Varying the matter terms with respect to the inverse metric and writing the Einstein equation as $G_{ab}=T_{ab}$ gives
$$
\boxed{\begin{aligned}
T_{ab}={}&\frac12\nabla_a\Phi\nabla_b\Phi
-\frac14g_{ab}(\nabla\Phi)^2\\
&+2e^{\alpha\Phi}\left(
F_{ac}F_b{}^c-\frac14g_{ab}F_{cd}F^{cd}
\right).
\end{aligned}}
$$
The 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 $G_{ab}=8\pi T^{\rm conventional}_{ab}$, the corresponding tensor is rescaled accordingly.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-309/4/b/ii/solution