From: Boudewijn Moonen Subject: Re: riemannian space Date: Tue, 02 Nov 1999 16:30:08 +0100 Newsgroups: sci.math To: Dany Denduyver Keywords: null geodesics, Koszul formula Dany Denduyver wrote: > > Hello to anyone who read this. > I'm having a hard time with the following problems: > > 1)In a space V(n) the metric tensor is a(mn). Show that the null > geodesics are unchanged if the metric tensor is changed to b(mn), > where b(mn) = ya(mn) with y being a function of the coordinates. > (n) and (mn) are subscript symbols. > > 2) Find the null geodesics of 4-space with line element: > ds^2= e y(dx^2 + dy^2 + dz^2 - dt^2) > > where y is an arbitrary function of x, y, z, and t.e = epsilon symbol > y = gamma symbol > > Thanks in advance. Let M be a pseudoriemannian space with metric g. Your Question 1 refers to a change g --> yg =: h, where y is a (nowhere vanishing) function on M. Now let D be the Levi-Civita-connection of g. We want to compute the Levi-Civita-connection D' of h. The connection D is uniquely determined by the following formula, attributed to Koszul: g(2D_X Y,Z) = Xg(Y,Z) - Yg(Z,X) - Zg(X,Y) - - g(X,[Y,Z]) + g(Y,[Z,X]) + g(Z,[X,Y]) for all vector fields X, Y and Z on M. Writing down the analogous formula for D' and using h = yg, there comes h(2D'_X Y,Z) = (1) yg(2D_X Y,Z) +(Xy)g(Y,Z) - (Yy)g(Z,X) - (Zy)g(X,Y) . Let c be a null geodesic for g, i.e. it satisfies the equations D_{dc/dt} dc/dt = 0 , g(dc/dt,dc/dt) = 0 . (2) Putting X := Y := dc/dt in (1) (this needs some justification, since in (1) X and Y are vector fields on M, whereas dc/dt is a vector field along c), we get h(2D'_{dc/dt} dc/dt,Z) = 0 because of (2) (remember X = Y) for all Z. This implies D'_{dc/dt} dc/dt = 0. Since h(dc/dt,dc/dt) = 0 anyway, this shows c is also a null geodesic for h. Your Question 2 should have a direct answer because of Question 1. Regards, -- Boudewijn Moonen Institut fuer Photogrammetrie der Universitaet Bonn Nussallee 15 D-53115 Bonn GERMANY e-mail: Boudewijn.Moonen@ipb.uni-bonn.de Tel.: GERMANY +49-228-732910 Fax.: GERMANY +49-228-732712 ============================================================================== [Minor typo in original post corrected above --djr] ==============================================================================