the matrix X is equal to . The special linear group acts on the space of such matrices via for each A ∈ SL(2,C), and this action of SL(2,C) preserves the determinant of X because . Since the determinant of X is identified with the quadratic form Q, SL(2,C) acts by Lorentz transformations. On dimensional grounds, SL(2,C) covers a neighborhood of the identity of SO(1,3). Since SL(2,C) is connected, it covers the entire restricted Lorentz group SO+(1,3). Furthermore, since the kernel of the action () is the subgroup {±I}, then passing to the quotient group gives the group isomorphism Focusing now attention on the case when (x0,x1,x2,x3) is null, the matrix X has zero determinant, and therefore splits as the outer product of a complex two-vector ξ with its complex conjugate: The two-component vector ξ is acted upon by SL(2,C) in a manner compatible with (). It is now clear that the kernel of the representation of SL(2,C) on hermitian matrices is {±I}. The action of PSL(2,C) on the celestial sphere may also be described geometrically using stereographic projection. Consider first the hyperplane in R4 given by x0 = 1. The celestial sphere may be identified with the sphere S+ of intersection of the hyperplane with the future null cone N+. The stereographic projection from the north pole (1,0,0,1) of this sphere onto the plane x3 = 0 takes a point with coordinates (1,x1,x2,x3) with to the point Introducing the complex coordinate the inverse stereographic projection gives the following formula for a point (x1, x2, x3) on S+: The action of SO+(1,3) on the points of N+ does not preserve the hyperplane S+, but acting on points in S+ and then rescaling so that the result is again in S+ gives an action of SO+(1,3) on the sphere which goes over to an action on the complex variable ζ. In fact, this action is by fractional linear transformations, although this is not easily seen from this representation of the celestial sphere. Conversely, for any fractional linear transformation of ζ variable goes over to a unique Lorentz transformation on N+, possibly after a suitable (uniquely determined) rescaling. A more invariant description of the stereographic projection which allows the action to be more clearly seen is to consider the variable ζ = z:w as a ratio of a pair of homogeneous coordinates for the complex projective line CP1. The stereographic projection goes