(a) Let $(\mathcal{M}, \boldsymbol{g})$ be a four-dimensional spacetime and let $\boldsymbol{T}$ denote the rank $\left(\begin{array}{l}1 \\ 1\end{array}\right)$ tensor defined by

$$\boldsymbol{T}: \mathcal{T}_{p}^{*}(\mathcal{M}) \times \mathcal{T}_{p}(\mathcal{M}) \rightarrow \mathbb{R}, \quad(\boldsymbol{\eta}, \boldsymbol{V}) \mapsto \boldsymbol{\eta}(\boldsymbol{V}), \quad \forall \boldsymbol{\eta} \in \mathcal{T}_{p}^{*}(\mathcal{M}), \quad \boldsymbol{V} \in \mathcal{T}_{p}(\mathcal{M})$$

Determine the components of the tensor $\boldsymbol{T}$ and use the general law for the transformation of tensor components under a change of coordinates to show that the components of $\boldsymbol{T}$ are the same in any coordinate system.

(b) In Cartesian coordinates $(t, x, y, z)$ the Minkowski metric is given by

$$d s^{2}=-d t^{2}+d x^{2}+d y^{2}+d z^{2} .$$

Spheroidal coordinates $(r, \theta, \phi)$ are defined through

$$\begin{aligned} x &=\sqrt{r^{2}+a^{2}} \sin \theta \cos \phi \\ y &=\sqrt{r^{2}+a^{2}} \sin \theta \sin \phi \\ z &=r \cos \theta \end{aligned}$$

where $a \geqslant 0$ is a real constant.

(i) Show that the Minkowski metric in coordinates $(t, r, \theta, \phi)$ is given by

$$d s^{2}=-d t^{2}+\frac{r^{2}+a^{2} \cos ^{2} \theta}{r^{2}+a^{2}} d r^{2}+\left(r^{2}+a^{2} \cos ^{2} \theta\right) d \theta^{2}+\left(r^{2}+a^{2}\right) \sin ^{2} \theta d \phi^{2}$$

(ii) Transform the metric ( $\dagger$ ) to null coordinates given by $u=t-r, R=r$ and show that $\partial / \partial R$ is not a null vector field for $a>0$.

(iii) Determine a new azimuthal angle $\varphi=\phi-F(R)$ such that in the new coordinate system $(u, R, \theta, \varphi)$, the vector field $\partial / \partial R$ is null for any $a \geqslant 0$. Write down the Minkowski metric in this new coordinate system.