guess and check in Maple or computer software parallel transport—that a tangent vector stays parallel geodesic equations with Christoffel symbols (also useful for Einstein’s field equations) How do we find geodesics? http://i.stack.imgur.com/La4Hj.png https://i1.creativecow.net/u/1027/geodesicvoxel_binding.jpg covering symmetry γ 00 proof on the sphere used the surface normal U = R in γ along with Frenet equations how about more generally? Dr. Sarah Math 4140/5530: Differential Geometry How do we find geodesics? http://i.stack.imgur.com/La4Hj.png https://i1.creativecow.net/u/1027/geodesicvoxel_binding.jpg covering symmetry γ 00 proof on the sphere used the surface normal U = R in γ along with Frenet equations how about more generally? guess and check in Maple or computer software parallel transport—that a tangent vector stays parallel geodesic equations with Christoffel symbols (also useful for Einstein’s field equations) Dr. Sarah Math 4140/5530: Differential Geometry 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ product rule 00 d ~ _ ~ ¨ d ~ _ ~ ¨ α =α ¨ = dt xuu + xuu + dt xv v + xv v chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt Dr. Sarah Math 4140/5530: Differential Geometry product rule d ~ _ ~ ¨ d ~ _ ~ ¨ = dt xuu + xuu + dt xv v + xv v chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ α00 =α ¨ Dr. Sarah Math 4140/5530: Differential Geometry chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ product rule 00 d ~ _ ~ ¨ d ~ _ ~ ¨ α =α ¨ = dt xuu + xuu + dt xv v + xv v Dr. Sarah Math 4140/5530: Differential Geometry (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ product rule 00 d ~ _ ~ ¨ d ~ _ ~ ¨ α =α ¨ = dt xuu + xuu + dt xv v + xv v chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + Dr. Sarah Math 4140/5530: Differential Geometry 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ product rule 00 d ~ _ ~ ¨ d ~ _ ~ ¨ α =α ¨ = dt xuu + xuu + dt xv v + xv v chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ Dr. Sarah Math 4140/5530: Differential Geometry Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ product rule 00 d ~ _ ~ ¨ d ~ _ ~ ¨ α =α ¨ = dt xuu + xuu + dt xv v + xv v chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Dr. Sarah Math 4140/5530: Differential Geometry u v = ~xu + ~xv + lU = Γuu~xu + Γuu~xv + lU c ~ u ~ v ~ ~ Γab called Christoffel symbols. xuv = Γuv xu + Γuv xv + mU = xvu. u v ~xvv = Γvv~xu + Γvv~xv + nU Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. α_ = ~xuu_ + ~xv v_ 2 2 α¨ = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ ~xuu = ~xu + ~xv + U Dr. Sarah Math 4140/5530: Differential Geometry u v = Γuu~xu + Γuu~xv + lU c ~ u ~ v ~ ~ Γab called Christoffel symbols. xuv = Γuv xu + Γuv xv + mU = xvu. u v ~xvv = Γvv~xu + Γvv~xv + nU Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. α_ = ~xuu_ + ~xv v_ 2 2 α¨ = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ ~xuu = ~xu + ~xv + U = ~xu + ~xv + lU Dr. Sarah Math 4140/5530: Differential Geometry u v ~xuv = Γuv~xu + Γuv~xv + mU = ~xvu. u v ~xvv = Γvv~xu + Γvv~xv + nU Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. α_ = ~xuu_ + ~xv v_ 2 2 α¨ = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ u v ~xuu = ~xu + ~xv + U = ~xu + ~xv + lU = Γuu~xu + Γuu~xv + lU c Γab called Christoffel symbols. Dr. Sarah Math 4140/5530: Differential Geometry Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. α_ = ~xuu_ + ~xv v_ 2 2 α¨ = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ u v ~xuu = ~xu + ~xv + U = ~xu + ~xv + lU = Γuu~xu + Γuu~xv + lU c ~ u ~ v ~ ~ Γab called Christoffel symbols. xuv = Γuv xu + Γuv xv + mU = xvu. u v ~xvv = Γvv~xu + Γvv~xv + nU Dr. Sarah Math 4140/5530: Differential Geometry ¨a a _ b _ c Or even more Einstein summation notation! x + Γbcx x = 0 http://scienceblogs.com/startswithabang/files/2013/07/einstein.jpg Geodesic Equations set ~xu; x~v components = 0. u 2 u u 2 1 1 b c u¨ + Γuuu_ + 2Γuv v_ u_ + Γvv v_ = 0 or x¨ + Γbcx_ x_ = 0 v 2 v v 2 2 2 b c v¨ + Γuuu_ + 2Γuv v_ u_ + Γvv v_ = 0 or x¨ + Γbcx_ x_ = 0 Dr. Sarah Math 4140/5530: Differential Geometry Geodesic Equations set ~xu; x~v components = 0. u 2 u u 2 1 1 b c u¨ + Γuuu_ + 2Γuv v_ u_ + Γvv v_ = 0 or x¨ + Γbcx_ x_ = 0 v 2 v v 2 2 2 b c v¨ + Γuuu_ + 2Γuv v_ u_ + Γvv v_ = 0 or x¨ + Γbcx_ x_ = 0 ¨a a _ b _ c Or even more Einstein summation notation! x + Γbcx x = 0 http://scienceblogs.com/startswithabang/files/2013/07/einstein.jpg Dr. Sarah Math 4140/5530: Differential Geometry ~xuu · ~xu + ~xu · ~xuu = 2~xuu · ~xu u v Also from a few slides up, ~xuu = Γuu~xu + Γuu~xv + lU, so u u ~xuu · ~xu = Γuu~xu · ~xu = ΓuuE Eu ~ ~ u u Eu Thus 2 = xuu · xu = ΓuuE so Γuu = 2E v Ev u Ev v Gu u Gu v Gv Similarly Γuu = − 2G ; Γuv = 2E ; Γuv = 2G ; Γvv = − 2E ; Γvv = 2G How do we find the Christoffel symbols? Elwin Bruno Christoffel and Albert Einstein http://www.ethbib.ethz.ch/aktuell/galerie/christoffel/Portr_gross.jpg http://scienceblogs.com/startswithabang/files/2013/07/einstein.jpg Rewrite ~xuu by taking uth partial of E = ~xu · ~xu Eu = Dr. Sarah Math 4140/5530: Differential Geometry u v Also from a few slides up, ~xuu = Γuu~xu + Γuu~xv + lU, so u u ~xuu · ~xu = Γuu~xu · ~xu = ΓuuE Eu ~ ~ u u Eu Thus 2 = xuu · xu = ΓuuE so Γuu = 2E v Ev u Ev v Gu u Gu v Gv Similarly Γuu = − 2G ; Γuv = 2E ; Γuv = 2G ; Γvv = − 2E ; Γvv = 2G How do we find the Christoffel symbols? Elwin Bruno Christoffel and Albert Einstein http://www.ethbib.ethz.ch/aktuell/galerie/christoffel/Portr_gross.jpg http://scienceblogs.com/startswithabang/files/2013/07/einstein.jpg Rewrite ~xuu by taking uth partial of E = ~xu · ~xu Eu = ~xuu · ~xu + ~xu · ~xuu = 2~xuu · ~xu Dr.