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Deriving The Dyer-Roeder Equation From The Geodesic Deviation Equation Via The Newman Penrose Null Tetrad, Aly Aly
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In this paper we examine the geodesic deviation equation using the Newman-Penrose (N-P) formalism for a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric (Carroll, S. (2004); Ryden, B. (2003); Newman & Penrose (1962)). We solved the geodesic deviation equation for angular diameter distance, using the relevant N-P components, and the resulting expression was the Dyer-Roeder equation of cosmology (Ryden, B. (2003)) (Schneider et al. (1992)). This leads us to believe that we can apply the N-P formalism to a perturbed FLRW metric and find a solvable equation for angular diameter distance (Kling & Campbell (2008)). The perturbed FLRW metric incorporates clumps of matter …