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Help with Diffeomorphisms

by latentcorpse
Tags: diffeomorphisms
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fzero
#55
Apr18-11, 05:39 PM
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Quote Quote by latentcorpse View Post
Also, for the [itex]\nabla_\mu T^{\mu \nu}[/itex] question, can I take [itex]u^\mu=(1,0,0,0)[/itex] as we are assuming a comoving observer. And apparently they have [itex]u^\alpha=\delta^\alpha{}_0[/itex].
However, even though our notes say we can do this for a comoving observer, I find that
[itex]\nabla_\mu T^{\mu \nu}=0[/itex]
[itex]\nabla_\mu ( \rho u^\mu u^\nu ) + \nabla_\mu ( p u^\mu u^\nu) - \nabla_\mu (pg^{\mu \nu})=0[/itex]
Taking [itex]\nu=0[/itex] we get
[itex]\nabla_0 ( \rho u^0) + \nabla_i ( \rho u^i) + \nabla_0 ( pu^0) + \nabla_i ( p u^i) - \nabla_0p[/itex]
[itex]\frac{\partial \rho}{\partial t}=0[/itex]
which is clearly incomplete so something isn't right...

Thanks.
If you take the calculation another line you should find something like [tex]\dot{\rho} + (\rho +p){\Gamma^0}_{00}=0[/tex].
latentcorpse
#56
Apr19-11, 04:20 AM
P: 1,443
Quote Quote by fzero View Post
If you take the calculation another line you should find something like [tex]\dot{\rho} + (\rho +p){\Gamma^0}_{00}=0[/tex].
How did you manage to get any [itex]p[/itex] terms surviving? All of mine cancel.


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