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Weyl tensor in 2 dimensions- confused !

  1. Jul 5, 2013 #1
    hello,
    The Weyl tensor is:

    http://ars.els-cdn.com/content/image/1-s2.0-S0550321305002828-si53.gif

    In 2 dimensions , the Riemann tensor is (see MTW ex 14.2):
    Rabcd = K( gacgbd - gadgbc ) [R]

    Now the Weyl tensor must vanish in 2 dimensions. However, working with the g

    g =
    [-1 0 0 1 ]
    [ 0 1 0 0 ]
    [ 0 0 1 0 ]
    [ 0 0 0 1 ]

    Will yield a null Weyl tensor indeed with a Ricci scalar of 12K (from formula R). (see solution of Pb 9.27 in Lightmann)
    But in 2 dimensions with the g :

    g =
    [-1 0 ]
    [ 0 1 ]

    Will not give a null Weyl tensor ! with a Ricci scalar 2K (from formula R).

    Now I'm confused. The issue is very easy though.
    Where did I go wrong ?
     
  2. jcsd
  3. Jul 5, 2013 #2
    All N = 2 spaces are conformally flat.
    This would mean that since the Weyl tensor vanishes for the conformal space whose Riemann tensor has the form [R], thus one can conclude that for N=2, the Weyl tensor is null.
    This might make sense. But i do not know why the computation above did not yield a null Weyl tensor. For N=2 , the Weyl tensor is undefined, thus the link that I provided is not valid for N=2.
     
  4. Jul 5, 2013 #3
    This is related to the last question of MTW ex 21.21
     
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