勻速旋轉參考系的情形，則稱為旋轉圓柱座標，或稱玻恩座標（英语：Born coordinates）。

此條目目前正依照其他维基百科上的内容进行翻译。 (2018年6月21日) 如果您擅长翻译，並清楚本條目的領域，欢迎协助翻譯、改善或校对本條目。 此外，长期闲置、未翻譯或影響閱讀的内容可能会被移除。

更多資訊請參見馮·勞厄^[2]、包立^[3]、米勒^[48]、Zahar^[49]、Gourgoulhon^[47]，以及狹義相對論發現史中的歷史資料。

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1899年：在一靜止靜電粒子系統${\displaystyle S_{0}}$（靜止於乙太（英语：Lorentz ether theory）中）及具有相對平移的另一系統${\displaystyle S}$之間，亨德里克·勞侖茲^{[H 5]}在包含一因子${\displaystyle \epsilon }$的情況下，推導出了加速度、力、質量之間的正確關係，下式中${\displaystyle k}$為勞侖茲因子：

(7a)中${\displaystyle \mathbf {f} /\mathbf {f} ^{0}}$項：${\displaystyle {\frac {1}{\epsilon ^{2}}}}$, ${\displaystyle {\frac {1}{k\epsilon ^{2}}}}$, ${\displaystyle {\frac {1}{k\epsilon ^{2}}}}$；

(4a)中${\displaystyle \mathbf {a} /\mathbf {a} ^{0}}$項：${\displaystyle {\frac {1}{k^{3}\epsilon }}}$, ${\displaystyle {\frac {1}{k^{2}\epsilon }}}$, ${\displaystyle {\frac {1}{k^{2}\epsilon }}}$；

(5b)中${\displaystyle \mathbf {f} /(m\mathbf {a} )}$項：${\displaystyle {\frac {k^{3}}{\epsilon }}}$, ${\displaystyle {\frac {k}{\epsilon }}}$, ${\displaystyle {\frac {k}{\epsilon }}}$，因此為縱向與橫向質量；

勞侖茲提到了他無法決定${\displaystyle \epsilon }$的值。若當時他設${\displaystyle \epsilon =1}$，則他的關係式會跟相對論關係式一模一樣。

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1904年：勞侖茲^{[H 6]}以更詳盡的方法推導初上述關係式，採用了靜止於系統${\displaystyle \Sigma '}$及移動於系統${\displaystyle \Sigma }$之粒子的性質，搭配上新的輔助變數${\displaystyle l}$，相當於1899年推導中的${\displaystyle 1/\epsilon }$，而得到：

(7a)中，${\displaystyle \mathbf {f} }$為${\displaystyle \mathbf {f} ^{0}}$之函數，可得${\displaystyle {\mathfrak {F}}(\Sigma )=\left(l^{2},\ {\frac {l^{2}}{k}},\ {\frac {l^{2}}{k}}\right){\mathfrak {F}}(\Sigma ')}$；

(7b)中，${\displaystyle m\mathbf {a} }$為${\displaystyle m\mathbf {a} ^{0}}$之函數，可得${\displaystyle m{\mathfrak {j}}(\Sigma )=\left(l^{2},\ {\frac {l^{2}}{k}},\ {\frac {l^{2}}{k}}\right)m{\mathfrak {j}}(\Sigma ')}$；

(4a)中，${\displaystyle \mathbf {a} }$為${\displaystyle \mathbf {a} ^{0}}$之函數，可得${\displaystyle {\mathfrak {j}}(\Sigma )=\left({\frac {l}{k^{3}}},\ {\frac {l}{k^{2}}},\ {\frac {l}{k^{2}}}\right){\mathfrak {j}}(\Sigma ')}$；

(5b, 7b)中，縱向與橫向質量為靜質量之函數，可得${\displaystyle m(\Sigma )=\left(k^{3}l,\ kl,\ kl\right)m(\Sigma ')}$。

這次勞侖茲可以展示${\displaystyle l=1}$，而他的數學式與相對論形式完全相符。他也推導了運動方程式：

${\displaystyle {\displaystyle {\mathfrak {F}}={\frac {d{\mathfrak {G}}}{dt}}}}$而${\displaystyle {\displaystyle {\mathfrak {G}}={\frac {e^{2}}{6\pi c^{2}R}}kl{\mathfrak {w}}}}$

