Diffeomorphisms

latentcorpse

Surely from Killing's equation, [latex]\nabla_a \xi_b = - \nabla_b \xi_a[/latex] will be antisymmetric? Do I need to do anything about the two surfaces then?

I have been trying to take the [latex]\nu=0[/latex] component of [latex]T^{\mu \nu}{}_{; \mu}=0[/latex] but it simply will not work! Is this the way I should go about it?

But just from a plot of sinh, it looks like a smiley face (i.e. positive second derivatie), no?


So the particle horizon is the boundary of the past light cone?

fzero

OK that'll do. As for the two surfaces, you need to figure out what your surface [tex]\Omega[/tex] is. Is [tex]\partial \Omega[/tex] necessarily connected?

It looks like you need to compute a trace of one of the Christoffel symbols.

[tex]\ddot{a}[/tex] is the 2nd derivative w.r.t. coordinate time, not [tex]\eta[/tex]. I can't see the figures, so I don't know if they should be helpful to you.

The figure is probably clearer than I would put into words. The horizon is a boundary on spacetime at our present expansion parameter. To determine the points that make up the horizon, you need to follow the worldlines of the particles that crossed our past light cone at some earlier time.

latentcorpse

I thought S is the surface given in the formula for J_S. Then [latex]\partial \Omega=S[/latex] and [latex]\Omega[/latex] would just be the enclosed 3-sphere?


Well from the figure, it appears to me that the particle horizon actually is the boundary of the past light cone. But if this is so, why don't they just define it as this? This is why I'm having doubts about my interpretation. But I can't think of an example where they wouldn't be equal.

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

Thanks.

fzero

If you take the calculation another line you should find something like [tex]\dot{\rho} + (\rho +p){\Gamma^0}_{00}=0[/tex].

latentcorpse

How did you manage to get any [latex]p[/latex] terms surviving? All of mine cancel.