Question

You are given that a 4-dimensional pseudo-Riemannian space-time has the interval ds2dudvf (u) dx2 g?(u) dy*, (u, v, x, y) in terms of the coordinates x^ = (i) Use the standard variational principle 2 ds dt = 0 dt ti to find the r-equation governing the geodesic, with parameter t, between given points t and t2 (ii) Deduce from the x-geodesic equation obtained in (i) that f' T.. T. =. ur f where a prime denotes differentiation with respect to u (iii) Show that other non-zero Christoffel symbols are I yy = g (iv) Show that RD RT Ry uyu = 0, R°aßy= 0. uru uvu

Answer 1

du dr + Then 2. ds 2 dt where ffenentiadion sdet denotes ime w.r. t action The is dt L dt t i kara tha Lagramgiam is 2 (Eder-Lagrongeson euation The - dt +( 23 ) tha geodesie equaton with Compasing Se X

Similandy reads eauafion dt 23 aeodesic equadion uoith Cempasing this abtain we yu eqation eation tha om Finaby are ne 2 23- ctt 2- - 23a 293 x. Or,

Cenn paning eqation enaton wite Hhe endesic T Cknistoffembds inal 1Y 13 = 2a IV - GD 2. - - 2 Luu 2 + 1L 2

1) R uVu - ' R. + eaton gince