Exact Spacetimes, Black Holes, and Causality Source Map

Exact solutions and theorem boundaries

This map keeps each result attached to its hypotheses. Exact Solutions of Einstein’s Equations develops metric constructions; Black-Hole Geometry, Global Structure, and Mathematical Relativity develop horizon mechanics, causal geometry, and proofs. A construction paper is not a global analysis, and a review is not each branch’s primary proof. Use Research for dated judgments about open theorem status.

Question	Source class and exact locator	What the source supports	Decisive exclusion
Which sources construct and interpret the standard exact black-hole families?	Orientation lecture notes: Reall 2013, §12, printed pp. 131–148, Open PDF. Historical survey: Bičák 2000, §§2–4, printed pp. 19–51, Open PDF. Primary construction paper: Newman et al. 1965, printed pp. 918–919.	Reall analyzes Schwarzschild charts; Bičák interprets Schwarzschild, Reissner–Nordström, Kerr, and their limits; Newman et al. construct Kerr–Newman and its electromagnetic field.	The notes and survey orient rather than prove. The 1965 paper does not supply a modern global, causal, or uniqueness analysis.
Where is the maximal Schwarzschild extension explained?	Orientation lecture notes: Reall 2013, §12, printed pp. 131–148, Open PDF. Historical survey: Bičák 2000, §2, printed pp. 19–30, Open PDF.	Schwarzschild, horizon-regular, and Kruskal charts, their domains, and the extended causal structure.	The eternal maximal extension is not a one-sided collapse spacetime. Chart extension alone proves neither geodesic completeness nor inextendibility in a declared regularity class.
How are conformal boundaries and causal diagrams defined?	Authoritative review: Frauendiener 2004, §§1–2.	Conformal completion, physical versus unphysical metrics, boundary character, null infinity, and compactified diagrams.	A radial diagram suppresses spherical orbits and metric scale; a drawn edge need not be a regular conformal boundary.
Which horizon notions are genuinely distinct?	Authoritative reviews: Booth 2005, §§2–5, especially §2.1, printed pp. 1075–1078 and Ashtekar and Krishnan 2004, §§2–4.	Event, apparent, trapping, nonexpanding, isolated, and dynamical horizons differ by causal, foliation, signature, expansion, and flux data.	Local marginal data do not locate an event horizon. Equilibrium mechanics for a Killing or isolated horizon does not transfer unchanged to a generic marginal tube or dynamical horizon.
What do the three classical singularity-theorem branches actually conclude?	Primary theorem papers: Penrose 1965, complete proof, printed pp. 57–59; Hawking 1967, complete paper, printed pp. 187–201; and Hawking and Penrose 1970, printed pp. 531–544. Authoritative survey: Steinbauer 2023, §§2.1, 2.3, and 3.1–3.5, especially Thms. 3.8, 3.9, and 3.11.	In the shared classical smooth four-dimensional setting, Penrose uses a closed trapped surface, null convergence, and a noncompact Cauchy surface for future null incompleteness. Hawking uses a compact Cauchy hypersurface, uniform expansion, and timelike convergence for past timelike incompleteness. Hawking–Penrose use chronology, causal convergence and genericity, and one of three initial alternatives for causal incompleteness without fixed direction.	None proves a singular point, curvature blowup, or inextendibility. Causal type, time orientation, convergence, genericity, and initial alternatives cannot be exchanged.
What supports area nondecrease without an everywhere smooth event horizon?	Primary theorem papers: Hawking 1972, §3, pp. 156–160 and Chruściel et al. 2001, Thm. 1.1, pp. 111–112; Def. 4.3, p. 132; §§3–6, pp. 121–158.	Hawking gives the smooth focusing argument; Chruściel et al. give global or H-regularity packages and low-regularity area comparison.	Focusing alone is insufficient. Cross-section, generator, null-convergence, regularity, and endpoint-control hypotheses remain; generic marginal tubes are excluded.
