201902210247 Exercise 5.5.1

Check the cases where $\alpha =t$ and $\beta =x$ , and where $\alpha =\beta =x$ .

Roughwork.

This exercise is to check the transformation law for the metric tensor of flat spacetime, given by Equation (5.16):

$\eta'_{\alpha\beta}=\displaystyle{\frac{\partial x^\mu}{\partial x'^\alpha}}\displaystyle{\frac{\partial x^\nu}{\partial x'^\beta}} \eta_{\mu\nu}=\eta_{\mu\nu}(\varLambda^{-1})^\mu_{\enspace \alpha}(\varLambda^{-1})^\nu_{\enspace \beta}=\eta_{\alpha\beta}$

(where the second equality is by Equation (5.13), and the third equality by Equation 4.19)

The textbook has already had a checking for $\eta'_{tt}=\eta_{tt}$ through Equations (5.29) and (5.30). I will do the remaining.

Proof. ( $\eta'_{tx}=\eta_{tx}$ )

$\begin{aligned} \eta'_{tx} & = (\varLambda^{-1})^\mu_{\enspace t}(\varLambda^{-1})^\nu_{\enspace x}\eta_{\mu\nu} \\ & = (\varLambda^{-1})^t_{\enspace t}(\varLambda^{-1})^\nu_{\enspace x}\eta_{t\nu} + (\varLambda^{-1})^x_{\enspace t}(\varLambda^{-1})^\nu_{\enspace x}\eta_{x\nu} + (\varLambda^{-1})^y_{\enspace t}(\varLambda^{-1})^\nu_{\enspace x}\eta_{y\nu} \\ & \qquad\quad + (\varLambda^{-1})^z_{\enspace t}(\varLambda^{-1})^\nu_{\enspace x}\eta_{z\nu}\\ \end{aligned}$

But the metric tensor given by Equation (4.6):

$\begin{bmatrix} \eta_{tt} & \eta_{tx} & \eta_{ty} & \eta_{tz} \\ \eta_{xt} & \eta_{xx} &\eta_{xy} & \eta_{xz} \\ \eta_{yt} & \eta_{yx} & \eta_{yy} & \eta_{yz} \\ \eta_{zt} & \eta_{zx} & \eta_{zy} & \eta_{zz} \end{bmatrix}\equiv \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

tells that

$\eta_{t\nu}$ is nonzero only when $\nu =t$ ;

$\eta_{x\nu}$ is nonzero only when $\nu =x$ ;

$\eta_{y\nu}$ is nonzero only when $\nu =y$ ;

$\eta_{z\nu}$ is nonzero only when $\nu =z$ .

So, the summation continues as follows:

$\begin{aligned} \eta'_{tx} & = (\varLambda^{-1})^t_{\enspace t}(\varLambda^{-1})^t_{\enspace x}\eta_{tt} + (\varLambda^{-1})^x_{\enspace t}(\varLambda^{-1})^x_{\enspace x}\eta_{xx} + (\varLambda^{-1})^y_{\enspace t}(\varLambda^{-1})^y_{\enspace x}\eta_{yy} \\ &\qquad\quad + (\varLambda^{-1})^z_{\enspace t}(\varLambda^{-1})^z_{\enspace x}\eta_{zz}\\ & = (\gamma )(\gamma\beta)(-1) + (\gamma\beta)(\gamma)(1) + (0)(0)(1) + (0)(0)(1) \\ & = -\gamma^2\beta + \gamma^2\beta \\ & = 0 \\ & \equiv \eta_{tx} \end{aligned}$

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