What is the metric of a surface defined by $0=\phi^2 t^4-x^2-y^2-z^2$?

Question

This is a follow up to this post. We've worked out the formula for a surface (still haven't got a name for it), and now I want to know if there's a metric that can be made from:$$0=\phi^2 t^4+x^2+y^2+z^2$$Unlike the Minkowski metric, this one has a dependence on the absolute value of $t$, so it's almost like the metric needs to be some function of $t$ (e.g. $ds^2(t)= <some formula >$), but that's where I get stuck. Ideally, I'd like to find an expression for the curvature of the surface we've defined.

Answer 1

OK, I worked this out on a spreadsheet and it seems to give the desired result (a formula for the distance between two points on this surface). Let me know if there's something I got wrong:$$ds^2(t)=dt^2a_0^2(dt-2t)^2-dx^2-dy^2-dz^2$$I don't know if the notiation is right, but I know the metric depends on the time. If there's a way to express this independent of the time, I'd love to see it.