Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 3

Put $h_{ab}=\delta g_{ab}$. Varying $g^{ac}{g}_{cb}={\delta }_b^a$ givesThe determinant identity $\delta \log |\det g|=g^{ab}{h}_{ab}$ givesFinally, varying the formula for the Christoffel symbols and rewriting partial derivatives covariantly gives the tensorThese are the basic metric variation identities.

Varying the Ricci tensor and contracting givesSubstitution of part (i), followed by metric compatibility, reduces the divergence toThus the metric variation of scalar curvature isand consequently

Vary the gravitational volume factor and use part (a)(ii):Twice applying integration by parts moves both derivatives in $\delta R$ from the compactly supported $h_{ab}$ onto $f^{\prime }(R)$. Combining the result with the stated matter variation and requiring every coefficient of $h_{ab}$ to vanish gives the f(R) gravity equationTherefore

For Minkowski spacetime, $R_{ab}=0$ and $R=0$, while $f^{\prime }(R)$ is constant. In vacuum $T_{ab}=0$, so every derivative term vanishes and the remaining term is $-\frac{1}{2}{\eta }_{ab}f(0)=0$. Hence Minkowski spacetime is a vacuum solution of this modified gravity theory.

Because $R=O(ϵ)$ andthe Taylor series gives $f(R)=R+O({ϵ}^3)$ and $f^{\prime }(R)=1+O({ϵ}^2)$. The extra derivative terms in the f(R) equation are therefore at least second order. At first order the theory reduces to the Linearized Einstein equationsIn harmonic coordinates, equivalently the Lorenz gauge in linearized gravity ${\partial }_{\mu }{\bar{h}}^{\mu }{}_{\nu }=0$, the trace-reversed metric perturbationsatisfiesThus Linearized f(R) gravity about Minkowski spacetime has the same weak-field predictions as general relativity under these assumptions. Differences can first appear through nonlinear curvature terms or in backgrounds about which $f^{\prime \prime }(R)$ is nonzero.