Mathematical and Computational Relativity Source Map

Proofs, formulations, and computational evidence

This map routes a precise question to the smallest source set that can answer it. The proof-focused entries support the theorem statements developed in Mathematical Relativity; the formulation, benchmark, and evidence entries lead into implementation and testing in Computational Relativity. A cited formulation is not a theorem about its discretization, and a successful benchmark is not validation of an astrophysical model.

Question	Source class and exact locator	What the source supports	Decisive exclusion
What is the universal development of Einstein initial data?	Primary theorem paper: Choquet-Bruhat and Geroch 1969, theorem and construction, pp. 329–335.	Existence and uniqueness, up to an initial-data-fixing isometry, of the maximal globally hyperbolic development. Every globally hyperbolic development embeds in that maximal one in the stated solution category.	Maximality is only among globally hyperbolic developments. It implies neither geodesic completeness nor absolute inextendibility, and it does not exclude a non-globally-hyperbolic extension.
Can data on intersecting null hypersurfaces determine a local spacetime?	Primary theorem paper: Rendall 1990, main local theorem and reduction, pp. 221–239.	Local reduction of compatible characteristic data on two intersecting null hypersurfaces to a Cauchy problem for the Einstein equations.	This is not global null evolution or a theorem at null infinity. The intersection and its corner compatibility conditions cannot be omitted.
Is there a vacuum Einstein problem with a timelike boundary?	Primary theorem paper: Friedrich and Nagy 1999, §§3–7, pp. 625–650, and Thm. 8.1, pp. 650–652.	Smooth local vacuum evolution for a compact initial manifold with timelike boundary, prescribed gauge sources, admissible boundary data, and consistent corner data.	It does not establish an unrestricted, gauge-independent geometric uniqueness theorem. The gauge, boundary, and corner package is part of the result.
How rough may vacuum initial geometry be in a local evolution theorem?	Primary theorem paper: Klainerman, Rodnianski, and Szeftel 2015, §2.2, Thm. 2.2.	A controlled local existence time from the stated initial $L^2$ Ricci-curvature and $L^2\nabla k$ bounds, together with a positive lower volume radius, in the maximal asymptotically flat vacuum setting.	It is not a theorem for arbitrary matter, arbitrary foliations, or distributional initial data.
Which source proves nonlinear stability of Minkowski spacetime?	Primary research monograph: Christodoulou and Klainerman 1993, Thms. 1.0.1 and 1.0.3, pp. 12–18; §10.2, pp. 298–310; Ch. 17, pp. 491–512.	Global existence, geodesic completeness, and detailed asymptotics for the stated small, strongly asymptotically flat vacuum data.	The result does not cover large data, matter, Kerr, or later weaker-decay formulations.
What is proved about gravitational perturbations of Schwarzschild?	Primary theorem paper: Dafermos, Holzegel, and Rodnianski 2019, §2.2, pp. 20–24; §§10.3–10.4; §§13–14.	Boundedness and decay of solutions of the linearized Einstein equations on Schwarzschild toward a linearized Kerr solution after controlled pure-gauge normalization.	This is linear stability of Schwarzschild, not nonlinear stability and not stability of rotating Kerr.
What are the two classical positive-mass proof routes?	Two primary theorem papers: the minimal-surface route in Schoen and Yau 1979, §1, Thms. 1–2, pp. 47–48, with §§2–4, pp. 50–70, and the spinorial route in Witten 1981, §III, Eqs. (57) and (72), pp. 395–399, with the note on p. 402.	Schoen–Yau establish the three-dimensional Riemannian positivity and zero-mass rigidity result under their asymptotic, completeness, regularity, and boundary hypotheses. Witten proves the future-causal ADM four-momentum result using a global spin structure, the dominant energy condition, and an elliptic spinor construction.	The hypotheses and conclusions must not be merged. These papers do not by themselves cover arbitrary dimensions, charged or asymptotically AdS refinements, horizon boundaries, Bondi mass, or quasi-local mass.
Which source proves the Riemannian Penrose inequality by conformal flow?	Primary theorem paper: Bray 2001, Thm. 1, pp. 185–186; §3; §13.	The inequality and rigidity for complete smooth asymptotically flat Riemannian three-manifolds with nonnegative scalar curvature and outer-minimizing minimal boundary.	This is the time-symmetric Riemannian result, not the full Lorentzian Penrose conjecture. Its boundary and equality hypotheses remain essential.
Where is the BSSN-type evolution system specified and tested?	Primary formulation paper: Baumgarte and Shapiro 1999, §§II–IV.	The conformal variables, connection functions, implementation choices, and reported small-amplitude-wave comparison with ADM evolution.	A formulation and a reported test do not prove general well-posedness, universal numerical stability, or strong-field performance.
What supports constraint damping in Z4 and generalized harmonic evolution?	Primary formulation and analysis papers: Gundlach et al. 2005, §§II–IV and VI and Lindblom et al. 2006, §§2.1–2.2, §3, §§4.1–4.4, and §5.	Z4 and harmonic constraint-damping analysis, a first-order symmetric-hyperbolic generalized-harmonic reduction, boundary construction, and the reported tests.	Preserve the linearized, frozen-coefficient, short-wavelength, and constant-mode qualifications. These analyses are not a general nonlinear initial-boundary theorem.
Which problems are standard numerical-relativity testbeds?	Primary benchmark-definition paper: Alcubierre et al. 2004, §§II–V.	Definitions, common inputs, outputs, and comparison rationale for robust-stability, linear-wave, gauge-wave, and related periodic-vacuum tests.	These are code-verification tests, not astrophysical validation; they do not cover matter, adaptive refinement, realistic outer boundaries, or production binary simulations.
Which sources define verification, reproducibility, provenance, and packaging practice?	Authoritative technical report and institutional standards: Salari and Knupp 2000, Chapters 1–5; National Academies 2019, Summary, pp. 6–8, and Ch. 3; W3C PROV-DM 2013, §§2–5; and Kunze et al. 2018, RFC 8493, §§1–3.	Manufactured-solution code verification; reproducibility and replicability terminology; Entity–Activity–Agent provenance; and file-list-and-checksum packaging semantics.	These practices neither validate a relativity model nor prove convergence or correctness of a particular code. BagIt is an informational RFC, not an Internet Standards Track specification.

