Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Calculation involving angular momentum

  1. Aug 28, 2009 #1
    First of all, I'm sorry but I couldn't find a better title. And I'm sorry if I'm using the wrong forum.

    I'm going through W. G. Dixon "Extended Bodies in general relativity" (Appearing in "Isolated Gravitating Systems in General Relativity", J. Ehlers)

    To summarize the problem, we have an (antisymmetric) angular momentum tensor [tex]S^{\kappa \lambda}[/tex] and a timelike unit vector [tex]n^\kappa[/tex] such that [tex]n_\kappa S^{\kappa \lambda} =0[/tex]

    I'm trying to get
    Code (Text):
     [tex]S^{\kappa [ \lambda} S^{\rho ] \sigma} = \frac{1}{2}S^{\rho  \lambda} S^{\kappa \sigma} [/tex]
    where square brackets denote antisymmetrization.

    Dixon says "by [tex]n_\kappa S^{\kappa \lambda} =0[/tex], [tex]S^{\kappa \lambda}[/tex] has matrix rank at most 2 since it cannot have 3 in virtue of antisymmetry. Consequently [tex] S^{[\kappa \lambda} S^{\rho ] \sigma}[/tex] must be identically zero, which implies that ...." (above equation)

    I can see that [tex]S^{\kappa \lambda}[/tex] has 3 linearly independent components, and (only by explicit calculation) showed that it has at most rank 2, but I was not able to show the rest.
    Is it possible to show the first claim without calculation? And how should I proceed for the rest of the problem?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Calculation involving angular momentum
Loading...