# Singularities and Kasner solution

1. Feb 1, 2007

### Logarythmic

1. The problem statement, all variables and given/known data
Investigate the possible behaviour of the singularity as $$t \rightarrow 0$$ in the Kasner solution.

2. Relevant equations
The metric for the Kasner solution is given by

$$ds^2 = c^2dt^2 - X_1^2(t)dx_1^2 - X_2^2(t)dx_2^2 - X_3^2(t)dx_3^2$$

3. The attempt at a solution
I have no clue...

2. Feb 1, 2007

### Dick

Try writing down a more specific form of the Kasner soln. The spatial metric functions can be written as powers of t. What are the conditions on the exponents coming from Einsteins eqns?

3. Feb 1, 2007

### Logarythmic

I don't get it.

4. Feb 1, 2007

### Dick

You can't do anything with the metric as written - it's too general. What ARE X_1,X_2 and X_3 in the Kasner soln. You may have to look it up...

5. Feb 1, 2007

### Logarythmic

$$\frac{\ddot{X}_i}{X_i} - \left(\frac{\dot{X}_i}{X_i}\right)^2 +3\left(\frac{\dot{X}_i}{X_i}\right)\left(\frac{\dot{a}}{a}\right) = \frac{4\pi G}{c^4}\left(\rho - \frac{p}{c^2}\right)$$

in which $$a^3 = X_1X_2X_3$$.

6. Feb 1, 2007

### Dick

That's a start. You shouldn't have a loose index i floating around though. Assuming you can get the correct Einstein equations (and there should be two), Kasner is a vacuum solution, so put rho=p=0. Put X_i=t^p_i. Turn this into equations in the constants p_i. Are you supposed to actually derive Kasner or just describe it's properties? It might be a good idea to just look up the solution to see what you are aiming for.

7. Feb 1, 2007

### Logarythmic

I think I'm just supposed to describe the behaviour of the singularity. I don't like this book, it has too many errors. ;) Why should I use $$X_i = t^{p_i}$$? According to my book this is a perfect fluid model.

8. Feb 1, 2007

### Logarythmic

Ok, I found that this is a particulary simple behaviour. Also

$$\frac{\dot{X}_1 \dot{X}_2}{X_1X_2} + \frac{\dot{X}_2 \dot{X}_3}{X_2X_3} + \frac{\dot{X}_3 \dot{X}_1}{X_3X_1} = \frac{8 \pi G}{c^4}\rho$$

Putting in $$X_i = t^{p_i}$$ and using $$\rho=0$$ gives me

$$0 \propto \frac{1}{t^2}$$

Is this true?

9. Feb 1, 2007

### Dick

It gives you an algebraic condition on the p's that must vanish. What is it? Again there is another Einstein equation. It will give you another algebraic condition.

10. Feb 1, 2007

### Logarythmic

So $$p_1p_2 + p_2p_3 + p_3p_1 = 0$$?
I also got that

$$\frac{\dot{a}}{a} = \frac{1}{3} \left( \frac{\dot{X}_1}{X_1} + \frac{\dot{X}_2}{X_2} + \frac{\dot{X}_3}{X_3} \right)$$

but then I get some a aswell..?

11. Feb 1, 2007

### Dick

That is one alright. Since we are not actually trying to derive this you should find

$$p_1+p_2+p_3=1$$ and
$${p_1}^2+{p_2}^2+{p_3}^2=1$$.

12. Feb 1, 2007

### Logarythmic

Yeah that I got..

13. Feb 1, 2007

### Dick

Good. So there are actually lots of Kasner solutions corresponding to different values of the p's. Find a specific example and describe it's behavior.

14. Feb 1, 2007

### Logarythmic

Well, what I don't understand is HOW to describe it's behaviour.

15. Feb 1, 2007

### Dick

Ok, here's a set of p's.

$$p_1=1/3, p_2=(1+\sqrt{3})/3, p_3=(1-\sqrt{3})/3$$

How would you describe this behavior as t->0?

16. Feb 1, 2007

### Logarythmic

Well, I will always get

$$\frac{\dot{a}}{a} = \frac{1}{3t}$$

so for $$t \rightarrow 0$$

$$\dot{a} \rightarrow \infty$$

17. Feb 1, 2007

### Dick

Sure. But describe qualitatively the behavior of the scale factors themselves (the X_i's).

18. Feb 1, 2007

### Logarythmic

Sorry, I don't know what to say about them.

19. Feb 1, 2007

### Dick

That's ok. In a matter dominated universe (a=t^(2/3), I think) a->0 as t->0. Is that true here?

20. Feb 1, 2007

### Logarythmic

Humm, if I solve

$$\frac{\dot{a}}{a} = \frac{1}{3t}$$

I get

$$a = t^{1/3} \rightarrow 0$$

for $$t \rightarrow 0$$.

But if I use $$a_0$$ instead, I get

$$a = a_0 \frac{1}{3} ln(t) \rightarrow - \infty$$.