Spacetimes and Coordinates

Coordinates reveal different structures of the same geometry, but a chart is never the spacetime itself. This directory is a calculation-first index of six compact cards: choose a background and purpose, then check the chart domain, parameter normalization, boundary coverage, and invariant geometry before using its metric components.

Helpful background. Lorentzian Metrics and Causal Cones supplies causal classification and time orientation; Exact Spacetimes supplies the chart-domain and invariant-check method.

Choose the chart by the calculation

• Minkowski Spacetime and Standard Charts compares inertial, polar, null, Rindler, and Milne coordinates. Use it to distinguish an axis, an accelerated-observer horizon, and the edge of a light-cone chart inside one flat spacetime.
• Schwarzschild Spacetime and Horizon-Penetrating Charts compares static, null, freely falling, and Kruskal–Szekeres coordinates. Use it when a calculation must approach or cross a specified horizon branch.
• Kerr Spacetime and Common Charts compares Boyer–Lindquist, ingoing and outgoing Kerr, and Kerr–Schild descriptions. Use it to keep the horizons, stationary-limit surfaces, axis, and ring singularity distinct.
• FLRW Spacetime and Cosmological Coordinates translates between cosmic and conformal time and between the standard constant-curvature spatial charts. Use it before comparing distances, Hubble variables, or cosmological redshifts.
• de Sitter and Anti-de Sitter Charts compares global and partial patches, embedding descriptions, horizons, conformal boundaries, and the anti-de Sitter universal cover.
• Charged Black-Hole Spacetimes fixes one Einstein–Maxwell charge convention and compares Reissner–Nordström and Kerr–Newman horizons, ergosurfaces, regular ingoing charts, and limiting families.

Derivations and scope

Each card links to the exact-spacetime collection for complete derivations and geometric properties. Causality and Global Structure explains global causal completions, while Black-Hole Geometry develops horizon mechanics. The cards describe mathematical spacetimes; cosmological and astrophysical parameter estimates require separate data and inference.

Coordinate-domain checklist

Before importing a metric, identify:

1. the spacetime dimension, theory, matter content, and parameter range;
2. the range of every coordinate and the connected region represented;
3. the transformation and inverse transformation on their common domain;
4. where the metric, inverse metric, and Jacobian are nondegenerate;
5. which boundary is an axis, chart edge, horizon, ergosurface, conformal boundary, or genuine curvature singularity;
6. whether an extension, quotient, or covering space has changed the global spacetime; and
7. which time orientation and asymptotic normalization fix the physical interpretation.

A vanishing $g_{tt}$, divergent $g_{rr}$, zero Jacobian, or finite-coordinate edge is not by itself a singularity or an event horizon. Cross-check the candidate surface in a regular overlapping chart and with invariant geometry.