狭义相对论中的加速度

狭义相对论中的加速度类似于牛顿力学中的概念，乃速度对于时间的微分。因为相对论中的洛伦兹转换及时间膨胀，时间与距离的概念变为复杂，因此“加速度”的定义也变得复杂。狭义相对论为平直闵可夫斯基时空的理论，即使加速度存在依然有效，前提是能量动量张量所造成的重力场效应可以忽略。否则，则需用到广义相对论以及弯曲时空来诠释。在地球表面附近，时空弯曲程度不明显，因此实务上采用狭义相对论来诠释物理现象仍是合宜作法，比如粒子加速器实验。^[1]

如同在外界惯性坐标系中的测量，三维空间中的普通加速度（称为“三维加速度”或“坐标加速度”）的转换式可以推导得出。此外作为一特例，也可用共动（comoving）的加速规来测量固有加速度。另一种有用的形式是四维加速度，其分量可透过洛伦兹转换在不同参考系中做连结。连结加速度与力的运动方程也可得到。几种特殊形式的加速物体运动方程以及它们的弯曲世界线可以透过对上述方程的积分求得。知名的特例如双曲运动（英语：hyperbolic motion (relativity)），适用于常数值纵向固有加速度的例子，以及等速率圆周运动。最后，在狭义相对论的架构下，描述加速参考系中的物理现象亦为可行。

历史演进上，在相对论发展的早年即已出现包含加速度的相对论性方程，在早年的教科书中有整理，如马克斯·冯·劳厄（1911年、1921年）^[2]或沃尔夫冈·泡利（1921年）。^[3]举例来说，运动方程以及加速度转换式于以下学者的论文中建立起来：亨德里克·洛伦兹（1899年、1904年）、儒勒·昂利·庞加莱（1905年）、阿尔伯特·爱因斯坦（1905年）、马克斯·普朗克（1906年）；四维加速度、固有加速度与双曲运动的分析参见赫尔曼·闵可夫斯基 （1908年）、马克斯·玻恩（1909年）、古斯塔夫·赫格洛茨（英语：Gustav Herglotz）（1909年）、阿诺·索末菲（1910年）、冯·劳厄（1911年）。

三维加速度

在牛顿力学与狭义相对论中，三维加速度或坐标加速度的定义保持一致。${\displaystyle \mathbf {a} =\left(a_{x},\ a_{y},\ a_{z}\right)}$是速度${\displaystyle \mathbf {u} =\left(u_{x},\ u_{y},\ u_{z}\right)}$对坐标时间的一阶导数，亦即是位置${\displaystyle \mathbf {r} =\left(x,\ y,\ z\right)}$对坐标时间的二阶导数：

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

然而在另一相异的惯性参考系中做三维加速度测量时，两项理论的预测就出现重大歧异。牛顿力学中，时间是绝对的（${\displaystyle t'=t}$），采用的惯性系转换式为伽利略转换。因此，从伽利略转换推导而得的三维加速度在所有惯性系中皆相同：^[4]

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

相反地，在狭义相对论中，${\displaystyle \mathbf {r} }$与${\displaystyle t}$两者皆与洛伦兹转换相依，因此三维加速度${\displaystyle \mathbf {a} }$及其分量在不同惯性系也各不相同。当惯性系间的相对速度是沿着x轴，即${\displaystyle v=v_{x}}$（${\displaystyle \gamma _{v}=1/{\sqrt {1-v^{2}/c^{2}}}}$为相对应的洛伦兹因子），洛伦兹转换式为：

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

1a

或是对于一长度${\displaystyle v}$及任意方向的速度矢量${\displaystyle \mathbf {v} =\left(v_{x},\ v_{y},\ v_{z}\right)}$（其中${\displaystyle |\mathbf {v} |=v={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}$），洛伦兹转换式为：^[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

为了求得三维加速度的转换式，必须分别对洛伦兹转换式中的空间坐标${\displaystyle \mathbf {r} }$及${\displaystyle \mathbf {r} '}$做时间${\displaystyle t}$与${\displaystyle t'}$的微分。首先是得到三维速度${\displaystyle \mathbf {u} }$及${\displaystyle \mathbf {u} '}$的转换式（亦称为速度加成式）；尔后再次做时间${\displaystyle t}$与${\displaystyle t'}$的微分运算而得到三维加速度${\displaystyle \mathbf {a} }$及${\displaystyle \mathbf {a} '}$的转换式。从式(1a)出发，所得到的转换式为平行（x方向）与垂直（y、z方向）于速度${\displaystyle v=v_{x}}$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

