Since we assume that each null normal [l.sub.u] of the family of hypersurfaces F is

null geodesic, using above result of Perlick, we state the following corollary of Theorem 1 (proof is easy).

In addition,

null geodesic curvature intuitively says that geodesics have no curvature other than the curvature of the manifold itself.

We may now fix an open subinterval I of I', containing 0, and a

null geodesic I [contains as member] t [??] x(t) in M with x(0) = y, parametrized by the function t (in the sense that the function t restricted to the geodesic coincides with the geodesic parameter).

The conditions for being class-2 can be best described in terms of properties of the

null geodesic congruences spanned by Newman-Penrose vectors (NP) [1,8,11,35,55,71].

In other words any

null geodesic initially tangent to the photon sphere hypersurface will remain tangent to it.

Singh, "

Null geodesic expansion in spherical gravitational collapse," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol.

A radial light wave or radial

null geodesic (RNG) satisfies

For example, the metric (25) can be obtained from (24) by introducing a new time coordinate [bar.t] = t + 2M ln(r - 2M) in which the radial

null geodesics (see Section 5) become straight lines.

The events ([psi] = 0, t = T) and ([psi] = [[psi].sub.T], t = 0) are connected by a

null geodesics. The first event is relative to the fundamental Observer, while the second event refers to the emission of the CMB photons at t [approximately equal to] 0 as explained above.

It is known that the only null curves lying on pseudosphere [S.sup.2.sub.1] are the null straight lines, which are the

null geodesics.

A more oblique approach has to be used, defining a singularity as an 'ideal' point providing an edge or boundary to the spacetime, which prevents the extendibility of timelike or

null geodesics through the singular point.

To work out the lens equation we have to calculate the

null geodesics in z = const-planes.