# Raychaudhuri equation (and focusing theorem) The Raychaudhuri equation is derived from Einstein's equation. It is a useful equation in establishing proofs in General Relativity. For example, it is used in the derivation of [[0225 Singularity theorems|singularity theorems]]. For an affine parameter $\lambda$: $\frac{d \theta}{d \lambda}=-\frac{\theta^{2}}{D-2}-\sigma_{a b} \sigma^{a b}-R_{a b} k^{a} k^{b}.$ For a non-affine parameter $v$: $\frac{d \hat{\theta}}{d v}=\kappa \hat{\theta}-\frac{\hat{\theta}^{2}}{D-2}-\hat{\sigma}_{ab} \hat{\sigma}^{ab}-R_{ab} \chi^{a} \chi^{b}.$ ## Refs - see e.g. [[2007#Amsel, Marolf, Virmani]] Appendix A for a derivation. - general null surfaces: - [[2023#Ciambelli, Freidel, Leigh]] - review - [[2024#Chakraborty, Chakraborty (Review)]] - discrete notion of nonexpansion - [[2024#Bousso, Tabor]] - higher-spin - [[2024#Yan]] ## In modified gravity - it will change with [[0006 Higher-derivative gravity|HDG]] - [[ChoudhuryDasguptaBanerjee2021]][](https://arxiv.org/abs/2103.08869): [[0140 Scalar-tensor theory]]