Penrose-Carter diagrams
(a) In the Penrose-Carter (PC) diagrams for Minkowski spacetime, plot lines of constant $t$ and constant $r$
(where $t$, $r$ are Minkowski coordinates).
(b) Consider two metrics, $g_{ab}$ and $\tilde{g}_{ab} = f(x^k)^2 g_{ab}$, with $f(x^k)$ a real function. Show that the light cone
structure of their PC diagrams are similar as long as $f(x^k) \neq 0$. Explain the difference that arises when
there exists a domain where $f(x) = 0$ by comparing the PC diagrams of Schwarzschild geometry, Minkowski
spacetime, and of Schwarzschild spacetime with negative mass (that is, with $M < 0$ in the metric).
(c) Consider a class of metrics $g_{ab} = f(x^k)^2 \eta_{ab}$, with $f(x^k)$ a real function. Obtain the Christoffel symbols
for $g_{ab}$ and show that timelike/spacelike geodesics of $\eta_{ab}$ are not necessarily timelike/spacelike geodesics of
$g_{ab}$.
