# Singularities and Kasner solution

## Homework Statement

Investigate the possible behaviour of the singularity as $$t \rightarrow 0$$ in the Kasner solution.

## Homework Equations

The metric for the Kasner solution is given by

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

## The Attempt at a Solution

I have no clue...

Dick
Homework Helper
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?

I don't get it.

Dick
Homework Helper
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...

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

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

Dick
Homework Helper
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.

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.

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?

Dick
Homework Helper
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.

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..?

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

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

Yeah that I got..

Dick
Homework Helper
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.

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

Dick
Homework Helper
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?

Well, I will always get

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

so for $$t \rightarrow 0$$

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

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

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

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

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

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

Dick
Homework Helper
You don't have to solve a DE for the scale factors. You already did that. a_i=t^p_i. Where the p_i's are what I sent you a few posts back. Note there are three different scale factors - the universe is anisotropic. And, hint, one of the scale factors is not like the others.

I don't get it. What about $$t \rightarrow 0$$?

And how do I get $$a_i = t^{p_i}$$?

Dick
Homework Helper
And how do I get $$a_i = t^{p_i}$$?

Huh? It's a Kasner solution. The scale factors are t^p1, t^p2 and t^p3. I sent you a sample set of p's. What's the behavior of each as t->0?

Well then every $$a_i \rightarrow 0$$ as for the matter dominated universe.

Dick
Homework Helper
Are all of the p's positive?

No the third one is not. So this one behaves like $$\frac{1}{t}$$ and thus goes to infinity. Right?

Dick
Well wouldn't $$\vec{p} = [1,0,0]$$ be a solution?