Mathematical and Global Relativity

Mathematical and global relativity asks what the Einstein equations force once local differential geometry meets global causal structure. The hard questions are rarely yes-or-no statements without qualifications: the answer can change with the matter model, asymptotics, regularity class, topology on initial data, horizon definition, or boundary condition. This directory organizes six finite research questions around those distinctions. Each record is a dated evidence assessment; the stable definitions and proof machinery remain on the linked Library pages.

The common discipline is to identify the exact theorem cell before comparing results. A continuous extension need not be twice differentiable, a trapped-surface construction need not be generic, and an all-dimensional Riemannian inequality need not address non-time-symmetric data. Those are not verbal caveats. They determine whether two results are genuinely in tension or simply answer different questions.

Six bounded questions in mathematical relativity

Record	Finite boundary	Decisive evidence
Weak Cosmic Censorship	A named Einstein–matter system, asymptotics, topology on data, and definition of generic visibility	Stable or unstable naked-singularity formation in the same data topology
Strong Cosmic Censorship and Regularity	A named model and a specified extension class, from continuous metrics to curvature-level regularity	Nonlinear extendibility or inextendibility at that exact threshold
The Penrose Inequality Beyond Time Symmetry	General initial data with a precise enclosing-horizon prescription and rigidity statement	An unconditional deformation theorem or a valid counterexample to the careful formulation
Low-Regularity Einstein Evolution	A geometric norm, gauge, solution concept, and separate claims of existence, uniqueness, and continuous dependence	A threshold estimate or gauge-invariant ill-posedness mechanism
Generic Singularity Formation	A topology on complete data and a distinction between trapped surfaces and the later singular boundary	An open or prevalent collapse class with a controlled endpoint
Global Dynamics of Asymptotically AdS Spacetimes	A fixed Einstein–matter model, timelike-boundary condition, symmetry class, and long-time quantifier	A long-time theorem or independently converged evolution family

The records deliberately do not collapse into a single “status of general relativity.” They cover different mathematical objects: maximal developments, trapped sets, asymptotic boundaries, quasi-local surfaces, and weak metrics. The useful comparison is across methods and hypotheses, not across headline conclusions.

How the questions connect

Low-regularity evolution sits closest to the foundations. Any claim about generic behavior presupposes a solution map and a topology in which perturbations can be compared. Its distinction between wave-coordinate failure and geometric failure also recurs throughout the atlas: a gauge obstruction is not automatically a spacetime obstruction.

Generic singularity formation supplies one route from regular data to trapped surfaces. Weak censorship then asks whether the resulting singular behavior is hidden from infinity for generic data. Strong censorship asks a different downstream question: whether evolution remains uniquely determined beyond an inner boundary, and at what regularity. The two censorship questions share causal language but neither implies the other.

The Penrose-inequality record connects initial geometry to the mass expected after collapse. It touches censorship heuristics, yet its central obstruction is geometric: which horizon or enclosure carries the correct area when the extrinsic curvature is nonzero? Counterexamples to convenient surface choices narrow the conjecture without refuting every physical formulation.

Asymptotically AdS dynamics changes the causal experiment. A timelike conformal boundary returns radiation unless the boundary condition removes it, so asymptotically flat dispersal intuition cannot simply be imported. Its collapse, periodic, and stability-island results should be compared only after the matter model and boundary condition agree.

Choose an entry route

Start with the work you can already do, then choose a frontier whose missing step uses those skills. The projects below begin with a known result. Completing one establishes preparation and a reproducible baseline; it does not establish novelty or guarantee a thesis topic.

If your strength is differential geometry, begin with the Penrose-inequality project. You should be able to compute null expansions and distinguish the mass at infinity from a quasi-local mass. Read the Riemannian statement for its equality case, then the counterexample for the surface choice that fails. The first deliverable is an explicit mass-and-area calculation. An adviser in geometric analysis can help turn a surviving formulation into a precise question about deformation, boundary behavior, or rigidity.

If your strength is analysis, choose between the endpoint-obstruction calculation and the strong-censorship regularity comparison. The former needs Sobolev integrability and wave equations; the latter needs double-null geometry and weighted endpoint estimates. Read the model calculation before the full theorem. A short derivation that identifies precisely where a norm diverges is a better first deliverable than a survey of all weak solutions. Both need access to a PDE expert, even though their computational cost is low.

If your strength is computation, begin with the AdS scalar baseline. Recover a linear normal mode and its energy before attempting nonlinear transfer. The first useful result is a resolved interval with quantified error, not the longest available run. A spherical solver is a manageable entry; vacuum evolution and long cascades require more infrastructure and supervision.

The weak-censorship topology comparison and formation scaling project are good next steps once characteristic data and the constraints are familiar. They reward careful comparison of admissible data more than broad speculation about singularities.

Turn a baseline into a research question

Before adopting a direction, take a prospective supervisor one finite question, the exact baseline theorem or computation, and one proposed change of hypothesis. Agree on a deliverable that remains useful if the hoped-for extension fails: an explicit obstruction, a corrected norm comparison, or a trustworthy convergence study. Verify that the group has the relevant proof expertise or maintained numerical infrastructure.

Use the record’s cutoff to delimit what this atlas checked, then search later papers, corrections, and citing work before asserting a gap. A topic becomes a candidate for original research only after that novelty check. If the proposed change simultaneously alters matter, regularity, asymptotics, and boundary conditions, reduce it to one axis. If the baseline itself cannot be reproduced, work on that difficulty before enlarging the claim.

Stable background and changing verdicts

The Library pages define the durable objects: maximal globally hyperbolic development, extendibility, null expansion, asymptotic flatness, timelike conformal boundary, weak curvature, and the Riemannian mass–area inequality. These records instead answer what is known at a stated cutoff. A new theorem can change a leaf assessment without changing the architecture of this directory.

When reading any record, first locate its domain row, then inspect the adverse evidence and the proposed discriminating result. A result outside the stated matter model, symmetry class, regularity, or boundary condition may be mathematically important, but it cannot silently decide the finite question. That discipline is what makes apparently incompatible frontiers navigable.