APPENDIX D
Where to go nextWhere to go next
D.1 Foundational lecture notes#D.1 Foundational lecture notes
Two substantial, freely accessible courses make good companions after this tutorial:Two substantial, freely accessible courses make good companions after this tutorial:
- David Tong, General Relativity: a connected course spanning geometry, Einstein’s equation, weak fields, waves, and black holes, with problem sheets.David Tong, General Relativity: a connected course spanning geometry, Einstein’s equation, weak fields, waves, and black holes, with problem sheets.
- Sean Carroll, Lecture Notes on General Relativity: a full set of introductory graduate notes, including special relativity, manifolds, curvature, dynamics, and applications. Carroll’s page also flags that the older notes are not maintained as comprehensively as the later textbook.Sean Carroll, Lecture Notes on General Relativity: a full set of introductory graduate notes, including special relativity, manifolds, curvature, dynamics, and applications. Carroll’s page also flags that the older notes are not maintained as comprehensively as the later textbook.
Do not compare a single curvature equation across books before comparing their signature, Riemann definition, Ricci contraction, and units. The physics can agree while several intermediate signs differ.Do not compare a single curvature equation across books before comparing their signature, Riemann definition, Ricci contraction, and units. The physics can agree while several intermediate signs differ.
D.2 Choose the next subject by the question that bothers you#D.2 Choose the next subject by the question that bothers you
| If your question is… | Study next | What to learn to do |
|---|---|---|
| “What does curvature do beyond symmetric examples?” | Differential geometry and Jacobi fields | Move between abstract, coordinate, and frame formulations |
| “How do we evolve a binary black hole?” | Initial-value GR and numerical relativity | Solve constraints, choose gauge, understand stability and waveform extraction |
| “How do I calculate what an observer sees?” | Relativistic astrophysics, ray tracing, and geometric optics | Connect tetrads, null geodesics, emission models, and detector quantities |
| “How do waves carry energy?” | Perturbation theory, asymptotic structure, and radiation theory | Distinguish local approximations from asymptotic flux definitions |
| “How can a horizon have entropy?” | Quantum fields in curved spacetime and black-hole thermodynamics | Separate classical horizon geometry from quantum state and detector effects |
| “Why this action rather than another?” | Effective field theory and gravitational amplitudes | State symmetry and scale assumptions and calculate controlled corrections |
| “What is conserved in a universe without a preferred time?” | Symmetries, Hamiltonian GR, and covariant phase space | Construct charges with explicit boundary and symmetry assumptions |
| “Why did this take Einstein so long?” | History based on notebooks and correspondence | Separate a modern textbook derivation from the actual process of discovery |
D.3 Research and historical sources cited in the chapters#D.3 Research and historical sources cited in the chapters
This list gathers the sources linked at the point of use. A link to a paper does not mean every interpretation of its subject is settled. The chapter text specifies whether a statement is classical, perturbative, semiclassical, observational, or an open problem.This list gathers the sources linked at the point of use. A link to a paper does not mean every interpretation of its subject is settled. The chapter text specifies whether a statement is classical, perturbative, semiclassical, observational, or an open problem.
