# Comoving fluids - radiation and matter

1. May 21, 2014

### befj0001

$$0 = ({\rho}_m + P_m)u^{m}_iu^{m}_j + \frac{4}{3}{\rho}_ru^{r}_iu^{r}_j$$

where i,j = 1,2,3 and different. That is the off-diagonal elements of the tresstensor for matter fluid and radiation fluid.

The energy conditions imply that

$\rho_m + p_m > 0$ and $\rho_r > 0$

This implies that

$$u^{m}_1=u^{m}_2=u^{m}_3=u^{r}_1=u^{r}_2=u^{r}_3=0$$

But how do one conclude the last equality?

edit: Tried to write in latex-code, but it doesn't seem to work (I don't know how to do).

Last edited: May 21, 2014
2. May 21, 2014

### Bill_K

3. May 21, 2014

### befj0001

4. May 21, 2014

### George Jones

Staff Emeritus
Put $$both at the start and end of stand-alone latex math; put ## both at the start and end of stand-alone latex math. I have edited your original post. Do you mean$$u^{m}_1=u^{m}_2=u^{m}_3=u^{r}_1=u^{r}_2=u^{r}_3 = 0?

5. May 21, 2014

### befj0001

Yes I do. My mistake.

I also wonder, since this imply that matter and radiation are both comoving. It means that
L.R.S (locally rotational symmetric) space-times does not admit two-fluid models where one of the perfect models is tilted.

What does it mean for a fluid to be tilted? What does "locally rotational symmetric means" in the context of a cosmological model?

6. May 21, 2014

### George Jones

Staff Emeritus
Suppose spacetime is foliated into spatially homogeneous spatial sections. A fluid is tilted if its flow lines are not orthogonal to the spatial sections.

7. May 21, 2014

### befj0001

So it just means that it is not stationary in space? It changes coordinates in the x,y,z direction?

But I still don't understand the reasoning from the statement:

"If matter and radiation are both comoving, it means that
L.R.S (locally rotational symmetric) space-times does not admit two-fluid models where one of the perfect models is tilted."

How can matter and radiation be comoving in the first place? Radiation moves with the speed of light.

8. May 21, 2014

### George Jones

Staff Emeritus
So I suppose u^m is timelike and u^r is lightlike. Gotta run for my bus.

9. May 22, 2014

### George Jones

Staff Emeritus
From what article or book have you taken this?

10. May 25, 2014