Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 49 / 4

Treat the printed $e^{\mu }$ as an orthonormal coframe, with frame metric $\eta =\mathrm{diag}(-1,1,1,1)$. Work on a patch $z\ne 0$ and $F=1-\alpha z^3>0$, so the given real coframe exists. Define $f=\sqrt{F}$ and $A=z{f}^{\prime }-f$. The exterior derivatives areThe connection 1-forms satisfying Cartan's first structure equation and metric compatibility areAll other forms vanish. With a Lorentzian frame, it is the lowered forms ${\omega }_{ab}={\eta }_{ac}{\omega }^c{}_b$ that are antisymmetric; the two mixed time–space forms are equal. These displayed forms solve the torsion-free structure equation, and uniqueness of the Levi-Civita connection identifies them as the required connection.

Apply Cartan's second structure equation. For example,The remaining derivatives give ${\Theta }^1{}_3=Af{e}^1\wedge e^3$ and ${\Theta }^1{}_2=-f^2{e}^1\wedge e^2$, with analogous forms for index 2. Put $q=\alpha z^3$ and $B=1+q/2$. From $f^2=1-q$,Therefore all six independent curvature 2-forms areThe other six are fixed by ${\Theta }^b{}_a=-{\eta }_{aa}{\eta }_{bb}{\Theta }^a{}_b$ for $a\ne b$, with no sum: equal for time–space pairs and opposite for spatial pairs. All diagonal forms are zero. The Lorentzian connection-form antisymmetry is essential to these signs.

Use the convention ${\Theta }^a{}_b=\frac{1}{2}{R}^a{}_{bcd}{e}^c\wedge e^d$ and $R_{ab}=R^c{}_{acb}$. Each independent two-form has only one coordinate-plane wedge, so the Ricci tensor is diagonal. Contracting the six coefficients givesBut $2B+F=2(1+q/2)+(1-q)=3$, soIn coordinate-independent form, $R_{\mu \nu }=-3g_{\mu \nu }$. Thus the vacuum Einstein field equations $G_{\mu \nu }+\Lambda g_{\mu \nu }=0$ hold withThe curvature radius is one in the metric's normalization. The parameter $\alpha$ changes the individual curvature 2-forms but cancels from their Ricci contraction. For $\alpha =0$ this is Anti-de Sitter spacetime; for nonzero $\alpha$ it is the planar Einstein metric with cubic radial function, with a nontrivial Weyl tensor rather than universally constant sectional curvature. The result is local on regular coordinate patches; a zero of $F$ invalidates this particular static coframe, not the tensor equation in a suitable regular extension.