Question

(a) Equation 12.40 defines proper velocity in terms of ordinary velocity. Invert that equation to get the formula for u in terms of $\mathbf{\eta }$.

(b) What is the relation between proper velocity and rapidity (Eq. 12.34)? Assume the velocity is along the x direction, and find as a function of $\mathit{\theta }$.

Short answer

(a) The formula for uin terms of $\mathbf{\eta }$is $\mathbf{u}=\left(\frac{\begin{array}{c}\begin{array}{c}\\ 1\end{array}\end{array}}{\sqrt{\left(1+\frac{{\mathrm{\eta }}^2}{{\mathrm{c}}^2}\right)}}\right)\mathbf{\eta }$

(b) The relation between proper velocity and rapidity of the value of $\mathbf{\eta }$asa function of$\theta$islocalid="1654674617679" $\mathbf{\eta }\mathbf{=}c\sinh \theta$.

Step-by-step solution

Expression for the relationship between proper velocity and ordinary velocity:

Write the relationship between proper and ordinary velocity.

$\mathbf{\eta }\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{1}\mathbf{-}{\displaystyle \frac{{\mathbf{u}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}}}\mathbf{u}$ …… (1)

Here, $\mathbf{\eta }$ is the proper velocity, u is the ordinary velocity, and c is the speed of light.

Determine the formula for in terms of η:

(a)

Squaring on both sides in equation (1).

$$\begin{gathered} {\mathbf{\eta }}^2{\left[\sqrt{\left(1-\frac{{\mathrm{u}}^2}{{\mathrm{c}}^2}\right)}\right]}^2={\mathbf{u}}^2 \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{\eta }}^2\left(1-\frac{{\mathrm{u}}^2}{{\mathrm{c}}^2}\right)={\mathbf{u}}^{\mathbf{2}} \end{gathered}$$

Invert the above equation to get the formula for uin terms of $\mathbf{\eta }$.

localid="1654675851625" ${\mathbf{\eta }}^2\left(\frac{\begin{array}{c}\begin{array}{c}\\ 1\end{array}\end{array}}{\left(1-\frac{{\eta }^2}{c^2}\right)}\right)={\mathbf{u}}^2$ ….. (2)

Here, localid="1654675673648" $\left(1-\frac{u^2}{c^2}\right)=\left(1+\frac{{\eta }^2}{c^2}\right)$.

Substitute $\left(1-\frac{u^2}{c^2}\right)=\left(1+\frac{{\eta }^2}{c^2}\right)$in equation (2).

localid="1654675808046" ${\mathbf{\eta }}^2\left(\frac{\begin{array}{c}\begin{array}{c}\\ 1\end{array}\end{array}}{\left(1+\frac{{\mathrm{\eta }}^2}{{\mathrm{c}}^2}\right)}\right)={\mathbf{u}}^2$

localid="1654678079311" $\mathbf{u}=\left(\frac{\begin{array}{c}\begin{array}{c}\\ 1\end{array}\end{array}}{\sqrt{\left(1+\frac{{\mathrm{\eta }}^2}{{\mathrm{c}}^2}\right)}}\right)\mathbf{\eta }$

Therefore, the formula for uin terms oflocalid="1654678116724" $\mathbf{\eta }$islocalid="1654678106817" $\mathbf{u}=\left(\frac{\begin{array}{c}\begin{array}{c}\\ 1\end{array}\end{array}}{\sqrt{\left(1+\frac{{\mathrm{\eta }}^2}{{\mathrm{c}}^2}\right)}}\right)\mathbf{\eta }$.

Determine the relationship between proper velocity and rapidity and find η as a function of θ:

(b)

Write the formula for the rapidity.

$$\begin{gathered} \theta ={\tanh }^{-1}\left(\frac{\mathbf{u}}{c}\right) \\ \tanh \theta =\left(\frac{\mathbf{u}}{c}\right) \end{gathered}$$

From equation (1), it is known that:

$\mathbf{\eta }=\frac{1}{\sqrt{1-{\displaystyle \frac{{\mathrm{u}}^2}{{\mathrm{c}}^2}}}}\mathbf{u}$

Substitute $\tan \theta =\left(\frac{\mathbf{u}}{c}\right)$in the above formula.

role="math" localid="1654677620862" $$\begin{gathered} \mathbf{\eta }=\frac{1}{\sqrt{1-{\tanh }^2\mathrm{\theta }}}\mathbf{u} \\ \mathbf{\eta }=\frac{\mathbf{1}}{\sqrt{1-{\displaystyle \frac{{\sinh }^2\theta }{{\cosh }^2\theta }}}}\mathbf{u} \\ \mathbf{\eta }=\frac{\mathrm{cosh\theta }}{\sqrt{{\cos }^2\theta \mathit{-}{\sinh }^2\theta }}\mathbf{u} \\ \mathbf{\eta }=\cosh \theta \mathbf{u} \end{gathered}$$

On further solving,

$$\begin{gathered} \mathbf{\eta }=\cosh \theta \mathbf{u} \\ \mathbf{\eta }=\cosh \theta \left(c\tanh \theta \right) \\ \mathbf{\eta }=\cosh \theta \mathit{\left(}\mathit{c}\frac{\mathit{sinh}\mathit{\theta }}{\mathit{cosh}\mathit{\theta }}\mathit{\right)} \\ \mathbf{\eta }=c\sinh \theta \end{gathered}$$

Therefore, the relation between proper velocity and rapidity or the value of $\mathbf{\eta }$as a function of $\theta$isrole="math" localid="1654677536539" $\mathbf{\eta }=c\sinh \theta \mathit{.}$