Equations

The equations collected here are compact, convention-explicit records for calculations in gravity. Each record states the domain in which an equation applies, identifies the assumptions that change its form, and links to the Library page where the result is derived. Use these cards for lookup; use the linked derivations when a sign, normalization, or hypothesis is consequential.

Core geometric and dynamical equations

Record	Immediate lookup	Full treatment
Geodesic and Jacobi Equations	affine and nonaffine motion; geodesic deviation; Killing integrals; null-screen caveat	Geodesics, Exponential Maps, and Jacobi Fields
Einstein Field Equation	$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}$; stress–energy sign; trace reversal; Bianchi consistency	Einstein Field Equations
Raychaudhuri Equations	timelike rest-space and null screen-space evolution; acceleration and nonaffinity; focusing hypotheses	Congruences, Focusing, and Energy Conditions
ADM Constraints and Evolution	lapse and shift; extrinsic-curvature sign; matter projections; constraints versus evolution	The 3+1 Decomposition
Teukolsky Master Equation	Kerr tetrad and spin weight; Fourier and separation conventions; radial, angular, source, and boundary data	Kerr Perturbations and Teukolsky Theory

The site uses metric signature $(-,+,+,+)$ and

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

That convention fixes the Einstein tensor, the signs in the Jacobi and Raychaudhuri equations, and the curvature projections entering ADM. The site also uses $e^{-i\omega t}$, which fixes the Teukolsky horizon and infinity phases. A source using a different curvature, extrinsic-curvature, tetrad, or Fourier convention must be translated as a complete package rather than corrected one sign or symbol at a time.

Reading a formula safely

Before substituting numbers or components, check four items.

1. Domain. Is the connection Levi–Civita? Is the curve or null generator affinely parametrized? Is the hypersurface spacelike? Is the perturbation linear and the Kerr background subextremal?
2. Normalization. Is $G$ explicit, how is $T_{\mu\nu}$ defined, which sign defines $K_{ij}$, and which tetrad component or separated variable is being evolved?
3. Geometric and spectral data. Which curvature convention, index order, orientation, coordinate domain, spin weight, Fourier phase, and boundary branch are being used?
4. Boundary of the claim. A local identity does not supply a focusing theorem, global boundary conditions, a maximal extension, constraint preservation, well-posedness, or a reconstruction theorem whose additional hypotheses have not been stated.

For notation used across the site, consult Conventions and Symbols. For a derivation, proof, or worked application, follow the full-treatment link on the individual record rather than treating this directory as a substitute textbook.