In linearized general relativity, we consider spacetime metrics that are perturbatively close to Minkowski, $g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}$, where $\eta_{\mu \nu}=\operatorname{diag}(-1,1,1,1)$ and $h_{\mu \nu}=\mathcal{O}(\epsilon) \ll 1$. In the Lorenz gauge, the Einstein tensor, at linear order, is given by

$$G_{\mu \nu}=-\frac{1}{2} \square \bar{h}_{\mu \nu}, \quad \bar{h}_{\mu \nu}=h_{\mu \nu}-\frac{1}{2} \eta_{\mu \nu} h$$

where $\square=\eta^{\mu \nu} \partial_{\mu} \partial_{\nu}$ and $h=\eta^{\mu \nu} h_{\mu \nu}$.

(i) Show that the (fully nonlinear) Einstein equations $G_{\alpha \beta}=8 \pi T_{\alpha \beta}$ can be equivalently written in terms of the Ricci tensor $R_{\alpha \beta}$ as

$$R_{\alpha \beta}=8 \pi\left(T_{\alpha \beta}-\frac{1}{2} g_{\alpha \beta} T\right), \quad T=g^{\mu \nu} T_{\mu \nu}$$

Show likewise that equation $(\dagger)$ can be written as

$$\square h_{\mu \nu}=-16 \pi\left(T_{\mu \nu}-\frac{1}{2} \eta_{\mu \nu} T\right)$$

(ii) In the Newtonian limit we consider matter sources with small velocities $v \ll 1$ such that time derivatives $\partial / \partial t \sim v \partial / \partial x^{i}$ can be neglected relative to spatial derivatives, and the only non-negligible component of the energy-momentum tensor is the energy density $T_{00}=\rho$. Show that in this limit, we recover from equation $(*)$ the Poisson equation $\vec{\nabla}^{2} \Phi=4 \pi \rho$ of Newtonian gravity if we identify $h_{00}=-2 \Phi$.

(iii) A point particle of mass $M$ is modelled by the energy density $\rho=M \delta(r)$. Derive the Newtonian potential $\Phi$ for this point particle by solving the Poisson equation.

[You can assume the solution of $\vec{\nabla}^{2} \varphi=f(\boldsymbol{r})$ is $\varphi(\boldsymbol{r})=-\int \frac{f\left(\boldsymbol{r}^{\prime}\right)}{4 \pi\left|\boldsymbol{r}-\boldsymbol{r}^{\prime}\right|} d^{3} r^{\prime} .$ ]

(iv) Now consider the Einstein equations with a small positive cosmological constant, $G_{\alpha \beta}+\Lambda g_{\alpha \beta}=8 \pi T_{\alpha \beta}, \Lambda=\mathcal{O}(\epsilon)>0$. Repeat the steps of questions (i)-(iii), again identifying $h_{00}=-2 \Phi$, to show that the Newtonian limit is now described by the Poisson equation $\vec{\nabla}^{2} \Phi=4 \pi \rho-\Lambda$, and that a solution for the potential of a point particle is given by

$$\Phi=-\frac{M}{r}-B r^{2}$$

where $B$ is a constant you should determine. Briefly discuss the effect of the $B r^{2}$ term and determine for which range of the radius $r$ the weak-field limit is a justified approximation. [Hint: Absorb the term $\Lambda g_{\alpha \beta}$ as part of the energy-momentum tensor. Note also that in spherical symmetry $\vec{\nabla}^{2} f=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r f)$.]