Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 37D (General Relativity)

The Einstein tensor is $G_{\alpha \beta }=R_{\alpha \beta }-\frac{1}{2}{g}_{\alpha \beta }R$. Taking the metric trace of the Einstein field equations in four dimensions givesand substitution back gives the Trace-reversed Einstein field equations in D dimensions specialized to $D=4$:For linearized gravity, the given equation and $G_{\mu \nu }=8\pi T_{\mu \nu }$ implySince $\bar{h}=-h$, its trace is $\square h=16\pi T$. Using ${\bar{h}}_{\mu \nu }=h_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }h$ then yields

In the Newtonian limit of general relativity, time derivatives are negligible and $T_{00}=\rho$ is the only leading component. With signature $(-,+,+,+)$,The $00$ component of the trace-reversed equation becomesIdentifying $h_{00}=-2\Phi$ therefore gives the Poisson equation of Newtonian gravity:

For the Dirac delta distribution source $\rho (\mathbf{r})=M\delta (\mathbf{r})$, the stated Green function of the Laplacian givesThis is the Newtonian potential of a point mass; its negative gradient gives the inverse-square gravitational acceleration $-M\hat{\mathbf{r}}/r^2$.

Move the positive cosmological constant to the source side and defineTo first order about Minkowski spacetime, $T^{(\Lambda )}=-\Lambda /(2\pi )$. Its contribution to the trace-reversed $00$ source isRepeating the Newtonian limit of general relativity calculation givesFor a point mass, ${\nabla }^2{r}^2=6$, soThe radial acceleration isThus positive $\Lambda$ supplies an outward acceleration and balances the point-mass attraction at the static radiusThe weak-field approximation requires both $M/r\ll 1$ and $\Lambda r^2\ll 1$, giving parametrically