1. Sep 14, 2014

### space-time

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 er 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:

g00 = -1
g11 = 1
g22= r2
and g33= r2sin2(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?

2. Sep 14, 2014

### ChrisVer

are you sure that's the metric of the 4-sphere? this looks like Minkowski metric in spherical coords.

$ds^{2} = g_{ab} dx^{a} dx^{b}= dt^{2} - dr^2 - r^2 d \theta^2 - r^2 sin^2 \theta d \phi^2$

Also what do you mean by applying a component to n-sphere? or temporal component?
Also what do you mean by "Minkowski version of spherical coordinates ... for 4D spherical polar coordinates"?

Last edited: Sep 14, 2014
3. Sep 14, 2014

### Staff: Mentor

You can, but the resulting metric will describe a different geometry than the surface of an n-sphere in space (unless, of course, you can find a coordinate transformation under which the metric components transform into that new form - and you cannot).

4. Sep 14, 2014

### space-time

Ok. Now I have another question. If I derive the metric tensor for the 3-sphere instead and then put an extra row and column within the matrix for a -1 (as is done with the Minkowski version of 3D spherical coordinates), then will that particular metric describe the 3D surface of a 4D sphere with the 4th dimension being time (or in other words, spherically curved space time)? Here is what I am talking about:

g00= -1
g11=r2sin2(ø)sin2(ψ)
g22=r2sin2(ψ)
g33=r2

As for my coordinate labels:
x0=t (my temporal component)
x1
x2
x3

All other elements are 0. You may notice that I simply added a negative 1 element to the matrix that would have otherwise simply been the 3-sphere (just as the Minkowski version of spherical coordinates just adds a -1 to what would otherwise just be 3D spherical polar coordinates).

This is what I meant when I asked: Would the metric tensor above describe spherically curved space time?

5. Sep 14, 2014

### space-time

It is the Minkowski metric of spherical coordinates. I did not actually post my 4-sphere metric.

6. Sep 15, 2014

### ChrisVer

Yes. Such a metric would describe a spacetime where the space would form a 3Sphere.