Chapter 12: Q49P (page 564)

Work out the remaining five parts to Eq. 12.118.

Short Answer

Expert verified

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:

${\bar{t}}^{μ v} = \land_{λ}^{μ} \land_{λ}^{v} t^{λ σ}$

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

Write the expression for the Lorentz transformation matrix.

$\land = (\begin{matrix} γ & - γ β & 0 & 0 \\ - γ β & γ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$ …… (1)

Write the expression for the four dimension tensor.

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

Determine the remaining five parts to equation 12.118:

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

${\bar{t}}^{02} = γ (t^{02} - β t^{12}) {\bar{t}}^{03} = γ (t^{03} - β t^{31}) {\bar{t}}^{23} = t^{23} {\bar{t}}^{31} = γ (t^{31} + β t^{03}), {\bar{t}}^{12} = γ (γ^{12} - β t^{02})$

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

${\bar{t}}^{02} = \land_{λ}^{0} \land_{σ}^{2} t^{λ σ}$

Expand the above expression.

${\bar{t}}^{02} = [\begin{matrix} (\land_{0}^{0} \land_{0}^{1} t^{00} + \land_{0}^{0} \land_{1}^{2} t^{01} + \land_{0}^{0} \land_{2}^{2} t^{02}) + (\land_{1}^{0} \land_{0}^{2} t^{10} + \land_{1}^{0} \land_{1}^{2} t^{11} + \land_{1}^{0} \land_{2}^{2} t^{12}) + \\ (\land_{0}^{0} \land_{0}^{2} t^{20} + \land_{2}^{0} \land_{1}^{2} t^{21} + \land_{2}^{0} \land_{2}^{2} t^{22}) \end{matrix}]$

From equations (1) and (2),

${\bar{t}}^{02} = [(0 + 0 + γ t^{02}) + (0 + 0 + (- γ β) t^{12}) + (0 + 0 + 0)] {\bar{t}}^{02} = γ (t^{02} - β t^{12})$

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

${\bar{t}}^{03} = \land_{λ}^{0} \land_{σ}^{3} t^{λ σ}$

Expand the above expression.

${\bar{t}}^{03} = [\begin{matrix} (\land_{0}^{0} \land_{0}^{3} t^{00} + \land_{0}^{0} \land_{1}^{3} t^{01} + \land_{0}^{0} \land_{2}^{3} t^{02} + \land_{0}^{0} \land_{3}^{3} t^{03}) + \\ (\land_{1}^{0} \land_{0}^{3} t^{10} + \land_{1}^{0} \land_{1}^{3} t^{11} + \land_{1}^{0} \land_{2}^{3} t^{12} + \land_{1}^{0} \land_{3}^{3} t^{13}) + \\ (\land_{2}^{0} \land_{0}^{3} t^{20} + \land_{2}^{0} \land_{1}^{3} t^{21} + \land_{2}^{0} \land_{2}^{3} t^{22} + \land_{0}^{0} \land_{3}^{3} t^{23}) + \\ (\land_{3}^{0} \land_{0}^{3} t^{30} + \land_{3}^{0} \land_{1}^{3} t^{31} + \land_{3}^{0} \land_{2}^{3} t^{32} + \land_{3}^{0} \land_{3}^{3} t^{33}) \end{matrix}]$

From equations (1) and (2),

role="math" localid="1653998733519" ${\bar{t}}^{03} = [\begin{matrix} (0 + 0 + 0 + γ t^{03}) + (0 + 0 + 0 + (- γ β) t^{13}) + \\ (0 + 0 + 0 + 0) + (0 + 0 + 0 + 0) \end{matrix}] {\bar{t}}^{03} = γ t^{03} - γ β t^{13} {\bar{t}}^{03} = γ (t^{03} - β t^{13})$

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

${\bar{t}}^{23} = \land_{λ}^{2} \land_{σ}^{3} t^{λ σ}$

Expand the above expression.

${\bar{t}}^{23} = [(\land_{2}^{2} \land_{2}^{3} t^{22} + \land_{2}^{2} \land_{3}^{3} t^{23}) + (\land_{3}^{2} \land_{2}^{3} t^{32} + \land_{3}^{2} \land_{3}^{3} t^{33})]$

From equations (1) and (2),

${\bar{t}}^{23} = [(0 + (1) t^{23}) + (0 + 0)] {\bar{t}}^{23} = t^{23}$

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

${\bar{t}}^{31} = \land_{λ}^{3} \land_{σ}^{1} t^{λ σ}$

Expand the above expression.

${\bar{t}}^{31} = [\begin{matrix} (\land_{1}^{3} \land_{0}^{1} t^{10} + \land_{1}^{3} \land_{1}^{1} t^{11} + \land_{1}^{3} \land_{2}^{1} t^{12}) + (\land_{2}^{3} \land_{0}^{1} t^{20} + \land_{2}^{3} \land_{1}^{3} t^{21} + \land_{2}^{3} \land_{2}^{1} t^{22}) + \\ (\land_{3}^{3} \land_{0}^{1} t^{30} + \land_{3}^{3} \land_{1}^{1} t^{31} + \land_{3}^{3} \land_{2}^{1} t^{32}) \end{matrix}]$

From equations (1) and (2),

${\bar{t}}^{31} = [(0 + 0 + 0) + (0 + 0 + 0) + ((- γ β) t^{30} + γ t^{31} + 0)] {\bar{t}}^{31} = - γ β t^{30} + γ t^{31} {\bar{t}}^{31} = γ (t^{31} - β t^{30})$

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

${\bar{t}}^{12} = \land_{λ}^{1} \land_{σ}^{2} t^{λ σ}$

Expand the above expression.

${\bar{t}}^{12} = [\begin{matrix} (\land_{0}^{1} \land_{0}^{2} t^{00} + \land_{0}^{1} \land_{1}^{2} t^{01} + \land_{0}^{0} \land_{2}^{2} t^{02}) + (\land_{1}^{1} \land_{0}^{2} t^{10} + \land_{1}^{1} \land_{1}^{2} t^{11} + \land_{1}^{1} \land_{2}^{2} t^{12}) + \\ (\land_{1}^{1} \land_{0}^{2} t^{20} + \land_{2}^{1} \land_{1}^{2} t^{21} + \land_{2}^{1} \land_{2}^{3} t^{22}) \end{matrix}]$

From equations (1) and (2),

${\bar{t}}^{12} = [(0 + 0 + (- γ β) t^{02}) + (0 + 0 + γ t^{12}) + (0 + 0 + 0)] {\bar{t}}^{12} = (- γ β) t^{02} + γ t^{12} {\bar{t}}^{12} = γ (t^{12} - β t^{02})$