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The Newman-Penrose tetrade

Source: The Holographic Principle and the Renormalization Group, Enrique Alvarez and Cesar Gomez, arXiv:hep-th/9810102v1, 1998.10.14 – http://arxiv.org/abs/hep-th/9810102.

In terms of a real orthonormal tetrad e^a:

  • l :\equiv l^\mu\partial_\mu \equiv e^+ \equiv \frac1{\sqrt{2}}(e^0+e^3),
  • n :\equiv n^\mu\partial_\mu \equiv e^- \equiv \frac1{\sqrt{2}}(e^0-e^3),
  • m :\equiv m^\mu\partial_\mu \equiv e_T \equiv \frac1{\sqrt{2}}(e^1-i e^2),
  • \bar{m} :\equiv \bar{m}^\mu\partial_\mu \equiv \bar{e}_T \equiv \frac1{\sqrt{2}}(e^1+i e^2).


  • l^2=n^2=m^2=\bar{m}^2=0,
  • l\cdot n =1, m\cdot\bar{m}=-1,
  • l\cdot m=l\cdot\bar{m}=n\cdot m=n\cdot\bar{m}=0.

Some optical scalars:

  • \rho :\equiv - \nabla_\mu l_\nu m^\nu\bar{m}^\mu = - (\theta+i\omega),
  • \theta :\equiv \frac12\nabla_\alpha l^\alpha \ \ (expansion),
  • \omega^2 :\equiv \frac12\omega_{\alpha\beta}\omega^{\alpha\beta}, \ (rotation),
  • \omega_{\alpha\beta} :\equiv \nabla_{[\alpha}l_{\beta]}.

Raychadhuri theorem:

  • l^\mu\nabla_\mu\theta = \omega^2-\frac12R_{\mu\nu}l^\mu l^\nu - \sigma\bar{\sigma} - \theta^2.
  • Here the shear \sigma\bar{\sigma} is given by: \sigma\bar{\sigma} :\equiv  \frac12\nabla_{[\alpha}l_{\beta]}\nabla^{[\alpha}l^{\beta]} - \frac14(\nabla_\alpha l^\alpha)^2.

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