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.

More varieties of spherically symmetric and axially symmetric solutions are found, such as the Reissner–Nordström black hole as well as de Sitter and anti-de Sitter variations thereof. Rotating black holes are also given a healthy dose of attention. An old but relatively less-known fact invoked in this chapter is how the rotating Kerr solution can be extracted from an analytic continuation of the spherically symmetric Schwarzschild solution. The same relation is known between the Kerr–Newman and the Reissner–Nordström. Maximally symmetric solutions with the cosmological constant, de Sitter, and anti-de Sitter, are also explored under various coordinate choices.

In preparation for the ADM formulation of General Relativity, we quickly scan Dirac's theory of constrained systems. How to deal with dynamics when the number of variables is larger than the true degrees of freedom is at issue. Starting from a familiar classical mechanics with Lagrange multipliers, we classify constraints into the first class and the second class. The former is particularly relevant for field theories with gauge redundancies, as is the case with General Relativity. Again, the Maxwell theory is invoked as a prototype, with the Gauss constraint given a unique meaning as the generator of the gauge redundancy.

Although the metric is clearly one of the minimal necessities for physics in curved spacetimes, the orthonormal frame is often more sensible as the bearer of the Riemannian geometry. A hallmark of the covariantly constant metric is how the Riemann curvature can at best rotate tensors, whose characteristics are lost in the Christoffel symbol. The Maurer–Cartan alternative addresses this cleanly by introducing a bigger set of variables, the vielbein, which defines an SO-valued connection 1-form, also known as the spin connection, and leads us to the curvature 2-form on par with the ordinary Yang–Mills field strength. Related issues, such as how the Riemann tensor in a general basis differs from the common commutator definition, are also addressed. Several highly symmetric geometries are offered as examples.

We model the Einstein equation, which eventually determines the spacetime metric, after the Maxwell equations. The Bianchi identity of the electromagnetic field strength is required by the charge–current conservation, which inspires the conserved energy-momentum and a symmetric rank-2 tensor that should be divergence-free as a mathematical identity. The universal Bianchi identity of the curvature 2-form is shown to build the divergence-free Einstein tensor as the requisite symmetric tensor, leading us to the Einstein equation. The Newtonian limit fixes the relative coefficient, via the weak field approximation that also leads to gravitational waves. Some rudimentary explorations of the latter are offered.

Once the proper time is recognized as the only viable notion of time, relativistic gravity as an external force arises naturally via the analogy of how one introduces the metric in Newtonian dynamics in curvilinear coordinates. The resulting action principle comes with a key property that the time parameter choice should be entirely irrelevant to the dynamics, which is, in turn, used to simplify the action by choosing the parameter to be the proper time of the particle in question. With the metric supplied later by the gravitational field equation, we discover that the Kepler problem elevates to a fully relativistic one straightforwardly. This chapter closes with the application of all these to the light-bending phenomena.

This chapter considers the analog of the time-dependent Hartree–Fock (mean-field) decoupling treated in Part I and extends it to the broken-symmetry phase for superfluid fermions. Two coupled equations for the “normal” and “anomalous” time-dependent single-particle Green’s functions are obtained, which extend to nonequilibrium situations the equations originally obtained at equilibrium by Gor’kov, soon after the BCS original article on the theory of superconductivity. Accordingly, the time-dependent gap (order) parameter is also introduced.

This chapter derives from first principles the time-dependent Gross–Pitaevskii equation, which describes the time-dependent behavior of the condensate wave function associated with the composite bosons that form on the BEC side of the BCS–BEC crossover at sufficiently low temperature. The derivation relies on the Green’s functions method for nonequilibrium problems developed before and explores the assumption that the fermionic chemical potential, associated with the initial preparation of the system at thermodynamic equilibrium, is the largest energy scale in the problem. The relation between the scattering length for composite bosons and the scattering length for the constituent fermions is also discussed.

This chapter gives a concise overview about a number of specific physical problems, which are of recent, current, and possibly future interest, problems that can be ideally dealt with in terms of the nonequilibrium Schwinger–Keldysh Green’s functions technique developed at a formal level in Parts I and II. Accordingly, this chapter aims at providing a synthetic demonstration of the versatility of the Schwinger–Keldysh technique, especially in the view of possible future applications to scientific problems as well as to technological issues. In particular, it considers the main features associated with closed and driven open quantum systems, spectroscopic problems related to pump and probe photoemission, metastable photo-induced superconductivity, dynamics induced by quenches and rumps in “closed” quantum systems with emphasis on thermalization, and driven “open” quantum systems with emphasis on dissipation. A more detailed treatment of these topics is deferred to the following chapters.

This chapter considers the extension of the t-matrix approximation to the superfluid phase, for which it is convenient to restrict from the outset to a contact-type interparticle interaction. This is because, when addressing nonequilibrium (time-dependent) situations, the extension of the fermionic t-matrix approach from the normal to the superfluid phase requires a careful account for the Nambu indices in the two-particle channels, owing to the presence of the “anomalous” single-particle Green’s functions. The ladder approximation for the many-particle T-matrix is specifically considered.