Raychaudhuri

Raychaudhuri refers to Amal Kumar Raychaudhuri, an Indian physicist who formulated the Raychaudhuri equation in 1955. The equation describes how the expansion of a congruence of nearby geodesics evolves under gravity and is a foundational result in general relativity and cosmology. It provides a mathematical account of how matter and energy influence the focusing and convergence of geodesics, which is central to understanding spacetime structure and singularities.

For a congruence of timelike worldlines with tangent vector u^a, the expansion θ is defined as θ = ∇_a u^a. The Raychaudhuri equation is

dθ/dτ = - (1/3) θ^2 - σ_{ab} σ^{ab} + ω_{ab} ω^{ab} - R_{ab} u^a u^b + ∇_a a^a,

where σ_{ab} is the shear, ω_{ab} the vorticity, R_{ab} the Ricci tensor, and a^a = u^b ∇_b u^a the acceleration. In the special case of geodesic and hypersurface-orthogonal flow (a^a = 0 and ω_{ab} = 0), the equation simplifies to

dθ/dτ = - (1/3) θ^2 - σ_{ab} σ^{ab} - R_{ab} u^a u^b.

There is also a null form of the equation for congruences of null geodesics, which plays a key role in analyzing light ray focusing. The Raychaudhuri equation is widely used to derive focusing theorems and, together with energy conditions, underpins Hawking and Penrose’s singularity theorems. It remains a central tool in the study of gravitational collapse, cosmology, and the global structure of spacetime.