Null cones in Schwarzschild geometry

Abstract

In this work we investigate aspects of light cones in a Schwarzschild geometry, making connections to gravitational lensing theory and to a new approach to general relativity, the null surface formulation. By integrating the null geodesics of our model, we obtain the light cone from every space-time point. We study three applications of the light cones. First, by taking the intersection of the light cone from each point in the space-time with null infinity, we obtain the light cone cut function, a four parameter family of cuts of null infinity, which is central to the null surface formulation. We examine the singularity structure of the cut function. Second, we give the exact gravitational lens equations, and their specialization to the Einstein ring. Third, as an application of the cut function, we show that the recently introduced coordinate system, the ''pseudo Minkowski'' coordinates, are a valid coordinate system for the space-time.

Spacetime perspective of Schwarzschild lensing

“…where Eq. (7) has been used for the source's space-time location. Equation (15) gives x 1 implicitly in terms of the frequency of emission, the received frequency and the observed image angle.…”

“…(37) represents one of the two patches. The integration of the angular coordinates of the lightcone is carried out in [7]. Representing the angular coordinates (θ, φ) in terms of the complex stereographic variables ζ ≡ cot(θ/2)e iφ the integration yields…”

Study of Errors in Strong Gravitational Lensing

“…Image distortion and weak gravitational lensing were considered from this perspective in Frittelli et al (2001bFrittelli et al ( , 2002. An application of this no-lens-plane approach in (noncosmological) spacetimes representing spherically symmetric nonsingular matter distributions is presented in Kling & Newman (1999). The case of Schwarzschild black holes is developed in Frittelli et al (2001a), and the resulting lensing predictions are compared to Virbhadra & Ellis (2000) where a lens-plane approximation is applied to light rays that undergo large bending.…”

Foundations of Gravitational Lens Theory (Geometry of Light Cones)

“…Except for the special, ideal case of spherical symmetry ( [9], [10], λ presumably cannot be computed exactly. Except for the special, ideal case of spherical symmetry ( [9], [10], λ presumably cannot be computed exactly.…”

“…The map λ defined in section 6 depends on the spacetime curvature between sources and observer, which in turn is related to the energy-momentum distribution of matter. Except for the special, ideal case of spherical symmetry ( [9], [10], λ presumably cannot be computed exactly. The best one can do is find, under special assumptions on the matter distribution in (part of) spacetime, approximations to λ, analytically or numerically.…”