對應於(5c)裡的${\displaystyle \mathbf {f} ={\frac {d\mathbf {p} }{dt}}={\frac {d(m\gamma \mathbf {u} )}{dt}}}$，其中${\displaystyle l=1}$、${\displaystyle {\mathfrak {F}}=\mathbf {f} }$、${\displaystyle {\mathfrak {G}}=\mathbf {p} }$、${\displaystyle {\mathfrak {w}}=\mathbf {u} }$、${\displaystyle k=\gamma }$，以及視為電磁靜質量（英语：electromagnetic mass）的${\displaystyle m=e^{2}/(6\pi c^{2}R)}$。他更進一步地闡述：這些數學式不只適用於帶電粒子的力與質量，也適用於其他過程，因此使得乙太中地球運動的影響無法被偵測出來。

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1905年：儒勒·昂利·庞加莱^{[H 9]}引入了三維力的轉換式(6a):

${\displaystyle X_{1}^{\prime }={\frac {k}{l^{3}}}{\frac {\rho }{\rho ^{\prime }}}\left(X_{1}+\epsilon \Sigma X_{1}\xi \right),\quad Y_{1}^{\prime }={\frac {\rho }{\rho ^{\prime }}}{\frac {Y_{1}}{l^{3}}},\quad Z_{1}^{\prime }={\frac {\rho }{\rho ^{\prime }}}{\frac {Z_{1}}{l^{3}}}}$

其中${\displaystyle {\frac {\rho }{\rho ^{\prime }}}={\frac {k}{l^{3}}}(1+\epsilon \xi )}$，而${\displaystyle k}$為勞侖茲因子，${\displaystyle \rho }$為電荷密度。或以現代符號表記：${\displaystyle \epsilon =v}$，${\displaystyle \xi =u_{x}}$，${\displaystyle \left(X_{1},\ Y_{1},\ Z_{1}\right)=\mathbf {f} }$，以及${\displaystyle \Sigma X_{1}\xi =\mathbf {f} \cdot \mathbf {u} }$。與勞侖茲相同，他設定${\displaystyle l=1}$。

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1905年：阿爾伯特·愛因斯坦^{[H 10]}以其狹義相對論為基礎，推導出運動方程式。此表示出等價慣性系之間的關係，而不需要用到機械式乙太。愛因斯坦總結到，在一瞬時慣性系${\displaystyle k}$中，運動方程式維持牛頓力學形式：

${\displaystyle \mu {\frac {d^{2}\xi }{d\tau ^{2}}}=\epsilon X',\quad \mu {\frac {d^{2}\eta }{d\tau ^{2}}}=\epsilon Y',\quad \mu {\frac {d^{2}\zeta }{d\tau ^{2}}}=\epsilon Z'}$。

此關係式對應到${\displaystyle \mathbf {f} ^{0}=m\mathbf {a} ^{0}}$，因為${\displaystyle \mu =m}$，${\displaystyle \left({\frac {d^{2}\xi }{d\tau ^{2}}},\ {\frac {d^{2}\eta }{d\tau ^{2}}},\ {\frac {d^{2}\zeta }{d\tau ^{2}}}\right)=\mathbf {a} ^{0}}$，以及${\displaystyle \left(\epsilon X',\ \epsilon Y',\ \epsilon Z'\right)=\mathbf {f} ^{0}}$。透過轉換式轉換至一相對移動之系統${\displaystyle K}$，他得到了在新參考系中能觀察到之電磁分量方程式：

${\displaystyle {\frac {d^{2}x}{dt^{2}}}={\frac {\epsilon }{\mu }}{\frac {1}{\beta ^{3}}}X,\quad {\frac {d^{2}y}{dt^{2}}}={\frac {\epsilon }{\mu }}{\frac {1}{\beta }}\left(Y-{\frac {v}{V}}N\right),\quad {\frac {d^{2}z}{dt^{2}}}={\frac {\epsilon }{\mu }}{\frac {1}{\beta }}\left(Z+{\frac {v}{V}}M\right)}$。

