We take another simplifying assumption of spherical symmetry and derive the Schwarzschild geometry as a solution to the Einstein equation with no source. Although we assume time-independence for convenience, the Birkhoff theorem states that the latter follows from the Ricci flatness combined with spherical symmetry. After exploring the resulting black hole geometry, we return to the relativistic Kepler problem with the Hamilton–Jacobi approach. The formation of black holes via gravitational collapse is then studied in a very idealized form known as the Vaidya metric.

The Einstein–Hilbert action may be formulated in the canonical form once a time foliation is introduced. The resulting ADM formulation shows that the bulk part of the Hamiltonian consists only of the Lagrange multipliers, the lapse function, and the shift vector, multiplied by the analogs of the Gauss constraint, namely the Hamiltonian constraint and the Momentum constraints. The on-shell value of the Hamiltonian resides entirely in some boundary expression, half of which originates from the Gibbons–Hawking–York term. The resulting total energy is called the ADM mass. Much of this chapter is devoted to the computational detail that leads to this final fact. Along the way, we revisit the question of the propagating degrees of freedom for gravity and understand why d = 4 graviton has only two helicities and also why the Birkhoff theorem is valid.

We trace how a theorist would eventually discover Special Relativity as an inevitable consequence of the Maxwell theory, as was probably the case with pioneers, including Einstein. After rewriting the Maxwell equations in a manifestly relativistic form, we arrive at the Lorentz transformation and the relativistic free particles. Along the way, we bypass much of the confusing discussion of Lorentz contraction, time-dilation, and the so-called Twin Paradox, focusing on the proper time as the only absolute measure of time.

Canonical quantization of matter fields admits a surprisingly simple extension into curved spacetime as long as there exists a suitable time foliation. The main conceptual difficulty arises when multiple time foliations compete, with nontrivial Bogoliubov transformations mixing up the notions of particle and antiparticle. With the Minkowski spacetime written in the Rindler coordinates as a prototype, we explore how various distinct vacua appear and how to choose one based on physics considerations. For eternal black hole geometry, smooth event horizons demand the Hartle–Hawking vacuum, while, for black holes made from gravitational collapse, the radiation vacuum of Hawking naturally emerges. After a brief stop on black hole thermodynamics, we close the volume with a simple observation of how all these are connected to the primordial density perturbation of the cosmic inflation scenario.