# What happens for a "closed" universe without any content?

Let an universe with no content and positive curvature. Friedmann-Lemaître equation $$H^2=\frac{8\pi G}{3}\left(\rho_m+\rho_r+\rho_{\Lambda}\right)-\frac{k\, c^2}{a^2},$$ where $$a$$ corresponds to the scale factor of Friedmann-Lemaître-Robertson-Walker Metric, and the $$\rho$$ to the density of the contents, will become $$H^2=-\frac{\, c^2}{a^2},$$ as positive curvature (closed universe) corresponds to $$k=+1$$.

So, as $$H=\frac{\dot{a}}{a}$$ with $$\dot{a}=\frac{da}{dt}$$,

$$da=\pm i \, c \, dt$$

$$a(t)=\pm i \, c \, t + cst$$

what am I doing wrong ?

• FYI a native English speaker would say "a universe". Sep 28, 2018 at 4:04