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

$$ds^2 =−(\phi^2 t^2)dt^2+dx^2+dy^2+dz^2$$Where ##\phi## has units of ##km s^{-2}##. The space in this universe grows quadratically with time (and, as I understand it, probably isn’t Minkowski space). A particle travelling at the speed of causality, c, will follow this contour from point O to point P.

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

$$ds^2 =−(\phi^2 t^2)dt^2+dx^2+dy^2+dz^2$$Where ##\phi## has units of ##km s^{-2}##. The space in this universe grows quadratically with time (and, as I understand it, probably isn’t Minkowski space). A particle travelling at the speed of causality, c, will follow this contour from point O to point P.

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

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