- #1
Pouramat
- 28
- 1
- Homework Statement
- Using Tetrad formalism, I am trying to solve exercise 16 carroll.
In final steps I cannot read elements of Riemann tensor correctly. please help.
- Relevant Equations
- $$
ds^2 = d \psi^2 + \sin^2 \theta (d \theta^2 + \sin^2 \theta d \phi^2)
$$
My attempt at solution:
in tetrad formalism:
$$ds^2=e^1e^1+e^2e^2+e^3e^3≡e^ae^a$$
so we can read vielbeins as following:
$$
\begin{align}
e^1 &=d \psi;\\
e^2 &= \sin \psi \, d\theta;\\
e^3 &= \sin \psi \,\sin \theta \, d\phi
\end{align}
$$
componets of spin connection could be written by using ##de^a+{\omega^a}_b∧e^b=0## as:
$$
\begin{align}
de^1+ {\omega^1}_2 \wedge e^2 + {\omega^1}_3 \wedge e^3 = 0 \\
de^2+ {\omega^2}_1 \wedge e^1 + {\omega^2}_3 \wedge e^3 = 0 \\
de^3+ {\omega^3}_1 \wedge e^1 + {\omega^3}_2 \wedge e^2 = 0
\end{align}
$$
Now we can lower the indices of ##\omega##s and write:
$$
\begin{align}
{\omega^1}_2 = \omega_{12} = -\omega_{21} = \omega_1 \\
{\omega^1}_3 = \omega_{13} = -\omega_{31} = \omega_3 \\
{\omega^2}_3 = \omega_{23} = -\omega_{32} = \omega_2
\end{align}
$$
Now I can rewrite torsionless equations using :
$$
\begin{align}
de^1+ \omega_1 \wedge e^2 + \omega_3 \wedge e^3 = 0 \\
de^2 - \omega_1 \wedge e^1 + \omega_2 \wedge e^3 = 0 \\
de^3 - \omega_3 \wedge e^1 - \omega_2 \wedge e^2 = 0
\end{align}
$$
Now we can calculate exterior derivative of e's:
$$
\begin{align*}
de^1 &= d(d \psi) = 0 \\
de^2 &= \cos \psi \, d\psi \wedge d\theta = \cot \psi \,e^1 \wedge e^2 \\
de^3 &= \cos \psi \, \sin \theta \, d\psi \wedge d \phi+ \sin \psi \, \cos \theta \, d\theta \wedge d \phi = \cot \psi \, e^1 \wedge e^3 + \frac{\cot \theta}{\sin \psi} e^2 \wedge e^3
\end{align*}
$$
We can read ωs from the equations:
$$
\begin{align}
\omega_1 &= \cot \psi \, d \psi \\
\omega_2 &= -\cos \theta \, d \phi \\
\omega_3 &= \cos \psi \, \sin \theta \, d \psi
\end{align}
$$
Using Cartan's structure equation for ##{R^a}_b=d{\omega^a}_b+{\omega^a}_c{\omega^c}_b##:
$$
\begin{align}
{R^1}_2 &= d{\omega^1}_2 + {\omega^1}_3 \wedge{\omega^3}_2 \\
{R^1}_3 &= d{\omega^1}_3 + {\omega^1}_2 \wedge{\omega^2}_3 \\
{R^2}_3 &= d{\omega^2}_3 + {\omega^2}_1 \wedge{\omega^1}_3
\end{align}
$$$$
\begin{align}
{R^1}_2 &= 0 \\
{R^1}_3 &= \sin \theta \, d \theta \wedge d\phi - \cot \psi \, \cos \psi \, \sin \theta \, d \psi \wedge d\phi \\
{R^2}_3 &= - \sin \psi \, \sin \theta \, d \psi \wedge d\phi + \cos \psi \, \cos \theta \, d\theta \wedge d\phi - \cot \psi \, \cos \theta \, d\psi \wedge d\phi
\end{align}
$$
The Problem is now begining, I cannot read elements of Reimann tensor using these equations. Can anyone help?
Last edited: