What can be said about this metric? I've developed a metric that seems to work pretty well for predicting the magnitude of supernovae.$$a\left(t\right)=\frac{t^2}{t_0^2}$$$$ds^2=i^2\left(c_3+A_3 t \right)^2 dt^2+\left(\frac{1+a\left(t\right)}{2} \right)^2\left(dr^2+\left(\frac{sin\left(r\sqrt{k}\right)}{\sqrt{k}} \right)^2 \left(d\theta^2+sin^2 \theta\space d \phi^2 \right) \right)$$Where $c_3$ is a constant velocity, $A_3$ is a constant acceleration, and $t_0$ is a constant time and $t<t_0$.  What can we say about this geometry?
For example, it appears that there's an acceleration at every point on this manifold.  So is this manifold locally flat anywhere?
It's obviously not Lorentzian, is there a category for this kind of a metric?
 A: With the coordinate transformation $2A_3\tau = (c_3+A_3t)^2\implies 2A_3d\tau = 2A_3(c_3+A_3t)dt$ we have that your metric becomes
$$-\left[(c_3+A_3t)dt\right]^2 + \tilde{a}(t)\Bigr(\cdots\Bigr) = -d\tau^2+\tilde{a}\left(\frac{\sqrt{2A_3\tau}-c_3}{A_3}\right)\Biggr(\cdots\Biggr) \equiv -d\tau^2 + \tilde{a}'(\tau)\Bigr(\cdots\Bigr)$$
FRW with a rescaled time parameter. In other words it's just constituting a different coordinate system on the same underlying geometry, just like how imposing spherical coordinates on a flat space does not magically curve it.

You mentioned that this was good at predicting some kind of behavior with supernovas, let's analyze this further. In the notation from my work we have the formula for the scaling parameter is
$$\tilde{a}'(\tau) = \frac{1}{2}\left(1+\frac{(\sqrt{2A_3\tau}-c_3)^2}{A_3^2t_0^2}\right) = k_0 + k_{1/2}\sqrt{\tau} + k_1\tau$$
Notice that at different times $\tau$ the growth of this scale factor is either dominated by the square root term or the linear term. Square root growth is associated with a radiation dominated universe (filled with highly energetic, relativistically moving particles) and linear growth is associated with a "curvature dominated" universe (that is, an empty universe with extreme spatial curvature).
