Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 4

Write $h_{ab}=\delta g_{ab}$ and $h=g^{ab}{h}_{ab}$. Standard metric variation givesand variation of metric compatibility gives

Varying the coordinate expression for the curvature and replacing partial derivatives by covariant ones givesThereforewhereThis is the metric variation of scalar curvature; its divergence contributes only a boundary term to the variation of the Einstein-Hilbert action.

Varying $\Phi$ and integrating by parts givesSince $\delta F_{ab}=2{\nabla }_{[a}\delta A_{b]}$, variation of the potential and another integration by parts giveThese are the scalar and Maxwell equations of Einstein-Maxwell-dilaton theory.

Varying the matter terms with respect to the inverse metric and writing the Einstein equation as $G_{ab}=T_{ab}$ givesThe factors reflect that the complete action, including its matter terms, carries the common prefactor $1/(16\pi )$. With the conventional definition in which a separate matter action satisfies $G_{ab}=8\pi T_{ab}^{\mathrm{conventional}}$, the corresponding tensor is rescaled accordingly.

Let $k^a$ be null and define $v_c=k^a{F}_{ac}$. The metric-proportional terms vanish after contraction with $k^a{k}^b$, leavingAntisymmetry of $F$ gives $v_c{k}^c=0$. A vector orthogonal to a null vector in Lorentzian signature has nonnegative squared norm, so $v_c{v}^c\ge 0$; also $e^{\alpha \Phi }>0$. Henceand the theory satisfies the null energy condition.