In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, and never with a definite quadratic form.
- First system:
- Second system:
Laguerre then interpreted these lines as geodesics:
- An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line situated at a finite distance in the plane is zero. In other terms, these lines satisfy the differential equation ds2 = 0. On an arbitrary surface one can study curves that satisfy this differential equation; these curves are the geodesic lines of the surface, and we also call them isotropic lines.: 90
In terms of the affine subspace x3 = 1, an isotropic line through the origin is
In projective geometry, the isotropic lines are the ones passing through the circular points at infinity.
In the real orthogonal geometry of Emil Artin, isotropic lines occur in pairs:
- A non-singular plane which contains an isotropic vector shall be called a hyperbolic plane. It can always be spanned by a pair N, M of vectors which satisfy
- We shall call any such ordered pair N, M a hyperbolic pair. If V is a non-singular plane with orthogonal geometry and N ≠ 0 is an isotropic vector of V, then there exists precisely one M in V such that N, M is a hyperbolic pair. The vectors x N and y M are then the only isotropic vectors of V.
Isotropic lines have been used in cosmological writing to carry light. For example, in a mathematical encyclopedia, light consists of photons: "The worldline of a zero rest mass (such as a non-quantum model of a photon and other elementary particles of mass zero) is an isotropic line." For isotropic lines through the origin, a particular point is a null vector, and the collection of all such isotropic lines forms the light cone at the origin.
- Edmond Laguerre (1870) "Sur l’emploi des imaginaires en la géométrie", Oeuvres de Laguerre 2: 89
- C. E. Springer (1964) Geometry and Analysis of Projective Spaces, page 141, W. H. Freeman and Company
- Emil Artin (1957) Geometric Algebra, page 119 via Internet Archive
- Encyclopedia of Mathematics World line
- Cartan, Élie (1981) , The theory of spinors, New York: Dover Publications, p. 17, ISBN 978-0-486-64070-9, MR 0631850