Find the invariant product of the 4-velocity with itself, ${\mathit{\eta }}^{\mathbf{\mu }}{\mathit{\eta }}^{\mathbf{\mu }}$. Is localid="1654516875655" ${\mathit{\eta }}^{\mathbf{\mu }}$timelike, spacelike, or lightlike?

Write the expression for the proper velocity 4 vector.

${\overline{\mathit{\eta }}}^{\mathit{\mu }}\mathbf{=}{\mathbf{𝚲}}_{\mathit{\nu }}^{\mathit{\mu }}\mathbf{}{\mathit{\eta }}^{\mathit{v}}$

Here,$𝚲$is the Lorentz transformation matrix, $\mu$ is the row matrix, and v is the column matrix.

Take the invariant product of the 4-velocity.

$$\begin{gathered} {\eta }^{\mu }{\eta }_{\mu }=\left(\begin{array}{c}{𝚲}_{\nu }^{\mu }\end{array}{\eta }^{\nu }\right)\left(\begin{array}{c}{𝚲}_{\mu }^{\nu }\end{array}{\eta }_v\right) \\ {\eta }^{\mu }{\eta }_{\mu }=-{\left(\begin{array}{c}{\eta }^0\end{array}\right)}^2+{\eta }^2 \end{gathered}$$ …… (1)

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

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

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

${\eta }^0=\frac{d{x}^{\mu }}{d\tau }$

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

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

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

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

As ${\eta }^0>{\eta }^{\mu }{\eta }_{\mu }$the value of ${\eta }^{\mu }$ will be timelike.

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