洛倫茨變換,各參照系物理量之轉換關係也,數學方程組也。
名於創立者荷蘭物理學家亨德里克·洛倫茲。洛倫茲變換初以調經典電動力學牛頓力學,後乃狹義相對論基本方程組也。
始[纂]
麥克斯韋方程組之經典電動力學於經典力學伽利略變換非協變也。
數學形式[纂]
加速觀者世界線之時空。豎時橫距,虛劃時空軌也。
洛倫茲變換視以太存也,然今未見有也。據光速不變原理,光恆速也。愛因斯坦遂提之狹義相對論,時空乃一也,遂曰:
![{\displaystyle {\begin{cases}x'={\frac {x-vt}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\\y'=y\\z'=z\\t'={\frac {t-{\frac {v}{c^{2}}}x}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\end{cases}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85zNsPoqdEzNi5zjmNagaNzArDztG0yts5oDCNyjvAnqeQzga4ajm5)
箇中:x、y、z、t,慣性坐標系Σ之位也;x'、y'、z'、t'慣性坐標系Σ'之位也;v,Σ'系對Σ沿x軸之速也。
v、x'、y'、z'、t'換之-v、x、y、z、t可得之洛倫茲變換反變換式:
![{\displaystyle {\begin{cases}x={\frac {x'+vt'}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\\y=y'\\z=z'\\t={\frac {t'+{\frac {v}{c^{2}}}x'}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\end{cases}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Aa2vBoNeQzAnEaDs1aAw4aDs0njJFagi1yqe4nta3zqs3atK0zDBF)
v遠小光c乃退之經典力學伽利略變換:
![{\displaystyle {\begin{cases}x'=x-vt\\y'=y\\z'=z\\t'=t\end{cases}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82oDlEzAzFoNCQztdDytlFaDrBnjKPoqoOnDGOyto2otdFoAdBaDdE)
遂狹義相對論經典力學不矛盾,差之不大。高速如電子,方須慮修之以相對論。
四維形式[纂]
狹義相對論時空坐標四參數((t,x,y,z))也。洛倫茲變換可得四維間隔不變之變。
若x、y、z化x1、x2、x3曰:
![{\displaystyle {\begin{cases}x^{0}=ct\\x^{\prime }{}^{0}=ct^{\prime }\end{cases}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ztwPnjw3zge2zqs1zNw5oNs3otrAaNC0zgdFoqrCate1zje1nDe4)
可矩陣之:
![{\displaystyle {\begin{bmatrix}x^{\prime }{}^{0}\\x^{\prime }{}^{1}\\x^{\prime }{}^{2}\\x^{\prime }{}^{3}\end{bmatrix}}={\begin{bmatrix}\gamma &-\beta \gamma &0&0\\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}}{\begin{bmatrix}x^{0}\\x^{1}\\x^{2}\\x^{3}\end{bmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnAaOntFDoDKQaDe5z2eQyqw4ytBBnjCQo2vByjsOyqdEotK0aNsO)
箇中
,曰洛倫茲因子。
勞侖茲變換之推導[纂]
愛因斯坦初推之勞侖茲變換以光速不變之物理原則作始點。實,勞侖茲變換不決於電磁波之物理性質;粒子定域性原理之弗能瞬傳,此最高速巧光速也。
群論之推導[纂]
作乘組群符之公理曰:
- 閉合:以
寫
到
。![{\displaystyle [K\to K^{\prime \prime }]=[K\to K^{\prime }][K^{\prime }\to K^{\prime \prime }]}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DajC5nqzEoNnEyjJBnDhDajm3o2eNzDs0aga1zDK4oAoNytaNnjw5)
- 結合律:
![{\displaystyle [K\to K^{\prime }]\left([K^{\prime }\to K^{\prime \prime }][K^{\prime \prime }\to K^{\prime \prime \prime }]\right)=\left([K\to K^{\prime }][K^{\prime }\to K^{\prime \prime }]\right)[K^{\prime \prime }\to K^{\prime \prime \prime }]}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnqdEytnCa2i5ztG5otmQajmNzto1nDiOagrCaDFAaAvCoDG1nDw2)
- 單位元:
![{\displaystyle [K\to K]}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EytFBnArFztaPoDo5nga4aDlAotC2a2hEoNe0athDnjeOnAe5nDK3)
- 逆元:
