Past exam of the general relativity II course of the University of Oxford / 2017 / 1

The coordinate basis vector $K={\partial }_v$ has constant components $K^a={\delta }_v^a$. Its norm isso it is a nonzero null vector everywhere. The Lie derivative of the metric tensor isThus $\boxed{K={\partial }_v\text{ is a null Killing vector field}.}$ In fact it is a parallel vector field: $K_b=g_{bv}={\delta }_b^u$ and the Christoffel symbols found below give ${\nabla }_a{K}_b=-{\Gamma }^u{}_{ab}=0$. This stronger property is characteristic of this plane-fronted gravitational wave.

In the order $(u,v,x,y)$ the metric tensor and inverse metric areOnly $g_{uu}$ has nonzero partial derivatives. Substitution into the formula for the Levi-Civita connection,therefore givesEvery other Christoffel symbol vanishes. For example $g^{ud}$ selects $d=v$, and the metric tensor is independent of $v$ with constant $g_{av}$; hence ${\Gamma }^u{}_{bc}=0$.

Use the paper's Riemann curvature tensor conventionFor the independent components,The quadratic Christoffel symbol terms vanish here: any potentially contributing intermediate $u$ index gives a factor ${\Gamma }^u{}_{ab}=0$, and the remaining possible products have a lower $v$ index and vanish. Since the symmetric matrix $M$ gives $F_{,ij}=2M_{ij}$, lowering the first index givesAll remaining nonzero components follow from the Riemann curvature tensor symmetriesThere are no other independent components: terms involving a $v$ index vanish, while nonzero terms require one $u$ in each antisymmetric pair.

The Ricci tensor is consequentlyThe Ricci scalar vanishes even when the trace is nonzero, since $g^{uu}=0$. With zero cosmological constant, the vacuum Einstein field equations reduce to $R_{ab}=0$. Therefore the necessary and sufficient condition isThis is the Ricci tensor of a Brinkmann plane wave: a nonzero, trace-free profile can have nonzero Riemann curvature tensor despite vanishing Ricci tensor.

The geodesic deviation equation compares neighboring geodesics by covariant derivatives. Replacing those derivatives by ordinary derivatives of coordinate components requires a parallel transport frame along the reference geodesic. There is also an affine parameter normalization to specify: the argument of $M$ is $u$, and it becomes $\tau$ only after choosing $u=\tau$.

For the intended local tidal calculation, take the reference timelike geodesicHere $g(V,V)=-1$, so $\tau$ is proper time. Both $F$ and its first partial derivatives vanish on $\gamma$. All Christoffel symbols therefore vanish along this curve, and its coordinate basis is parallel transport along it. With the Riemann curvature tensor from part (b),The Jacobi field equation thus becomes, in this adapted frame,A purely transverse initial Jacobi field, with ${\sigma }^u={\sigma }^v={\dot{\sigma }}^u={\dot{\sigma }}^v=0$ initially, remains transverse. This transverse tidal equation of a Brinkmann plane wave describes the relative tidal acceleration of the freely falling bodies.

In vacuum write the symmetric matrix asIts instantaneous eigenvalues are $\pm \sqrt{A^2+B^2}$. The tidal acceleration stretches along one principal axis and squeezes along the perpendicular axis, with the axes allowed to vary with $u$. This describes accelerations; it does not imply monotonically growing and shrinking separations for arbitrary initial velocities. For the simple constant profile $A={\omega }^2$, $B=0$,Thus the geodesic deviation can be defocusing in one direction and oscillatory in the other. For a varying profile, diagonalizing at each instant does not decouple the ordinary differential equations without also including derivatives of the rotating eigenvectors.

Qualification of the coordinate claim. The displayed equation for ${\sigma }^v$ is not true for arbitrary coordinate displacements along arbitrary reference geodesics. For synchronized neighboring curves with $u=\tau$ and ${\sigma }^u=0$, the exact geodesic equations areTheir variation givesThe second expression need not vanish. A concrete vacuum counterexample is $M=\mathrm{diag}(1,-1)$ and the one-parameter family of timelike geodesicsEach has $g(V,V)=-1$. Differentiating with respect to $A$ produces a Jacobi fieldThis even has $g(V,\sigma )=0$, so it is a physical orthogonal separation. The covariant derivative form of geodesic deviation remains correct; the unrestricted assertion about ordinary coordinate derivatives needs the adapted reference and frame used above. If instead $\dot{u}=c$ is a constant affine parameter normalization, the transverse tidal coefficient is $c^{2}M(u)$.