Talk:Mathematics of general relativity/to do
- partial derivatives to covariant derivatives; explain what this means.
- discussion of levels of structure and local vs. global,
- exterior calculus, including covector and Hodge dual,
- intuition for Spinors,
- Cartan-Karlhede algorithm,
- Newman-Penrose formalism, GHP formalism, ADM formulation, other initial value formulations, Cauchy evolution, characteristic evolution, Cauchy characteristic matching - relation to computing wave generation.
- matching/junction conditions.
- EIH approximation (geodesics and inertial motion of test particles),
- geodesic equations and Papapetrou-Dixon equations (motion of spinning test-particles),
- relativistic multipole moments,
- link to article on computation of connection and curvature via Cartan's method using exterior calculus,
- congruence (general relativity) and discussion of kinematical decomposition for timelike and null vector fields,
- the Ricci tensor directly couples to immediate presence of mass-energy-momentum,
- via Lanczos tensor potential for Weyl tensor, Weyl indirectly couples to mass-energy,
- local versus global isometries, conserved currents,
- isotropy subgroup, holonomy subgroup, new article on spacetime holonomy (groups, classification of spacetimes etc.) .
- integration by parts in curved spacetimes,
- nonlocality of gravitational field energy-momentum in general relativity,
- discussion and links to article explaining triangularization of Killing equations (etc.),
- conformal structure, e.g. Carter-Penrose diagrams,
- list various methods used in establishing Penrose-Hawking singularity theorems,
- list open problems in gtr having more to do with math than physics, e.g. rigorous classification of spacetime singularities,
- links to general references, point out specialized articles contain specialized references.
Throughout, article should stress
- intuitive meaning,
- levels of structure,
- degree to which a given mathematical technique/concept is special to gtr, Lorentzian manifolds, etc. (for example, triangularization via Gröbner basis methods is very widely applicable in applied mathematics, as are perturbation theory methods, appropriate bundling procedure very important for smooth manifolds).