Proving Homogeneous & Isotropic FRW Universe Energy-Momentum Tensor

[Kamikaze_951]

Hi everyone,

It's not a real homework problem, but something I am trying to do that I haven't found in the literature. I am still stating the problem as if it was a homework

Homework Statement

Consider a FRW Universe. That is, ℝ x M, where M is a maximally symmetric 3-manifold, with a RW metric: ds[itex]^{2}[/itex] = dt[itex]^{2}[/itex] - R[itex]^{2}[/itex](t) dσ[itex]^{2}[/itex], where

dσ[itex]^{2}[/itex] = dχ[itex]^{2}[/itex] + f(χ)[itex]^{2}[/itex](dθ[itex]^{2}[/itex] + sin[itex]^{2}[/itex]θd[itex]\varphi^2[/itex])

Consider also a system that is homogeneous and isotropic in space.

Prove that :
1) T[itex]^{t}_{r = sinχ}[/itex] = T[itex]^{t}_{θ}[/itex] = T[itex]^{t}_{\varphi}[/itex] = 0
2) T[itex]^{r}_{θ}[/itex] = T[itex]^{r}_{\varphi}[/itex] = T[itex]^{θ}_{\varphi}[/itex] = 0
3) T[itex]^{r}_{r}[/itex] = T[itex]^{θ}_{θ}[/itex] = T[itex]^{\varphi}_{\varphi}[/itex]

where T is the energy-momentum tensor.

Homework Equations

1) We can use known Killing vectors related to underlying symmetries without having to prove that they indeed are Killing vectors (ex : the 3 Killing vectors related to spherical symmetry).

For instance, spherical symmetry implies that there are 3 Killing vectors R, S, T satisfying :
i) [R,S] = T
ii) [S,T] = R
iii) [T,R] = S

and in polar coordinates (θ,[itex]\varphi[/itex]), these 3 Killing vectors can be written as follow :
R = [itex]\partial_{\phi}[/itex]
S = [itex]\cos\phi\partial_{\theta} - \cot\theta\sin\phi\partial_\phi[/itex]
T = [itex]-\sin\phi\partial_{\theta} - \cot\theta\cos\phi\partial_\phi[/itex]

We can also use the Christoffel symbols for the RW metric (that can be found in most GR textbooks).

Finally, the equation that defines a conserved charge :
K is a Killing vector implies [itex]T_{\mu\nu}K^\nu = P_\mu[/itex] and
[itex]\nabla_\mu P^\mu = \nabla_\mu( T^{\mu\nu}K_{\nu}) = 0 [/itex]

The Attempt at a Solution

I considered only R for the moment (if I succeed for the simplest Killing vector, I should succeed for the others). The components of R are :[itex]R^{\nu} = \delta^{\nu}_\phi[/itex].

By plugging this into [itex]\nabla_\mu T^{\mu\nu}K_{\nu} = 0[/itex], I got that
[itex]\nabla_\mu T^{\mu\phi} = 0[/itex].

By direct calculation of the divergence of that vector (using Christoffel symbols), I found :

[itex]0 = \nabla_\mu T^{\mu\phi} = (\partial_t + 3\frac{\dot{R}}{R})T^{t\phi} +
(\partial_\chi + 2\frac{f'(\chi)}{f(\chi)})T^{\chi\phi}
+ (\partial_\theta + \cot\theta)T^{\theta\phi} + \partial_\phi T^{\phi\phi}
[/itex]

This equation should probably imply that some components of T are zero. However, I don't see any way of stating this from that equation. It probably implies (from the problem statement) that [itex]T^{t\phi} = 0[/itex]. Do I have the right methodology to solve this problem? Could someone help me JUST with the R Killing vector (since I will probably be able to do the others when the R one is solved...I just need an example of how to do)?

Thank you a lot for considering my request.

Kami

 

[clamtrox]

I am a bit confused by what you are doing here... For starters, recall that in GR, energy-momentum is always conserved: [itex] \nabla_\mu T^{\mu \nu} = 0 [/itex].

I think it's possible to show that for all maximally symmetric spacetimes, [itex] R_{\mu \nu} \propto g_{\mu \nu} [/itex]. Then all you need to do is to use Einstein equation and you find what you were asked to prove. Of course you can also just use raw power to calculate the components of LHS of [itex] R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 8\pi G T^{\mu \nu}[/itex].

 

[Kamikaze_951]

Hi clamtrox,

I thank you a lot for your reply. It doesn't solve my problem, but it allows me to clarify what I am trying to do.

We can compute [itex]R_{\mu\nu}[/itex] easily once we know that the FRW metric applies. However, it is the energy-momentum tensor that I am looking for (i.e. I have to work the other way around). For a perfect fluid, [itex]T_{\mu\nu} = (\rho + p)U_\mu U_\nu + pg_{\mu\nu}[/itex], where ρ is the energy density, p is the pressure and [itex]U_{\mu}[/itex] is the four-velocity (with its index lowered). We need the assumption of matter being modeled by a perfect fluid (which leads to Universe being spatially homogeneous and isotropic) in order to derive the FRW metric and the Friedmann equations.

What I am trying to prove is that a spatially homogeneous and isotropic system (that determines the FRW universe via the Einstein equations) is a perfect fluid. To do that, I must relate symmetries in the system (example : isotropy) to determine the form of [itex]T_{\mu\nu}[/itex].

This makes me realize that my methodology was wrong, since I implicitly assumed the FRW metric and via the Christoffel connection. In fact, I think that the problem is deeper than I thought : How is the energy-momentum tensor defined and how symmetries affects it?

Thanks a lot!

Kami

 

[voko]

I do not have any requisite literature at hand, but from my recollections, any text that intends to derive FRW relations begins by explaining why the energy-stress tensor must be in the form of a perfect fluid. Which is exactly what you are trying prove, if I read you correctly.

 

[clamtrox]

It of course depends on what you assume...

-Assume maximally symmetric spacetime -> [itex] R_{\mu \nu} \propto g_{\mu \nu} [/itex]
-Assume homogeneity and isotropy -> you can either show that this implies maximal symmetry, or talk away the non-perfect terms in [itex] T_{\mu \nu} [/itex].
-Assume FRW -> you can calculate everything explicitly