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-----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