Geometry and Einstein Gravity Source Map

From local geometry to Cauchy development

This map routes a sharply stated question to the smallest source set that can answer it. General Relativity develops Einstein dynamics; Physical Foundations develops equivalence and operational measurement, Spacetime Geometry supplies the underlying Lorentzian geometry, and GR Formulations develops variational and initial-value formulations. Lecture notes and reviews below orient the reader and fix conventions; where a theorem or formal construction is at issue, the located primary paper carries the claim.

Question	Source class and exact locator	What the source supports	Decisive exclusion
What does the equivalence principle say about local measurement?	Orientation lecture notes: Reall 2013, Ch. 1, printed pp. 1–10.	The clock, redshift, free-fall, and tidal comparisons that separate a locally inertial experiment from curvature measured across a finite region.	Local cancellation of connection coefficients is not removal of curvature on a finite neighborhood, a global inertial frame, or a derivation of the Einstein equation.
Where are Lorentzian metrics, the Levi-Civita connection, and curvature defined in the site’s convention?	Orientation lecture notes: Tong 2019, §3.1, printed pp. 91–99, and §§3.2–3.4, pp. 100–132; Wendl 2021–2022, §§22–24, printed pp. 184–211, and §§25–31, pp. 212–278.	Precise definitions of a pseudo-Riemannian metric, its unique torsion-free metric connection, the curvature operator and contractions, and their differential-geometric context. Tong uses the site’s $(-,+,+,+)$ signature and displayed curvature convention; Wendl supplies the broader geometric construction.	These notes do not make a Lorentzian metric into a solution of a gravitational field equation, and convention-compatible formulas do not eliminate the need to translate sources that reverse the Riemann sign or metric signature.
Which sources connect geodesics with relative acceleration?	Orientation lecture notes: Carroll 1997, §3; Wendl 2021–2022, §§22–24, printed pp. 184–211; Reall 2013, §§4.2–4.4 and Chs. 5–6, printed pp. 49–66.	Affinely parametrized geodesics, normal coordinates, Jacobi fields, Killing integrals, and the tidal interpretation of geodesic deviation, with the deviation sign fixed by the stated curvature convention.	A geodesic equation does not establish geodesic completeness, global causal behavior, or the motion of an accelerated or extended body outside the test-body approximation.
Where are the Einstein equation and Einstein–Hilbert action developed with compatible signs?	Orientation lecture notes: Reall 2013, Chs. 5–6, printed pp. 49–66; Carroll 1997, §4, printed pp. 97–120.	The field equation $G_{ab}+\Lambda g_{ab}=8\pi G T_{ab}$, stress-energy conservation, and the bulk variational route from the Einstein–Hilbert action, using the site’s signature and compatible curvature sign.	The bulk variation alone does not supply a complete boundary-value action, a canonical energy for every boundary condition, or quantum and higher-curvature corrections.
What is the covariant variational and Noether-charge construction?	Primary formalism paper: Iyer and Wald 1994, §§2–4 and appendices.	For a diffeomorphism-covariant Lagrangian, the symplectic potential and current, Noether current and charge, Hamiltonian variation, exact-form ambiguities, and the bifurcation-surface entropy construction.	A Noether form is not automatically an integrable or uniquely normalized physical charge. Boundary conditions, orientations, generator normalization, and Lagrangian or symplectic-potential ambiguities remain part of the problem.
How are proper time and radar coordinates operationally defined?	Orientation lecture notes and a specialist methodological chapter: Reall 2013, Ch. 1, printed pp. 1–10, and §3.2, pp. 35–36; Perlick 2008, §§1–4, printed PDF pp. 1–12.	Proper time along a specified observer worldline and radar time and distance from unique outgoing and returning null signals, including the local comparison with Fermi and optical coordinates.	Radar coordinates are not a unique global simultaneity or distance prescription. Multiple null rays, caustics, or horizons can destroy uniqueness or prevent the return signal.
Which assumptions define the weak-field and Newtonian limits?	Orientation lecture notes and a review: Reall 2013, §§8.1–8.2, printed pp. 77–83; Will 2014, §§3.1–3.2, printed pp. 26–33.	The separate roles of small metric perturbations, slow source motion, quasistatic derivative ordering, gauge freedom, and the Newtonian potential, with explicit $G$ restored.	Weak amplitude alone is neither the Newtonian limit nor a statement of small curvature in every gauge. The cited review is not used here for time-sensitive experimental bounds, and the approximation does not cover strong-field or generic radiative dynamics.
Where are the $3+1$ constraints and evolution equations fixed with the site’s extrinsic-curvature sign?	Orientation lecture notes: Gourgoulhon 2007, §§2.2–2.5, printed pp. 15–38; §§3.3–3.4, pp. 41–50; and Chs. 4–5, pp. 51–82.	Hypersurface projection, Gauss–Codazzi relations, lapse and shift, the Hamiltonian and momentum constraints, and ADM evolution with $n^a n_a=-1$ and $K_{ij}=-\tfrac12\mathcal L_n\gamma_{ij}$.	A geometric split and algebraically correct evolution system do not by themselves prove well-posedness, select a gauge, control constraint growth, or guarantee numerical stability.
How do local Einstein evolution and maximal Cauchy development differ?	Orientation survey for the local reduction: Friedrich and Rendall 2000, §2.4, arXiv v1 PDF pp. 21–24, and §§5.1–5.3, pp. 75–81. Primary theorem paper for the global geometric result: Choquet-Bruhat and Geroch 1969, theorem and construction, printed pp. 329–335.	The survey explains wave-gauge reduction, subsidiary propagation, local existence, geometric uniqueness, and stability for the sufficient classical class $h\in H^s_{\mathrm{loc}}$, $K\in H^{s-1}_{\mathrm{loc}}$, $s\gt5/2$, with $g\in C^0_{\mathrm{loc}}H^s_{\mathrm{loc}}\cap C^1_{\mathrm{loc}}H^{s-1}_{\mathrm{loc}}$ and the required uniform or asymptotic qualifications. The primary theorem gives a maximal globally hyperbolic development, unique up to an initial-data-fixing isometry, into which every globally hyperbolic development in the same solution category embeds isometrically while fixing the data.	A local reduced-system theorem is not the maximal-development theorem. Maximality is only among globally hyperbolic developments and implies neither geodesic completeness nor inextendibility through a non-globally-hyperbolic extension.

