Raychaudhuri equation

The Raychaudhuri equation describes how the expansion of a family of nearby geodesics changes along their paths. Its terms connect expansion to spacetime curvature, shear, and rotation.

Geodesic congruence: A family of curves whose members follow geodesics through a space or spacetime. The equation tracks the changing expansion of precisely such a family.
Affine parameter: A parameter along a geodesic for which its tangent vector is parallel transported along itself. For geodesic flows, affine parameterization gives the standard form of the evolution equation.
Focusing theorem: A result showing that suitable energy and initial conditions force geodesics to focus within finite parameter distance. It follows from the equation when rotation vanishes and curvature satisfies an appropriate positivity condition.
Amal Kumar Raychaudhuri: An Indian physicist whose 1955 work derived the equation now bearing his name. His derivation established the relation between congruence expansion, curvature, shear, and rotation.
Expansion scalar: A scalar measuring the fractional rate at which the cross-sectional volume of a geodesic congruence changes. Its derivative along the geodesics is the quantity governed by the equation.
Covariant derivative: A derivative that accounts for how vector and tensor components vary across curved spaces. The equation differentiates congruence properties using spacetime's covariant derivative.
Penrose singularity theorem: A general-relativistic theorem showing that trapped surfaces and suitable global conditions imply null geodesic incompleteness. Raychaudhuri focusing is a key local ingredient in its singularity argument.
General relativity: A theory of gravity in which spacetime geometry is determined by matter and energy. The equation is a geometric tool for studying families of paths in relativistic spacetime.
Shear tensor: The trace-free part of a congruence's deformation rate, describing shape change without volume change. Shear contributes a focusing term even when the congruence's volume does not change locally.
Riemann curvature tensor: A tensor encoding the intrinsic curvature of a manifold and the failure of covariant derivatives to commute. Its contraction appears in the equation as the Ricci-curvature focusing term.

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