# 四维加速度

## 惯性坐标系中的四维加速度

${\displaystyle \mathbf {A} ={\frac {d\mathbf {U} }{d\tau }}=\left(\gamma _{u}{\dot {\gamma }}_{u}c,\gamma _{u}^{2}\mathbf {a} +\gamma _{u}{\dot {\gamma }}_{u}\mathbf {u} \right)=\left(\gamma _{u}^{4}{\frac {\mathbf {a} \cdot \mathbf {u} }{c}},\gamma _{u}^{2}\mathbf {a} +\gamma _{u}^{4}{\frac {\left(\mathbf {a} \cdot \mathbf {u} \right)}{c^{2}}}\mathbf {u} \right)=\left(\gamma _{u}^{4}{\frac {\mathbf {a} \cdot \mathbf {u} }{c}},\gamma _{u}^{4}\left(\mathbf {a} +{\frac {\mathbf {u} \times \left(\mathbf {u} \times \mathbf {a} \right)}{c^{2}}}\right)\right)}$,

${\displaystyle \mathbf {a} ={d\mathbf {u} \over dt}}$ ，应用到三维加速度${\displaystyle \mathbf {a} }$与三维速度${\displaystyle \mathbf {u} }$

${\displaystyle {\dot {\gamma }}_{u}={\frac {\mathbf {a\cdot u} }{c^{2}}}\gamma _{u}^{3}={\frac {\mathbf {a\cdot u} }{c^{2}}}{\frac {1}{\left(1-{\frac {u^{2}}{c^{2}}}\right)^{3/2}}}}$

${\displaystyle \mathbf {A} =\left(0,\mathbf {a} \right)}$

${\displaystyle F^{\mu }=mA^{\mu },}$

## 非惯性坐标系中的四维加速度

${\displaystyle A^{\lambda }:={\frac {DU^{\lambda }}{d\tau }}={\frac {dU^{\lambda }}{d\tau }}+\Gamma ^{\lambda }{}_{\mu \nu }U^{\mu }U^{\nu }}$

## 参考文献

1. ^ Tsamparlis M. Special Relativity Online. Springer Berlin Heidelberg. 2010: 185. ISBN 978-3-642-03837-2.
2. ^ Pauli W. Theory of Relativity 1981 Dover. B.G. Teubner, Leipzig. 1921: 74. ISBN 978-0-486-64152-2.
3. ^ Synge J.L.; Schild A. Tensor Calculus 1978 Dover. University of Toronto Press. 1949: 149, 153 and 170. ISBN 0-486-63612-7.
• Papapetrou A. Lectures on General Relativity. D. Reidel Publishing Company. 1974. ISBN 90-277-0514-3.
• Rindler, Wolfgang. Introduction to Special Relativity (2nd). Oxford: Oxford University Press. 1991. ISBN 0-19-853952-5.