Part III: Numerical Relativity
The three chapters that constitute Part III of Projects in Scientific Computing survey the literature of numerical relativity and black hole physics from a perspective that completes the arc begun in Parts I and II. Where Part I built outward from mathematical and computational infrastructure toward financial applications, and Part II took quantum field theory itself as its subject while tracing the mathematical structures that unify its branches, Part III arrives at the physical setting where computation is not merely useful but indispensable: the nonlinear dynamics of spacetime geometry. Einstein’s field equations, $G_{\mu\nu} = 8\pi T_{\mu\nu}$, admit no closed-form solutions for most physically interesting configurations. The enterprise of solving them numerically—and the theoretical physics of the objects those solutions describe—is the subject of these notes.
The organizing principle remains the same as in Parts I and II: follow the mathematical structures rather than the conventional disciplinary boundaries. In practice, this means that the three chapters are arranged not as a linear progression through a single subject but as three complementary views of the same physical territory. Chapter III.1 treats the computational methods and community infrastructure. Chapter III.2 treats the theoretical physics of the objects being computed. Chapter III.3 treats a class of speculative solutions that push both theory and computation to their current limits. Together, the three chapters survey roughly 106 primary sources spanning from Richardson’s 1910 finite-difference methods to 2024 preprints on gravitational wave signatures of exotic spacetimes.
Chapter III.1 (Numerical Relativity) traces the development of the field through its literature, organized around four principal themes: the discretization of continuum equations, the reformulation of Einstein’s equations into computationally tractable form and the community infrastructure built to solve them, gravitational collapse and the discovery of critical phenomena, and the problem of locating black hole horizons in numerically evolved spacetimes. The intellectual arc of the first theme—from Richardson’s 1910 extrapolation technique through Kreiss and Oliger’s stability-aware discretizations of the 1970s to Berger and Oliger’s 1984 adaptive mesh refinement algorithm—constitutes a remarkably coherent development in which each generation built directly on the preceding one. The second theme addresses the central difficulty that delayed the field for three decades: the ADM formulation of Einstein’s equations, while conceptually clean, is only weakly hyperbolic and therefore numerically unstable. The reformulations that resolved this—the BSSN conformal decomposition and the generalized harmonic formulation—together with the gauge conditions that make them work (1+log slicing, Gamma-driver shift), are treated alongside the community infrastructure (the Einstein Toolkit, GRChombo) that embodies them. The discussion of Choptuik’s 1993 discovery of critical phenomena in gravitational collapse provides a clear example of numerical computation revealing genuinely new physics: power-law scaling of black hole mass at formation threshold, discrete self-similarity, and universality classes that parallel those of statistical mechanics. The chapter closes with the problem of finding horizons—both event horizons (global, teleological, found in postprocessing) and apparent horizons (quasi-local, found during evolution)—whose algorithmic development is a microcosm of the field’s broader trajectory.
Chapter III.2 (Black Holes) shifts from computational methods to the theoretical physics of the objects being computed. The treatment is organized into four sections reflecting the logical structure of the subject: general theorems (thermodynamic laws, exact solutions, uniqueness results, the formation problem), the event horizon (membrane paradigm, null hypersurface geometry, Raychaudhuri focusing, horizon merger dynamics), scattering from black hole backgrounds (from Matzner’s 1968 analysis through modern exact methods), and gravitational radiation (binary dynamics, the self-force program, data analysis methods, and the detection of GW150914). The bibliography of 31 entries spans 1968 to 2024 and emphasizes mathematical rigor and geometric insight. Three texts anchor the mathematical framework at different levels of difficulty and emphasis: Wald (1984) for global methods and causal structure, Chandrasekhar (1983) for exhaustive perturbation-theoretic calculations of exact solutions, and Chru'sciel (2020) for the modern rigorous geometric approach. The Kerr solution receives extended treatment because it is the astrophysically relevant exact solution: Teukolsky’s master equation, the hidden symmetry encoded in the Carter constant, and the separability that reduces partial differential equations to ordinary differential equations are the theoretical backbone of gravitational wave template construction. The chapter’s gravitational wave section traces the complete pipeline from post-Newtonian inspiral through numerical merger to perturbation-theoretic ringdown, culminating in the detection of GW150914—a result that confirmed the accuracy of numerical relativity waveform predictions and vindicated the computational program that produced them.
