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 into a metric that is on average homogeneous and isotropic. Deriving a solvable equation for angular diameter distance, in a perturbed FLRW metric, could prove useful to astronomers. By including clumps of matter along the line-of-sight into the math, we can calculate distances to light emitting objects that are obstructed by weak gravitational fields. This paper should serve as a test of both our tetrad and our methodology, by showing that they work in the flat unperturbed metric.
Deriving the Dyer-Roeder Equation from the Geodesic Deviation Equation via the Newman-Penrose Null Tetrad.
Undergraduate Review, 11, 6-16.
Available at: https://vc.bridgew.edu/undergrad_rev/vol11/iss1/4
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