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Raychaudhuri equation in Jacobson's paper (gr-qc/9504004) ?

  1. May 18, 2010 #1
    In Jacobson's famous paper "Thermodynamics of Spacetime: The Einstein Equation of State" (gr-qc/9504004) Phys. Rev. Lett. 75, 1260–1263 (1995) , he wrote the Raychaudhuri equation as (Eq.(4) in his paper):

    [tex]\frac{d\theta}{d\lambda}=-\frac{1}{2}\theta^{2}-\sigma^{2}-R_{ab}k^{a}k^{b}[/tex]

    However, in General Relativity textbooks, Raychaudhuri equation is written as: (for example, Eq.(9.2.32) in Wald's GR book)

    [tex]\frac{d\theta}{d\lambda}=-\frac{1}{2}\theta^{2}-\hat{\sigma}_{ab}\hat{\sigma}^{ab}+\hat{\omega}_{ab}\hat{\omega}^{ab}-R_{cd}k^{c}k^{d}[/tex]

    Why is the term [tex]\hat{\omega}_{ab}\hat{\omega}^{ab}[/tex] missing in Jacobson's paper, or is it equal to Zero?
     
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  3. May 18, 2010 #2

    haushofer

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    How are \sigma and \hat{\sigma} defined?
     
  4. May 18, 2010 #3

    George Jones

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    The link to the paper is

    http://arxiv.org/abs/gr-qc/9504004.

    Because a lightlike vector is orthogonal to itself, the null generators of the horizon are orthogonal to the horizon (hypersurface orhtogonal), and thus, by Frobenius' theorem, the vorticity tensor vanishes.
     
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