Classic material: photon sphere, event horizon, Eddington–Finkelstein coordinates. Their discussion of “river model” of black holes is unique.
Quadrupole formula, TT gauge, energy flux. Still a benchmark. Classic material: photon sphere
The core: Riemann tensor, Bianchi identities, Einstein equation. Pay special attention to Box 14.5 (derivation from action principle). Classic material: photon sphere