Riemannian Curvature and the Petrov Classification
نویسنده
چکیده
Let M be a Lorentzian space-time manifold. If p G 31, let TV(M) denote the tangent space to M at p and let I e Tp (M ) be a null vector. Because of the identification of radiation in General Relativity with null geodesic congruences of curves in M, one is led to study the geometrical properties of the wave surfaces of I. These are the two dimensional subspaces of TP(M), each member of which is spacelike and orthogonal to I. There is in fact a two parameter family of such wave surfaces for a given null vector leTv(M) which, from the physical viewpoint, might be thought of as the totality of instantaneous wave surfaces of all observers with all possible velocities at p.* This two parameter family of wave surfaces can also be described as the orbit of one particular such surface under the action of a two parameter null rotation subgroup of the proper Lorentz group about the null vector I. These transformations are those null rotations about I for which I is the only fixed null direction and they constitute an abelian subgroup of the proper Lorentz group, being in fact isomorphic to the (translation) subgroup of the Möbius group which have only one fixed point (the point at infinity) on the extended complex plane. (See for example [2].)
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