91影视

Write an expression for the second rank tensor:

${\stackrel{\mathbf{炉}}{\mathbf{t}}}^{\mathbf{渭}\mathbf{v}}\mathbf{=}{\mathbf{鈭�}}_{\mathbf{位}}^{\mathbf{渭}}{\mathbf{鈭�}}_{\mathbf{位}}^{\mathbf{v}}{\mathit{t}}^{\mathbf{位}\mathbf{蟽}}$

Here, $鈭�$ is the Lorentz transformation matrix.

Write the expression for the Lorentz transformation matrix.

$鈭�=\left(\begin{array}{cccc}纬& -纬尾& 0& 0\\ -纬尾& 纬& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$ 鈥︹�� (1)

Write the expression for the four dimension tensor.

$t^{渭v}=\left(\begin{array}{cccc}{t}^{00}& t^{01}& t^{02}& t^{03}\\ t^{10}& t^{11}& t^{12}& t^{13}\\ t^{20}& t^{21}& t^{22}& t^{23}\\ t^{30}& t^{31}& t^{32}& t^{33}\end{array}\right)$ 鈥︹�� (2)

Using equation 12.118, write the complete set of transformation rules.

$$\begin{gathered} {\stackrel{炉}{t}}^{02}=纬\left(t^{02}-尾t^{12}\right) \\ {\stackrel{炉}{t}}^{03}=纬\left(t^{03}-尾t^{31}\right) \\ {\stackrel{炉}{t}}^{23}=t^{23} \\ {\stackrel{炉}{t}}^{31}=纬\left(t^{31}+尾t^{03}\right),{\stackrel{炉}{t}}^{12}=纬\left({纬}^{12}-尾t^{02}\right) \end{gathered}$$

Write the expression for${\stackrel{炉}{t}}^{02}$.

${\stackrel{炉}{t}}^{02}={鈭�}_{位}^0{鈭�}_{蟽}^2{t}^{位蟽}$

Expand the above expression.

${\stackrel{炉}{t}}^{02}=\left[\begin{array}{c}\left({鈭�}_0^0{鈭�}_0^1{t}^{00}+{鈭�}_0^0{鈭�}_1^2{t}^{01}+{鈭�}_0^0{鈭�}_2^2{t}^{02}\right)+\left({鈭�}_1^0{鈭�}_0^2{t}^{10}+{鈭�}_1^0{鈭�}_1^2{t}^{11}+{鈭�}_1^0{鈭�}_2^2{t}^{12}\right)+\\ \left({鈭�}_0^0{鈭�}_0^2{t}^{20}+{鈭�}_2^0{鈭�}_1^2{t}^{21}+{鈭�}_2^0{鈭�}_2^2{t}^{22}\right)\end{array}\right]$

From equations (1) and (2),

$$\begin{gathered} {\stackrel{炉}{t}}^{02}=\left[\left(0+0+纬t^{02}\right)+\left(0+0+\left(-纬尾\right)t^{12}\right)+\left(0+0+0\right)\right] \\ {\stackrel{炉}{t}}^{02}=纬\left(t^{02}-尾t^{12}\right) \end{gathered}$$

Write the expression for ${\stackrel{炉}{t}}^{03}$.

${\stackrel{炉}{t}}^{03}={鈭�}_{位}^0{鈭�}_{蟽}^3{t}^{位蟽}$

Expand the above expression.

${\stackrel{炉}{t}}^{03}=\left[\begin{array}{c}\left({鈭�}_0^0{鈭�}_0^3{t}^{00}+{鈭�}_0^0{鈭�}_1^3{t}^{01}+{鈭�}_0^0{鈭�}_2^3{t}^{02}+{鈭�}_0^0{鈭�}_3^3{t}^{03}\right)+\\ \left({鈭�}_1^0{鈭�}_0^3{t}^{10}+{鈭�}_1^0{鈭�}_1^3{t}^{11}+{鈭�}_1^0{鈭�}_2^3{t}^{12}+{鈭�}_1^0{鈭�}_3^3{t}^{13}\right)+\\ \left({鈭�}_2^0{鈭�}_0^3{t}^{20}+{鈭�}_2^0{鈭�}_1^3{t}^{21}+{鈭�}_2^0{鈭�}_2^3{t}^{22}+{鈭�}_0^0{鈭�}_3^3{t}^{23}\right)+\\ \left({鈭�}_3^0{鈭�}_0^3{t}^{30}+{鈭�}_3^0{鈭�}_1^3{t}^{31}+{鈭�}_3^0{鈭�}_2^3{t}^{32}+{鈭�}_3^0{鈭�}_3^3{t}^{33}\right)\end{array}\right]$

