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

Vary the gravitational volume factor and use part (a)(ii):

δ (f (R) d vol_{g}) = [f^{'} (R) δ R + \frac{1}{2} f (R) g^{ab} h_{ab}] d vol_{g} .

Twice applying integration by parts moves both derivatives in

δ R

from the compactly supported

h_{ab}

onto

f^{'} (R)

. Combining the result with the stated matter variation and requiring every coefficient of

h_{ab}

to vanish gives the f(R) gravity equation

f^{'} (R) R_{ab} - \frac{1}{2} g_{ab} f (R) + (g_{ab} \nabla_{c} \nabla^{c} - \nabla_{a} \nabla_{b}) f^{'} (R) = 8 π T_{ab} .

Therefore

α^{'} = 1, β^{'} = - 1 .

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