- #1
space-time
- 218
- 4
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?
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?