From equations (1) and (2),

role="math" localid="1653998733519" $$\begin{gathered} {\stackrel{炉}{t}}^{03}=\left[\begin{array}{c}\left(0+0+0+纬t^{03}\right)+\left(0+0+0+\left(-纬尾\right)t^{13}\right)+\\ \left(0+0+0+0\right)+\left(0+0+0+0\right)\end{array}\right] \\ {\stackrel{炉}{t}}^{03}=纬t^{03}-纬尾t^{13} \\ {\stackrel{炉}{t}}^{03}=纬\left(t^{03}-尾t^{13}\right) \end{gathered}$$

Write the expression for ${\stackrel{炉}{t}}^{23}$.

${\stackrel{炉}{t}}^{23}={鈭�}_{位}^2{鈭�}_{蟽}^3{t}^{位蟽}$

Expand the above expression.

${\stackrel{炉}{t}}^{23}=\left[\left({鈭�}_2^2{鈭�}_2^3{t}^{22}+{鈭�}_2^2{鈭�}_3^3{t}^{23}\right)+\left({鈭�}_3^2{鈭�}_2^3{t}^{32}+{鈭�}_3^2{鈭�}_3^3{t}^{33}\right)\right]$

From equations (1) and (2),

$$\begin{gathered} {\stackrel{炉}{t}}^{23}=\left[\left(0+\left(1\right)t^{23}\right)+\left(0+0\right)\right] \\ {\stackrel{炉}{t}}^{23}=t^{23} \end{gathered}$$

Write the expression for ${\stackrel{炉}{t}}^{31}$.

${\stackrel{炉}{t}}^{31}={鈭�}_{位}^3{鈭�}_{蟽}^1{t}^{位蟽}$

Expand the above expression.

${\stackrel{炉}{t}}^{31}=\left[\begin{array}{c}\left({鈭�}_1^3{鈭�}_0^1{t}^{10}+{鈭�}_1^3{鈭�}_1^1{t}^{11}+{鈭�}_1^3{鈭�}_2^1{t}^{12}\right)+\left({鈭�}_2^3{鈭�}_0^1{t}^{20}+{鈭�}_2^3{鈭�}_1^3{t}^{21}+{鈭�}_2^3{鈭�}_2^1{t}^{22}\right)+\\ \left({鈭�}_3^3{鈭�}_0^1{t}^{30}+{鈭�}_3^3{鈭�}_1^1{t}^{31}+{鈭�}_3^3{鈭�}_2^1{t}^{32}\right)\end{array}\right]$

From equations (1) and (2),

$$\begin{gathered} {\stackrel{炉}{t}}^{31}=\left[\left(0+0+0\right)+\left(0+0+0\right)+\left(\left(-纬尾\right)t^{30}+纬t^{31}+0\right)\right] \\ {\stackrel{炉}{t}}^{31}=-纬尾t^{30}+纬t^{31} \\ {\stackrel{炉}{t}}^{31}=纬\left(t^{31}-尾t^{30}\right) \end{gathered}$$

Write the expression for ${\stackrel{炉}{t}}^{12}$.

${\stackrel{炉}{t}}^{12}={鈭�}_{位}^1{鈭�}_{蟽}^2{t}^{位蟽}$

Expand the above expression.

${\stackrel{炉}{t}}^{12}=\left[\begin{array}{c}\left({鈭�}_0^1{鈭�}_0^2{t}^{00}+{鈭�}_0^1{鈭�}_1^2{t}^{01}+{鈭�}_0^0{鈭�}_2^2{t}^{02}\right)+\left({鈭�}_1^1{鈭�}_0^2{t}^{10}+{鈭�}_1^1{鈭�}_1^2{t}^{11}+{鈭�}_1^1{鈭�}_2^2{t}^{12}\right)+\\ \left({鈭�}_1^1{鈭�}_0^2{t}^{20}+{鈭�}_2^1{鈭�}_1^2{t}^{21}+{鈭�}_2^1{鈭�}_2^3{t}^{22}\right)\end{array}\right]$

From equations (1) and (2),

$$\begin{gathered} {\stackrel{炉}{t}}^{12}=\left[\left(0+0+\left(-纬尾\right)t^{02}\right)+\left(0+0+纬t^{12}\right)+\left(0+0+0\right)\right] \\ {\stackrel{炉}{t}}^{12}=\left(-纬尾\right)t^{02}+纬t^{12} \\ {\stackrel{炉}{t}}^{12}=纬\left(t^{12}-尾t^{02}\right) \end{gathered}$$

Therefore, all the remaining five parts to equation 12.118 is proved.