A note on behaviour at an isotropic singularity (Q2718520)

A spacetime \((\overline{M},\overline{g})\) is said to admit an isotropic singularity if there exists a spacetime \((M,g)\), a smooth cosmic time function \(t\) defined on \(M\) and a function \(\Omega(t)\) such that: NEWLINENEWLINENEWLINE(a) \(\overline{M}\) is the open submanifold \(t>0\); NEWLINENEWLINENEWLINE(b) \(\overline{g} =\Omega^2(t)g\) on \(\overline{M}\), with \(g\in C^3\) and non-degenerate on an neighbourhood of \(t=0\); NEWLINENEWLINENEWLINE(c) \(\Omega\in C[0,b]\cap C^3(0,b]\) for some \(b>0\) and \(\Omega (t)>0\) for \(t\in (0,b]\); NEWLINENEWLINENEWLINE(d) the limit \(\lambda=\lim_{t\to 0+}\Omega\Omega''/(\Omega')^2\neq 1\) exists, where the prime indicates differention with respect to \(t\). NEWLINENEWLINENEWLINEThe author shows that every Jacobi field along a timelike geodesic running into an isotropic singularity is crushed to zero length at the same rate as the singularity is approached. The rate at which this crushing takes place depends only on the index \(\lambda\).