Null hypersurface

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In relativity, a null surface is a 3-surface whose normal vector is everywhere null (zero length with respect to the local Lorentz metric), but the vector is not identically zero. For example, light cones are null surfaces.

An alternative characterization is that the tangent space at any point contains vectors that are all space-like except in one direction, in which vectors have a null "length". The metric applied to such a vector and any other vector in the tangent space (including the vector itself) is null. Another way of saying this is that the pullback of the metric onto the tangent space is degenerate.

Physically, the tangent space at a given point on a null surface is space-like except that it contains one line corresponding to the world-line of a particle moving at the speed of light. For example, a light cone is a null hypersurface. Another example is a Killing horizon or the event horizon of a black hole.


  • Galloway, Gregory (2000), "Maximum Principles for Null Hypersurfaces and Null Splitting Theorems", Annales Poincare Phys.Theor. 1: 543–567, arXiv:math/9909158, Bibcode:1999math......9158G .
  • James B. Hartle, Gravity: an Introduction To Einstein's General Relativity.