U^{b} \nabla_{b} U_{a} = 0

gives the optical evolution equation

U^{c} \nabla_{c} B_{ab} = - B_{a c} B^{c}_{b} - R_{a c b d} U^{c} U^{d} .

Taking its screen trace yields

\frac{d θ}{d λ} = - B_{ab} B^{ba} - R_{ab} U^{a} U^{b} .

The optical decomposition and the symmetry or antisymmetry of its pieces imply

B_{ab} B^{ba} = \frac{1}{2} θ^{2} + w i d e ha t σ_{ab} σ^{ab} - ω_{ab} ω^{ab} .

Thus the four-dimensional Null Raychaudhuri equation is

\frac{d θ}{d λ} = - \frac{1}{2} θ^{2} - σ_{ab} σ^{ab} + ω_{ab} ω^{ab} - R_{ab} U^{a} U^{b} .

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.