Riemannian from Pseudo-Riemannian?

The biggest obstacle that I am facing in pursuit of completion of my work on acceleration of light and its consequences is that, to get the correct result, I'm enforcing the signature and correct numerical value of acceleration of null rays `by hand'.

The problem is to get the Riemannian metric

$$du^2=(cH)^2 dt^2 + R^2(t) d\textbf{v}^2=j_{\alpha\beta}dv^\alpha dv^\beta,$$ of velocity-time space from the Psuedo-Riemannian FLRW metric of spacetime $$ds^2=-c^2 dt^2 + R^2(t) d\textbf{x}^2=g_{\mu\nu}dx^\mu dx^\nu.$$

From the spacetime metric (FLRW) we have

$$(\frac{dv}{dt})^2_{\text{FLRW}}=(\frac{cH}{R})^2,$$ while the metric of velocity-time requires  $$(\frac{dv}{dt})^2=-(\frac{cH}{R})^2.$$