Open slicing of anti-de Sitter spacetime ($a(t)={\alpha }^{-1}\sin (\alpha t),k=-1$)

An FLRW metric with $p=-{\varrho }_0$ and constant negative energy density ${\varrho }_0$ has ${\alpha }^2=-8\pi G{\varrho }_0/3>0$ and obeys ${\dot{a}}^2+k=-{\alpha }^2{a}^2$. Positive solutions exist only for $k=-1$. They are $a={\alpha }^{-1}\sin [\alpha (t-t_B)]$ on $0<\alpha (t-t_B)<\pi$. This is an open slicing of Anti-de Sitter spacetime, whose Ricci scalar is $-12{\alpha }^2$ and Kretschmann scalar is $24{\alpha }^4$. The zeros mark boundaries of this slicing rather than curvature singularities.