Nullkegel

Nullkegel, literally "null cone" in German, is a term used in mathematics and physics to denote the set of vectors for which a given quadratic form vanishes, forming a conical surface through the origin. In relativity, the null cone is the light cone at a spacetime event: the collection of events connected to that event by light signals. More generally, for a vector space V with a nondegenerate quadratic form Q, the null cone N = {v ∈ V | Q(v) = 0} comprises all null directions.

In Minkowski space with signature (-,+,+,+), Q(t,x,y,z) = -t^2 + x^2 + y^2 + z^2, and the null cone is given by t^2 = x^2 + y^2 + z^2, dividing time-like and space-like regions. The cone is invariant under the Lorentz group and plays a central role in describing causal structure and conformal geometry. In projective geometry, the projective null cone is the set of directions [v] with Q(v) = 0, forming a projective quadric.

Applications include analyzing wave propagation, causal relations, and the geometry of quadratic forms. The term "Nullkegel" is predominantly used in German-language texts; English sources typically use "null cone" or "light cone" depending on context. Related concepts include light-like vectors, causal structure in spacetime, and the study of quadratic forms in linear and differential geometry.