# What class of metric is this?

1. Nov 22, 2014

### Breo

Hello,

this is the metric I am talking about:

$$ds^2= (dt - A_idx^i)^2 - a^2(t)\delta_{ij}dx^idx^j$$

I never see one like this. How the metric tensor matrix would be?

2. Nov 22, 2014

### Matterwave

How do you think you can obtain the metric tensor from the space-time interval that you gave?

3. Nov 22, 2014

### Breo

I think the matrix would be:
$$g_{\mu\nu} = \left( \begin{array}{ccc} 1 & 2A_1 & 2A_2 & 2A_3 \\ 0 & A_1^2 - a^2 & 0 & 0 \\ 0 & 0 & A_2^2 - a^2 & 0 \\ 0 & 0 & 0 & A_3^2 - a^2 \end{array} \right)$$

4. Nov 22, 2014

### Matterwave

I think you are missing some more off-diagonal terms. The first parenthesis looks like it will also give terms that look like $dx^1 dx^2$ etc. Make sure to foil correctly. :)

Also note that the metric must be a symmetric matrix, and your matrix is definitely not symmetric.

5. Nov 22, 2014

### Breo

Do you mean:

$$(A_i dx^i)^2 = A_i^2 dx^idx^j$$ or $$2 . A_i dt . dx^i \longrightarrow 2A_i dtdx^i + 2A_i dx^i dt$$

So with the latter I get:

$$g_{\mu\nu} = \left( \begin{array}{ccc} 1 & 2A_1 & 2A_2 & 2A_3 \\ 2A_1 & A_1^2 - a^2 & 0 & 0 \\ 2A_2 & 0 & A_2^2 - a^2 & 0 \\ 2A_3 & 0 & 0 & A_3^2 - a^2 \end{array} \right)$$

6. Nov 22, 2014

### Matterwave

Notice that $A_i dx^i=A_1 dx^1+A_2 dx^2+A_3 dx^3$, then you want to take $(dt-A_1 dx^1-A_2 dx^2-A_3 dx^3)^2$ what do you get? You can see that there will be not only terms like $dt dx^1$ etc in there, there will be terms like $dx^1 dx^2$, which are all 0 in your matrix.

7. Nov 23, 2014

### Mentz114

Are you sure ? I think $a^2(t)\delta_{ij}dx^idx^j$ means diagonal spatial elements only.

The metric an FLRW type expanding cosmology with anisotropic matter flow.

8. Nov 23, 2014

### Matterwave

I don't see how you can avoid the off diagonal terms when clearly the squaring of the first term will give you terms like $dx^1 dx^2$. There's a sum inside the square. I'm talking about the term $(dt-A_i dx^i)^2$

9. Nov 23, 2014

### Breo

Mmm!! nice! I just only have a doubt, as I know the metric tensor is symmetric, when I obtain, for example, -2A2A3dx²dx³ should I write also in the ds² formula the symmetric term -2A3A2dx³dx² ? i think it is not necesary, right?

Well the matrix I obtained now is:

$$\left( \begin{array}{ccc} 1 & -2A_1 & -2A_2 & -2A_3 \\ -2A_1 & A_1^2 - a^2 & 2A_1 A_2 & 2A_1 A_3 \\ -2A_2 & 2A_1 A_2 & A_2^2 - a^2 & 2A_2 A_3 \\ -2A_3 & 2A_1 A_3 & 2A_2 A_3 & A_3^2 - a^2 \end{array} \right)$$

10. Nov 23, 2014

### Mentz114

Yes, I missed that sum. Very weird. Otherwise the metric is not unusual.

11. Nov 23, 2014

### Matterwave

To check your answer, try to obtain your original $ds^2=(dt-A_i dx^i)^2-a^2 \delta_{ij} dx^i dx^j$ by using $ds^2=g_{\mu\nu}dx^\mu dx^\nu$. Match the two sides to see if they give you the same expression. :)

12. Nov 23, 2014

### Breo

So I must write explicitly the 16 terms in the ds² expression and the matrix seems right :)

13. Nov 23, 2014

### Matterwave

To check to make sure. But notice that since $g_{\mu\nu}=g_{\nu\mu}$ when you take the sum $g_{\mu\nu}dx^\mu dx^\nu$ you will get terms like $dx^1 dx^2$ and then another term like $dx^2 dx^1$ repeated, with the same factor in front.

What I'm getting at is I think you maybe have a factor of 2 off on some of your off diagonal terms, so you might want to double check.

14. Nov 23, 2014

### Breo

Oh, I did not notice. If my intuition does not fail, the off-diagonal terms in the ds² equation when you obtain something like: -2A2A3dx²dx³ must be splitted in two terms dividing by two? so you would have: -A2A3dx²dx³ - A3A2dx³dx² ?

15. Nov 23, 2014

### Matterwave

Yeah, basically.

16. Nov 23, 2014

### Breo

:D

$$\left( \begin{array}{ccc} 1 & -A_1 & -A_2 & -A_3 \\ -A_1 & A_1^2 - a^2 & A_1 A_2 & A_1 A_3 \\ -A_2 & A_1 A_2 & A_2^2 - a^2 & A_2 A_3 \\ -A_3 & A_1 A_3 & A_2 A_3 & A_3^2 - a^2 \end{array} \right)$$

17. Nov 23, 2014

### Breo

So now in order to define a natural vierbein I must diagonalize this matrix:

$$g_{\mu\nu} = e^{\alpha}_{\mu}\eta_{\alpha\beta}e^{\beta}_{\nu}$$

right?

18. Nov 23, 2014

### Matterwave

You must diagonalize the metric into the form diag(-1,1,1,1) or diag(1,-1,-1,-1) depending on the signature.

19. Nov 23, 2014

### DrGreg

Yes. (In the sense that Matterwave just said.)

But here is a hint. You should find it easier to work with the original form of the metric in post #1, rather than the matrix you have just calculated.

20. Nov 23, 2014

### Breo

Mmm interesting.

I must find an analytical tranform to obtain something like $(\alpha dt^2 +\beta_i (dx^i)²)$ from $(dt - A_i dx^i)^2$... I am wondering how. Maybe second grade equations... roots... ?