# Only at the vertex of the light-cone the vectors, that are tangent to the cone are all and only light-cone ones

Tangent vectors to a light-cone in the spacetime are not always vector timelike. The condition of tangency to a submanifolds is a linear equation in the tangent space, and this also applies to the light-cone at any point other than the vertex that is a singular point,while the equation of the light-cone in tangent space is quadratic. Considering a light-cone with vertex in $x_{o}^\mu$a: $$\eta_{\mu \nu}(x^\mu-x^\mu_{0})(x^\nu-x^\nu_{0})=0$$ Only at the vertex of the light-cone the vectors, that are tangent to the cone are all and only light-cone ones The tangent vectors to the remainder of the cone may be timelike or spacelike. Someone cuold give me an explanation?

• It is correct, tangent vectors to the cone may be spacelike, timelike or null. At the vertex the are no tangent vectors defined. What is wrong with all of this? – magma Aug 15 '17 at 22:07
• I would like a mathematical clarification explaining why this is true. – Stefano Barone Aug 16 '17 at 7:31