### 7.4 Description in terms of the 3 + 1 variables

In terms of the 3 + 1 variables, a MOTS (and thus an apparent horizon) satisfies the
condition
where is the outward-pointing unit 3-vector normal to the
surface.
Assuming the Strahlkörper surface parameterization (4), Equation (15) can be rewritten in terms of
angular 1st and 2nd derivatives of the horizon shape function ,
where is a complicated nonlinear algebraic function of the arguments shown. (Shibata [146] and
Thornburg [153, 156] give the function explicitly.)