My understanding is that extra space dimensions are added in a way such that the line element in flat spacetime is calculated by the Pythagorean theorem:
Now I know that the all space dimensions are linked to time, that's why it is called spacetime. But I'm wondering about a case when the change in a space dimension is equal to the change in time. Let's say a change of 1m in $dx_4$ would mean a change of 1s in $dt$. How would you add such a dimension to the metric? And is there any logic to adding such a dimension?
If I just replace $dx_4$ with $cdt$ then I have removed time from my metric and have suddenly a 4+0 metric instead of 3+1. But I can also go back if I want. Does this make sense?