此關係式對應到(5b)，其中${\displaystyle \mathbf {a} ={\frac {\mathbf {f} }{m}}\left({\frac {1}{\gamma ^{3}}},\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)}$，因為${\displaystyle \mu =m}$，${\displaystyle \left({\frac {d^{2}x}{dt^{2}}},\ {\frac {d^{2}y}{dt^{2}}},\ {\frac {d^{2}z}{dt^{2}}}\right)=\mathbf {a} }$，${\displaystyle \left[\epsilon X,\ \epsilon \left(Y-{\frac {v}{V}}N\right),\ \epsilon \left(Z+{\frac {v}{V}}M\right)\right]=\mathbf {f} }$，以及${\displaystyle \beta =\gamma }$。也因此，愛因斯坦決定了縱向與橫向質量，儘管他將之與瞬時慣性系${\displaystyle k}$中的力${\displaystyle \left(\epsilon X',\ \epsilon Y',\ \epsilon Z'\right)=\mathbf {f} ^{0}}$（可透過共動的彈簧秤測量）以及在系統${\displaystyle K}$中之三維加速度${\displaystyle \mathbf {a} }$做了關聯：^[39]

${\displaystyle {\begin{array}{c|c}{\begin{aligned}\mu \beta ^{3}{\frac {d^{2}x}{dt^{2}}}&=\epsilon X=\epsilon X'\\\mu \beta ^{2}{\frac {d^{2}y}{dt^{2}}}&=\epsilon \beta \left(Y-{\frac {v}{V}}N\right)=\epsilon Y'\\\mu \beta ^{2}{\frac {d^{2}z}{dt^{2}}}&=\epsilon \beta \left(Z+{\frac {v}{V}}M\right)=\epsilon Z'\end{aligned}}&{\begin{aligned}{\frac {\mu }{\left({\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}\right)^{3}}}&\ {\text{縱 向 質 量}}\\\\{\frac {\mu }{1-\left({\frac {v}{V}}\right)^{2}}}&\ {\text{橫 向 質 量}}\end{aligned}}\end{array}}}$

此關係式對應到(7b)，其中${\displaystyle m\mathbf {a} \left(\gamma ^{3},\ \gamma ^{2},\ \gamma ^{2}\right)=\mathbf {f} \left(1,\ \gamma ,\ \gamma \right)=\mathbf {f} ^{0}}$。

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1905年：龐加萊^{[H 1]}引入了三維加速度轉換式（1c）：

${\displaystyle {\frac {d\xi ^{\prime }}{dt^{\prime }}}={\frac {d\xi }{dt}}{\frac {1}{k^{3}\mu ^{3}}},\quad {\frac {d\eta ^{\prime }}{dt^{\prime }}}={\frac {d\eta }{dt}}{\frac {1}{k^{2}\mu ^{2}}}-{\frac {d\xi }{dt}}{\frac {\eta \epsilon }{k^{2}\mu ^{3}}},\quad {\frac {d\zeta ^{\prime }}{dt^{\prime }}}={\frac {d\zeta }{dt}}{\frac {1}{k^{2}\mu ^{2}}}-{\frac {d\xi }{dt}}{\frac {\zeta \epsilon }{k^{2}\mu ^{3}}}}$

其中${\displaystyle \left(\xi ,\ \eta ,\ \zeta \right)=\mathbf {u} }$，以及${\displaystyle k=\gamma }$，${\displaystyle \epsilon =v}$，${\displaystyle \mu =1+\xi \epsilon =1+u_{x}v}$。

他更進一步地引入了四維力，採如下形式：

${\displaystyle k_{0}X_{1},\quad k_{0}Y_{1},\quad k_{0}Z_{1},\quad k_{0}T_{1}}$

其中${\displaystyle k_{0}=\gamma _{0}}$ and ${\displaystyle \left(X_{1},\ Y_{1},\ Z_{1}\right)=\mathbf {f} }$，以及${\displaystyle T_{1}=\Sigma X_{1}\xi =\mathbf {f} \cdot \mathbf {u} }$.

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1906年：馬克斯·普朗克^{[H 8]}導出了運動方程式：

${\displaystyle {\frac {m{\ddot {x}}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}=e{\mathfrak {E}}_{x}-{\frac {e{\dot {x}}}{c^{2}}}\left({\dot {x}}{\mathfrak {E}}_{x}+{\dot {y}}{\mathfrak {E}}_{y}+{\dot {z}}{\mathfrak {E}}_{z}\right)+{\frac {e}{c}}\left({\dot {y}}{\mathfrak {H}}_{z}-{\dot {z}}{\mathfrak {H}}_{y}\right)\ {\text{等 等。}}}$