若从式(1b)出发，则得到通解，速度与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

此转换式表示：若有两惯性系${\displaystyle S}$与${\displaystyle S'}$，两者相对速度${\displaystyle \mathbf {v} }$，则${\displaystyle S}$系中测到一物体瞬时速度为${\displaystyle \mathbf {u} }$、加速度为${\displaystyle \mathbf {a} }$，该物体在${\displaystyle S'}$系中则具有瞬时速度${\displaystyle \mathbf {u} '}$、加速度${\displaystyle \mathbf {a} '}$。一如速度加成式，这些加速度转换式可保证一物体无法加速到光速，遑论超过光速。

四维加速度

主条目：四维加速度

若改采用四维矢量，即${\displaystyle \mathbf {R} }$乃四维位置，${\displaystyle \mathbf {U} }$乃四维速度，则一物体的四维加速度${\displaystyle \mathbf {A} =\left(A_{t},\ A_{x},\ A_{y},\ A_{z}\right)=\left(A_{t},\ \mathbf {A} _{r}\right)}$可透过对固有时${\displaystyle \mathbf {\tau } }$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${\displaystyle \mathbf {a} }$为物体的三维加速度；${\displaystyle \mathbf {u} }$为物体的瞬时三维速度，长度为${\displaystyle |\mathbf {u} |=u}$，所对应的洛伦兹因子为${\displaystyle \gamma =1/{\sqrt {1-u^{2}/c^{2}}}}$。若只考虑空间分量且速度是沿着x方向（即${\displaystyle u=u_{x}}$），且只考虑与速度平行（x方向）或垂直（y、z方向）的加速度，则关系式可简化为：^[15][16]

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

与前述的三维加速度不同，四维加速度不需要推导新的转换关系式，因为所有四维矢量（包括四维加速度）在两个具有相对速度${\displaystyle v}$的惯性系之间都呈现洛伦兹协变性。因此只要将式(1a)中的${\displaystyle x,\ y,\ z,\ ct}$代换为${\displaystyle A_{x},\ A_{y},\ A_{z},\ A_{t}}$，即为四维加速度在两惯性系之间的转换式：^[17]

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

又或将式(1b)中的${\displaystyle \mathbf {r} ,\ ct}$代换为${\displaystyle \mathbf {A} _{r},\ A_{t}}$，亦可得到任意相对速度${\displaystyle \mathbf {v} }$情形的转换式：

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

此外，四维加速度自身内积${\displaystyle \mathbf {A} ^{2}=-A_{t}^{2}+\mathbf {A} _{r}^{2}}$及矢量大小${\displaystyle |\mathbf {A} |={\sqrt {\mathbf {A} ^{2}}}}$为不变量（这里所采用的度规标记（metric signature）为(−,+,+,+)），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

固有加速度

主条目：固有加速度

在无限小的瞬间，总有一惯性系${\displaystyle S'}$与一加速物体（加速参考系）相对静止，即两者相对速度为0。在这样的惯性系中，洛伦兹转换成立。相对应的三维加速度${\displaystyle \mathbf {a} ^{0}=\left(a_{x}^{0},\ a_{y}^{0},\ a_{z}^{0}\right)}$可透过加速规直接测量，称之为固有加速度^{[19][H 3]}或静止加速度。^{[20][H 4]}此瞬时惯性系${\displaystyle S'}$中的${\displaystyle \mathbf {a} ^{0}}$与外界另一惯性系${\displaystyle S}$所测到的${\displaystyle \mathbf {a} }$之间的关系式为(1c)与(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（${\displaystyle u=u_{x}=v=v_{x}}$），且只考虑与速度平行（x方向）或垂直（y、z方向）的加速度，则关系式为：^{[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

或将固有加速度此一特例条件代入通式(1d)，其中任意方向的速度${\displaystyle \mathbf {u} }$其长度为${\displaystyle |\mathbf {u} |=u}$：^[21][22][18]

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

另外固有加速度与四维加速度的长度也有密切关系：如前述，四维加速度的长度为不变量，可在瞬间共动惯性系${\displaystyle S'}$中被测定，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

因此四维加速度长度对应到固有加速度长度。将此结果与式(3)结合，可得到将${\displaystyle S'}$系中${\displaystyle \mathbf {a} ^{0}}$与${\displaystyle S}$系中${\displaystyle \mathbf {a} }$连结的另一种关系式求法，亦即：^[13][18]

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

从这里可得到式(4a)，只要再次采用如下条件：速度沿着x方向（${\displaystyle u=u_{x}}$），只考虑与速度平行（x方向）或垂直（y、z方向）的加速度。

