What is the Unique Metric Tensor in this Line Element?

[Breo]

I'm not sure what other hints I can give other than just telling you the answer haha. Recall what you did for the spherical coordinates? What did the metric look like there, and how did you find a triad?

But using $dp^2$ ?

 

[Matterwave]

But using $dp^2$ ?

Hmm, let's try this, if you had a metric $ds^2=dr^2+r^2d\theta^2$ (polar coordinates! This space is flat by the way), what are the two ortho-normal forms one could make? (do not transform back to x and y, just turn this into an orthonormal set).

 

[Breo]

$$e^1 = dr \\ e^2 = rd\theta $$

But I still do not see the point which you are aiming to. The only relation with the minkowskian metric I could find was $g_{\mu\nu} = \eta_{ab} e^a_{\mu}e^b_{\nu}$ which is satisfied by the metric I already found and it is diagonal aswell. My mind will blow-up xD

I think the problem should be something that I made which is wrong...

EDIT: Are, maybe, you talking about this: $ds^2 = \eta_{ab} e^a \otimes e^b$ ?

 

[Matterwave]

So you were able to find really quickly $e^1=dr,~e^2=rd\theta$, why not apply this same idea to $ds^2=dp^2-a^2[(dx^1)^2+(dx^2)^2+(dx^3)^2]$? Do you see the similarity? What happens if I switch $r\leftrightarrow a$?

 

[Breo]

$$e^1= dp \\
e^2= adx^1 \\
e^2 = adx^2 \\
e^3 = adx^3 \\

$$

Oh wait. Are you trying to tell me that if I make a transformation that involves some coordinates like dp does with $dx^i$ should I change also the rest of $dx^i$ in the metric? So I can not write $a^2(dx^i)^2$ ?

 

[Matterwave]

No, you're good. You found all 4 (but you numbered them wrong).

 

[Breo]

$$e^0= dp \\
e^i= adx^i

$$

I noticed this error. Still do not find what is that next step you are talking about. I have just found almost the same metric in Carroll's book: pg.490 and still do not know what to do.

 

[Matterwave]

? I think you are done...you asked to find the 4 orthonormal forms, and you have.

 

[Breo]

Maybe my last post sounds a bit aggresive. Was not my intention, I am not good at english. Sorry if so.

Is just I can not find the next step which you referred to.

 

[Matterwave]

Maybe my last post sounds a bit aggresive. Was not my intention, I am not good at english. Sorry if so.

Is just I can not find the next step which you referred to.

I don't think I referred to any "next step". You asked to find an ortho-normal set of one-forms basis, and I think you have in post #42.

 

[Breo]

You have diagonalized the metric, but you still need to put it in terms of diag(1,-1,-1,-1) so you have to do one more transformation.

You still have the $a^2$ there. What you wrote is just explicitly what you had in post #30, you just wrote out every term. It is identical with the one in #30 though. You need to go 1 more step.

Must be all a misscomunication haha I thought I must to do something to the line element

 

[Matterwave]

Must be all a misscomunication haha I thought I must to do something to the line element

You already did. Now the line element is $ds^2=(e^0)^2-(e^1)^2-(e^2)^2-(e^3)^2$ there is no longer that $a^2$ because you subsumed them into $e^i$.