Comoving fluids - radiation and matter

[befj0001]

$$\begin{equation}

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

\end{equation}$$

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

$$\begin{equation}

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

\end{equation}$$

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).

 

[Bill_K]

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

It's explained in one of our FAQs: https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

 

[befj0001]

It's explained in one of our FAQs: https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

Ok, I will try that next time, thanks!

 

[George Jones]

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.

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

Do you mean

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

 

[befj0001]

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

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

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?

 

[George Jones]

What does it mean for a fluid to be tilted?

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

 

[befj0001]

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

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.

 

[George Jones]

S"If matter and radiation are both comoving,
Radiation moves with the speed of light.

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

 

[George Jones]

From what article or book have you taken this?

 

[befj0001]

From what article or book have you taken this?

http://www.google.se/url?sa=t&rct=j...=aySpE30FnfpNgLRbY8JWjA&bvm=bv.67720277,d.bGQ