加速度与力

更多信息：相对论力学 § 力和狭义相对论中的质量

四维力${\displaystyle \mathbf {F} }$可写为三维力${\displaystyle \mathbf {f} }$的函数：${\displaystyle \mathbf {F} =\gamma \left((\mathbf {f} \cdot \mathbf {u} )/c,\ \mathbf {f} \right)}$。四维力、四维加速度（式(2)）以及不变质量${\displaystyle m}$则具有如下关系式：${\displaystyle \mathbf {F} =m\mathbf {A} }$^[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，三维力与三维加速度的关系式则可写成：^[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

当速度沿着x方向，即${\displaystyle u=u_{x}}$，且仅考虑平行（x方向）或垂直（y、z方向）于速度方向的加速度与力，则三维力与三维加速度的关系式为：^{[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

牛顿力学中，将质量简单定义为三维力与三维加速度的比值；此想法在狭义相对论中变为拙劣，因为这样定义的质量将与速度的大小及方向相依。是故如下曾出现在旧版教科书中的质量定义在当代已舍弃不用：^{[28][29][H 6]}

${\displaystyle m_{\Vert }={\frac {f_{x}}{a_{x}}}=m\gamma ^{3}}$，称为“纵向质量”；

${\displaystyle m_{\perp }={\frac {f_{y}}{a_{y}}}={\frac {f_{z}}{a_{z}}}=m\gamma }$，称为“横向质量”。

式(5a)中三维加速度与三维力的关系式也可透过运动方程求得：^{[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${\displaystyle \mathbf {p} }$是三维动量。若惯性参考系${\displaystyle S}$与${\displaystyle S'}$间的相对速度为沿着x方向，即${\displaystyle v=v_{x}}$，且仅考虑平行（x方向）或垂直（y、z方向）于速度方向的情形时，${\displaystyle S}$中的三维力${\displaystyle \mathbf {f} }$与${\displaystyle S'}$中的三维力${\displaystyle \mathbf {f} '}$之间的转换关系式可透过对${\displaystyle \mathbf {u} }$、${\displaystyle \mathbf {a} }$、${\displaystyle m\gamma }$、${\displaystyle d(m\gamma )/dt}$等相关转换式做代换，或透过四维力进行洛伦兹转换后取其分量，而得到以下结果：^{[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

又或将之推广到任意方向的${\displaystyle \mathbf {u} }$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

固有加速度与固有力

透过一共动弹簧秤来测量一瞬时惯性系中的力${\displaystyle \mathbf {f} ^{0}}$可称为固有力。^[34][35]从式(6a)与式(6b)，设定${\displaystyle \mathbf {f} '=\mathbf {f} ^{0}}$、${\displaystyle \mathbf {u} '=0}$、${\displaystyle \mathbf {u} =\mathbf {v} }$、${\displaystyle \gamma =\gamma _{v}}$等条件，可得到固有力的关系式。当速度沿x轴，${\displaystyle u=u_{x}=v=v_{x}}$，且仅考虑平行（x方向）或垂直（y、z方向）之加速度，可采用式(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、大小为${\displaystyle |\mathbf {u} |=u}$的速度${\displaystyle \mathbf {u} }$之通则采用式(6b)：^[36][37]

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

因为${\displaystyle \gamma =1}$，牛顿力学关系式${\displaystyle \mathbf {f} ^{0}=m\mathbf {a} ^{0}}$在瞬时惯性系中成立，是故式(4a)、式(5b)、式(7a)可归结为：^[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

透过这些式子，历史上对横向质量${\displaystyle m_{\perp }}$定义中的明显矛盾可以得到解释。^[39]爱因斯坦（1905年）的定义是固有力与三维加速度的比值：^{[H 10]}

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

而洛伦兹（1899年与1904年）、普朗克（1906年）的定义则是三维力与三维加速度的比值^{[H 6]}

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

弯曲世界线

对运动方程做积分，可得到一加速物体的世界线，对应到一连串的瞬时惯性系。如此则需考虑相关的“时钟假设”：^[40][41]共动时钟的固有时与加速度无关。也就是说，对外部惯性系而言，这些时钟的时间膨胀只相依于和外部惯性系之间的相对速度。以下是两个简单的弯曲世界线范例，透过对式(4a)中固有加速度的积分而得：

a) 双曲运动（英语：Hyperbolic motion (relativity)）：式(4a)中为恒定的纵向固有加速度${\displaystyle \alpha =a_{x}^{0}=a_{x}\gamma ^{3}}$，造成世界线^{[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

