As you may know, the metric tensor for 3D spherical coordinates is as follows:(adsbygoogle = window.adsbygoogle || []).push({});

g_{11}= 1

g_{22}= r^{2}

g_{33}= r^{2}sin^{2}(θ)

Now, the Minkowski metric tensor for spherical coordinates is this:

g_{00}= -1

g_{11}= 1

g_{22}= r^{2}

g_{33}= r^{2}sin^{2}(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:

g_{11}=r^{2}sin^{2}(ø)sin^{2}(ψ)

g_{22}=r^{2}sin^{2}(ψ)

g_{33}=r^{2}

The rest of the elements are 0.

As for my coordinate labels:

x^{1}=θ

x^{2}=ø

x^{3}=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":

g_{00}= -1

g_{11}=r^{2}sin^{2}(ø)sin^{2}(ψ)

g_{22}=r^{2}sin^{2}(ψ)

g_{33}=r^{2}

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?

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# Temporal components in metric tensors

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