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
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[Minor typo in original post corrected above --djr]
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