Tham khảo tài liệu 'intro to differential geometry and general relativity - s. warner episode 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | gnn d dxn d dxn 2 . 21. 2 2 r sin x sin x . sin2 x 1 gij 0 if i j so that 2 r 0 0 .0 1 0 r2sin2x1 0 .0 g 0 0 r2sin2x1 sin2 x . 0 0 0 0 0 .o r2sin2x1 sin2 x2 . . sin2 x 1 _ f Diagonalizing the Metric Let G be the matrix of g in some local coordinate system evaluated at some pointp on a Riemannian manifold. Since G is symmetric it follows from linear algebra that there is an invertible matrix P Pj such that T r 1 0 0 1 0 0 0 1 0 PGP1 0 0 0 . at the point p. Let us call the sequence 1 1 . . . 1 the signature of the metric atp. Thus in particular a Minkowski metric has signature 1 1 1 -1 . If we now define new coordinates x by x PjÁ so that we are using the inverse of P for this then ix X Pji and so _ ox dx gij x gat I PiagabPjb Piagab bj PGPT ij showing that at the point p r 1 0 0 01 0 1 0 0 g L 0 0 0 1J Thus in the eyes of the metric the unit basis vectors Cị d dx are orthogonal that is 41 ei ej Note The non-degeneracy condition in Definition is equivalent to the requirement that the locally defined quantities g det gij are nowhere zero. Here are some things we can do with a Riemannian manifold. Definition If X is a contravariant vector field on M then define the square norm norm of X by IIXII2 X X jX Note that IIXII2 may be negative. If IIXII2 0 we call X timelike if IIXII2 0 we call X spacelike and if IIXII2 0 we call X null. If X is not spacelike then we can define IIXII n ĨlXĨ2 VgjX Xj . In the exercise set you will show that null need not imply zero. Note Since X X is a scalar field so is IIXII is a scalar field if it exists and satisfies II0XII I0I-IIXII for every contravariant vector field X and every scalar field Ộ. The expected inequality IIX YII IIXII IIYII need not hold. See the exercises. Arc Length One of the things we can do with a metric is the following. A path C given by x x is non-null if Ildx d II2 0. It follows that Ildx7d ll2 is either always positive spacelike or negative timelike . Definition If C is a non-null path in M then define
