Methods

A method is useful only after its mathematical object, domain, data, and success test have been fixed. This index starts from the operation a gravity calculation must perform, then routes to the shortest preparation and to a physical treatment where the method is actually used. It is a map, not a substitute for those derivations or for problem-specific verification.

Choose by mathematical task

Task	Start here	See it used in gravity	Minimum check before trusting the result
Manipulate vectors, duals, tensors, bilinear forms, or eigenproblems	Linear Algebra and Tensor Computation	Tensor Fields and Natural Operations	Identify the vector spaces, basis dependence, inner product, index type, and conditioning; verify one invariant or basis-independent reconstruction.
Differentiate discontinuities or represent concentrated sources	Distributions and Generalized Functions	Distributional Curvature	Name the test-function space and topology, check density and sign conventions, and do not form a product or pullback until its hypotheses are known.
Use weak limits, compactness, or elliptic regularity in geometric PDE	Analysis for Mathematical Relativity	Low-Regularity Spacetimes and Weak Solutions	Name the space, topology, measure, operator, domain, and boundary hypotheses; do not pass weak limits through nonlinear terms without compactness or a defect term.
Decompose waves into modes or analyze a continuous spectrum	Fourier and Spectral Analysis	Mode Green Functions and Wronskians	State transform signs and $2\pi$ factors, function space, boundary conditions, contour or sheet, and the reconstruction or Parseval-type check.
Solve an initial-value, boundary-value, or eigenvalue ODE	ODE Initial-, Boundary-, and Eigenvalue Problems	ODE, Boundary-Value, and Spectral Computation	Count free data against boundary conditions, inspect singular endpoints, and test residuals, conditioning, and convergence rather than a solver success flag.
Propagate fields, impose causal support, or invert a PDE	PDEs and Green Functions	Retarded and Advanced Green Functions	Classify the operator, state domain and boundary or initial data, choose retarded, advanced, elliptic, or other support, and verify the defining distributional equation.
Connect regions governed by different scales	Asymptotics and Matched Expansions	Matched Asymptotic Expansions Across the Throat	Identify the small parameter and overlap domain, match the same variables and gauge, and test the remainder in both outer and inner limits.
Establish that a computation solves the intended equations	Numerical Analysis, Verification, and Validation	Solution Verification and Convergence	Separate code verification from model validation; report residuals, constraints, resolution or tolerance studies, conserved quantities, and failure cases.
Infer parameters or compare models from noisy data	Probability and Statistical Inference	Bayesian and Frequentist Inference	Declare likelihood, priors or sampling law, selection effects, calibration and nuisance models, uncertainty convention, and predictive or coverage checks.

The rows are deliberately operation-first. “Spectral,” for example, can mean an analytic decomposition, an operator spectrum, or a numerical collocation basis; those uses share vocabulary but not automatically a domain, norm, or error estimate.

Match the method to the question

Start with the quantity you need, not with a favorite technique.

1. State the output. Is it an identity, a weak solution, an eigenfrequency, a causal response, an asymptotic coefficient, a numerical field, or a posterior statement?
2. Write the operator and its domain. Include the measure, inner product, regularity class, coordinate or gauge choices, and boundary or initial conditions. The same differential expression on two domains can define different problems.
3. Identify the controlled approximation. Record the expansion parameter, truncation order, regulator, discretization, or statistical model. A formal series and a converged approximation are different claims.
4. Choose an independent diagnostic. Useful examples are a conserved quantity, limiting case, distributional test, residual norm, refinement rate, alternative formulation, synthetic-data recovery, or comparison with an exact benchmark.
5. Route interpretation to the physical application. A method establishes what the calculation does; the linked page supplies the geometry, observable, regime, and scientific conclusion.

Common category errors

Formal equality versus defined operation. Symbolic manipulation does not establish that a product of distributions, an unbounded-operator composition, or a divergent integral exists. The function space and limiting prescription are part of the result.

Small residual versus small physical error. A discretized equation can have a tiny residual while solving the wrong boundary-value problem or a poorly conditioned formulation. Pair residuals with convergence, constraints, and a problem-level benchmark.

Mode completeness versus useful approximation. Locating poles or eigenvalues does not by itself prove completeness, dominance, or an accurate time-domain reconstruction. Check branch cuts, continuous spectra, prompt response, late-time behavior, and the norm in which an expansion is claimed.

Posterior precision versus information in the data. A narrow posterior may reflect prior support, a fixed calibration, an incomplete noise model, or unmodeled selection. Report sensitivity to those choices before interpreting width as measurement precision.

Detailed method records

Each row in the task table now opens a concise record for that method family. Use those records to fix conventions, domains, data, approximations, and independent checks; use the application column for the full physical or mathematical treatment.