Calculating RiemannScalar in 2-D: Where to Start

[ssamsymn]

Where should I start from to show that curvature scalar (RiemannScalar) is

2$\frac{R_1212}{det (g_μ√)}$

?

 

[ssamsymn]

I am asked to show it by the symmetries of the Riemann tensor by the way.

 

[ssamsymn]

Ok here is my thoughts, I try to stay away from the connection coefficients. So, I don't write the R tensors in form of $\Gamma$'s. So I am trying:

R= g$^{αβ}$ R$_{αβ}$
= g$^{αβ}$ R$^{c}$$_{αcβ}$
= g$^{αβ}$ g$^{αb}$ R$_{bαcβ}$

but now I can't have the R$_{αβαβ}$ form. Since it is 2-d, I put α=1 and β=2, but the c and b contractions doesn't give me what I want.

How can get that 2R$_{1212}$ ?

 

[Bill_K]

remember that g^αβ is the inverse matrix of g_αβ. And it's easy to take the inverse of a 2 x 2 matrix.

Write out completely the individual components of g^αβ in terms of the g_αβ components and see what you get.

 

[pervect]

My thoughts are to start by looking at the only completely anti-symmetric 2-form , which must have components

$$\left[ \begin{array}{cc}<br /> 0 & R \\<br /> -R & 0 \\<br /> \end{array} \right]$$

Now we know that R_abcd = 0 if a=b or c=d by the anti-symmetry properties, and also that R_abcd = R_bacd and that R_abcd = -R_abdc

This, and a little thought, gives us the value of all the components of R, which can be described as an anti-symmetric 2d array of two-forms, i.e. it looks like the array above, but the members of the array are the anti-symmetric two-forms.

Next we just have to compute the contractions to get the Riemann tensor and scalar, which I'm too lazy to do by hand.

 

[ssamsymn]

Thank you very much.

Yes I rewrited everything with g lower index.

I am not sure about that symmetry:

R$^{αβ}$$_{αβ}$= - R$^{βα}$$_{βα}$

But I feel I am close to it. Thank you again.