Geodesics and parallel transport I need to develop all the procedures and missing steps from the textbook. I would reall

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Geodesics and parallel transport
I need to develop all the procedures and missing steps from the textbook. I would really apreciate the help

Geodesics And Parallel Transport I Need To Develop All The Procedures And Missing Steps From The Textbook I Would Reall 1 (209.65 KiB) Viewed 244 times

Geodesics, Parallel Transport The covariant derivative of vectors, tensors, and the Christoffel symbols m also be approached from geodesics. A geodesic in Euclidean space is a straig line. In general, it is the curve of shortest length between two points. Hence tl geodesic equation can be obtained from Fermat's variational principle of opti (see Chapter 17 for Euler's equation) 8[₁ ds = 0, (2.14 where ds² is the metric, Eq. (2.123), of our space. Using the variation of ds², 2dsdds = dq dq¹8gij +8ijdqisq³ + gijdqisqi (2.14 in Eq. (2.145) yields 1 dqi d [d [dq' dq¹ -8gij + gij ds ds dq¹ d -8q³ + Bij ds ds -8q' ds = 0, (2.14 ds ds where ds measures the length on the geodesic. Expressing the variations agij 8gij = -8q M aqk (əksij)8qk in terms of the independent variations 8qk, shifting their derivatives in the oth two terms of Eq. (2.147) upon integrating by parts and renaming dummy sui mation indices, we obtain 1 d dqi dq¹ [da da x8- (82 +84)] sq* ds = -Əksij ds ds gik gkj 0. (2.14 ds ds ds The integrand of Eq. (2.148), set equal to zero, is the geodesic equation. It is t Euler equation of our variational problem. Upon expanding dgik = (@jgik) dq dgkj ds ds dqi ds (di gkj) (2.14 ds along the geodesic we find 1 dq¹ dq¹ 2 ds ds k8ijdj8ik-digkj) - - Bik d²q¹ = 0. ds² (2.15 Multiplying Eq. (2.150) with gk and using Eq. (2.125), we find the geode: equation d²q² dq dq 1 + ds2 ds ds 24 gkl (di 8kj+j8ik-aksij) = 0, (2.15 where the coefficient of the velocities is the Christoffel symbol I; of Eq. (2.13 2.10 Non-Cartesian Tensors 157 Geodesics are curves that are independent of the choice of coordinates. They can be drawn through any point in space in various directions. Since the length ds measured along the geodesic is a scalar, the velocities dq' /ds form a contravariant vector. Hence Vidqk/ds is a well-defined scalar on any geodesic, which we can differentiate in order to define the covariant derivative of any covariant vector Vk. Using Eq. (2.151) we obtain from the scalar d dqk dVk dqk & (v₁ do²) = dve det + v, d²gt Vk ds ds ds ds ƏVk dq dqk Verk dq dqi (2.152) aq ds ds ds ds - da da (av - riv). ds ds When the quotient theorem is applied to Eq. (2.152) it tells us that avk (2.153) aqi - TV is a covariant tensor that defines the covariant derivative of Vk, consistent with Eq. (2.144). Similarly higher-order tensors may be derived. The second term in Eq. (2.153) defines the parallel transport or displacement, 8Vk = Iki Vidq¹, (2.154) of the covariant vector V from the point with coordinates q to q +8q'. The par- allel transport SUk of a contravariant vector Uk may be found from the invariance of the scalar product Uk Vk under parallel transport, 8(Uk Vk) = 8Uk Vk + Uk8Vk = 0, (2.155) in conjunction with the quotient theorem. In summary, when we shift a vector to a neighboring point, parallel transport prevents it from sticking out of our space. This can be clearly seen on the surface of a sphere in spherical geometry, where a tangent vector is supposed to remain a tangent upon translating it along some path on the sphere. This explains why the covariant derivative of a vector or tensor is naturally defined by translating it along a geodesic in the desired direction.