id author title date pages extension mime words sentence flesch summary cache txt american_scientific_journal-1767 Wimaladharma, Sashinka; de Silva, Nalin A Static Solution to Einstein’s Field Equations for a Spherical Distribution of Electrically Counterpoised Dust with a Set of New Boundary Conditions 2016 15 .pdf application/pdf 4823 149 64 (2016) Volume 21, No 1, pp 163-177 173 The radial null geodesics within the sphere are given by the metric (18) as 0 = 1 οΏ½πœƒπœƒοΏ½π‘Ÿπ‘Ÿπ‘™π‘™οΏ½οΏ½ 2 𝑐𝑐2𝑑𝑑𝑑𝑑2 βˆ’ οΏ½πœƒπœƒ οΏ½π‘Ÿπ‘Ÿ 𝑙𝑙 οΏ½οΏ½ 2 π‘‘π‘‘π‘Ÿπ‘Ÿ2 , Taking the plus sign since π‘Ÿπ‘Ÿ increases with time 𝑑𝑑 as the photons are going away from the centre of the sphere gives π‘‘π‘‘π‘Ÿπ‘Ÿ 𝑑𝑑𝛿𝛿 = 𝑐𝑐 οΏ½πœƒπœƒοΏ½π‘Ÿπ‘Ÿπ‘™π‘™οΏ½οΏ½ 2. (28) Equation (28) can be integrated considering the front of the pulse as ∫ 𝑐𝑐 𝑑𝑑𝑑𝑑𝛿𝛿𝑒𝑒′ 𝛿𝛿𝑒𝑒 = ∫ οΏ½πœƒπœƒ οΏ½π‘Ÿπ‘Ÿ 𝑙𝑙 οΏ½οΏ½ 2π‘Žπ‘Ž π‘Ÿπ‘Ÿπ‘’π‘’ π‘‘π‘‘π‘Ÿπ‘Ÿ (29) Then the metrics (6) become 𝑑𝑑𝑑𝑑2 = 1 οΏ½πœƒπœƒοΏ½π‘…π‘…π‘™π‘™ οΏ½οΏ½ 2 𝑐𝑐2𝑑𝑑𝑑𝑑2 βˆ’ οΏ½πœƒπœƒ �𝑅𝑅 𝑙𝑙 οΏ½οΏ½ 2 (π‘‘π‘‘π‘Ÿπ‘Ÿ2 + π‘Ÿπ‘Ÿ2𝑑𝑑Ω2) 0 ≀ π‘Ÿπ‘Ÿ ≀ π‘Žπ‘Ž 𝑑𝑑𝑑𝑑2 = 1 οΏ½1βˆ’1𝑅𝑅� π‘Žπ‘Ž2 𝑙𝑙 πœƒπœƒβ€²οΏ½ π‘Žπ‘Ž 𝑙𝑙��� 2 𝑐𝑐2𝑑𝑑𝑑𝑑2 βˆ’ οΏ½1 βˆ’ 1 𝑅𝑅 οΏ½π‘Žπ‘Ž 2 𝑙𝑙 πœƒπœƒβ€² οΏ½π‘Žπ‘Ž 𝑙𝑙 οΏ½οΏ½οΏ½ 2 (𝑑𝑑𝑑𝑑2 + 𝑑𝑑2𝑑𝑑Ω2) 𝐴𝐴 < 𝑑𝑑 (18) where we have replaced 𝐴𝐴 by (π‘Žπ‘Ž πœƒπœƒ οΏ½π‘Žπ‘Ž 𝑙𝑙 οΏ½ + π‘Žπ‘Ž2 𝑙𝑙 πœƒπœƒβ€² οΏ½π‘Žπ‘Ž 𝑙𝑙 οΏ½) . cache/american_scientific_journal-1767.pdf txt/american_scientific_journal-1767.txt