快度

歷史

1910年，[3][4]提出用此參數來取代速度的觀念。而這個參數被阿爾弗雷德·羅伯英语Alfred Robb (1911)[5]命名為快度，並隨後被許多筆者所採用，如盧迪威格·席柏斯坦 (1914)，愛德華·莫立 (1936)和沃夫岡·潤德勒 (2001)。

在一維空間中

${\displaystyle {\begin{pmatrix}ct'\\x'\end{pmatrix}}={\begin{pmatrix}\cosh w&-\sinh w\\-\sinh w&\cosh w\end{pmatrix}}{\begin{pmatrix}ct\\x\end{pmatrix}}=\mathbf {\Lambda } (w){\begin{pmatrix}ct\\x\end{pmatrix}}}$

${\displaystyle \mathbf {Z} ={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}$

${\displaystyle \mathbf {\Lambda } (w_{1}+w_{2})=\mathbf {\Lambda } (w_{1})\mathbf {\Lambda } (w_{2})}$

${\displaystyle w_{\text{AC}}=w_{\text{AB}}+w_{\text{BC}}}$

${\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\equiv \cosh w}$

${\displaystyle u=(u_{1}+u_{2})/(1+u_{1}u_{2}/c^{2})}$

${\displaystyle \beta _{i}={\frac {u_{i}}{c}}=\tanh {w_{i}}}$

{\displaystyle {\begin{aligned}\tanh w&={\frac {\tanh w_{1}+\tanh w_{2}}{1+\tanh w_{1}\tanh w_{2}}}\\&=\tanh(w_{1}+w_{2})\end{aligned}}}

βγ的乘積時常出現，從先前的討論可知

${\displaystyle \beta \gamma =\sinh w\,}$

指數和對數關係

${\displaystyle e^{w}=\gamma (1+\beta )=\gamma \left(1+{\frac {v}{c}}\right)={\sqrt {\frac {1+{\tfrac {v}{c}}}{1-{\tfrac {v}{c}}}}}}$

${\displaystyle e^{-w}=\gamma (1-\beta )=\gamma \left(1-{\frac {v}{c}}\right)={\sqrt {\frac {1-{\tfrac {v}{c}}}{1+{\tfrac {v}{c}}}}}}$

${\displaystyle w=\ln \left[\gamma (1+\beta )\right]=-\ln \left[\gamma (1-\beta )\right]\,}$

在多維空間中

${\displaystyle {\mathfrak {so}}(3,1)\supset \mathrm {span} \{K_{1},K_{2},K_{3}\}\approx \mathbb {R} ^{3}\ni \mathbf {w} ={\boldsymbol {\hat {\beta }}}\tanh ^{-1}\beta ,\quad {\boldsymbol {\beta }}\in \mathbb {B} ^{3},}$

${\displaystyle \mathbf {w} ={\boldsymbol {\hat {\beta }}}\tanh ^{-1}\beta ,\quad {\boldsymbol {\beta }}={\boldsymbol {\beta }}_{1}\oplus {\boldsymbol {\beta }}_{2},}$

${\displaystyle \cosh w=\cosh w_{1}\cosh w_{2}+\sinh w_{1}\sinh w_{2}\cos \theta }$

${\displaystyle \Lambda =e^{-i{\boldsymbol {\theta }}\cdot \mathbf {J} }e^{-i\mathbf {w} \cdot \mathbf {K} }}$

${\displaystyle [K_{i},K_{j}]=-i\epsilon _{ijk}J_{k}}$

在粒子物理中

${\displaystyle E=\gamma mc^{2}}$
${\displaystyle |\mathbf {p} |=\gamma mv}$

${\displaystyle w=\operatorname {artanh} {\frac {v}{c}}}$

${\displaystyle \cosh w=\cosh \left(\operatorname {artanh} {\frac {v}{c}}\right)={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=\gamma }$
${\displaystyle \sinh w=\sinh \left(\operatorname {artanh} {\frac {v}{c}}\right)={\frac {\frac {v}{c}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=\beta \gamma }$

${\displaystyle E=mc^{2}\cosh w}$
${\displaystyle |\mathbf {p} |=mc\,\sinh w}$

${\displaystyle w=\operatorname {artanh} {\frac {|\mathbf {p} |c}{E}}={\frac {1}{2}}\ln {\frac {E+|\mathbf {p} |c}{E-|\mathbf {p} |c}}}$

${\displaystyle y={\frac {1}{2}}\ln {\frac {E+p_{z}c}{E-p_{z}c}}}$

注釋

1. ^ 這可以被理解成，欲求給定兩個速度所對應到的快度和，實際上就是在對原速度作相對論性的求和，再求出該速度對應的快度。此外，快度從${\displaystyle \mathbb {R} ^{3}}$ 上也繼承了三維向量加法的求和性質，這是與上述快度和不同的一種和。在下文提到「快度求和」時，請依照上下文判斷是哪一種求和。

參考文獻

1. ^ 赫爾曼·閔考斯基 (1908) Fundamental Equations for Electromagnetic Processes in Moving Bodies via Wikisource
2. ^ Sommerfeld, Phys. Z 1909
3. ^ (1910)Application of Lobachevskian Geometry in the Theory of Relativity Physikalische Zeitschrift 經由維基文庫
4. ^ (1910) A History of the Theories of the Aether and Electricity, 第441頁，經由互聯網檔案館.
5. ^ 阿爾弗雷德·羅伯英语Alfred Robb (1911) Optical Geometry of Motion p.9
6. ^ Jackson 1999，第547页
7. ^ Rhodes & Semon 2003
8. ^ Robb 1910, Varićak 1910，Borel 1913
9. ^ Landau & Lifshitz 2002，Problem p. 38
10. ^ Rhodes & Semon 2003
11. ^ Amsler, C. et al., "The Review of Particle Physics", Physics Letters B 667 (2008) 1, Section 38.5.2
• (1910), (1912), (1924) See
• Whittaker, E. T. A history of the theories of aether and electricity: 441. 1910 [22 January 2016].
• Robb, Alfred. Optical geometry of motion, a new view of the theory of relativity. Cambridge: Heffner & Sons. 1911.
• 埃米爾·博雷爾 (1913) La théorie de la relativité et la cinématique, Comptes Rendus Acad Sci Paris 156 215-218; 157 703-705
• Silberstein, Ludwik. The Theory of Relativity. London: Macmillan & Co. 1914.
• (1936)"Restricted relativity in terms of hyperbolic functions of rapidities", 赫爾曼·邦迪 43:70.
• 法蘭克·莫雷 (1936) "When and Where", The Criterion, edited by T.S. Eliot, 15:200-2009.
• 沃夫岡·潤德勒 (2001) Relativity: Special, General, and Cosmological, page 53, 牛津大學出版社.
• Shaw, Ronald (1982) Linear Algebra and Group Representations, v. 1, page 229, ISBN 0-12-639201-3.
• Walter, Scott. The non-Euclidean style of Minkowskian relativity (PDF). (编) J. Gray. The Symbolic Universe: Geometry and Physics. Oxford University Press. 1999: 91–127.(see page 17 of e-link)
• Rhodes, J. A.; Semon, M. D. Relativistic velocity space, Wigner rotation, and Thomas precession. Am. J. Phys. 2004, 72: 93–90. Bibcode:2004AmJPh..72..943R. arXiv:gr-qc/0501070. doi:10.1119/1.1652040.
• Jackson, J. D. Chapter 11. Classical Electrodynamics 3d. John Wiley & Sons. 1999 [1962]. ISBN 0-471-30932-X.