Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 35D (General Relativity)

The coordinate transformation law for a affine connection has a homogeneous term with one upper and two lower index factors, and an inhomogeneous term involving second coordinate derivatives. The latter is the same for both Levi-Civita connections, so it cancels in their difference. Consequentlywhich is exactly the tensor component transformation law. Equivalently, $(\tilde{\nabla }-\nabla {)}_{X}Y$ is linear over functions in both $X$ and $Y$.

At any chosen point take normal coordinates for $g$, so $\Gamma =0$ there. Substitute $\tilde{\Gamma }=\Gamma +T$ into the stated Riemann curvature tensor formula and subtract the original curvature. At that point the mixed products with $\Gamma$ vanish and ordinary derivatives of $T$ equal its covariant derivatives. The result isIn unit-weight antisymmetrization notation this is $\boxed{2T^a{}_{b[d;c]}-2T^a{}_{e[d}{T}^e{}_{c]b}}$. Every term is a tensor, so the identity established in normal coordinates holds in every coordinate system.

Substitute $T^a{}_{bc}={\ell }^a{g}_{bc}$ into the preceding identity, using metric compatibility. Set $M_{bd}={\ell }_{b;d}+{\ell }_b{\ell }_d$ and $S=g^{bd}{M}_{bd}={\nabla }_a{\ell }^a+{\ell }_a{\ell }^a$. Contract the first and third curvature indices, in the stated curvature convention, to obtainIf the two Ricci tensors agree, then $M_{bd}=S{g}_{bd}$. Taking the trace in dimension $n>1$ gives $S=nS$, so $S=0$ and $M=0$. Conversely $M=0$ makes the Ricci difference vanish. Thus $\boxed{{\tilde{R}}_{bd}=R_{bd}⟺{\ell }_{b;d}+{\ell }_b{\ell }_d=0}$.

The preceding condition is symmetric in $b,d$, so ${\ell }_{[b;d]}=0$. On a patch where the potential representation supplied in the question holds, write ${\ell }_a={\partial }_a\varphi$. DefineThen $k_{a;b}=e^{\varphi }({\ell }_{a;b}+{\ell }_a{\ell }_b)$. Since $e^{\varphi }\ne 0$, the condition on $\ell$ is exactly $\boxed{k_{a;b}=0}$, so $k$ is a parallel covector field. Closed covectors have this scalar-potential representation locally by the Poincare lemma; a global potential additionally requires exactness, for example on a simply connected domain.