The Penrose-Rindler Equation

When applying the GHP calculus to spherically symmetric spacetimes (see the second part of the book), we have encountered several times that one specific commutator is extremely useful. Remembering that commutators of covariant derivatives give place to curvature, one is tempted to somehow assign to [ ð, ð^′] the (intrinsic) Gaussian curvature of any spacelike surface (remember that ð and ð^′ are GHP-covariant derivatives in the directions of m^a and m¯ ^a ; i.e., they should describe the intrinsic curvature of, let us say, the spacelike sector of the geometry. Although equivalent assertions relating with the intrinsic curvature of a timelike surface can be stated, here we will not go deep along this line). In fact, this is exactly what occurs, as the following proposition Penrose et al.