可返原系![{\displaystyle [K^{\prime }\to K]}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NntJCzqi4zDhBzqsNo2dCo2vFyji5zNnEo2zFytzEztJBajm1nqdC)
符合群公理之轉矩陣[纂]
、
,
之原點相對
原點速
(設向
無
、
方向)。出時空之均勻性勞侖茲變換必保慣性,必一綫性轉換,可矩陣之:
![{\displaystyle {\begin{pmatrix}t^{\prime }\\z^{\prime }\end{pmatrix}}={\begin{pmatrix}\Lambda _{11}&\Lambda _{12}\\\Lambda _{21}&\Lambda _{22}\end{pmatrix}}{\begin{pmatrix}t\\z\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DzNGOotGOaDmQoNzAntlAaqdAnDdDotmQoDrAzgaQnjw0nDzBaNeP)
箇中
乃待算之矩陣元。相對速
之函數。
參照系
之原點
於參照系
之運動曰:
![{\displaystyle {\begin{pmatrix}t^{\prime }\\0\end{pmatrix}}={\begin{pmatrix}\Lambda _{11}&\Lambda _{12}\\\Lambda _{21}&\Lambda _{22}\end{pmatrix}}{\begin{pmatrix}t\\vt\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NyjrDzNiOotw1ygvBnqzBaqo1otnEoDo2o2i4ngsPzNw0ztvDotFF)
得
![{\displaystyle \Lambda _{21}+v\,\Lambda _{22}=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NzghBaAiPzjdEzAhFoNe5ztaQzgo4o2s3zDC4zjvCngzBajvEotCQ)
同,參照系
之原點
於參照系
之運動曰:
![{\displaystyle {\begin{pmatrix}t^{\prime }\\-vt^{\prime }\end{pmatrix}}={\begin{pmatrix}\Lambda _{11}&\Lambda _{12}\\\Lambda _{21}&\Lambda _{22}\end{pmatrix}}{\begin{pmatrix}t\\0\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EnAnDzNrDzqeOaNdBnjK0zAaQotvAygs5a2nFngdEntnAzqa3otm1)
得
![{\displaystyle \Lambda _{21}+v\,\Lambda _{11}=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qyqo2yje2zjK0yjw1ygaQyjaQoDGOoqe5oqrDa2s2ytm2ngrDatlC)
主斜同且可曰
。
:
![{\displaystyle {\begin{pmatrix}t^{\prime }\\z^{\prime }\end{pmatrix}}={\begin{pmatrix}\gamma &\Lambda _{12}\\-v\gamma &\gamma \end{pmatrix}}{\begin{pmatrix}t\\z\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ngzCytlCaNKPoDa3aDvEnqa2zNlEzge1zNa2ajFBoqa2zNvByjo2)
因
,
乃時間膨脹之因子。各向同性
僅決速即
。
群元可逆故取逆矩陣:
![{\displaystyle {\begin{pmatrix}t\\z\end{pmatrix}}={\frac {1}{\gamma ^{2}+\Lambda _{12}v\gamma }}{\begin{pmatrix}\gamma &-\Lambda _{12}\\v\gamma &\gamma \end{pmatrix}}{\begin{pmatrix}t^{\prime }\\z^{\prime }\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84yghAaqw4aNK3oNKNotoQzDnBztlEnjzEoDG1ntm4zDnFytG5njlA)
以
之性質:
![{\displaystyle {\frac {1}{\gamma ^{2}+\Lambda _{12}v\gamma }}{\begin{pmatrix}\gamma &-\Lambda _{12}\\v\gamma &\gamma \end{pmatrix}}={\begin{pmatrix}\gamma &-\Lambda _{12}\\v\gamma &\gamma \end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Bzqs4zDa5otdFago1ytwOytzBotaPotdDyjm2atoOyjG0o2wPzjGP)
每較得:
![{\displaystyle \gamma ^{2}+\Lambda _{12}v\gamma =1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QngzAotvAyqaNnAs3z2dAzgzCathDnDG5ztdAaDC1ajGOnDGQaNnA)
閉合性求兩轉換等速度和之單次轉換。即兩矩陣之積:
