Re: The constrain equations for Einstein equations

[Joel Rauber <Joel_Rauber@SDSTATE.EDU>]

A somewhat recent reference is "Relativity and Scientific Computing",
published by Springer, edited by Hehl, Puntigam, and Ruder.

Particularly the article by Ed Seidel, "Numerical Relativity and Black-Hole
Collisions".  Note the many references in that article as well as other
articles.

Joel Rauber

> -----Original Message-----
> From: phys-l@lists.nau.edu: Forum for Physics Educators
> [mailto:PHYS-L@lists.nau.edu]
> Sent: Thursday, May 18, 2000 10:04 AM
> To: PHYS-L@lists.nau.edu
> Subject: The constrain equations for Einstein equations
>
>
>
> For Einstein equation R_{uv}=0, there are 10 independent
> equations. Due to 4 Bianchi equations, there are only 6
> independent equations left to solve 10 variationals in
> g_{uv}. Four constrain equations must be chosen to fix
> four freedom of Einstein equation.
>
> I have known two sets to constrains equations. One is
>
>                  diff[(g^{1/2}*g^{uv}),x_v]=0              (1)
>
> The other base on the spacetime metric in the form
>
>              ds^2=-(alpha^2 - beta^a*beta_b)  * dt^2
>                                   + 2*beta_a  * dx^a*dt
>                                     + r_{ab}  * dx^a*dx^b  (2)
>
> The four constrains may be subdivided into one Hamiltonian
> constrain equation
>
>                   R + (tr K)^2 - K^{ab}K_{ab}=0            (3)
>
> and three momentum constraint equations
>
>                   D_b[K^{ab}-r^{ab}*trK]=0                 (4)
>
> In these equations K_{ab} is the extrinsic curvature of the slice,
>
>   K_{ab}=-(diff[r_{ab},t] - D_a*beta_b - D_b*beta_a)/(2*alpha)   (5)
>
> I want to know the detail of the second type of the constraint
> equations. Who can tell me the books or references in which I can
> find them?
>
> What I have written may contain syntex errors. I wish some
> friends can correct it and send it to my mail address.
> correct them,
>
> --
> Best regards,
> Qiang Lu
> mailto:luqiang@nankai.edu.cn