Tham khảo tài liệu 'intro to differential geometry and general relativity - s. warner episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 1 2Ộ 0 0 0 g 0 1 2Ộ 0 0 0 0 1 2Ộ L 0 0 0 -1 2Ộ J I or _ 9 _ . _ 9 _ 9 _ 9. _ . _ 9 J 2 z1 . -LX z J- .2 _7_ 2 J2x 1 s _ J 2 ds 1 20 dx dy dz - 1-20 dt . Notes 1. We are not in an inertial frame modulo scaling since Ộ need not be constant but we are in a frame that is almost inertial. 2. The metric g is obtained from the Minkowski g by adding a small multiple of the identity matrix. We shall see that such a metric does arise to first order of approximation as a consequence of Einstein s field equations. Now we would like to examine the behavior of a particle falling freely under the influence of this metric. What do the timelike geodesics look like Let us assume we have a particle falling freely with 4-momentum P m0U where U is its 4-velocity dxl dx. The paramaterized path x r must satisfy the geodesic equation by A2. Definition gives this as d2x _ i dxr dxs 0 dx2 rsdx dx . Multiplying both sides by m02 gives d2 mữxi ị d moxr d moxs m drF rrs 0-or m - rrisPrPs 0 since Pi d m0xi dx dx where by the ordinary chain rule note that we are not taking covariant derivatives here. that is dPl dx is not a vector see Section 7 on covariant differentiation dP P dx_ dx k dx so that Pi dm0xk TrisPrPs 0 101 or P p I j p p 0 . I Now let us do some estimation for slowly-moving particles v 1 the speed of light where we work in a frame where g has the given form. First since the frame is almost inertial Lorentz we are close to being in SR so that p m0U m0 v1 v2 v3 1 we are taking c lhere 0 0 0 m0 since v 1 in other words the frame is almost comoving Thus I reduces to p .p r 4 m02 0 . II Let us now look at the spatial coordinates i 1 2 3. By definition 4 4 2 g g4j 4 gj4 4 - g44j - We now evaluate this at a specific coordinate i 1 2 or 3 where we use the definition of the metric g recalling that g g -1 and obtain 1 1 20 -1 0 0 - 20 2 1-20 -20 i -0 . Why don t we work in an inertial frame the frame of the particle Well in an inertial frame we adjust the coordinates to make g diag 1 1