其中

${\displaystyle e\left({\dot {x}}{\mathfrak {E}}_{x}+{\dot {y}}{\mathfrak {E}}_{y}+{\dot {z}}{\mathfrak {E}}_{z}\right)={\frac {m\left({\dot {x}}{\ddot {x}}+{\dot {y}}{\ddot {y}}+{\dot {z}}{\ddot {z}}\right)}{\left(1-{\frac {q^{2}}{c^{2}}}\right)^{3/2}}}}$ and ${\displaystyle e{\mathfrak {E}}_{x}+{\frac {e}{c}}\left({\dot {y}}{\mathfrak {H}}_{z}-{\dot {z}}{\mathfrak {H}}_{y}\right)=X\ {\text{等 等。}}}$

以及

${\displaystyle {\frac {d}{dt}}\left\{{\frac {m{\dot {x}}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}\right\}=X\ {\text{等 等。}}}$

這些方程式對應到(5c)，其中${\displaystyle \mathbf {f} ={\frac {d\mathbf {p} }{dt}}={\frac {d(m\gamma \mathbf {u} )}{dt}}=m\gamma ^{3}\left({\frac {(\mathbf {a} \cdot \mathbf {u} )\mathbf {u} }{c^{2}}}\right)+m\gamma \mathbf {a} }$，以及${\displaystyle X=f_{x}}$，${\displaystyle q=v}$，${\displaystyle {\dot {x}}{\ddot {x}}+{\dot {y}}{\ddot {y}}+{\dot {z}}{\ddot {z}}=\mathbf {u} \cdot \mathbf {a} }$，與勞侖茲（1904年）所給的相應。

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1907年：愛因斯坦^[50]分析了一均勻加速參考系，得到與座標相依的時間膨脹及光速之關係式，類同於Kottler-Møller-Rindler座標。

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1907年：赫爾曼·閔考斯基^{[H 12]}定義了四維力（他稱之為「移動力」）與四維加速度之間的關係：

${\displaystyle m{\frac {d}{d\tau }}{\frac {dx}{d\tau }}=R_{x},\quad m{\frac {d}{d\tau }}{\frac {dy}{d\tau }}=R_{y},\quad m{\frac {d}{d\tau }}{\frac {dz}{d\tau }}=R_{z},\quad m{\frac {d}{d\tau }}{\frac {dt}{d\tau }}=R_{t}}$

對應到${\displaystyle m\mathbf {A} =\mathbf {F} }$。

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1908年：閔考斯基^{[H 13]}將${\displaystyle x,y,z,t}$對原時作微分的二次導數稱之為「加速向量」（四維加速度）。他展示了：在世界線上任一點${\displaystyle P}$，此向量的大小為${\displaystyle c^{2}/\varrho }$，其中${\displaystyle \varrho }$為從相對應「曲率雙曲線」（德語：Krümmungshyperbel）之中心點指向點${\displaystyle P}$所成之向量的大小。

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1909年：馬克斯·玻恩^{[H 11]} denotes the motion with constant magnitude of Minkowski's acceleration vector as "hyperbolic motion" (德語：Hyperbelbewegung), in the course of his study of rigidly accelerated motion. He set ${\displaystyle p=dx/d\tau }$ (now called proper velocity) and ${\displaystyle q=-dt/d\tau ={\sqrt {1+p^{2}/c^{2}}}}$ as Lorentz factor and ${\displaystyle \tau }$ as proper time, with the transformation equations

${\displaystyle x=-q\xi ,\quad y=\eta ,\quad z=\zeta ,\quad t={\frac {p}{c^{2}}}\xi }$.

which corresponds to (8) with ${\displaystyle \xi =c^{2}/\alpha }$ and ${\displaystyle p=c\sinh(\alpha \tau /c)}$. Eliminating ${\displaystyle p}$ Born derived the hyperbolic equation ${\displaystyle x^{2}-c^{2}t^{2}=\xi ^{2}}$, and defined the magnitude of acceleration as ${\displaystyle b=c^{2}/\xi }$. He also noticed that his transformation can be used to transform into a "hyperbolically accelerated reference system" (德語：hyperbolisch beschleunigtes Bezugsystem).

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1909年：古斯塔夫·黑格洛茲（英语：Gustav Herglotz）^{[H 14]} extends Born's investigation to all possible cases of rigidly accelerated motion, including uniform rotation.