Chapter III.3 (Wormholes) extends the scope from established astrophysical objects to speculative solutions that probe the boundary between classical general relativity and quantum implications to gravity. The subject of rotating traversable wormholes sits at an intersection of exact-solution theory, numerical relativity, spinor field theory, and gravitational-wave phenomenology. The chapter surveys roughly three decades of literature organized into five thematic blocks: foundational solutions (Ellis-Bronnikov drainhole, Visser’s framework, the Teo rotating ansatz), rotating solution construction (slow-rotation approximations, fully nonlinear numerical solutions, electrovac constructions), the black-bounce interpolation program connecting regular black holes to traversable wormholes, the Einstein-Dirac-Maxwell (EDM) wormhole program (including Kain’s numerical relativity results demonstrating dynamical instability), and supporting infrastructure (energy conditions, axisymmetric formalism, Dirac fields in curved spacetime, gravitational lensing). The 46 references represent the largest bibliography in Part III and reflect the interdisciplinary character of the subject. The EDM program receives particular emphasis as the most significant recent theoretical development: The construction of traversable wormhole solutions using fermion fields rather than phantom scalars, and Kain’s subsequent demonstration that these solutions are dynamically unstable to gravitational collapse, illustrate a recurring tension in wormhole physics between the ease of constructing static solutions and the difficulty of making them dynamically viable.
Several structural features of Part III as a whole merit attention. First, the relationship among the three chapters is one of progressive conceptual expansion. Chapter III.1 provides the computational methods; Chapter III.2 provides the theoretical physics of the objects those methods compute; Chapter III.3 applies both the methods and the theoretical framework to objects whose physical existence remains speculative. This progression—from established computation to established theory to the speculative frontier—mirrors the structure of a research program rather than a textbook.
Second, the cross-referencing between chapters is substantial and intentional. The 3+1 decomposition, BSSN formulation, and gauge conditions developed in Chapter III.1 are the computational framework within which the black hole simulations of Chapter III.2 and the wormhole evolutions of Chapter III.3 are performed. The horizon-finding algorithms of Chapter III.1 are essential diagnostic tools for the event horizon physics of Chapter III.2 and the dynamical stability analysis of Chapter III.3. Choptuik’s critical collapse (Chapter III.1) reappears in the axisymmetric infrastructure discussion of Chapter III.3 through Choptuik’s own work on axisymmetric and Einstein-Dirac collapse. The Raychaudhuri equation appears in the black hole area theorem (Chapter III.2) and in the flare-out condition that forces NEC violation at wormhole throats (Chapter III.3). The perturbation theory of the Kerr solution (Chapter III.2) provides the quasi-normal mode spectrum against which wormhole observational signatures (Chapter III.3) must be compared.
Third, Part III maintains the explicit connections to Parts I and II that characterize the project as a whole. The finite-difference methods, convergence testing, and adaptive mesh refinement of Chapter III.1 draw on the scientific computing infrastructure developed in Part I, Chapter 2. The critical phenomena discovered by Choptuik—universality, scaling, renormalization group structure—are mathematically cognate to the critical phenomena and renormalization group ideas treated in Part II, Chapters 5 and 8. The heat-kernel methods that appear in short-maturity implied-volatility expansions (Part I, Chapter 8) and in QFT in curved spacetime (Part II, Chapter 6) are the same heat-kernel methods used in spectral geometry on black hole backgrounds. The Dirac equation in curved spacetime, central to the EDM wormhole program of Chapter III.3, connects to the spinor field theory treated in Part II. The stochastic quantization program of Part II, Chapter 7 points directly toward the numerical relativity methods of the present volume. And the path integral, which serves as the unifying mathematical thread across all three parts, appears here in the guise of the gravitational path integral that motivates both the Euclidean approach to quantum gravity and the semiclassical methods used to analyze Hawking radiation and wormhole stability.
Fourth, Part III exhibits the same methodological stance as the preceding parts—what might be called computation as a mode of theoretical physics. Choptuik’s discovery of critical phenomena is the clearest example: A genuinely new physical result, with precise parallels to condensed matter criticality, that could not have been obtained by any method other than numerical computation. The 2005 breakthrough simulations of binary black hole coalescence are another: the merger waveforms that LIGO uses for template matching exist only as outputs of numerical codes. And the dynamical instability of EDM wormholes, demonstrated by Kain’s numerical evolutions, resolved a theoretical question that static analysis alone could not answer. In each case, computation is not merely implementing a theory but extending the reach of theoretical physics into regimes where analytic methods fail.
The notes are intended for readers with graduate-level training in general relativity (at the level of Wald), partial differential equations, and some experience with numerical computation. Each chapter is designed to be readable independently, but the cumulative effect of reading them in sequence—methods, then theory, then frontier—traces the arc of a field that, over six decades, progressed from crashing simulations to routine production of gravitational waveforms, and from the first tentative numerical experiments to a mature computational discipline whose predictions have been confirmed by observation to remarkable precision.