广义相对论基础【1】狭义相对论中的张量

前言

我左手是修魔海短暂的因果，右手里百年一个漫长的孤陌。

洛伦兹变换

洛伦兹变换
( x 1 , x 2 , x 3 , x 4 ) → ( x 1 ′ , x 2 ′ , x 3 ′ , x 4 ′ ) (x_1,x_2,x_3,x_4)\rightarrow (x_1',x_2',x_3',x_4') (x1,x2,x3,x4)→(x1′,x2′,x3′,x4′)
x 4 = i c t x_4=ict x4=ict
x 4 x_4 x4是时间坐标， x 1 , x 2 , x 3 x_1,x_2,x_3 x1,x2,x3是空间坐标

{ x 1 ′ = x 1 − v t 1 − ( v / c ) 2 x 2 ′ = x 2 x 3 ′ = x 3 x 4 ′ = i c t − ( v / c ) 2 x 1 1 − ( v / c ) 2 \begin{cases} x_1'=\frac{ x_1 - vt }{ \sqrt{ 1 - (v/c)^2 } } \\x_2'=x_2\\x_3'=x_3\\x_4'=ic\frac{t-(v/c)^2 x_1 }{\sqrt{ 1 - (v/c)^2 }} \end{cases} ⎩⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎧x1′=1−(v/c)2 x1−vtx2′=x2x3′=x3x4′=ic1−(v/c)2 t−(v/c)2x1

{ x 1 ′ = a 11 x 1 + a 12 x 2 + a 13 x 3 + a 14 x 4 x 2 ′ = a 21 x 1 + a 22 x 2 + a 23 x 3 + a 24 x 4 x 3 ′ = a 31 x 1 + a 32 x 2 + a 33 x 3 + a 34 x 4 x 4 ′ = a 41 x 1 + a 42 x 2 + a 43 x 3 + a 44 x 4 \begin{cases}x_1'=a_{11}x_1+a_{12}x_2+a_{13}x_3+a_{14}x_4 \\x_2'=a_{21}x_1+a_{22}x_2+a_{23}x_3+a_{24}x_4 \\x_3'=a_{31}x_1+a_{32}x_2+a_{33}x_3+a_{34}x_4 \\x_4'=a_{41}x_1+a_{42}x_2+a_{43}x_3+a_{44}x_4 \end{cases} ⎩⎪⎪⎪⎨⎪⎪⎪⎧x1′=a11x1+a12x2+a13x3+a14x4x2′=a21x1+a22x2+a23x3+a24x4x3′=a31x1+a32x2+a33x3+a34x4x4′=a41x1+a42x2+a43x3+a44x4

x μ ′ = ∑ ν 4 a μ ν x ν x'_{\mu}=\sum^4_{\nu} a_{\mu\nu} x_{\nu} xμ′=ν∑4aμνxν

根据爱因斯坦求和约定，重复指标代表求和，上式可以写成

x μ ′ = a μ ν x ν x'_{\mu}=a_{\mu\nu}x_{\nu} xμ′=aμνxν

[ x 1 ′ x 2 ′ x 3 ′ x 4 ′ ] = [ γ 0 0 i β γ 0 1 0 0 0 0 1 0 − i β γ 0 0 γ ] [ x 1 x 2 x 3 x 4 ] \left[ \begin{array}{l} x_1'\\ x_2'\\ x_3'\\ x_4' \end{array}\right] = \left[ \begin{array}{l} \gamma & 0 & 0 & i\beta \gamma\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ -i\beta\gamma & 0 & 0 & \gamma \end{array}\right] \left[ \begin{array}{l} x_1\\ x_2\\ x_3\\ x_4 \end{array}\right] ⎣⎢⎢⎡x1′x2′x3′x4′⎦⎥⎥⎤=⎣⎢⎢⎡γ00−iβγ01000010iβγ00γ⎦⎥⎥⎤⎣⎢⎢⎡x1x2x3x4⎦⎥⎥⎤

其中 γ = ( 1 − β 2 ) ( − 1 / 2 ) , β = v / c \gamma = (1-\beta^2)^{(-1/2)},\beta=v/c γ=(1−β2)(−1/2),β=v/c

