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Topic: computation of two-point objects
Replies: 2   Last Post: Jun 19, 2013 1:21 AM

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 Roland Franzius Posts: 569 Registered: 12/7/04
Re: computation of two-point objects
Posted: Jun 19, 2013 1:21 AM

Am 15.06.2013 10:18, schrieb Mark Roberts:
> hello,
> I have been stuck for decades trying to calculate the world function for
>
> ds^2=-(1+2\sigma)dv^2+2dvdr+r(r-2\sigma v)(d\theta^2+\sin(\theta)^2d\phi^2)
> \phi=\n(1-2\sigma v/r)/2
> R_{ab}=2\phi_a\phi_b
>
> one gets elliptic functions if one try direct method, The trouble with
> approximations is that it is hard to tell if they converge.....
>
> bye,

Did you try Zimmerman/Olness chapter 10 methods in

http://library.wolfram.com/infocenter/Books/4539

The other simple way is to use the geometrical Lagrangian method

Define the Lagrangian

Lagrangian =1/2 ds2 /. dphi-> D[(1-2\sigma v/r)/2, v] dv + D[(1-2\sigma
v/r)/2, r] dr

and
momenta = {Pv -> D[Lagrangian,dv],
Pr-> D[Lagrangian,dr],
Ptheta -> D[L,dtheta} }

Then prepare all variables with a time argument [t] and read the table
of Christoffel symbols off from the Euler-Lagrange equations for geodesics

D[pv/.momenta, t] -D[L,dv[t]] == 0

and calulate Riemann and Ricci.

--

Roland Franzius

Date Subject Author
6/15/13 Mark Roberts
6/17/13 Alexei Boulbitch
6/19/13 Roland Franzius