I am trying to understand FRW universe. To do so I am following the link below: http://www.phys.washington.edu/users/dbkaplan/555/lecture_04.pdf I am confused at equation 74. I got R00 but for Rij part I am always getting a[itex]\ddot{a}[/itex]. I am trying to solve it for k =0. Can some please expand the Rij calculation from basics?
Did you do the calculation by hand ? It's hard to point out an error without seeing the working. You could list the Christoffel symbols you got.
I did by hand and the significant Christoffel symbols here are: [itex]\Gamma^{t}_{xx}[/itex] = a[itex]\ddot{a}[/itex] [itex]\Gamma^{x}_{tx}[/itex] = [itex]\frac{\dot{a}}{a}[/itex] I am following Sean's note too. I don't know when I try to calculate R[itex]_{xx}[/itex] i.e. R[itex]^{t}_{xtx}[/itex]. I am not getting the correct answer
Are you using this ? [tex] R^r_{mqs}=\Gamma ^{r}_{mq,s}-\Gamma ^{r}_{ms,q}+\Gamma ^{r}_{ns}\Gamma ^{n}_{mq}-\Gamma ^{r}_{nq}\Gamma ^{n}_{ms} [/tex]
OK, so you've got [tex] R^t_{xtx}=\Gamma ^{t}_{xt,x}-\Gamma ^{t}_{xx,t}+\Gamma ^{t}_{nx}\Gamma ^{n}_{xt}-\Gamma ^{t}_{nt}\Gamma ^{n}_{xx} [/tex] Are you doing the summation over n ?
My formula is actually: [tex] R^t_{xtx}=\Gamma ^{t}_{xx,t}-\Gamma ^{t}_{xt,x}+\Gamma ^{n}_{xx}\Gamma ^{t}_{nt}-\Gamma ^{n}_{xt}\Gamma ^{t}_{nx} [/tex]
Are you calculating the Ricci in a coordinate basis, or in an orthonormal frame? And it'd be helpful to get the line element (for the former) or the set of basis vectors (for the later) that you're using - IIRC there are a couple of (equivalent) ways of writing the metric for k=0.
Here's my line element: ds^{2} = -dt^{2} + a^{2}(t) (dx^{2} + dy^{2} + dz^{2}) Can someone please show couple of steps here?
[itex] R^t_{rtr} = \partial_t \Gamma^t_{rr} - \partial_r \Gamma^t_{rt} + \Gamma^t_{t \lambda} \Gamma^{\lambda}_{rr} - \Gamma^t_{r \lambda} \Gamma^{\lambda}_{tr} = \frac{\dot{a}^2+ a \ddot{a}}{1-kr^2} - 0 + 0 - \Gamma^t_{r r} \Gamma^{r}_{tr} = \frac{\dot{a}^2+ a \ddot{a}}{1-kr^2} - \frac{\dot{a}^2}{1-kr^2} = \frac{a \ddot{a}}{1-kr^2} [/itex] ... hopefully that's right, it's hard to get all the terms when doing calculations in latex... atleast a quick check gave the correct value for [itex] R_{rr}[/itex] so maybe it's right.
if k = 0 then you get only a[itex]\ddot{a}[/itex] But according to the notes, we should get a[itex]\ddot{a}[/itex] + 2[itex]\dot{a}[/itex]^{2}
I think you're talking about the Ricci tensor: [itex] R_{rr} = R^{\mu}_{r \mu r} = \frac{a \ddot{a} + 2 \dot{a}^2 + 2k}{1-kr^2} [/itex]
isn't it same as as [itex] R_{rr} = R^{t}_{r t r}[/itex] I am confused here. I am talking about (74) i.e R_{ij} from: http://www.phys.washington.edu/users/dbkaplan/555/lecture_04.pdf
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]