How Does the Singularity Behave as t Approaches 0 in the Kasner Solution?

[Logarythmic]

I get

$$p_1 = -\frac{p_3}{2} \pm \frac{1}{2} \sqrt{-3p_3^2+2p_3+1} + \frac{1}{2}$$

and

$$p_2 = \frac{1}{2} \left(-p_3 \pm \sqrt{-3p_3^2+3p_3+1} + 1 \right)$$

but what does this tell me? They cannot have the same sign but then what?

 

[Dick]

Unless I can use $$p_1p_2+p_2p_3+p_3p_1=0$$ aswell?

You are working way too hard. What does this say about the possibility that all of the p's are positive or all are negative?

 

[Logarythmic]

They cannot all be positive nor negative, but one positive and two negative or two positive and one negative. Is that right?

 

[Dick]

Almost. Except you can't have two negative p's either.
$$p_1+p_2+p_3=1$$. What would this tell you about p3 if p1 and p2 are negative?

 

[Logarythmic]

Then $$p_3 > 1$$?

 

[Dick]

You're catching on. But the sum of the squares should be one too!

 

[Logarythmic]

There we are. So it's either [1,0,0] or [+,+,-] and the behavior is strange. ;)

 

[Dick]

Yes. If you want to see an attempt to use this strange behavior look up Mixmaster cosmologies sometime.

 

[Logarythmic]

Can you briefly tell me something about it?

 

[Dick]

Sorry, better get to work here. Besides, other people tell the story better than I.

 

[Logarythmic]

I'll look it up tomorrow, now I need some sleep. Thanks for all your help.