What is the Unique Metric Tensor in this Line Element?

[Matterwave]

The same metric with different coords?

$$ ds^2 = dp^2 - \delta_{ij} a^2dx^idx^j $$

Which is diagonal. This seems too much easy :-/

I wrote this on mobile. Hope it worked xD

It seems too easy because Pervect basically gave it to you haha.

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.

 

[Breo]

Mmm as it seems to work with the signs I bet you are talking about this:

$$ ds^2 = dp^2 - a^2 [(dx^1)^2 + (dx^2)^2 + (dx^3)^2] $$

removing the delta expression?

 

[Matterwave]

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.

 

[Breo]

I am still wondering :-/

 

[Matterwave]

I am still wondering :-/

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?

 

[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 $e^i$.