Conventions and Symbols

Conventions are part of a formula, not decoration around it. This directory collects the five convention packages that most often produce silent disagreement between otherwise correct gravity calculations: metric and curvature signs, index and bracket notation, orientation and dualization, the constants hidden by a unit system, and the linked tetrad, Newman–Penrose, GHP, and two-spinor alphabets. Each card states the site default, shows how to translate a common alternative, and ends with checks that can distinguish a real convention change from a physical error.

Five convention questions

If you need to know…	Use this record	What it fixes
Which sign belongs in a commutator, Ricci tensor, or Einstein tensor?	Metric Signature and Curvature Signs	The $(-,+,+,+)$ signature, the complete Riemann convention, contractions, and a two-switch translation map
Whether an index labels a component or an abstract slot, and how brackets are weighted	Indices, Symmetrization, and Antisymmetrization	Index families, free and dummy indices, raising and lowering, $1/k!$ bracket weights, and excluded-index bars
Whether an epsilon is a symbol, tensor, form, or density, and which boundary sign is intended	Orientation, Levi-Civita Symbols, and Volume Forms	Positive orientation, raised and lowered volume tensors, Hodge signs, and non-null versus null boundary data
Whether a mass is an energy or a length, and which constants have been suppressed	Units and Dimensional Conventions	Natural, relativistic, geometrized, and SI units; dimensional restoration; and Planck-mass normalization
How orthonormal and null tetrads, Newman–Penrose operators and scalars, and a two-spinor dyad fit one declared package	Tetrad, Newman–Penrose, and Spinor Symbols	Tetrad normalization, directional derivatives, twelve spin coefficients, five Weyl scalars, the minimal dyad bridge, and common symbol collisions

These records are deliberately narrow. The fifth card bridges two deeper treatments: Spacetime Geometry develops orthonormal-frame, Clifford, and spin-structure foundations. The card adds the minimal two-spinor dyad bridge directly from the cited sources, while GR Formulations defines the null-tetrad, Newman–Penrose, and GHP quantities. It supplies a common symbol map without repeating either development. Likewise, a numerical constants table needs dated standards data and is not implied by the algebraic unit rules here.

Defaults inherited across the site

Unless a page announces a sustained exception, the site uses metric signature $(-,+,+,+)$ and defines curvature by

$$[ \nabla_\mu,\nabla_\nu ]V^\rho =R^\rho{}_{\sigma\mu\nu}V^\sigma, \qquad R_{\sigma\nu}=R^\rho{}_{\sigma\rho\nu}.$$

In a positively oriented chart, $\epsilon_{0123}=+\sqrt{-g}$. Symmetrization and antisymmetrization carry unit weight, including the factor $1/k!$ over $k$ enclosed slots. The default unit choice is $c=\hbar=k_{\mathrm B}=1$ with $G$ retained explicitly.

That compact declaration is enough only when every participating formula uses the same package. A source described merely as “mostly plus” may still reverse the Riemann tensor, contract different slots for Ricci, normalize brackets without $1/k!$, or use a different orientation. A page may adopt such a choice locally, but it must name what changed and translate every dependent formula together.

A safe translation workflow

First identify the object before comparing components. A mixed Riemann tensor and an all-lowered Riemann tensor do not respond identically to a reversal of the metric sign. An alternating symbol and a metric volume tensor do not have the same transformation law. A physical mass $M$ and the geometrized length $GM/c^2$ are related quantities, not two spellings of the same number.

Next make each switch explicit. Rewrite the source convention as a map from its metric, curvature, orientation, bracket weight, or unit scale to the site choice. Apply the map to contractions, raised indices, integration measures, couplings, and boundary data—not just to the line where a disagreement first appears.

Finally use a discriminating check. Flat space in curvilinear coordinates must still have zero curvature; a round sphere must have positive scalar curvature in the site convention; raising all indices of the Lorentzian volume tensor introduces a minus sign; a restored black-hole frequency must have units of inverse seconds. These tests are more reliable than matching the visual shape of a familiar formula.

For the full geometric constructions, continue to Spacetime Geometry. For the physical meaning of measurement scales and constants, use Special Relativity and Physical Foundations. For an executable data contract that carries conventions between programs, see Units, Conventions, and Data Contracts.