Question
Let ##t,x,y,z## be Lorentz coordinates in flat spacetime, and let $$r=\left(x^2+y^2+z^2\right)^\frac{1}{2}\ ,\ \theta=\arccos{\left(\frac{z}{r}\right)}\ ,\ \phi=\arctan{\left(\frac{y}{x}\right)}$$be the corresponding spherical coordinates. Then$$e_0=\frac{\partial\mathcal{P}}{\partial t}\ ,\ e_r=\frac{\partial\mathcal{P}}{\partial r}\ ,\ e_\theta=\frac{\partial\mathcal{P}}{\partial\theta}\ ,\ e_\phi=\frac{\partial\mathcal{P}}{\partial\phi}$$is a coordinate basis, and $$e_{\hat{0}}=\frac{\partial\mathcal{P}}{\partial t}\ ,\ e_{\hat{r}}=\frac{\partial\mathcal{P}}{\partial r}\ ,\ e_{\hat{\theta}}=\frac{1}{r}\frac{\partial\mathcal{P}}{\partial\theta}\ ,\ e_{\hat{\phi}}=\frac{1}{r\sin{\theta}}\frac{\partial\mathcal{P}}{\partial\phi}$$is a noncoordinate basis.
(a) Draw a picture of ##e_\theta,e_\phi,e_{\hat{\theta}},e_{\hat{\phi}}## at several different points on a sphere of constant ##t,r##.
(b) What are the oneform bases dual to these tangentvector bases?
(c) What is the transformation matrix linking the original Lorentz frame to the spherical coordinate frame ##\left\{e_a\right\}##?
(d) Use this transformation matrix to calculate the metric components ##g_{\alpha\beta}## in the spherical coordinate basis and invert the result to get ##g^{\alpha\beta}##.
(e) Show that the noncoordinate basis ##\left\{e_{\hat{a}}\right\}## is orthonormal everywhere; i.e. that ##g_{\hat{\alpha}\hat{\beta}}=\eta_{\alpha\beta}##; i.e. that $$g=\omega^{\hat{0}}\otimes\omega^{\hat{0}}+\omega^{\hat{r}}\otimes\omega^{\hat{r}}+\omega^{\hat{\theta}}\otimes\omega^{\hat{\theta}}+\omega^{\hat{\phi}}\otimes\omega^{\hat{\phi}}$$(f) Write the gradient of a function ##f## in terms of the spherical coordinate and noncoordinate bases.
(g) What are the components of the LeviCivita tensor in the spherical coordinate and noncoordinate bases?
Answers
This exercise seems mainly to be concerned with putting the rules in Box 8.4 into practice. We meet a noncoordinate (anholonomic) basis. One of the features of these is that coordinates cannot be used to describe positions but you can use components to describe other tensors. So$$p^{\hat{\mu}}=\left(a,b,c,d\right)\equiv ae^{\hat{0}}+be^{\hat{r}}+ce^{\hat{\theta}}+de^{\hat{\phi}}$$is at best meaningless but$$g_{\hat{\alpha}\hat{\beta}}=\left(\begin{matrix}a&0&0&0\\0&b&0&0\\0&0&c&0\\0&0&0&d\\\end{matrix}\right)$$is valid and is the answer to (d) when ##a=1,b=c=d=1## .
I found an excellent new way of creating animated gifs. Which I used for the answer to (a) below. The diagrams for ##e_{\hat{\theta}},e_{\hat{\phi}}## are a good indication of (e).
Coordinate bases going round equator

Coordinate bases going over pole

Noncoordinate bases going round equator 
Noncoordinate bases going round equator
