Temporal components in metric tensors

[space-time]

As you may know, the metric tensor for 3D spherical coordinates is as follows:

g₁₁= 1
g₂₂= r²
g₃₃= r²sin²(θ)

Now, the Minkowski metric tensor for spherical coordinates is this:
g₀₀= -1
g₁₁= 1
g₂₂= r²
g₃₃= r²sin²(θ)

In both of these metric tensors, all other elements are 0.

Now, the only obvious difference between the two metric tensors is the fact that the Minkowski version has a -1 in it and a temporal row and column. The first metric tensor represents just flat space, while the second one represents flat space time.

Now, in my recent studies of curvature, I was wondering if you could just add a -1 along with a temporal row and column to the metric tensor of the 3-sphere in order to make it represent a spherically curved 4 dimensional space time (just as adding a -1 and a temporal row/column to the 3D spherical coordinates metric tensor creates a 4D metric tensor that represents flat space-time).

Here is what I mean:

This is the metric tensor for the 3-sphere:

g₁₁=r²sin²(ø)sin²(ψ)
g₂₂=r²sin²(ψ)
g₃₃=r²

The rest of the elements are 0.

As for my coordinate labels:

x¹=θ
x²=ø
x³=ψ

And now, here is what I mean when I mention "adding a -1 and a temporal row/column to the metric tensor of the 3-sphere":

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

The rest of the elements are 0.

As you can see here, I just added a -1 to the 3-sphere metric tensor just as the Minkowski metric tensor for spherical coordinates adds a -1 to the metric tensor of 3D spherical coordinates.

Will adding this -1 to the 3-sphere's metric tensor make this metric represent spherically curved space time as a result?

 

[Simon Bridge]

It is not a good idea to do physics by analogy. Have you applied the definition of curvature to the resulting metric to see if it represents a curved spacetime?

 

[haushofer]

Only the Einstein equations can tell you that :P