Metric coefficient (signature) of an additional coordinate

 How do you know when you are adding $t$ and when $z$ so that you can treat them differently?!

Suppose you have $x^j, \ j=1,\cdots n$ coordinates. You add a new coordinate $x^{n+1}$ and want to know how it should be added to the metric.

There are two possibilities: 

* Either

$$\forall n \ \ \frac{\partial x^{n+1}}{\partial x^{n}}=0.$$

In this case you treat the new coordinate just like the previous ones. 

For example, assume you have $(x,y)$ and want to add $z$. 

Since 

$$ \frac{\partial z}{\partial x}=\frac{\partial z}{\partial y}=0 $$

you'll have $$ r^2=x^2+y^2+z^2, $$

or $$\exists m\ \ \frac{\partial x^m}{\partial x^{m+1}}\neq 0.$$

In such case let $$\theta=\arccos\langle x^m,x^{m+1}\rangle,$$

then 

$$g_{\hat{m+1}\hat{m+1}}=e^{2i\theta}.$$

For example, if

$$x^{m+1}=t,$$

and

$$x^i=(x,y,z),$$ we have $$\frac{\partial z}{\partial t}=\frac{dz}{dt}\neq 0,$$

so

$$g_{\hat{t}\hat{t}}=e^{2i\frac{\pi}{2}}=-1.$$