Re: gravitational field equation problem
- From: carlip-nospam@xxxxxxxxxxxxxxxxxxx
- Date: Thu, 10 May 2007 22:39:58 +0000 (UTC)
Barrow <GRseminar@xxxxxxxxx> 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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