Wald General Relativity: On the homogenous cosmology, Page 178

[qinglong.1397]

Hi, everybody. I have some problem with Wald's statement shown in the picture. This is from the last paragraph in Page 178.

He claimed that there are only solutions with two of the $p_{\alpha}$ positive and one negative. But it's easy to find out that if two of the $p_{\alpha}$ are negative while the third positive, there is no contradiction.

Can you guys help me with this? Why should all the solutions have two positive $p_{\alpha}$ and one negative? Thank you

(The picture is from http://books.google.com/books?id=9S...ce=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)

 

[WannabeNewton]

Well using (7.2.58) and (7.2.60) we have $p_2^2 + p_1^2 - p_1 - p_2 + p_1p_2 = 0$. Now if $p_3 < 0$ then $p_2 > 1 - p_1$. Plot these two and you will find that both $p_1,p_2 > 0$. If $p_3 >0$ then $1 - p_1> p_2$; plotting these two again you will find that $p_1 > 0,p_2 < 0$ or vice-versa. Finally if $p_3 = 0$ then either $p_1 = 1$ and $p_2 = 0$ or vice-versa which are just the trivial solutions.

 

[qinglong.1397]

Well using (7.2.58) and (7.2.60) we have $p_2^2 + p_1^2 - p_1 - p_2 + p_1p_2 = 0$. Now if $p_3 < 0$ then $p_2 > 1 - p_1$. Plot these two and you will find that both $p_1,p_2 > 0$. If $p_3 >0$ then $1 - p_1> p_2$; plotting these two again you will find that $p_1 > 0,p_2 < 0$ or vice-versa. Finally if $p_3 = 0$ then either $p_1 = 1$ and $p_2 = 0$ or vice-versa which are just the trivial solutions.

Thanks! Never thought of this. Great!

 

[WannabeNewton]

No problem! Make sure you do the problems at the end of that chapter; some of them are really fun (problems 7.1,7.4, and 7.5 in particular).

 

[qinglong.1397]

No problem! Make sure you do the problems at the end of that chapter; some of them are really fun (problems 7.1,7.4, and 7.5 in particular).

Sure. I'll try to solve all of them before the end of the next week.

 

[WannabeNewton]

Awesome, have fun with that!

 

[qinglong.1397]

Awesome, have fun with that!

Hi WannabeNewton, I know it's been late, but I haven't been able to figure out how to solve the problem 7.4. Can you help me out? Thank you!