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

1. May 18, 2010

### im_hammer

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):

$$\frac{d\theta}{d\lambda}=-\frac{1}{2}\theta^{2}-\sigma^{2}-R_{ab}k^{a}k^{b}$$

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

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

Why is the term $$\hat{\omega}_{ab}\hat{\omega}^{ab}$$ missing in Jacobson's paper, or is it equal to Zero?

2. May 18, 2010

### haushofer

How are \sigma and \hat{\sigma} defined?

3. May 18, 2010

### George Jones

Staff Emeritus
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.