![{\displaystyle {\begin{pmatrix}\gamma ^{\prime }&\Lambda _{12}^{\prime }\\-v^{\prime }\gamma ^{\prime }&\gamma ^{\prime }\end{pmatrix}}{\begin{pmatrix}\gamma &\Lambda _{12}\\-v\gamma &\gamma \end{pmatrix}}={\begin{pmatrix}\gamma ^{\prime }\gamma -\Lambda _{12}^{\prime }v\gamma &\gamma ^{\prime }\Lambda _{12}+\gamma \Lambda _{12}^{\prime }\\-\gamma ^{\prime }\gamma (v+v^{\prime })&\gamma ^{\prime }\gamma -v^{\prime }\gamma ^{\prime }\Lambda _{12}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QyjvEzjmOzjw5aqeOagwQoqo5oDK3nti4zti1aqhFnDsOngdDz2hE)
必擁同之矩陣型式。故曰:
![{\displaystyle \kappa \equiv {\frac {\Lambda _{12}}{v\gamma }}={\frac {\Lambda _{12}^{\prime }}{v^{\prime }\gamma ^{\prime }}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FaqsNz2rBzqrFzDvAytlCatsPoqo5otnDyji4nAo5atlBoAiPoDs1)
必一相對速
無關之常數。插入較前等式得
之定義:
![{\displaystyle \gamma ={\frac {1}{\sqrt {1+\kappa v^{2}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zjK1otG2ztiQajKOyjJCzgo2aAe2zqrBz2hFatmNoqdByqsNotC1)
而最廣泛之勞侖茲變換矩陣型式曰:
![{\displaystyle {\frac {1}{\sqrt {1+\kappa v^{2}}}}{\begin{pmatrix}1&\kappa v\\-v&1\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qotw3zAi2oDsPngaOytvEyqaOzNnEagaQoNCPoDrEajw2zgvBygo2)
至此
乃不變速。若
,c限之。未符實。故
。
可曰
、
兩:
伽利略轉換[纂]
得伽利略轉換矩陣:
![{\displaystyle {\begin{pmatrix}t^{\prime }\\z^{\prime }\end{pmatrix}}={\begin{pmatrix}1&0\\-v&1\end{pmatrix}}{\begin{pmatrix}t\\z\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnDlDaDi0zje3zDo2yqvCa2o4nqiQyts3atzFotnEnja5aqnEoNoP)
於此時乃絕對之:
。
勞侖茲變換[纂]
於更一般
之情况遂得前之勞侖茲變換矩陣:
![{\displaystyle {\begin{pmatrix}t^{\prime }\\z^{\prime }\end{pmatrix}}={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{\begin{pmatrix}1&-{\frac {v}{c^{2}}}\\-v&1\end{pmatrix}}{\begin{pmatrix}t\\z\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FytJAaAi2njw3yjBFnto4a2nDytFCatC0zqsNoAw5nAw1njrFatG5)
遂所有參照系不變之速限:
。
速度變換公式[纂]
設慣性坐標系Σ之各軸之速量ux、uy、uz;Σ'之各軸之速量u'x、u'y、u'z:
![{\displaystyle u'_{x}={\frac {u_{x}-v}{1-{\frac {vu_{x}}{c^{2}}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zjBFytaNzjo2yjC5otKQaNi4nAvEz2ePz2vDzqnAnje0nqnAoDo5)
![{\displaystyle u'_{y}={\frac {u_{y}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1-{\frac {vu_{x}}{c^{2}}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CatJCytm2oNhAaja3ztzEnDC3njFFags1atnEo2a3zti3ngdAnqe1)
![{\displaystyle u'_{z}={\frac {u_{z}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1-{\frac {vu_{x}}{c^{2}}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83o2vFzjnCajC4oAzCzje0oNs2agvBoDePzNsOntwNntwOyjrDoNrB)
v、x'、y'、z'、t'換之-v、x、y、z、t可得之洛倫茲變換反變換式:
v遠小光c乃退之經典力學伽利略變換:
![{\displaystyle u'_{x}=u_{x}-v}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FoDaOotzAnDhAzNeQnga5zNaNoAnAzDaNaqi1ngaOythBoAo5njeN)
