For generators of a null hypersurface, the <Frobenius theorem> gives $\widehat\omega_{ab}=0$. The <null energy condition> and the <Einstein field equations> imply $R_{ab}U^aU^b\geq0$, while the shear norm is nonnegative. With $A=D-2$, Raychaudhuri's equation gives
$$
\theta'\leq-\frac{\theta^2}{A}.
$$
As long as $\theta<0$ remains finite,
$$
\frac d{d\lambda}\left(\frac1\theta\right)
=-\frac{\theta'}{\theta^2}\geq\frac1A.
$$
If $\theta(0)=\theta_0<0$, integration gives
$$
\frac1{\theta(\lambda)}\geq\frac1{\theta_0}+\frac\lambda A.
$$
The right-hand side reaches zero at $\lambda=A/|\theta_0|$, which a finite negative $\theta$ cannot cross. Hence the <null focusing theorem> forces
$$
\boxed{\theta\longrightarrow-\infty
\quad\text{for some}\quad
\lambda\leq\frac A{|\theta_0|}}
$$
provided the generator extends that far.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 311 / 2 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-311/2/d/solution