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The Distance between two points in a hypothetical universe.

I have a hypothetical universe where the distance between two points is defined as:

ds²=−(ϕ²t²)dt²+dx²+dy²+dz²

Where t is time, x, y & z are space and ϕ is a constant has units of km / s². The space in this universe grows quadratically with time (and, as I understand it, probably isn’t Minkowski spacetime). A particle travelling at the speed of causality, c, will follow this contour from point O to point P (see attached image).

Given the time between point O and point Q, is it possible to find the distance from point P to point Q (e.g. does a function exist such that f(Δt)=dL)? If so, what is that formula?

Note that I've had some preliminary work done in Mathematica to describe this space (see attached also).

Skills: Mathematics, Matlab and Mathematica

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