APPENDIX C
A plain-language glossaryA plain-language glossary
| Term | Meaning |
|---|---|
| Affine parameter | A geodesic parameter for which the tangent transports parallel to itself; linear rescalings preserve this property |
| Atlas | A collection of overlapping coordinate charts covering a manifold |
| Boundary condition | A restriction on fields or their behavior at a boundary or asymptotic region |
| Causal curve | A timelike or null curve; with a chosen time orientation, it can represent future-directed causal influence |
| Chart | A smooth local system of coordinates; it need not cover the whole spacetime |
| Christoffel symbols | Coordinate-basis coefficients of the Levi-Civita connection; the coefficients alone are not tensor components |
| Connection | A rule for differentiation and infinitesimal parallel transport; in standard metric GR it is the torsion-free, metric-compatible one |
| Constraint | An equation restricting permissible initial data rather than independently specifying all their time derivatives |
| Covector | A linear map from a tangent vector to a scalar; the differential , with components , is an example. Obtaining the gradient vector requires raising its index with the metric |
| Covariant derivative | A derivative that accounts for the connection so its output transforms as the intended tensor |
| Diffeomorphism | A smooth invertible map with smooth inverse; it underlies coordinate changes and GR’s field-relabeling gauge symmetry |
| Effective field theory | An expansion valid below a specified scale, organized so that finitely many parameters control predictions to a chosen accuracy |
| Einstein tensor | The covariantly divergence-free combination of Ricci curvature and its scalar trace that appears in the field equation |
| Equivalence principle | A family of experimentally meaningful statements about universal free fall and the local behavior of nongravitational physics; its precise version matters |
| Event horizon | A global causal boundary separating events that can send signals to the relevant future infinity from those that cannot |
| Extrinsic curvature | How a hypersurface sits within a higher-dimensional geometry; in a spacetime slicing it encodes important information about the slices’ evolution |
| Gauge freedom | Redundancy in a mathematical description; gauge-related descriptions represent the same physical situation when the transformations are treated as redundancies with the appropriate boundary conditions |
| Geodesic | A curve whose tangent is parallel-transported along itself with an affine parameter |
| Geodesic deviation | The evolution of separation between neighboring geodesics, governed by curvature |
| Global hyperbolicity | A strong causality condition associated with a well-defined Cauchy initial-value formulation |
| Holonomy | The net transformation produced by parallel transport around a closed loop |
| Killing vector | A vector field generating a continuous metric symmetry, with vanishing Lie derivative of the metric |
| Lapse | In a spacetime slicing, the factor converting coordinate-time separation into proper separation along the slice normal |
| Lie derivative | Change of a field under the flow generated by a vector field; it is distinct from covariant differentiation along that vector |
| Manifold | A space locally describable by ordinary coordinate neighborhoods with compatible smooth overlaps |
| Metric | A smoothly varying nondegenerate bilinear form defining intervals, causal character, and index conversion |
| Nonmetricity | Failure of a connection to preserve the metric under covariant differentiation |
| Null | Having zero spacetime norm; a nonzero null vector is not the zero vector |
| On shell | Satisfying the equations of motion; off shell means not imposing them |
| Orthonormal frame | A local basis in which the metric has the standard diagonal Minkowski form |
| Parallel transport | Moving a vector along a path so its covariant derivative along that path vanishes |
| Proper acceleration | The invariant magnitude of four-acceleration, measured by an ideal accelerometer |
| Proper time | The time accumulated by an ideal clock along its timelike worldline |
| Ricci curvature | A contraction of Riemann curvature, directly tied to stress-energy through Einstein’s equation |
| Riemann curvature | The tensor encoding local curvature, including tidal and parallel-transport information |
| Scalar curvature | A further contraction of Ricci; useful but insufficient to describe all curvature |
| Shift | The tangential part of the coordinate-time flow relative to a chosen slicing |
| Stress-energy tensor | The local tensor encoding energy, momentum, their fluxes, and mechanical stresses |
| Tangent space | The vector space of possible tangent directions at one event |
| Tensor | A multilinear geometric object with components transforming by the appropriate Jacobian factors |
| Tetrad | A local orthonormal frame or its dual coframe in four-dimensional spacetime |
| Torsion | The antisymmetric failure of a connection’s directional derivatives to match the Lie bracket; it vanishes for the Levi-Civita connection |
| Weyl curvature | The trace-free part of Riemann curvature; in four dimensions it can remain nonzero in Ricci-flat vacuum |
| Worldline | A path through spacetime representing the history of a localized object or observer |