I just recently derived the metric tensor of the 4-sphere by embedding the coordinate system within 5D spherical coordinates, deriving the tangential vectors and then doing the dot product with all of the tangential vectors except for the e(adsbygoogle = window.adsbygoogle || []).push({}); _{r}vector since r stays constant. I then added a sign signature to this 4-sphere metric tensor which I chose to be (- + + +).

Then a crucial memory hit me. A long time ago, back when I first learned how to derive metric tensors, I remember looking up the metric tensor for 3D spherical coordinates online. When I did this, a page with the metric tensor for spherical coordinates in Minkowski space came up. This particular metric tensor looked like this:

g_{00}= -1

g_{11}= 1

g_{22}= r^{2}

and g_{33}= r^{2}sin^{2}(θ)

Every other term was 0.

Now I know that this particular space-time is flat while the 4-sphere that I recently derived is curved, but the Minkowski spherical metric tensor above is not even the same as the metric tensor for 4D spherical coordinates (the coordinate system you use when deriving the 3-sphere).

Since this Minkowski version of spherical coordinates was different from the metric tensor for 4D spherical polar coordinates, this made me wonder whether or not you can actually apply a temporal component to n-spheres.

Can someone please tell me if it is valid to apply a sign signature to an n-sphere (indicating that n-spheres can have temporal components) or if the metric tensors of n-spheres can only be purely spatial?

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# Question about n-spheres

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