How can Minkowski spacetime be expressed as a U(2) manifold?

[etotheipi]

Homework Statement

It's question 6 of Example Sheet 1, very near the bottom of this document:
https://arxiv.org/pdf/gr-qc/9707012.pdf
An image is attached below

Relevant Equations

N/A

Firstly, since $\{ \mathbb{I}, \sigma_x, \sigma_y, \sigma_z \}$ is a basis of the space of $2 \times 2$ Hermitian matrices, and because $X = t \mathbb{I} + x\sigma_x - y \sigma_y + z \sigma_z$, the map is one-to-one (because each matrix has unique decomposition). It's also easily checked the determinant of $dX$ is $-ds^2$.

Next, need to show that any unitary $U$ can be expressed as asked. Consider an arbitrary $U = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then by constraining $U U^{\dagger} = U^{\dagger} U = \mathbb{I}$ we may write $aa^* + bb^* = aa^* + cc^* + dd^* + cc^* = bb^* + dd^* = 1$ and $ac^* + bd^* = ab^* + cd^* = 0$.

It follows from the first four that $|a| = |d| \implies d = a^* e^{i \varphi_1}$ for some $\varphi_1$ and likewise $|b| = |c| \implies c = b^* e^{i \varphi_2}$ for some $\varphi_2$. Thus, substituting for $c$ and $d^*$ in the equation $ab^* + cd^*=0$ gives\begin{align*}
ab^*(1 + e^{i (\varphi_2 - \varphi_1)}) = 0 &\implies \varphi_2 = \varphi_1 + (2n+1)\pi \\

&\implies e^{i \varphi_2} = e^{i \varphi_1} e^{2n\pi i} e^{i \pi} = - e^{i \varphi_1}
\end{align*}Therefore, defining a new parameter $\tau$ by $\tau := \varphi_1 / 2$, and similarly defining $\alpha := a \text{exp}(-i \varphi_1 / 2)$ and $\beta := b \text{exp}(-i \varphi_1 / 2)$, the most general form is indeed $U = e^{i \tau} \begin{pmatrix} \alpha & \beta \\ -\beta^* & \alpha^* \end{pmatrix}$ which satisfies $|\alpha|^2 + |\beta|^2 = |a|^2 + |b|^2 = 1$.

I am confused how to do the last part, i.e. to express the metric in terms of the parameters of $U$. From the definition of the Cayley map we can write down $(1-iX)dU = i(1+U)dX = 2dU(1+U)^{-1}$, but I don't see how that helps. I'd be grateful for a hint about how to get started; thanks

 

[etotheipi]

Actually, I figured out how to do it. You can just take the determinant of the relation they asked you to use, and substitute for $\mathrm{det}(dU)$. Welp, sorry for posting