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1910年：阿諾·索末菲^{[H 15]} brought Born's formulas for hyperbolic motion in a more concise form with ${\displaystyle l=ict}$ as the imaginary time variable and ${\displaystyle \varphi }$ as an imaginary angle:

${\displaystyle x=r\cos \varphi ,\quad y=y',\quad z=z',\quad l=r\sin \varphi }$

He noted that when ${\displaystyle r,y,z}$ are variable and ${\displaystyle \varphi }$ is constant, they describe the worldline of a charged body in hyperbolic motion. But if ${\displaystyle r,y,z}$ are constant and ${\displaystyle \varphi }$ is variable, they denote the transformation into its rest frame.

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1911年：索末菲^{[H 3]} explicitly used the expression "proper acceleration" (德語：Eigenbeschleunigung) for the quantity ${\displaystyle {\dot {v}}_{0}}$ in ${\displaystyle {\dot {v}}={\dot {v}}_{0}\left(1-\beta ^{2}\right)^{3/2}}$, which corresponds to (4a), as the acceleration in the momentary inertial frame.

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1911年：黑格洛茲^{[H 4]} explicitly used the expression "rest acceleration" (德語：Ruhbeschleunigung) instead of proper acceleration. He wrote it in the form ${\displaystyle \gamma _{l}^{0}=\beta ^{3}\gamma _{l}}$ and ${\displaystyle \gamma _{t}^{0}=\beta ^{2}\gamma _{t}}$ which corresponds to (4a), where ${\displaystyle \beta }$ is the Lorentz factor and ${\displaystyle \gamma _{l}^{0}}$ or ${\displaystyle \gamma _{t}^{0}}$ are the longitudinal and transverse components of rest acceleration.

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1911年：馬克斯·馮·勞厄^{[H 2]} derived in the first edition of his monograph "Das Relativitätsprinzip" the transformation for three-acceleration by differentiation of the velocity addition

${\displaystyle {\begin{aligned}{\mathfrak {\dot {q}}}_{x}&=\left({\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}+v{\mathfrak {q}}_{x}^{\prime }}}\right)^{3}{\mathfrak {\dot {q}}}_{x}^{\prime },&{\mathfrak {\dot {q}}}_{y}&=\left({\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}+v{\mathfrak {q}}_{x}^{\prime }}}\right)^{2}\left({\mathfrak {\dot {q}}}_{x}^{\prime }-{\frac {v{\mathfrak {q}}_{y}^{\prime }{\mathfrak {\dot {q}}}_{x}^{\prime }}{c^{2}+v{\mathfrak {q}}_{x}^{\prime }}}\right),\end{aligned}}}$

equivalent to (1c) as well as to Poincaré (1905/6). From that he derived the transformation of rest acceleration (equivalent to 4a), and eventually the formulas for hyperbolic motion which corresponds to (8):

${\displaystyle \pm {\mathfrak {q}}_{x}=\pm {\frac {dx}{dt}}={\frac {cbt}{\sqrt {c^{2}+b^{2}t^{2}}}},\quad \pm \left(x-x_{0}\right)={\frac {c}{b}}{\sqrt {c^{2}+b^{2}t^{2}}},}$

thus

${\displaystyle x^{2}-c^{2}t^{2}=x^{2}-u^{2}=c^{4}/b^{2},\quad y=\eta ,\quad z=\zeta }$,

and the transformation into a hyperbolic reference system with imaginary angle ${\displaystyle \varphi }$:

${\displaystyle {\begin{array}{c|c}{\begin{aligned}X&=R\cos \varphi \\L&=R\sin \varphi \end{aligned}}&{\begin{aligned}R^{2}&=X^{2}+L^{2}\\\tan \varphi &={\frac {L}{X}}\end{aligned}}\end{array}}}$.

He also wrote the transformation of three-force as

${\displaystyle {\begin{aligned}{\mathfrak {K}}_{x}&={\frac {{\mathfrak {K}}_{x}^{\prime }+{\frac {v}{c^{2}}}({\mathfrak {q'K'}})}{1+{\frac {v{\mathfrak {q}}_{x}^{\prime }}{c^{2}}}}},&{\mathfrak {K}}_{y}&={\mathfrak {K}}_{y}^{\prime }{\frac {\sqrt {1-\beta ^{2}}}{1+{\frac {v{\mathfrak {q}}_{x}^{\prime }}{c^{2}}}}},&{\mathfrak {K}}_{z}&={\mathfrak {K}}_{z}^{\prime }{\frac {\sqrt {1-\beta ^{2}}}{1+{\frac {v{\mathfrak {q}}_{x}^{\prime }}{c^{2}}}}},\end{aligned}}}$

equivalent to (6a) as well as to Poincaré (1905).