此世界线对应到双曲方程${\displaystyle c^{4}/\alpha ^{2}=\left(x+c^{2}/\alpha \right)^{2}-c^{2}t^{2}}$，因此这样的移动物体世界线被称作双曲运动。此方程组常用来计算孪生子佯谬或贝尔太空船佯谬的不同版本案例，亦与等加速度太空旅行（英语：space travel using constant acceleration）有关。

b) 式(4a)中为恒定的横向固有加速度${\displaystyle a_{y}^{0}=a_{y}\gamma ^{2}}$，可视为向心加速度，^[13]造成一匀速旋转物体的世界线：^[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${\displaystyle v=r\Omega _{0}}$为切线速率，${\displaystyle r}$是轨道半径；角速度${\displaystyle \Omega _{0}}$为坐标时间的函数，另外${\displaystyle \Omega =\gamma \Omega _{0}}$为固有角速度。

加速参考系

主条目：固有参考系 (平直时空)、润德勒坐标和玻恩坐标

加速运动亦可透过加速坐标系或曲线坐标系（英语：curvilinear coordinates）来描述。以此方式建立的固有参考系与费米坐标（英语：Fermi coordinates）密切相关。^[46][47]举例而言，一双曲加速参考系的坐标有时称为润德勒坐标，而匀速旋转参考系的情形，则称为旋转圆柱坐标，或称玻恩坐标（英语：Born coordinates）。

历史

更多信息请参见冯·劳厄^[2]、泡利^[3]、米勒^[48]、Zahar^[49]、Gourgoulhon^[47]，以及狭义相对论发现史中的历史资料。

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1899年：在一静止静电粒子系统${\displaystyle S_{0}}$（静止于乙太（英语：Lorentz ether theory）中）及具有相对平移的另一系统${\displaystyle S}$之间，亨德里克·洛伦兹^{[H 5]}在包含一因子${\displaystyle \epsilon }$的情况下，推导出了加速度、力、质量之间的正确关系，下式中${\displaystyle k}$为洛伦兹因子：

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

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

(5b)中${\displaystyle \mathbf {f} /(m\mathbf {a} )}$项：${\displaystyle {\frac {k^{3}}{\epsilon }}}$, ${\displaystyle {\frac {k}{\epsilon }}}$, ${\displaystyle {\frac {k}{\epsilon }}}$，因此为纵向与横向质量；

洛伦兹提到了他无法决定${\displaystyle \epsilon }$的值。若当时他设${\displaystyle \epsilon =1}$，则他的关系式会跟相对论关系式一模一样。

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1904年：洛伦兹^{[H 6]}以更详尽的方法推导初上述关系式，采用了静止于系统${\displaystyle \Sigma '}$及移动于系统${\displaystyle \Sigma }$之粒子的性质，搭配上新的辅助变数${\displaystyle l}$，相当于1899年推导中的${\displaystyle 1/\epsilon }$，而得到：

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

(7b)中，${\displaystyle m\mathbf {a} }$为${\displaystyle m\mathbf {a} ^{0}}$之函数，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，${\displaystyle \mathbf {a} }$为${\displaystyle \mathbf {a} ^{0}}$之函数，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，纵向与横向质量为静质量之函数，可得${\displaystyle m(\Sigma )=\left(k^{3}l,\ kl,\ kl\right)m(\Sigma ')}$。

这次洛伦兹可以展示${\displaystyle l=1}$，而他的数学式与相对论形式完全相符。他也推导了运动方程：

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

对应于(5c)里的${\displaystyle \mathbf {f} ={\frac {d\mathbf {p} }{dt}}={\frac {d(m\gamma \mathbf {u} )}{dt}}}$，其中${\displaystyle l=1}$、${\displaystyle {\mathfrak {F}}=\mathbf {f} }$、${\displaystyle {\mathfrak {G}}=\mathbf {p} }$、${\displaystyle {\mathfrak {w}}=\mathbf {u} }$、${\displaystyle k=\gamma }$，以及视为电磁静质量（英语：electromagnetic mass）的${\displaystyle m=e^{2}/(6\pi c^{2}R)}$。他更进一步地阐述：这些数学式不只适用于带电粒子的力与质量，也适用于其他过程，因此使得乙太中地球运动的影响无法被侦测出来。

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1905年：儒勒·昂利·庞加莱^{[H 9]}引入了三维力的转换式(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${\displaystyle k}$为洛伦兹因子，${\displaystyle \rho }$为电荷密度。或以现代符号表记：${\displaystyle \epsilon =v}$，${\displaystyle \xi =u_{x}}$，${\displaystyle \left(X_{1},\ Y_{1},\ Z_{1}\right)=\mathbf {f} }$，以及${\displaystyle \Sigma X_{1}\xi =\mathbf {f} \cdot \mathbf {u} }$。与洛伦兹相同，他设定${\displaystyle l=1}$。

