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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 condition23
Θ ≡ ∇isi + Kijsisj − K = 0, (15 )
where i s is the outward-pointing unit 3-vector normal to the surface24. Assuming the Strahlkörper surface parameterization (4View Equation), Equation (15View Equation) can be rewritten in terms of angular 1st and 2nd derivatives of the horizon shape function h,
Θ ≡ Θ (h, ∂uh,∂uvh; gij,∂kgij,Kij) = 0, (16 )
where Θ is a complicated nonlinear algebraic function of the arguments shown. (Shibata [146Jump To The Next Citation Point] and Thornburg [153Jump To The Next Citation Point156Jump To The Next Citation Point] give the Θ (h,∂uh,∂uvh ) function explicitly.)
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