R sinr r c c s r r r sinhr r c c

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Unformatted text preview: 8 1 MeV = 1.6×10- 13 J UW, ASTR 323, SP13, Eric Agol ©2013 - 1/3 7 Geometry in curved spaces: ______ curvature α + β +γ = π β α γ rc Distance between (r,θ) and (r+dr,θ+dθ) is ds: ds2 = dr 2 + rc2 sin(r / rc )dθ 2 5/21/13 ds UW, ASTR 323, SP13, Eric Agol ©2013 8 Geometry in curved spaces: ______ curvature Distance between (r,θ) and (r+dr,θ+dθ) is ds: α + β +γ = π ds2 = dr 2 + r 2dθ 2 ds r 5/21/13 UW, ASTR 323, SP13, Eric Agol ©2013 dθ r+dr 9 Geometry in curved spaces:_______curvature ds2 = dr 2 + rc2 sinh(r / rc )dθ 2 α β γ α + β +γ = π 5/21/13 UW, ASTR 323, SP13, Eric Agol ©2013 10 Curvature of the universe Posi8ve curvature Nega8ve curvature Zero curvature 5/21/13 κ: curvature constant κ = 1 α + β +γ = π + κA rc2 ds2 = dr 2 + Sκ (r )2 dθ 2 κ = - 1 ! r sin(r / r ) c #c # Sκ (r ) = " r # r sinh(r / r ) c #c $ κ = +1 κ =0 κ = −1 κ = 0 UW, ASTR 323, SP13, Eric Agol ©2013...
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