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.