| Topic: |
Science > Physics |
| User: |
"Barrow" |
| Date: |
29 Apr 2007 11:30:22 AM |
| Object: |
gravitational field equation problem |
Dear all,
It is known that the Einstein field equations are obtained by the
variation with respect to the metric components. For example, the
G_{00} = xT_{00} is obtained from the calculus of variation with
respect to g_{00}, where x is some constant.
I wanna do the variation by myself, i.e. not to use G_{ab} = T_{ab}.
The problem is, I do the variation described in the above paragraph
and I can't get the correct field equations.
For example, in the Bianchi models, specifically, type two:
ds^2 = -dt^2 + a_1(t)^2 (dy^2+y^2dz^2) + a_2(t)^2 (dx + (y^2/2) dz)^2
I do the calculus of variation of R(scalar curvature) with respect to
a_1(t). What I get is the G_{yy} = 0 field equation! Moreover, I label
the a_1(t) in g_{zz} component to be a_1*, i.e. I let the a_1 in front
of dy^2 to be different from the one in front of dz^2, and then do the
variation with respect to a_1(t), this is exactly followed the
principle of the first paragraph and I should get the correct field
equation, but what strange is, I can't get the correct field equation!
Why I got the correct equation by direct variation to scale factors,
and I got the wrong equation by variation to g_{yy} ???
Thanks for your help! Sincerely Barrow
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| User: "" |
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| Title: Re: gravitational field equation problem |
10 May 2007 05:39:58 PM |
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Barrow <GRseminar@gmail.com> wrote:
Dear all,
It is known that the Einstein field equations are obtained by the
variation with respect to the metric components. For example, the
G_{00} = xT_{00} is obtained from the calculus of variation with
respect to g_{00}, where x is some constant.
I wanna do the variation by myself, i.e. not to use G_{ab} = T_{ab}.
The problem is, I do the variation described in the above paragraph
and I can't get the correct field equations.
I don't have time to look at the details of your example, but in
general, if you restrict the metric to a special form before you
do the variation, you won't get the right answer. The reason is
that the field equations come from extremizing the action under
*all* variations, and by restricting the metric, you are limiting
yourself to only a restricted class of variations.
This can lead to several bad outcomes:
1. You won't get all the equations, because you're not looking at
all variations.
2. You won't get all the solutions, because solutions might not be
of the special form you are assuming.
3. You might get wrong "solutions," because the special form you've
assumed means that you're not really varying the right quantities
to get the field equations.
There are special cases in which you can get away with assuming a
particular form of the metric before varying, but these are not
typical -- they usually involve an assumption of an exact symmetry
of the solution, and then require that you start with the most
general form of the metric consistent with the symmetry.
Steve Carlip
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