Past exam of the general relativity and cosmology course of the University of Oxford / 2012 / 1 / b / Solution

Use the curvature sign conventionwhich has the derivative ordering printed in the source for $R^i{}_{0j0}$. A Christoffel symbol is of order $\mu /r^2$, so its derivative is of order $\mu /r^3$, whereas a product of two connections is of order ${\mu }^2/r^4$. Their ratio is $\mu /r\ll 1$. The metric is stationary, hence the temporal derivative term vanishes. ConsequentlyHere $g_{00}=-1+2\mu /r$, and ${\partial }_i{\partial }_j(1/r)=(3x_i{x}_j-r^2{\delta }_{ij})/r^5$. The weak-field tidal curvature in a stationary metric is

Printed sign inconsistency. The source's positive half-Hessian and its final negative-prefactor expression are the opposites of this result. A concrete check at $(r,0,0)$ is ${\Gamma }^1{}_{00}\simeq \mu /r^2$, giving ${\partial }_1{\Gamma }^1{}_{00}=-2\mu /r^3$, while the printed final expression gives $+2\mu /r^3$. Thus both printed equalities cannot hold with its stated derivative ordering. Reversing the definition of the Riemann curvature tensor would give the printed final sign, but would also require reversing the curvature term in the geodesic deviation equation. The convention used here and in the next part gives the physical radial stretching and transverse compression.