Ricci Tensor of FRW


by psimeson
Tags: ricci, tensor
psimeson
psimeson is offline
#19
Mar31-12, 02:47 PM
P: 19
Quote Quote by clamtrox View Post
No, you sum over ALL the indices!
i.e t= μ and t = [itex]\nu[/itex]

How the does Ricci tensor equation looks like then?

[itex] R_{rr} = R^{\mu}_{r\mu r} + R^{\nu}_{r\nu r} [/itex]

Since [itex] R^{\mu}_{r\mu r} = a\ddot{a}[/itex]

and [itex] R^{\nu}_{r\nu r} = a\ddot{a}[/itex]

[itex] R_{rr} = 2 a\ddot{a}[/itex]

that's not correct. I don't know I am getting confused. I am not seeing how we get [itex]\dot{a}[/itex]2
clamtrox
clamtrox is offline
#20
Mar31-12, 03:07 PM
P: 937
Quote Quote by psimeson View Post
i.e t= μ and t = [itex]\nu[/itex]

How the does Ricci tensor equation looks like then?

[itex] R_{rr} = R^{\mu}_{r\mu r} + R^{\nu}_{r\nu r} [/itex]

Since [itex] R^{\mu}_{r\mu r} = a\ddot{a}[/itex]

and [itex] R^{\nu}_{r\nu r} = a\ddot{a}[/itex]

[itex] R_{rr} = 2 a\ddot{a}[/itex]

that's not correct. I don't know I am getting confused. I am not seeing how we get [itex]\dot{a}[/itex]2
no no no, Ricci tensor is the trace of Riemann tensor, so [itex] R^{\mu}_{r\mu r} = R^{t}_{rtr} +R^{r}_{rrr} + R^{\theta}_{r \theta r} + R^{\phi}_{r \phi r} [/itex]
psimeson
psimeson is offline
#21
Mar31-12, 03:19 PM
P: 19
Quote Quote by clamtrox View Post
no no no, Ricci tensor is the trace of Riemann tensor, so [itex] R^{\mu}_{r\mu r} = R^{t}_{rtr} +R^{r}_{rrr} + R^{\theta}_{r \theta r} + R^{\phi}_{r \phi r} [/itex]
So that means, for x, y and z, I have:

[itex]R^{\mu}_{x\mu x} = R^{t}_{xtx} +R^{x}_{xxx} + R^{y}_{x y x} + R^{z}_{xzx}[/itex] right?

But the second term is zero and 3rd and 4th term does not have time in it so I will not "a" contribution from them
psimeson
psimeson is offline
#22
Mar31-12, 03:21 PM
P: 19
I think I got it.. Thanks a lot


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