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Re: computation of two-point objects


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



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