References

• Carroll, Sean M. Lecture Notes on General Relativity. arXiv:gr-qc/9712019v1, 1997. arXiv.
• Choquet-Bruhat, Yvonne, and Robert Geroch. “Global Aspects of the Cauchy Problem in General Relativity.” Communications in Mathematical Physics 14 (1969) 329–335. DOI.
• Friedrich, Helmut, and Alan D. Rendall. “The Cauchy Problem for the Einstein Equations.” In Einstein’s Field Equations and Their Physical Implications, edited by Bernd G. Schmidt, Lecture Notes in Physics 540, 127–224. Springer, 2000. arXiv:gr-qc/0002074v1.
• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. arXiv:gr-qc/0703035v1, 2007. arXiv.
• Iyer, Vivek, and Robert M. Wald. “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy.” Physical Review D 50 (1994) 846–864. DOI; arXiv:gr-qc/9403028.
• Perlick, Volker. “On the Radar Method in General-Relativistic Spacetimes.” In Lasers, Clocks and Drag-Free Control: Exploration of Relativistic Gravity in Space, edited by Hansjörg Dittus, Claus Lämmerzahl, and Slava G. Turyshev, Astrophysics and Space Science Library 349, 131–152. Springer, 2008. DOI; arXiv:0708.0170.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. Open PDF.
• Tong, David. General Relativity. Cambridge Part III lecture notes, 2019. Official PDF.
• Wendl, Chris. Differential Geometry I and II. Lecture notes, Humboldt-Universität zu Berlin, 2021–2022. Open PDF.
• Will, Clifford M. “The Confrontation between General Relativity and Experiment.” Living Reviews in Relativity 17 (2014) 4. DOI; arXiv:1403.7377.