| Topic: |
Science > Physics |
| User: |
"mathlover" |
| Date: |
12 Nov 2006 02:39:45 AM |
| Object: |
constraints on the Riemann curvature tensor |
Dear all,
It's known that R_{abcd} = -R_{bacd} = -R_{abdc} = R_{cdab},
moreover, R_{abcd} + R_{acdb} + R_{adbc} = 0. So, one can calculate the
number of independent components of R_{abcd} to be 20.
Since, R^a_{bcd} = g^{ea}R_{ebcd}, the number of the independent
components of the Riemann curvature tensor (R^a_{bcd}) should be 20
too. So is R^{ab}_{cd}.
I wanna to prove the results of the second paragraph. I can prove
R^a_{bcd} = -R^a_{bdc}. As for R^a_{bcd} = -R^b_{acd}, my proof is that
to show R^a_{abc} = 0, then the antisymmetry of the first two indices
of the Riemann curvature tensor is established(am I right?). But I have
trouble in proving R^a_{bcd} = R^c_{dab}. Does the constrant hold? But
if it didn't hold, then there should be some other constraints on the
Riemann curvature tensor, since its independent components should be
20.
I can prove R^{ab}_{cd} = -R^{ba}_{cd} = -R^{ab}_{dc}. But
R^{ab}_{cd} and R^{cd}_{ab} seem not equal to each other! Thus, what's
the constraint corresponding to R_{abcd} = R_{cdab} in the case of two
upper indices and two lower indices? It must exist since the number of
independent components is only 20.
Thanks for your help in advance, this is helpful in my self-studying.
Thanks!!!
Sincerely
.
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| User: "Jim Black" |
|
| Title: Re: constraints on the Riemann curvature tensor |
12 Nov 2006 10:46:01 PM |
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mathlover wrote:
Dear all,
It's known that R_{abcd} = -R_{bacd} = -R_{abdc} = R_{cdab},
moreover, R_{abcd} + R_{acdb} + R_{adbc} = 0. So, one can calculate the
number of independent components of R_{abcd} to be 20.
Since, R^a_{bcd} = g^{ea}R_{ebcd}, the number of the independent
components of the Riemann curvature tensor (R^a_{bcd}) should be 20
too. So is R^{ab}_{cd}.
Bear in mind that there is only one rank-four Riemman curvature tensor.
R_{abcd}, R^a_{bcd}, and R^{ab}_{cd} are different ways of breaking up
the tensor into components.
I wanna to prove the results of the second paragraph.
Your second paragraph already contains half of a proof. Since only 20
of the R_{abcd} are independent, with the other 236 expressable as
linear combinations of those 20, and since each of the R^a_{bcd} can be
written as a linear combination of the R_{abcd}, specifically
R^a_{bcd} = g^{ea} R_{ebcd},
then at most 20 of the R^a_{bcd} can be independent. You can show that
the number is exactly 20 by reversing the argument using
R_{abcd} = g_{ea} R^e_{bcd}.
It looks like you are trying to find explicit expressions for each
R^a_{bcd} in terms of the 20 independent R^a_{bcd}. I suspect these
expressions could be uglier than you expect, and I'm fairly sure
they'll have to involve the components g_ab of the metric tensor in the
coordinate system you're working in.
I can prove
R^a_{bcd} = -R^a_{bdc}. As for R^a_{bcd} = -R^b_{acd},
I don't believe that
R^a_{bcd} = -R^b_{acd}
is correct. In fact, I don't think it's even invariant under
coordinate transformations, because you've used the index a as an upper
index on the left and as a lower index on the right, and vice versa for
the index b. I think that if an index is raised on one side of an
equation, it has to be raised on the other side, and vice versa, or the
equation may not be invariant.
It is, however, true that
R^a_{bcd} = - g^{ae} g_{bf} R^f_{ecd}.
my proof is that
to show R^a_{abc} = 0, then the antisymmetry of the first two indices
of the Riemann curvature tensor is established(am I right?).
Probably not. I don't see how that would follow.
But I have
trouble in proving R^a_{bcd} = R^c_{dab}. Does the constrant hold?
I don't think so, for the same reason I don't think
R^a_{bcd} = -R^b_{acd}
holds.
But
if it didn't hold, then there should be some other constraints on the
Riemann curvature tensor, since its independent components should be
20.
If you're only looking for constraints, then you already have enough
constraints:
g_{ae} R^e_{bcd} = -g_{be} R^b_{acd}
= -g_{ae} R^e_{bdc} = g_{ce} R^c_{dab}
and
R^a_{bcd} + R^a_{cdb} + R^a_{dbc} = 0.
The only task remaining would be to show that the constraints are
independent.
.
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