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˙ 00 product rule 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 {~xu, x~v , U} 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 {~xu, x~v , U} 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 {~xu, x~v , U} 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˙ 00 product rule 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 {~xu, x~v , U} 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˙ 00 product rule 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 {~xu, x~v , U} 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˙ 00 product rule 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 {~xu, x~v , U} 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˙ 00 product rule 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 {~xu, x~v , U} 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 {~xu, x~v , U} 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 {~xu, x~v , U} 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 {~xu, x~v , U} 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. Sarah Math 4140/5530: Differential Geometry u u 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 u v Also from a few slides up, ~xuu = Γuu~xu + Γuu~xv + lU, so ~xuu · ~xu =
Dr. Sarah Math 4140/5530: Differential Geometry 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 u v Also from a few slides up, ~xuu = Γuu~xu + Γuu~xv + lU, so u u ~xuu · ~xu = Γuu~xu · ~xu = ΓuuE
Dr. Sarah Math 4140/5530: Differential Geometry 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 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
Dr. Sarah Math 4140/5530: Differential Geometry 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 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 Dr. Sarah Math 4140/5530: Differential Geometry Solving the Geodesic Equations? ¨a a ˙ b ˙ c x + Γbcx x = 0
differential equations expressed in intrinsic coordinates theoretical importance in mathematics and physics in analytic treatments of geodesics in practice, these equations can rarely be solved, except approximately (numerically) conetorusgeos.mw adapted from demo by John Oprea
Dr. Sarah Math 4140/5530: Differential Geometry In Higher Dimensions? Tensors and the Metric Tensor gij
towardsdatascience.com/a-beginner-introduction-to-tensorflow-part-1-6d139e038278 lists #s vectors stack of matrices algebraic combinations of vectors, matrices, vector spaces, algebras, modules or other structures often geometrically meaningful not all tensors are inherently linear maps EF c g inner products of tangent vectors a b ij FG d −1 EF G −F gij = g−1 = = 1 ij FG EG−F 2 −FE
Dr. Sarah Math 4140/5530: Differential Geometry SpaceTime-Time: Special Relativity
Albert Einstein special relativity (1905) Hermann Minkowski 4D spacetime model (1908) T surfaces: gij inner products of tangent vectors w gij v EF c a b FG d spacetime: metric tensor is now 4x4 symmetric matrix acting as T w gij v on (t, x, y, z) vectors. Yardstick plus clock!
Dr. Sarah Math 4140/5530: Differential Geometry Minkowski SpaceTime Model
http://jokerific.com/wp-content/uploads/2014/08/einstein-space-time-theory-joke.jpg g11 g12 g13 g14 t 1 0 0 0 g21 g22 g23 g24 x 0 −1 0 0 t x y z g31 g32 g33 g34 y 0 0 −1 0 g41 g42 g43 g44 z 0 0 0 −1 c, c2, −
Dr. Sarah Math 4140/5530: Differential Geometry In Higher Dimensions? Keep on Summing!
metric form 2 a b ds = gabdx dx Christoffel symbols
1 Γa = gad (∂ g + ∂ g − ∂ g ). bc 2 b dc c db d bc Riemann curvature tensor or Riemann-Christoffel tensor a a a e a e a Rbcd = ∂cΓbd − ∂d Γbc + Γbd Γec − ΓbcΓed c cd Ricci tensor Rab = Racb = g Rdacb ab Scalar curvature R = g Rab 1 Einstein tensor Gab = Rab − 2 gabR
Dr. Sarah Math 4140/5530: Differential Geometry Christoffel Symbols and Curvatures The Christoffel symbols intrinsic quantities, how to take covariant derivatives coefficients of tangent vectors (connection coefficients) measure whether or not vectors are parallel transports in relativity, gravitational forces. geodesics and curvatures. Riemann curvature tensor: measures how much a manifold is not flat via 44 = 256 entries for spacetime. Ricci tensor: trace (sum of diagonal elements) relates to the p metric volume detgij . Scalar curvature: number. For surfaces—twice Gaussian curvature. For relativity—Lagrangian density. Einstein tensor describes curvature of spacetime due to the presence of energy or mass, has zero divergence
Dr. Sarah Math 4140/5530: Differential Geometry SpaceTime-Time: Other SpaceTimes
http://www.spacetime-model.com/img/mass/einstein.jpg
4D Manifold, gij , curvature satisfy Einstein field equation T gij can be other 4x4 symmetric matrices acting as w gij v. gij 1 eigenvalue > 0 and three eigenvalues < 0 at each point. T q w gij v T angle : cos θ = |v||w| , spacetime interval: |v| = v gij v
change gij and change physical properties of spacetime astrophysicist begins with what we observe and tries to construct a metric that models it enter any metric, then use Einstein’s field equation to read off the physical properties of the resulting universe
Dr. Sarah Math 4140/5530: Differential Geometry