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.

We finally come to the question of why the black hole horizon is said to allow only one-way traffic. When viewed from the Kruskal coordinates, suitable for freely falling observers, the horizon consists of several distinct causal components. The future event horizon is the one we usually refer to when describing the one-way nature of the black hole geometry; its “past” cousin allows the opposite flow of trajectories but is often an artifact of the “eternal” geometry. We derive and display Penrose diagrams for many of the solutions accumulated so far and offer cautionary tales on causal structures and singularities.

Although the geometry background from Chapter 3 to the early part of Chapter 6 is self-contained, we attach this appendix to make contact with the more modern language of differential geometry. Fiber bundles can be seen as an obvious generalization of (co-)tangent bundles and allow us to introduce the notion of connections and curvatures in a more invariant manner via the principal bundle. This, in turn, leads to the frame bundle and the spinor bundle, which were implicitly invoked in Chapter 5. A quick overview of G-structure and holonomy classification is followed by how one must deal with spinors in curved spacetime. Although this last part is not used in this volume, it would become an essential tool in the companion book “Geometric Quantum Field Theories.”

The simplest class of solutions to the Einstein equation is that of the expanding universe with homogeneous and isotropic spatial slices. This chapter covers the most basic aspects of the resulting FLRW cosmology, with most examples centered upon the flat spatial slices. After a standard treatise on the expansion of the universe, dominated by ideal fluids, we turn to various puzzles of old-fashioned Big Bang cosmology that all revolve around causality and the initial condition. These puzzles are addressed handsomely by the cosmic inflation scenario that wipes out the initial data, repopulates the universe with matter and radiation, and then also seeds the primordial density perturbation. One puzzle that survives this reinitialization is that of dark energy, and we close with various opinions on the latter, including a Keplerian evasion of the problem via the cosmological landscape.

The Einstein equation is reproduced by Hilbert’s action principle, with the action as a functional of metric or as a functional of metric and connection. We list three related approaches, distinct in detail, with the common outcome of the Einstein equation. One unusual aspect of this action principle is the introduction of the Gibbons–Hawking–York boundary term. We give a detailed description of the extrinsic curvature for this purpose and derive the boundary term. The action principle is advantageous in that it produces a clean derivation of the symmetric energy–momentum tensor on the right-hand side of the Einstein equation. The last part of this chapter addresses how this Hilbert energy–momentum tensor of the Einstein equation is inevitably the same as the Noether one, contrary to popular lore.

Once a differentiable manifold is given, one can equip it with the affine connection or the covariant derivative. The further structure of the metric to be preserved by the affine connection, favors the Levi-Civita connection, which is often expressed via the Christoffel symbols, already encountered for the relativistic particle mechanics. This, in turn, defines the Riemann curvature tensor and the Ricci tensor. Numerous additional structures that follow the covariantly constant metric are introduced, such as raising and lowering of indices, Killing vector fields, the volume form, the Hodge star map, geodesics, and geodesic normal coordinates.

A differentiable manifold, defined with the help of collections of charts, comes with basic notions of calculus before the introduction of the metric. We start with the definitions of vectors as directional derivatives and 1-forms via the natural dual pairing and build up general tensors from these two. Partial derivatives for functions extend to the Lie derivative, while a special subclass of tensors known as differential forms admits the so-called exterior derivative. We develop calculus based on these most basic structures, ending with the Stokes theorem. This sets the stage for the Riemannian geometry, given in two alternate forms in Chapters 4 and 5.