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
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.