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1912年-1914年：弗里德里希·科特勒（英语：Friedrich Kottler）^[51]obtained general covariance of Maxwell's equations, and used four-dimensional Frenet-Serret formulas to analyze the Born rigid motions given by Herglotz (1909). He also obtained the proper reference frames for hyperbolic motion and uniform circular motion.

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1913年：馮·勞厄^{[H 7]} replaced in the second edition of his book the transformation of three-acceleration by Minkowski's acceleration vector for which he coined the name "four-acceleration" (德語：Viererbeschleunigung), defined by ${\displaystyle {\dot {Y}}={\frac {dY}{d\tau }}}$ with ${\displaystyle Y}$ as four-velocity. He showed, that the magnitude of four-acceleration corresponds to the rest acceleration ${\displaystyle {\dot {\mathfrak {q}}}^{0}}$ by

${\displaystyle |{\dot {Y|}}={\frac {1}{c}}|{\dot {\mathfrak {q}}}^{0}|}$,

which corresponds to (4b). Subsequently, he derived the same formulas as in 1911 for the transformation of rest acceleration and hyperbolic motion, and the hyperbolic reference frame.

• 加速度
• 狹義相對論
• 速度加成式
• 四維加速度
• 固有加速度

• Ashtekar, A., Petkov, V. Springer Handbook of Spacetime. Springer. 2014. ISBN 3642419925. 

• Bini, D., Lusanna, L., & Mashhoon, B. Limitations of radar coordinates. International Journal of Modern Physics D. 2005, 14 (8): 1413–1429. arXiv:gr-qc/0409052 . doi:10.1142/S0218271805006961. 

• Ferraro, R. Einstein's Space-Time: An Introduction to Special and General Relativity. Spektrum. 2007. ISBN 0387699465. 

• Fraundorf, P. A traveler-centered intro to kinematics: IV–B. 2012. arXiv:1206.2877 . 

• French, A.P. Special Relativity. CRC Press. 1968. ISBN 1420074814. 

• Freund, J. Special Relativity for Beginners: A Textbook for Undergraduates. World Scientific. 2008. ISBN 981277159X. 

• Gourgoulhon, E. Special Relativity in General Frames: From Particles to Astrophysics. Springer. 2013. ISBN 3642372767. 

• von Laue, M. Die Relativitätstheorie, Band 1 fourth edition of "Das Relativitätsprinzip”. Vieweg. 1921. ; First edition 1911, second expanded edition 1913, third expanded edition 1919.

• Koks, D. Explorations in Mathematical Physics. Springer. 2006. ISBN 0387309438. 

• Kopeikin,S., Efroimsky, M., Kaplan, G. Relativistic Celestial Mechanics of the Solar System. John Wiley & Sons. 2011. ISBN 3527408568. 

• Miller, Arthur I. Albert Einstein’s special theory of relativity. Emergence (1905) and early interpretation (1905–1911). Reading: Addison–Wesley. 1981. ISBN 0-201-04679-2. 

• Misner, C. W., Thorne, K. S., and Wheeler, J. A,. Gravitation. Freeman. 1973. ISBN 0716703440. 

• Møller, C. The theory of relativity. Oxford Clarendon Press. 1955 [1952]. 

• Nikolić, H. Relativistic contraction and related effects in noninertial frames. Physical Review A. 2000, 61 (3): 032109. arXiv:gr-qc/9904078 . doi:10.1103/PhysRevA.61.032109. 

• Pauli, W. Die Relativitätstheorie. Encyclopädie der mathematischen Wissenschaften 5.2. 1921: 539–776.  New edition 2013: Editor: Domenico Giulini, Springer, 2013 ISBN 3642583555.

• Pauri, M., & Vallisneri, M. Märzke-Wheeler coordinates for accelerated observers in special relativity. Foundations of Physics Letters. 2000, 13 (5): 401–425. arXiv:gr-qc/0006095 . doi:10.1023/A:1007861914639. 

• Pfeffer, J. & Nir, S. Modern Physics: An Introductory Text. World Scientific. 2012. ISBN 1908979577. 

• Shadowitz, A. Special relativity Reprint of 1968. Courier Dover Publications. 1988. ISBN 0-486-65743-4. 

• Rahaman, F. The Special Theory of Relativity: A Mathematical Approach. Springer. 2014. ISBN 8132220803. 