Where are stationary mechanics and Noether charges derived?	Primary mechanics paper: Bardeen, Carter, and Hawking 1973, printed pp. 162–166 and 168–169. Primary covariant-phase-space paper: Iyer and Wald 1994, §§2–4 and §VI, Eqs. (83)–(90), arXiv PDF pp. 15–16.	The first fixes stationary horizon quantities and four laws; the second derives covariant variations, Noether and Hamiltonian charges, entropy, and the first law.	Generator normalization, boundary terms, and exact-form ambiguities matter. Bare Noether charge is not Hamiltonian mass; equilibrium is not dynamical flux balance.
Which source gives dynamical-horizon balance laws?	Authoritative review: Ashtekar and Krishnan 2004, §§2–4.	Isolated and dynamical horizons, quasi-local charges, and matter-plus-gravitational flux balance, checked against stationary Kerr.	The laws require a specified horizon, foliation, axial field, and normalization; they neither locate an event horizon nor define energy on every surface.
What does static vacuum uniqueness establish?	Primary theorem paper: Bunting and Masood-ul-Alam 1987, complete theorem and proof, pp. 147–154.	In four dimensions, a complete regular asymptotically flat static vacuum slice with compact horizon boundary has a Schwarzschild exterior; multiple regular static vacuum horizons are excluded.	Staticity, vacuum, asymptotic flatness, completeness, horizon regularity, and the slice hypotheses are essential. This is not stationary, electrovacuum, matter, or higher-dimensional uniqueness.
Which sources classify analytic and degenerate stationary electrovacuum black holes?	Primary theorem papers: Costa 2010, Def. 1.2 and Thm. 1.3, arXiv PDF pp. 2–3 and Chruściel and Nguyen 2010, Thm. 1.1, arXiv PDF p. 1. Authoritative synthesis: Chruściel, Costa, and Heusler 2012, Def. 2.1; §§3.1–3.3, especially Thm. 3.3; §§6–8.	$I^+$-regularity includes complete stationary flow, a globally hyperbolic exterior, and an acausal slice whose compact inner boundary meets each future-horizon generator once. In four dimensions, Costa classifies the connected mean-nondegenerate analytic stationary asymptotically flat electrovacuum branch with invariant Maxwell field as Kerr–Newman. Within the same analytic, stationary, asymptotically flat, $I^+$-regular electrovacuum package, Chruściel–Nguyen close the connected rotating degenerate branch. The review organizes the proof chain.	Analyticity, four-dimensionality, connectedness, electrovacuum, and global regularity remain. The degenerate theorem retains rotation; multiple components give a Weinstein candidate, not a regular single-family theorem. The review is not the primary proof.
What smooth rigidity is known without analyticity?	Primary theorem papers: Alexakis, Ionescu, and Klainerman 2010, “Local Rigidity,” Thms. 1.1–1.2, arXiv PDF pp. 1–3 and Alexakis, Ionescu, and Klainerman 2010, “Near-Kerr Uniqueness,” §1.1 and Main Theorem.	The local theorem constructs a horizon-generating Killing field near a regular bifurcate nonexpanding vacuum horizon and, when the stated stationary field is present, a commuting rotational field. Globally, four-dimensional smooth stationary asymptotically flat vacuum uniqueness follows from the stated global regularity, bifurcation sphere, and small Mars–Simon tensor.	Local symmetry need not globalize. Global uniqueness remains perturbative and requires vacuum, a connected nondegenerate bifurcate horizon, regularity, and near-Kerr smallness.
Why retain dimension and matter assumptions in uniqueness claims?	Primary constructions: Emparan and Reall 2002, complete construction and parameter comparison and Herdeiro and Radu 2014, main construction.	Five-dimensional asymptotically flat stationary vacuum black rings overlap spherical black holes in charges; four-dimensional asymptotically flat stationary spinning black holes with a massive complex scalar field overlap Kerr in charges.	Rings do not contradict four-dimensional theorems; scalar hair does not contradict vacuum or electrovacuum uniqueness.