![{\displaystyle u'_{y}=u_{y}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Noqo0yjmQaqzAygw1zDwOatoQaDCNzgi0nqa2aAo3ajdEnjzDz2nF)
![{\displaystyle u'_{z}=u_{z}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85zti3ztdEzNCNzjiQots5zqePntGPoDrCnga2o2a1ytrAyjm2aqnC)
其他物理量之變換[纂]
類時分量
、類空分量
之四維向量
,其閔考斯基範(Minkowski norm)乃勞倫茲不變量(Lorentz invariant):
。
仿寫:
![{\displaystyle {\begin{aligned}A'&=\gamma \left(A-\mathbf {Z} \cdot {\boldsymbol {\beta }}\right){\mbox{,}}\\\mathbf {Z} '&=\mathbf {Z} +(\gamma -1)(\mathbf {Z} \cdot \mathbf {n} )\mathbf {n} -\gamma A{\boldsymbol {\beta }}{\mbox{,}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnjzEyta4atFCoqe4zgdEyjeOztaNzgs3zNe4otw0zqzEaDnCnjwQ)
箇中
,
乃
方向上之單位向量。
、
分解成垂直
和平行
與位置向量之分解方法同。取逆變換與四維位置同,遂換
與
,後相反相對動向,即
。
常見之四維向量如下表:
四維向量
|
|
|
四維位置
|
時間(乘以 )
|
位置向量
|
四維動量
|
能量(除以 )
|
動量
|
四維波向量
|
角頻率(除以 )
|
波向量
|
四維自旋
|
(無名稱)
|
自旋
|
四維電流密度
|
電荷密度(乘以 )
|
電流密度
|
四維電磁位勢
|
電位(除以 )
|
磁向量位
|
洛倫茲變換之幾何理解[纂]
平面幾何向量某
於原點以
順旋之。新系同向量
曰:
![{\displaystyle {\begin{bmatrix}x^{\prime }\\y^{\prime }\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DzAa0ajBCaDe0a2e5ztiOnta2nqdFnDsQa2dCztJBzNFBz2s3nDFB)
長不變曰:
。
異角度
再旋之,向量新舊關係曰:
![{\displaystyle {\begin{bmatrix}x^{\prime \prime }\\y^{\prime \prime }\end{bmatrix}}={\begin{bmatrix}\cos(\theta +\phi )&\sin(\theta +\phi )\\-\sin(\theta +\phi )&\cos(\theta +\phi )\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82aDG5zNhDntvDatzCoDdDyga3zjhEnDeQoNFDzge3o2oNoNm5oqo2)
即:續旋可加。
相似,定義快度
(略
和
)公式可曰:
![{\displaystyle {\begin{bmatrix}x^{\prime }{}^{0}\\x^{\prime }{}^{1}\end{bmatrix}}={\begin{bmatrix}\cosh w&-\sinh w\\-\sinh w&\cosh w\end{bmatrix}}{\begin{bmatrix}x^{0}\\x^{1}\end{bmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ati0otnAntaPzDhDajmPoDs2aDvAzAhCaAiOoNo5nqhEzNC3ntKO)
即:洛倫兹變換數學同於雙曲角旋轉。
旋長不變乃:
。
換異速
之系,再換
之。使
、
。即原系座標
兩換
曰:
![{\displaystyle {\begin{bmatrix}x^{\prime \prime }{}^{0}\\x^{\prime \prime }{}^{1}\end{bmatrix}}={\begin{bmatrix}\cosh(w_{21}+w_{32})&-\sinh(w_{21}+w_{32})\\-\sinh(w_{21}+w_{32})&\cosh(w_{21}+w_{32})\end{bmatrix}}{\begin{bmatrix}x^{0}\\x^{1}\end{bmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AntK2oNrCathDoNmNzjlDo2rDajC4aAoQajmQajnEajdAyje5nqe0)
遂見直加之數非速
而乃角之
。
直加減惟因速遠小光(
)
速
。
終,直接轉換若兩速
即:
![{\displaystyle {\begin{aligned}w_{31}&=w_{21}+w_{32}\\\tanh w_{31}&=\tanh(w_{21}+w_{32})={\frac {\tanh w_{21}+\tanh w_{32}}{1+\tanh w_{21}\tanh w_{32}}}\\\beta _{31}&={\frac {\beta _{21}+\beta _{32}}{1+\beta _{21}\beta _{32}}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzAdByqiOaDi4yqvCygdEagwQzDsNaqe3aNG0zDi3yqi0zjGPoAa3)
得之相對論速率加法公式。
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