How do I determine whether this metric is flat or not?

[joshyp93]

Hello everyone

1. Homework Statement
I have a homework question where I need to find out if the geometry is flat or not. The metric is shown below.

Homework Equations

The Attempt at a Solution

So far I have written the metric in the form guv but and I am trying to find coordinates in which it can be written as the standard Euclidean space matrix ds^2=dx^2+dy^2. I have no idea where to start and cannot seem to find the answer anywhere I look! I just need to know a systematic way of how to check whether it can be expressed as a flat metric or not.

Thanks
Josh

 

[andrewkirk]

Do you know how to calculate the components of the Riemann curvature tensor? If so then you could calculate them and if it is zero then the geometry is flat. There should be at most eight calculations to do.

There may be a much more elegant way of approaching this problem, but at least this gives a straightforward approach that should give an answer.

 

[joshyp93]

I have seen the equation but I don't think it is necessary to use it. Isn't there a more simple way? The curvature tensor has lots of terms and this is only for 2 dimensions. How would I calculate the christoffel symbols for this metric? Would I have to use the euler lagrange equations in the form d/dt(dL/(du/dt)) - dl/du = 0 just to calculate the christoffel symbols? This seems like it would be a complete mess and there would be u's and v's all over the place

 

[joshyp93]

I just used R(abcd) = K(g(ac)g(bd) - g(ad)g(cb)) which is the Riemann curvature tensor for 2D from wikipedia. I got all of the terms to cancel expect the 1/v^2 terms which added instead... This left me with 4/v^2 altogether. Does this mean it is not flat?

 

[andrewkirk]

Does this mean it is not flat?

IIRC, if the Riemann tensor has any nonzero components in any coordinate system, the geometry is not flat.

 

[joshyp93]

Ok I understand that, but people are telling me that the metric is indeed flat when I am calculating that there is a 4/v^2 term when i do the calculation which implies it isn't. Could you possibly show how to perform the calculation to make sure I am doing it right? Thanks