又

d x μ ′ = a μ ν d x ν dx'_{\mu}=a_{\mu\nu}dx_{\nu} dxμ′=aμνdxν

狭义相对论不用上面这个式子，因为狭义相对论讨论的是平直空间中的运动。

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(洛伦兹变换下的) 张量

零阶张量是标量，一阶张量是矢量，二阶张量是矩阵。

所以上面两个关于a的矩阵的每一行都是一阶张量。

标量(不变量)是在坐标变换中不变的量 U ′ ( x ′ ) = U ( x ) U'(x')=U(x) U′(x′)=U(x)，不变量可能是常数或函数
矢量是在坐标变换中与坐标微分元一起变的量 V μ ′ = a μ α V α V'_{\mu}=a_{\mu\alpha}V_{\alpha} Vμ′=aμαVα
二阶张量是在坐标变换中以 T μ ν ′ = a μ α a ν β ⋅ T α β T'_{\mu\nu}=a_{\mu\alpha} a_{\nu\beta} \cdot T_{\alpha\beta} Tμν′=aμαaνβ⋅Tαβ 规律变换的量
N阶张量 T μ ν ⋅ ⋅ ⋅ λ ′ = a μ α a ν β ⋅ ⋅ ⋅ a λ σ ⋅ T α β ⋅ ⋅ ⋅ σ T'_{\mu\nu\cdot\cdot\cdot\lambda} = a_{\mu\alpha} a_{\nu\beta} \cdot \cdot \cdot a_{\lambda\sigma} \cdot T_{\alpha\beta\cdot\cdot\cdot \sigma} Tμν⋅⋅⋅λ′=aμαaνβ⋅⋅⋅aλσ⋅Tαβ⋅⋅⋅σ 有 4 n 4^n 4n个分量
二阶及以上统称张量

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张量性质

δ α β = { 1 α = β 0 α ≠ β \delta_{\alpha\beta}=\begin{cases} 1 \qquad \alpha = \beta\\ 0 \qquad \alpha \ne \beta \end{cases} δαβ={1α=β0α=β

a μ α ( a − 1 ) α ν = ( a − 1 ) μ α a α ν = δ μ ν = a μ α a ν α = a α μ a α ν a_{\mu\alpha}(a^{-1})_{\alpha \nu}=(a^{-1})_{\mu\alpha}a_{\alpha \nu}=\delta_{\mu\nu}=a_{\mu\alpha}a_{\nu\alpha}=a_{\alpha\mu} a_{\alpha\nu} aμα(a−1)αν=(a−1)μαaαν=δμν=aμαaνα=aαμaαν

a a − 1 = a − 1 a = I 单位矩阵 aa^{-1}=a^{-1}a=I 单位矩阵 aa−1=a−1a=I单位矩阵

转置矩阵： a ~ α μ = a μ α \widetilde{a}_{\alpha\mu}=a_{\mu\alpha} a αμ=aμα

转置矩阵等于逆矩阵： a ~ α μ = ( a − 1 ) α μ \widetilde{a}_{\alpha\mu}=(a^{-1})_{\alpha\mu} a αμ=(a−1)αμ 说明矩阵a是正交矩阵

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洛伦兹变换的张量性质

d x μ ′ d x μ ′ = ( d x 1 ′ ) 2 + ( d x 2 ′ ) 2 + ( d x 3 ′ ) 2 + ( d x 4 ′ ) 2 = a μ α d x α a μ β d x β = a μ α a μ β d x α d x β = δ α β d x α d x β = d x α d x α = ( d x 1 ) 2 + ( d x 2 ) 2 + ( d x 3 ) 2 + ( d x 4 ) 2 dx_{\mu}'dx_{\mu}'=(dx_1')^2+(dx_2')^2+(dx_3')^2+(dx_4')^2\\\\=a_{\mu\alpha}dx_{\alpha}a_{\mu\beta}dx_{\beta}\\\\=a_{\mu\alpha}a_{\mu\beta}dx_{\alpha}dx_{\beta}\\\\=\delta_{\alpha\beta}dx_{\alpha}dx_{\beta} \\\\=dx_{\alpha}dx_{\alpha}=(dx_1)^2+(dx_2)^2+(dx_3)^2+(dx_4)^2 dxμ′dxμ′=(dx1′)2+(dx2′)2+(dx3′)2+(dx4′)2=aμαdxαaμβdxβ=aμαaμβdxαdxβ=δαβdxαdxβ=dxαdxα=(dx1)2+(dx2)2+(dx3)2+(dx4)2

d s = d x μ ′ d x μ ′ = d x α d x α ds=dx_{\mu}'dx_{\mu}'=dx_{\alpha}dx_{\alpha} ds=dxμ′dxμ′=dxαdxα 表示四维时空中，坐标变换不会改变两点之间的距离
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又因为 d s = ( d x 1 ) 2 + ( d x 2 ) 2 + ( d x 3 ) 2 − c ( d t ) 2 ds=(dx_1)^2+(dx_2)^2+(dx_3)^2-c(dt)^2 ds=(dx1)2+(dx2)2+(dx3)2−c(dt)2

所以当 d s = 0 ds=0 ds=0，表示物体以光速传播