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Temporal components in metric tensors

  1. Sep 14, 2014 #1
    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
    x2
    x3

    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?
     
  2. jcsd
  3. Sep 15, 2014 #2

    Simon Bridge

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    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?
     
  4. Sep 15, 2014 #3

    haushofer

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    Only the Einstein equations can tell you that :P
     
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