Abstract
Starting from the main definitions, we review the rigging technique for null hypersurfaces theory and most of its main properties. We make some applications to illustrate it. On the one hand, we show how we can use it to show properties of null hypersurfaces, with emphasis in null cones, totally geodesic, totally umbilic, and compact null hypersurfaces. On the other hand, we show the interplay with the ambient space, including its influence in causality theory.
Highlights
Null hypersurfaces are present in general relativity in several fundamental theories
The thermodynamic of black holes stated an unexpected link between classical thermodynamic theory and black hole theory using the horizon of the black hole, which is a null hypersurface [1,2,3]
It appears in causality theory and it is a null hypersurface near the vertex, but maybe not far away because of the presence of null conjugate points to the vertex or self-intersections
Summary
Null hypersurfaces are present in general relativity in several fundamental theories. Another interesting point is the study of an individual null hypesurface, that is, its geometry as a submanifold For this purpose, the most important object is its null second fundamental form, obtained through the choice of a null vector field tangent to the null hypersurface. The most important object is its null second fundamental form, obtained through the choice of a null vector field tangent to the null hypersurface This allows us to classify them in totally umbilic, totally geodesic, or other. The second technique provides more geometric objects which allows us to ask new questions and solve new problems, but since the geometric data (ξ, S ) are chosen arbitrarily and independently, they are not tuned enough and it generates difficulties handling them They do not provide any reasonable metric and connection on the null hypersurface, so the questions that can be asked and solved are, in general, not well linked to other classical properties. We are interested in the ideas more than their details, so we will not include proofs of the results, which can be checked in the references
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