The null expansion of a spacelike codimension-two surface is the trace of its second fundamental
form in a chosen null-normal direction. For tangent vectors 
, induced metric 
, and null normal 
, it is
 |
It measures the first variation of area along the null congruence. Replacing 
by a positive multiple scales the expansion
by the same factor, so its vanishing and sign do not depend on that normalization.
For the future outward normal convention on an initial
data set, 
.
Here 
is the trace of the spatial second fundamental
form with respect to the outward unit normal, namely the sum of the principal
curvatures, and 
is the extrinsic curvature of the spatial
slice. Thus 
uses the trace convention rather than the averaged mean
curvature. A marginally outer trapped
surface has vanishing future outward null expansion.