For orientation rather than primary proof or benchmark evidence, use the research monograph Ringström 2009, Chs. 3–16, pp. 21–183, the survey Friedrich and Rendall 2000, §2.4 and §§5.1–5.3, arXiv v1 PDF pp. 21–24 and 75–81, and the numerical-methods review Grandclément and Novak 2009, §§2.1–2.6, §3.1.2, and §4. They connect techniques and terminology, but the located primary works above carry the theorem, formulation, benchmark, and standards claims.

References

• Alcubierre, Miguel, et al. “Toward Standard Testbeds for Numerical Relativity.” Classical and Quantum Gravity 21 (2004) 589–613. DOI; arXiv:gr-qc/0305023.
• Baumgarte, Thomas W., and Stuart L. Shapiro. “On the Numerical Integration of Einstein’s Field Equations.” Physical Review D 59 (1999) 024007. DOI; arXiv:gr-qc/9810065.
• Bray, Hubert L. “Proof of the Riemannian Penrose Inequality Using the Positive Mass Theorem.” Journal of Differential Geometry 59 (2001) 177–267. DOI.
• Choquet-Bruhat, Yvonne, and Robert Geroch. “Global Aspects of the Cauchy Problem in General Relativity.” Communications in Mathematical Physics 14 (1969) 329–335. DOI.
• Christodoulou, Demetrios, and Sergiu Klainerman. The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series 41. Princeton University Press, 1993. DOI.
• Dafermos, Mihalis, Gustav Holzegel, and Igor Rodnianski. “The Linear Stability of the Schwarzschild Solution to Gravitational Perturbations.” Acta Mathematica 222 (2019) 1–214. DOI; arXiv:1601.06467.
• Friedrich, Helmut, and Alan D. Rendall. “The Cauchy Problem for the Einstein Equations.” In Einstein’s Field Equations and Their Physical Implications, Lecture Notes in Physics 540, 127–224. Springer, 2000. arXiv:gr-qc/0002074v1.
• Friedrich, Helmut, and Gabriel Nagy. “The Initial Boundary Value Problem for Einstein’s Vacuum Field Equation.” Communications in Mathematical Physics 201 (1999) 619–655. DOI.
• Grandclément, Philippe, and Jérôme Novak. “Spectral Methods for Numerical Relativity.” Living Reviews in Relativity 12 (2009) 1. DOI.
• Gundlach, Carsten, José M. Martín-García, Gioel Calabrese, and Ian Hinder. “Constraint Damping in the Z4 Formulation and Harmonic Gauge.” Classical and Quantum Gravity 22 (2005) 3767–3774. DOI; arXiv:gr-qc/0504114.
• Klainerman, Sergiu, Igor Rodnianski, and Jérémie Szeftel. “The Bounded $L^2$ Curvature Conjecture.” Inventiones Mathematicae 202 (2015) 91–216. DOI; arXiv:1204.1767.
• Kunze, John, John Littman, Liz Madden, Jasmine Scancella, and Chris Adams. The BagIt File Packaging Format (V1.0). RFC 8493, Informational. RFC Editor, 2018. RFC; DOI.
• Lindblom, Lee, Mark A. Scheel, Lawrence E. Kidder, Robert Owen, and Oliver Rinne. “A New Generalized Harmonic Evolution System.” Classical and Quantum Gravity 23 (2006) S447–S462. DOI; arXiv:gr-qc/0512093.
• Moreau, Luc, and Paolo Missier, eds. PROV-DM: The PROV Data Model. W3C Recommendation. World Wide Web Consortium, 2013. Official specification.
• National Academies of Sciences, Engineering, and Medicine. Reproducibility and Replicability in Science. Washington, DC: National Academies Press, 2019. DOI.
• Rendall, Alan D. “Reduction of the Characteristic Initial Value Problem to the Cauchy Problem and Its Applications to the Einstein Equations.” Proceedings of the Royal Society A 427 (1990) 221–239. DOI.
• Ringström, Hans. The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics. European Mathematical Society, 2009. Publisher record.
• Salari, Kambiz, and Patrick J. Knupp. Code Verification by the Method of Manufactured Solutions. SAND2000-1444. Sandia National Laboratories, 2000. DOI.
• Schoen, Richard, and Shing-Tung Yau. “On the Proof of the Positive Mass Conjecture in General Relativity.” Communications in Mathematical Physics 65 (1979) 45–76. DOI.
• Witten, Edward. “A New Proof of the Positive Energy Theorem.” Communications in Mathematical Physics 80 (1981) 381–402. DOI.