Consider a fictitious line interval (metric) with 3 extra dimensions as follows:
$ds^2 = -2dt^2 + a^2(t) \left( \frac{dr^2}{1 - r^2} + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2 \right) + da_{1}^2+da_{2}^2+da_{3}^2 $
Let's put the condition that $da_{1}^2+da_{2}^2+da_{3}^2 = dt^2$
How can we interpret such a metric, geometrically?
Is this metric and the geometry equivalent to the usual metric and geometry in the 4-dim spacetime given that it always cancels out the 2 factor in the timelike component of the metric, unless the timelike condition on the distance in the extra dimensions is released?