Remember to show all calculations and always include a complete, clear explanation of your reasoning.

• 1. Conformal metrics: A conformal metric is an equivalence class of metrics which differ by a conformal transformation; two metrics \(g^{(1)}_{ab}\) and \(g^{(2)}_{ab}\) on a manifold M belong to the same equivalence class if there is a non-vanishing function \(\Omega: M\to{\mathbb R}\) such that \(g^{(1)}_{ab} = \Omega^2\,g^{(1)}_{ab}\). Show that at each point \(p\in M\) a conformal metric is equivalent to the shape of the light cone at that point, i.e., knowledge of which tangent vectors at p are timelike, null, or spacelike.
   
• 2. Wald, Problem 8.1.
   
• 3. Wald, Problem 9.1.