Month: April 2014

Der Wanderer

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Find all components of the Riemann curvature tensor, the Ricci tensor, and the Ricci scalar

The metric of the 2-sphere S^2 is

ds^2 = d \theta^2 + sin^2 \theta d \phi^2

Problem: Find all components of the Riemann curvature tensor, the Ricci tensor, and the Ricci scalar.

Solution: We order the coordinates as {θ, φ}. We have

\Gamma ^\theta_{ \phi \phi} = \frac{1}{2} g^ {\theta \theta}[g_{ \theta \phi, \phi} +g_{ \theta \phi, \phi} - g_{ \phi \phi, \theta}] = -\frac{1}{2} g^ {\theta \theta}g_{ \phi \phi, \theta}= -\frac{1}{2} (2sin \theta cos \theta)=-sin \theta cos \theta = -\frac{1}{2} sin (2 \theta)

 \Gamma ^\phi_{ \theta \phi} = \frac{1}{2} g^ {\theta \theta}[g_{ \phi \theta, \phi} +g_{ \phi \phi, \theta} - g_{ \theta \phi, \phi}] = \frac{1}{2} \frac{1}{sin^2 \theta} (2sin \theta cos \theta)= \frac{ cos \theta}{ sin \theta}= cot \theta

\Gamma ^\phi_{ \phi \theta} = cot \theta =\Gamma ^\phi_{ \theta \phi}

Let us first look at the Riemann tensor:  the only component to compute is R_{ \theta \phi \theta \phi}.

The component will be

R_{\theta \phi \theta \phi} = \Gamma^\theta_{ \phi \phi, \theta} - \Gamma^\theta_{ \phi \theta, \phi} + \Gamma^\theta_{ \theta f} \Gamma ^f_{ \phi \phi} - \Gamma^\theta_{ \phi f} \Gamma^f_{ \phi \theta}

Since \Gamma ^\theta_{ \theta f}=0 the first double \Gamma term vanishes. So, we get

R_{\theta \phi \theta \phi} = \Gamma^\theta_{ \phi \phi, \theta} -  \Gamma^\theta_{ \phi \phi} \Gamma^\phi_{ \phi \theta}= - cos( 2\theta) + cos (\theta^2)= - (2cos^2 \theta -1) + cos^2 \theta= sin^2 \theta