References

• Alexakis, Spyros, Alexandru D. Ionescu, and Sergiu Klainerman. “Hawking’s Local Rigidity Theorem Without Analyticity.” Geometric and Functional Analysis 20 (2010) 845–869. DOI; arXiv:0902.1173.
• Alexakis, Spyros, Alexandru D. Ionescu, and Sergiu Klainerman. “Uniqueness of Smooth Stationary Black Holes in Vacuum: Small Perturbations of the Kerr Spaces.” Communications in Mathematical Physics 299 (2010) 89–127. DOI; arXiv:0904.0982.
• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004) 10. DOI.
• Bardeen, J. M., B. Carter, and S. W. Hawking. “The Four Laws of Black Hole Mechanics.” Communications in Mathematical Physics 31 (1973) 161–170. DOI; Open PDF.
• Bičák, Jiří. “The Role of Exact Solutions of Einstein’s Equations in the Developments of General Relativity and Astrophysics: Selected Themes.” arXiv:gr-qc/0004016v1 (2000). Open PDF.
• Booth, Ivan. “Black-Hole Boundaries.” Canadian Journal of Physics 83 (2005) 1073–1099. DOI; arXiv:gr-qc/0508107.
• Bunting, Gary L., and A. K. M. Masood-ul-Alam. “Nonexistence of Multiple Black Holes in Asymptotically Euclidean Static Vacuum Space-Time.” General Relativity and Gravitation 19 (1987) 147–154. DOI.
• Chruściel, Piotr T., and Luc Nguyen. “A Uniqueness Theorem for Degenerate Kerr–Newman Black Holes.” Annales Henri Poincaré 11 (2010) 585–609. DOI; arXiv:1002.1737.
• Chruściel, Piotr T., Erwann Delay, Gregory J. Galloway, and Ralph Howard. “Regularity of Horizons and the Area Theorem.” Annales Henri Poincaré 2 (2001) 109–178. arXiv:gr-qc/0001003v2.
• Chruściel, Piotr T., João Lopes Costa, and Markus Heusler. “Stationary Black Holes: Uniqueness and Beyond.” Living Reviews in Relativity 15 (2012) 7. DOI; arXiv:1205.6112.
• Costa, João Lopes. “On the Classification of Stationary Electro-Vacuum Black Holes.” Classical and Quantum Gravity 27 (2010) 035010. DOI; arXiv:0912.0834.
• Emparan, Roberto, and Harvey S. Reall. “A Rotating Black Ring Solution in Five Dimensions.” Physical Review Letters 88 (2002) 101101. DOI; arXiv:hep-th/0110260.
• Frauendiener, Jörg. “Conformal Infinity.” Living Reviews in Relativity 7 (2004) 1. DOI.
• Hawking, Stephen W. “Black Holes in General Relativity.” Communications in Mathematical Physics 25 (1972) 152–166. DOI.
• Hawking, Stephen W. “The Occurrence of Singularities in Cosmology. III. Causality and Singularities.” Proceedings of the Royal Society A 300 (1967) 187–201. DOI.
• Hawking, Stephen W., and Roger Penrose. “The Singularities of Gravitational Collapse and Cosmology.” Proceedings of the Royal Society A 314 (1970) 529–548. DOI.
• Herdeiro, Carlos A. R., and Eugen Radu. “Kerr Black Holes with Scalar Hair.” Physical Review Letters 112 (2014) 221101. DOI; arXiv:1403.2757.
• 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.
• Newman, Ezra T., Evelyn Couch, K. Chinnapared, A. Exton, A. Prakash, and Robert Torrence. “Metric of a Rotating, Charged Mass.” Journal of Mathematical Physics 6 (1965) 918–919. DOI.
• Penrose, Roger. “Gravitational Collapse and Space-Time Singularities.” Physical Review Letters 14 (1965) 57–59. DOI.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. Open PDF.
• Steinbauer, Roland. “The Singularity Theorems of General Relativity and Their Low Regularity Extensions.” Jahresbericht der Deutschen Mathematiker-Vereinigung 125, no. 2 (2023) 73–119. DOI; arXiv:2206.05939v2.