• Rebhan, E. Theoretische Physik I. Heidelberg · Berlin: Spektrum. 1999. ISBN 3-8274-0246-8. 

• Rindler, W. Essential Relativity. Springer. 1977. ISBN 354007970X. 

• Tolman, R.C. The theory of the Relativity of Motion. University of California Press. 1917. OCLC 13129939. 

• Zahar, E. Einstein's Revolution: A Study in Heuristic. Open Court Publishing Company. 1989. ISBN 0-8126-9067-2. 

1. ^ ^{1.0 1.1} Poincaré, Henri. Sur la dynamique de l'électron [On the Dynamics of the Electron]. Rendiconti del Circolo matematico di Palermo. 1906, 21: 129–176 [1905]. 
2. ^ ^{2.0 2.1 2.2 2.3} Laue, Max von. Das Relativitätsprinzip. Braunschweig: Vieweg. 1911. 
3. ^ ^{3.0 3.1 3.2} Sommerfeld, Arnold. Über die Struktur der gamma-Strahlen. Sitzungsberichte der mathematematisch-physikalischen Klasse der K. B. Akademie der Wissenschaften zu München. 1911, (1): 1–60. 
4. ^ ^{4.0 4.1 4.2} Herglotz, G. Über die Mechanik des deformierbaren Körpers vom Standpunkte der Relativitätstheorie. Annalen der Physik. 1911, 341 (13): 493–533 [2017-06-07]. doi:10.1002/andp.19113411303. （原始内容存档于2019-05-22）. 
5. ^ ^{5.0 5.1} Lorentz, Hendrik Antoon. Simplified Theory of Electrical and Optical Phenomena in Moving Systems. Proceedings of the Royal Netherlands Academy of Arts and Sciences. 1899, 1: 427–442. 
6. ^ ^{6.0 6.1 6.2 6.3 6.4 6.5} Lorentz, Hendrik Antoon. Electromagnetic phenomena in a system moving with any velocity smaller than that of light. Proceedings of the Royal Netherlands Academy of Arts and Sciences. 1904, 6: 809–831. 
7. ^ ^{7.0 7.1} Laue, Max von. Das Relativitätsprinzip 2. Ausgabe. Braunschweig: Vieweg. 1913. 
8. ^ ^{8.0 8.1 8.2} Planck, Max. Das Prinzip der Relativität und die Grundgleichungen der Mechanik [The Principle of Relativity and the Fundamental Equations of Mechanics]. Verhandlungen Deutsche Physikalische Gesellschaft. 1906, 8: 136–141. 
9. ^ ^{9.0 9.1} Poincaré, Henri. Sur la dynamique de l'électron [On the Dynamics of the Electron]. Comptes rendus hebdomadaires des séances de l'Académie des sciences. 1905, 140: 1504–1508. 
10. ^ ^{10.0 10.1} Einstein, Albert. Zur Elektrodynamik bewegter Körper. Annalen der Physik. 1905, 322 (10): 891–921. ; See also: English translation （页面存档备份，存于互联网档案馆）.
11. ^ ^{11.0 11.1} Born, Max. Die Theorie des starren Körpers in der Kinematik des Relativitätsprinzips [The Theory of the Rigid Electron in the Kinematics of the Principle of Relativity]. Annalen der Physik. 1909, 335 (11): 1–56. doi:10.1002/andp.19093351102. 
12. ^ Minkowski, Hermann, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern [The Fundamental Equations for Electromagnetic Processes in Moving Bodies], Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1908: 53–111 [1907] 
13. ^ Minkowski, Hermann. Raum und Zeit. Vortrag, gehalten auf der 80. Naturforscher-Versammlung zu Köln am 21. September 1908. [Space and Time]. Jahresberichte der Deutschen Mathematiker-Vereinigung (Leipzig). 1909 [1908]. 
14. ^ Herglotz, G. Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper [On bodies that are to be designated as "rigid" from the standpoint of the relativity principle]. Annalen der Physik. 1910, 336 (2): 393–415 [1909]. doi:10.1002/andp.19103360208. 
15. ^ Sommerfeld, Arnold. Zur Relativitätstheorie II: Vierdimensionale Vektoranalysis [On the Theory of Relativity II: Four-dimensional Vector Analysis]. Annalen der Physik. 1910, 338 (14): 649–689. doi:10.1002/andp.19103381402.