I'm reading this paper (https://arxiv.org/abs/1611.02702), and on page 5 they come up with this bound on a parameter, $\theta$, which they call the 'expansion of light rays':

$$ \frac{\partial\theta}{\partial\lambda} + \frac{1}{2}\theta^2 \leq 0 $$

where $\lambda$ is the affine parameter for a past-directed light ray emanating from a point, $p_0$. They then go on to define a horizon at $\theta = 0$ and then use the worldsheet traced by this as a holographic screen.

I'm just not entirely sure where this $\theta$ thing came from, and why taking it to be zero is a useful choice of holographic screen or why it defines a horizon.

  • $\begingroup$ They assume that the reader knows what expansion is. Most advanced GR text explain the notion. $\endgroup$ – MBN Aug 20 '18 at 16:57
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    $\begingroup$ Section 9.5.2 of my SR book lightandmatter.com/sr has what I intended to be the most elementary possible presentation of the expansion scalar. It's restricted to 1+1 dimensions in a flat spacetime. The definitions do get more complicated when you generalize to GR in 3+1 dimensions, but this may be a helpful starting point. $\endgroup$ – Ben Crowell Aug 20 '18 at 20:20
  • $\begingroup$ Ah. I missed the 'congruence' bit which would have lead me to the correct Wikipedia article. I've only ever really studied differential geometry as maths so I guess I just never came across this decomposition. $\endgroup$ – gautampk Aug 21 '18 at 9:29

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