Question

Work out the remaining five parts to Eq. 12.118.

Short answer

All the remaining five parts to equation 12.118 are proved.

Step-by-step solution

Expression for the second rank tensor:

Write an expression for the second rank tensor:

${\overline{\mathbf{t}}}^{\mathbf{\mu }\mathbf{v}}\mathbf{=}{\mathbf{\wedge }}_{\mathbf{\lambda }}^{\mathbf{\mu }}{\mathbf{\wedge }}_{\mathbf{\lambda }}^{\mathbf{v}}{\mathit{t}}^{\mathbf{\lambda }\mathbf{\sigma }}$

Here, $\wedge$ is the Lorentz transformation matrix.

Write the expression for the Lorentz transformation matrix.

$\wedge =\left(\begin{array}{cccc}\gamma & -\gamma \beta & 0& 0\\ -\gamma \beta & \gamma & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$ …… (1)

Write the expression for the four dimension tensor.

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

Determine the remaining five parts to equation 12.118:

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

$$\begin{gathered} {\overline{t}}^{02}=\gamma \left(t^{02}-\beta t^{12}\right) \\ {\overline{t}}^{03}=\gamma \left(t^{03}-\beta t^{31}\right) \\ {\overline{t}}^{23}=t^{23} \\ {\overline{t}}^{31}=\gamma \left(t^{31}+\beta t^{03}\right),{\overline{t}}^{12}=\gamma \left({\gamma }^{12}-\beta t^{02}\right) \end{gathered}$$

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

${\overline{t}}^{02}={\wedge }_{\lambda }^0{\wedge }_{\sigma }^2{t}^{\lambda \sigma }$

Expand the above expression.

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

From equations (1) and (2),

$$\begin{gathered} {\overline{t}}^{02}=\left[\left(0+0+\gamma t^{02}\right)+\left(0+0+\left(-\gamma \beta \right)t^{12}\right)+\left(0+0+0\right)\right] \\ {\overline{t}}^{02}=\gamma \left(t^{02}-\beta t^{12}\right) \end{gathered}$$

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

${\overline{t}}^{03}={\wedge }_{\lambda }^0{\wedge }_{\sigma }^3{t}^{\lambda \sigma }$

Expand the above expression.

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

From equations (1) and (2),

role="math" localid="1653998733519" $$\begin{gathered} {\overline{t}}^{03}=\left[\begin{array}{c}\left(0+0+0+\gamma t^{03}\right)+\left(0+0+0+\left(-\gamma \beta \right)t^{13}\right)+\\ \left(0+0+0+0\right)+\left(0+0+0+0\right)\end{array}\right] \\ {\overline{t}}^{03}=\gamma t^{03}-\gamma \beta t^{13} \\ {\overline{t}}^{03}=\gamma \left(t^{03}-\beta t^{13}\right) \end{gathered}$$

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

${\overline{t}}^{23}={\wedge }_{\lambda }^2{\wedge }_{\sigma }^3{t}^{\lambda \sigma }$

Expand the above expression.

${\overline{t}}^{23}=\left[\left({\wedge }_2^2{\wedge }_2^3{t}^{22}+{\wedge }_2^2{\wedge }_3^3{t}^{23}\right)+\left({\wedge }_3^2{\wedge }_2^3{t}^{32}+{\wedge }_3^2{\wedge }_3^3{t}^{33}\right)\right]$

From equations (1) and (2),

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

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

${\overline{t}}^{31}={\wedge }_{\lambda }^3{\wedge }_{\sigma }^1{t}^{\lambda \sigma }$

Expand the above expression.

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

From equations (1) and (2),

$$\begin{gathered} {\overline{t}}^{31}=\left[\left(0+0+0\right)+\left(0+0+0\right)+\left(\left(-\gamma \beta \right)t^{30}+\gamma t^{31}+0\right)\right] \\ {\overline{t}}^{31}=-\gamma \beta t^{30}+\gamma t^{31} \\ {\overline{t}}^{31}=\gamma \left(t^{31}-\beta t^{30}\right) \end{gathered}$$

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

${\overline{t}}^{12}={\wedge }_{\lambda }^1{\wedge }_{\sigma }^2{t}^{\lambda \sigma }$

Expand the above expression.

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

From equations (1) and (2),

$$\begin{gathered} {\overline{t}}^{12}=\left[\left(0+0+\left(-\gamma \beta \right)t^{02}\right)+\left(0+0+\gamma t^{12}\right)+\left(0+0+0\right)\right] \\ {\overline{t}}^{12}=\left(-\gamma \beta \right)t^{02}+\gamma t^{12} \\ {\overline{t}}^{12}=\gamma \left(t^{12}-\beta t^{02}\right) \end{gathered}$$

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