- Janssen and Renn, Untying the KnotJanssen and Renn, Untying the Knot
- Norton, How Einstein Found His Field Equations: 1912–1915Norton, How Einstein Found His Field Equations: 1912–1915
- David Tong’s differential-geometry chapterDavid Tong’s differential-geometry chapter
- Einstein’s 1905 paper, in English translationEinstein’s 1905 paper, in English translation
- Tong’s discussion of the metric volume formTong’s discussion of the metric volume form
- Clifford Will’s review of tests of gravitationClifford Will’s review of tests of gravitation
- Tong’s treatment of the equivalence principle and Rindler motionTong’s treatment of the equivalence principle and Rindler motion
- Sean Carroll’s notes, in the geodesics sectionSean Carroll’s notes, in the geodesics section
- Sean Carroll’s university lecture notes, “Curvature”Sean Carroll’s university lecture notes, “Curvature”
- David Tong’s general relativity notesDavid Tong’s general relativity notes
- Manasse and Misner’s original Fermi-coordinate paperManasse and Misner’s original Fermi-coordinate paper
- Pravda, Pravdova, Coley, and MilsonPravda, Pravdova, Coley, and Milson
- Misner and Putnam in “Active Gravitational Mass”Misner and Putnam in “Active Gravitational Mass”
- Lehner, Myers, Poisson, and Sorkin, “Gravitational action with null boundaries”Lehner, Myers, Poisson, and Sorkin, “Gravitational action with null boundaries”
- “Quasilocal Energy and Conserved Charges Derived from the Gravitational Action”“Quasilocal Energy and Conserved Charges Derived from the Gravitational Action”
- David Tong’s black-hole lecture notesDavid Tong’s black-hole lecture notes
- Chruściel and Costa, On uniqueness of stationary vacuum black holesChruściel and Costa, On uniqueness of stationary vacuum black holes
- Isaacson, Gravitational Radiation in the Limit of High Frequency. IIIsaacson, Gravitational Radiation in the Limit of High Frequency. II
- LIGO Scientific Collaboration and Virgo Collaboration, Observation of Gravitational Waves from a Binary Black Hole MergerLIGO Scientific Collaboration and Virgo Collaboration, Observation of Gravitational Waves from a Binary Black Hole Merger
- Riess and collaborators, Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological ConstantRiess and collaborators, Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant
- Éric Gourgoulhon’s author-written notes on the 3+1 formalismÉric Gourgoulhon’s author-written notes on the 3+1 formalism
- Choquet-Bruhat and Geroch’s original Cauchy-problem paperChoquet-Bruhat and Geroch’s original Cauchy-problem paper
- David Tong’s author-written chapter on connections and Cartan geometryDavid Tong’s author-written chapter on connections and Cartan geometry
- Dadhich and Pons’s paper on Einstein-Hilbert and Einstein-Palatini formulationsDadhich and Pons’s paper on Einstein-Hilbert and Einstein-Palatini formulations
- Penrose’s 1965 paper, “Gravitational Collapse and Space-Time Singularities”Penrose’s 1965 paper, “Gravitational Collapse and Space-Time Singularities”
- Wald’s research review of black-hole thermodynamicsWald’s research review of black-hole thermodynamics
- Donoghue’s original work on general relativity as an effective field theoryDonoghue’s original work on general relativity as an effective field theory
- Donoghue’s review of quantum GR and its effective-theory limitsDonoghue’s review of quantum GR and its effective-theory limits
- Solomon and Trodden’s research on higher derivatives in EFTSolomon and Trodden’s research on higher derivatives in EFT
- Deser’s “Self-Interaction and Gauge Invariance”Deser’s “Self-Interaction and Gauge Invariance”
- “Lovelock’s theorem revisited”“Lovelock’s theorem revisited”
- Jérôme Martin’s review of the cosmological constant problemJérôme Martin’s review of the cosmological constant problem
- LVK’s primary GWTC-5.0 tests paperLVK’s primary GWTC-5.0 tests paper
- David Tong, General RelativityDavid Tong, General Relativity
- Sean Carroll, Lecture Notes on General RelativitySean Carroll, Lecture Notes on General Relativity
One final challenge: explain the Einstein equation to a friend without saying “mass bends a rubber sheet.” Use a clock, two neighboring freely falling laboratories, a rule for comparing their directions, and an action whose stationary points determine the geometry. If you can do that—and explain why empty spacetime can still carry waves—you have moved well beyond recognizing the symbols.One final challenge: explain the Einstein equation to a friend without saying “mass bends a rubber sheet.” Use a clock, two neighboring freely falling laboratories, a rule for comparing their directions, and an action whose stationary points determine the geometry. If you can do that—and explain why empty spacetime can still carry waves—you have moved well beyond recognizing the symbols.