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1905年：阿尔伯特·爱因斯坦^{[H 10]}以其狭义相对论为基础，推导出运动方程。此表示出等价惯性系之间的关系，而不需要用到机械式乙太。爱因斯坦总结到，在一瞬时惯性系${\displaystyle k}$中，运动方程维持牛顿力学形式：

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

此关系式对应到${\displaystyle \mathbf {f} ^{0}=m\mathbf {a} ^{0}}$，因为${\displaystyle \mu =m}$，${\displaystyle \left({\frac {d^{2}\xi }{d\tau ^{2}}},\ {\frac {d^{2}\eta }{d\tau ^{2}}},\ {\frac {d^{2}\zeta }{d\tau ^{2}}}\right)=\mathbf {a} ^{0}}$，以及${\displaystyle \left(\epsilon X',\ \epsilon Y',\ \epsilon Z'\right)=\mathbf {f} ^{0}}$。透过转换式转换至一相对移动之系统${\displaystyle K}$，他得到了在新参考系中能观察到之电磁分量方程：

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

此关系式对应到(5b)，其中${\displaystyle \mathbf {a} ={\frac {\mathbf {f} }{m}}\left({\frac {1}{\gamma ^{3}}},\ {\frac {1}{\gamma }},\ {\frac {1}{\gamma }}\right)}$，因为${\displaystyle \mu =m}$，${\displaystyle \left({\frac {d^{2}x}{dt^{2}}},\ {\frac {d^{2}y}{dt^{2}}},\ {\frac {d^{2}z}{dt^{2}}}\right)=\mathbf {a} }$，${\displaystyle \left[\epsilon X,\ \epsilon \left(Y-{\frac {v}{V}}N\right),\ \epsilon \left(Z+{\frac {v}{V}}M\right)\right]=\mathbf {f} }$，以及${\displaystyle \beta =\gamma }$。也因此，爱因斯坦决定了纵向与横向质量，尽管他将之与瞬时惯性系${\displaystyle k}$中的力${\displaystyle \left(\epsilon X',\ \epsilon Y',\ \epsilon Z'\right)=\mathbf {f} ^{0}}$（可透过共动的弹簧秤测量）以及在系统${\displaystyle K}$中之三维加速度${\displaystyle \mathbf {a} }$做了关联：^[39]

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

此关系式对应到(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：庞加莱^{[H 1]}引入了三维加速度转换式（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${\displaystyle k=\gamma }$，${\displaystyle \epsilon =v}$，${\displaystyle \mu =1+\xi \epsilon =1+u_{x}v}$。

他更进一步地引入了四维力，采如下形式：

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

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：马克斯·普朗克^{[H 8]}导出了运动方程：

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

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

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

这些方程对应到(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${\displaystyle X=f_{x}}$，${\displaystyle q=v}$，${\displaystyle {\dot {x}}{\ddot {x}}+{\dot {y}}{\ddot {y}}+{\dot {z}}{\ddot {z}}=\mathbf {u} \cdot \mathbf {a} }$，与洛伦兹（1904年）所给的相应。

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1907年：爱因斯坦^[50]分析了一均匀加速参考系，得到与坐标相依的时间膨胀及光速之关系式，类同于Kottler-Møller-Rindler坐标。

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1907年：赫尔曼·闵可夫斯基^{[H 12]}定义了四维力（他称之为“移动力”）与四维加速度之间的关系：

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

对应到${\displaystyle m\mathbf {A} =\mathbf {F} }$。

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1908年：闵可夫斯基^{[H 13]}将${\displaystyle x,y,z,t}$对固有时作微分的二次导数称之为“加速矢量”（四维加速度）。他展示了：在世界线上任一点${\displaystyle P}$，此矢量的大小为${\displaystyle c^{2}/\varrho }$，其中${\displaystyle \varrho }$为从相对应“曲率双曲线”（德语：Krümmungshyperbel）之中心点指向点${\displaystyle P}$所成之矢量的大小。

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1909年：马克斯·玻恩^{[H 11]} denotes the motion with constant magnitude of Minkowski's acceleration vector as "hyperbolic motion" (德语：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" (德语：hyperbolisch beschleunigtes Bezugsystem).

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1909年：古斯塔夫·黑格洛兹（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：阿诺·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" (德语：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：黑格洛兹^{[H 4]} explicitly used the expression "rest acceleration" (德语：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：马克斯·冯·劳厄^{[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：冯·劳厄^{[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" (德语：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.

相关条目

• 加速度
• 狭义相对论
• 速度加成式
• 四维加速度
• 固有加速度