91影视

Write the expression for the proper velocity 4 vector.

${\overline{\mathit{畏}}}^{\mathit{渭}}\mathbf{=}{\mathbf{饾毑}}_{\mathit{谓}}^{\mathit{渭}}\mathbf{}{\mathit{畏}}^{\mathit{v}}$

Here,$饾毑$is the Lorentz transformation matrix, $渭$ is the row matrix, and v is the column matrix.

Take the invariant product of the 4-velocity.

$$\begin{gathered} {畏}^{渭}{畏}_{渭}=\left(\begin{array}{c}{饾毑}_{谓}^{渭}\end{array}{畏}^{谓}\right)\left(\begin{array}{c}{饾毑}_{渭}^{谓}\end{array}{畏}_v\right) \\ {畏}^{渭}{畏}_{渭}=-{\left(\begin{array}{c}{畏}^0\end{array}\right)}^2+{畏}^2 \end{gathered}$$ 鈥︹�� (1)

Here ${畏}^0$, is the zeroth component of the spatial part of a 4-vector and $畏$is the proper velocity.

$畏=\frac{u}{\sqrt{1-{\displaystyle \frac{u^2}{c^2}}}}$

Write the expression of the spatial part of a 4-vector.

${畏}^0=\frac{d{x}^{渭}}{d蟿}$

Write the zeroth component of the spatial part of a 4-vector.

$$\begin{gathered} {畏}^0=\frac{d{x}^0}{d蟿} \\ {畏}^0=c\frac{dt}{d蟿} \\ {畏}^0=\frac{c}{\sqrt{1-{\displaystyle \frac{u^2}{c^2}}}} \end{gathered}$$

Substitute $\frac{c}{\sqrt{1-{\displaystyle \frac{u^2}{c^2}}}}$for ${畏}^0$and $\frac{u}{\sqrt{1-{\displaystyle \frac{u^2}{c^2}}}}$for $畏$equation (1)

role="math" localid="1654680404953" $$\begin{gathered} {畏}^{渭}{畏}_{渭}=-\left(\frac{\begin{array}{c}\begin{array}{c}\\ c\end{array}\end{array}}{\sqrt{1-{\displaystyle \frac{u^2}{c^2}}}}\right)+\left(\frac{\begin{array}{c}\begin{array}{c}\\ u\end{array}\end{array}}{\sqrt{1-{\displaystyle \frac{u^2}{c^2}}}}\right) \\ {畏}^{渭}{畏}_{渭}=\frac{1}{\left(1-{\displaystyle \frac{u^2}{c^2}}\right)}\left(-c^2+u^2\right) \\ {畏}^{渭}{畏}_{渭}=-c^2\frac{\left(1-\frac{u^2}{c^2}\right)}{\left(1-\frac{u^2}{c^2}\right)} \\ {畏}^{渭}{畏}_{渭}=-c^2 \end{gathered}$$

As ${畏}^0>{畏}^{渭}{畏}_{渭}$the value of ${畏}^{渭}$ will be timelike.

Therefore, the invariant product of the 4-velocity is${畏}^{渭}{畏}_{渭}=-c^2$and${畏}^{渭}$is timelike.