Write down the differential equations governing geodesic curves $x^{a}(\lambda)$ both when $\lambda$ is an affine parameter and when it is a general one.

A conformal transformation of a spacetime is given by

$$g_{a b} \rightarrow \tilde{g}_{a b}=\Omega^{2}(x) g_{a b}$$

Obtain a formula for the new Christoffel symbols $\tilde{\Gamma}_{b c}^{a}$ in terms of the old ones and the derivatives of $\Omega$. Hence show that null geodesics in the metric $g_{a b}$ are also geodesic in the metric $\tilde{g}_{a b}$.

Show that the Riemann tensor has only one independent component in two dimensions, and hence derive

$$R=2 \operatorname{det}\left(g^{a b}\right) R_{0101}$$

where $R$ is the Ricci scalar.

It is given that in a 2-dimensional spacetime $M, R$ transforms as

$$R \rightarrow \tilde{R}=\Omega^{-2}(R-2 \square \log \Omega)$$

where $\square \phi=g^{a b} \nabla_{a} \nabla_{b} \phi$. Assuming that the equation $\square \phi=\rho(x)$ can always be solved, show that $\Omega$ can be chosen to set $\tilde{g}$ to be the metric of 2-dimensional Minkowski spacetime. Hence show that all null curves in $M$ are geodesic.

Discuss the null geodesics if the line element of $M$ is

$$d s^{2}=-t^{-1} d t^{2}+t d \theta^{2}$$

where $t \in(-\infty, 0)$ or $(0, \infty)$ and $\theta \in[0,2 \pi]$.