# Expansion of a trapped surface - clarification needed

1. Aug 9, 2011

### TomCurious

1. The problem statement, all variables and given/known data

I was just reading this article on the arXiv: http://arxiv.org/abs/0711.0313 , which discusses trapped surfaces and black holes.
There is a simple qualm I have, though, but it is persistent, and I cannot seem to come to terms with it.

Looking at equation's (1) and (2) of the document, we find (forgive me, I do not know latex:

Expansion along n = metric dually contracted with covariant derivative of one null vector +n*l*covariant derivative of n + l*n*covariant derivative of n,

where n and l are the two null vectors which are normal to the trapped surface.

2. Relevant equations

On the surface, we can reconstruct the metric as Qab = Gab + NaLa
(taken from Eric Poisson's book)

Expansion can easily be defined as the derivative of the cross-sectional area of the geodesic congruence, divided by the area. Going infinitesimal, and using some Differential Geometry, this is equivalent to the surface metric (Qab) fully contracted with the covariant derivative of one of null vectors.

3. The attempt at a solution

Perhaps the error lies in the definition, that is to say, Qab = Gab + NaLa + LaNa ? It is a fine point, but one that I cannot seem to find the answer to.

2. Aug 10, 2011

### George Jones

Staff Emeritus
Where is this in Poisson's book?
See equation (2.28) from (the hard-copy version of) Poisson's book.

3. Aug 10, 2011

### TomCurious

Oh, I see. Thank you very much! I was looking at equation (2.14) - my bad!